text:
<?php namespace RainLab\Blog;

use Backend;
use Controller;
use RainLab\Blog\Models\Post;
use System\Classes\PluginBase;
use RainLab\Blog\Classes\TagProcessor;
use RainLab\Blog\Models\Category;
use Event;

class Plugin extends PluginBase
{
    public function pluginDetails()
    {
        return [
            'name'        => 'rainlab.blog::lang.plugin.name',
            'description' => 'rainlab.blog::lang.plugin.description',
            'author'      => 'Alexey Bobkov, Samuel Georges',
            'icon'        => 'icon-pencil',
            'homepage'    => 'https://github.com/rainlab/blog-plugin'
        ];
    }

    public function registerComponents()
    {
        return [
            'RainLab\Blog\Components\Post'       => 'blogPost',
            'RainLab\Blog\Components\Posts'      => 'blogPosts',
            'RainLab\Blog\Components\Categories' => 'blogCategories',
            'RainLab\Blog\Components\RssFeed'    => 'blogRssFeed'
        ];
    }

    public function registerPermissions()
    {
        return [
            'rainlab.blog.access_posts' => [
                'tab'   => 'rainlab.blog::lang.blog.tab',
                'label' => 'rainlab.blog::lang.blog.access_posts'
            ],
            'rainlab.blog.access_categories' => [
                'tab'   => 'rainlab.blog::lang.blog.tab',
                'label' => 'rainlab.blog::lang.blog.access_categories'
            ],
            'rainlab.blog.access_other_posts' => [
                'tab'   => 'rainlab.blog::lang.blog.tab',
                'label' => 'rainlab.blog::lang.blog.access_other_posts'
            ],
            'rainlab.blog.access_import_export' => [
                'tab'   => 'rainlab.blog::lang.blog.tab',
                'label' => 'rainlab.blog::lang.blog.access_import_export'
            ],
            'rainlab.blog.access_publish' => [
                'tab'   => 'rainlab.blog::lang.blog.tab',
                'label' => 'rainlab.blog::lang.blog.access_publish'
            ]
        ];
    }

    public function registerNavigation()
    {
        return [
            'blog' => [
                'label'       => 'rainlab.blog::lang.blog.menu_label',
                'url'         => Backend::url('rainlab/blog/posts'),
                'icon'        => 'icon-pencil',
                'iconSvg'     => 'plugins/rainlab/blog/assets/images/blog-icon.svg',
                'permissions' => ['rainlab.blog.*'],
                'order'       => 300,

                'sideMenu' => [
                    'new_post' => [
                        'label'       => 'rainlab.blog::lang.posts.new_post',
                        'icon'        => 'icon-plus',
                        'url'         => Backend::url('rainlab/blog/posts/create'),
                        'permissions' => ['rainlab.blog.access_posts']
                    ],
                    'posts' => [
                        'label'       => 'rainlab.blog::lang.blog.posts',
                        'icon'        => 'icon-copy',
                        'url'         => Backend::url('rainlab/blog/posts'),
                        'permissions' => ['rainlab.blog.access_posts']
                    ],
                    'categories' => [
                        'label'       => 'rainlab.blog::lang.blog.categories',
                        'icon'        => 'icon-list-ul',
                        'url'         => Backend::url('rainlab/blog/categories'),
                        'permissions' => ['rainlab.blog.access_categories']
                    ]
                ]
            ]
        ];
    }

    public function registerFormWidgets()
    {
        return [
            'RainLab\Blog\FormWidgets\Preview' => [
                'label' => 'Preview',
                'code'  => 'preview'
            ]
        ];
    }

    /**
     * Register method, called when the plugin is first registered.
     */
    public function register()
    {
        /*
         * Register the image tag processing callback
         */
        TagProcessor::instance()->registerCallback(function($input, $preview) {
            if (!$preview) {
                return $input;
            }

            return preg_replace('|\]*)\/>|m',
                '

                        Click or drop an image...

                ',
            $input);
        });
    }

    public function boot()
    {
        /*
         * Register menu items for the RainLab.Pages plugin
         */
        Event::listen('pages.menuitem.listTypes', function() {
            return [
                'blog-category'       => 'rainlab.blog::lang.menuitem.blog_category',
                'all-blog-categories' => 'rainlab.blog::lang.menuitem.all_blog_categories',
                'blog-post'           => 'rainlab.blog::lang.menuitem.blog_post',
                'all-blog-posts'      => 'rainlab.blog::lang.menuitem.all_blog_posts',
            ];
        });

        Event::listen('pages.menuitem.getTypeInfo', function($type) {
            if ($type == 'blog-category' || $type == 'all-blog-categories') {
                return Category::getMenuTypeInfo($type);
            }
            elseif ($type == 'blog-post' || $type == 'all-blog-posts') {
                return Post::getMenuTypeInfo($type);
            }
        });

        Event::listen('pages.menuitem.resolveItem', function($type, $item, $url, $theme) {
            if ($type == 'blog-category' || $type == 'all-blog-categories') {
                return Category::resolveMenuItem($item, $url, $theme);
            }
            elseif ($type == 'blog-post' || $type == 'all-blog-posts') {
                return Post::resolveMenuItem($item, $url, $theme);
            }
        });
    }
}

redpajama_set_name: RedPajamaGithub


This prospect company is government owned/controlled, employ a government official or close family members of government officials.
Contact me before communication with this decision maker.
Are you working with a Salesforce account executive?
Is this lead associated with a salesforce compaign?
We plan to use third parties(subcontractor,intermediary,etc.)to assist with the sale of salesforce.com products and services.


Simply put, revenue sharing is the process by which a company or corporation shares or distributes profits to employees, investors or other stakeholders and affiliates. A company experiences quarterly or annual profits and shares that with relevant and qualified individuals or organizations via cash, shares and other benefits. An affiliate – that is, an individual or company – supports and promotes a particular product or brand in exchange for a percentage of each sale or order placed via their link, referral or recommendation.
Commonly used in online advertising, revenue sharing is an effective way to increase revenue for a parent company without shouldering the cost of increased marketing and advertising. The use of social media and blogs in affiliate marketing provides passive income for millions of content creators and managers all over the world.
Amazon, Ebay and ShareASale are all popular providers of affiliate marketing programs. Some blogging and social media professionals make posting and profiting from affiliate links a full-time job; many report earnings of six figures (and more!) annually. Most providers of affiliate services provide graphics and other digital media to make online advertising easy for marketers to use and implement. These graphics include logos, pictures of products formatted for high-quality display and web banners of various sizes and formats to make the posting process quick and convenient.
Email is also a popular method of taking advantage of the benefits of revenue sharing. Many bloggers and online content providers offer email subscription lists, newsletters and other free material that is sent weekly, monthly or sporadically. In these messages are often links to products, services and content that subscribers may find useful or want to share with friends and family. To make sharing these messages easy, productive and more efficient, many emails and other links contain buttons that provide quick ways to post this information on a user's social media profile.
Whether a person owns stock in a company or works hard to promote its product or service, revenue sharing is a benefit for all parties involved. Sharing revenue is essential to the growth of at-home bloggers, professional ad agencies, Fortune 500 companies and even the governments of many countries with an interest in their nation's stock exchange(s). In whatever form revenue is paid out, invested or shared, profit is important in measuring success, and sharing that profit in the right way and at the right time helps to create and perpetuate a cycle of benefits for marketers and businesses alike.


Explore the campuses by browsing through the detailed profiles and key information of each college.
Use the College Finder for a quick and easy way to identify campuses by selecting preferences such as college type, location, size, cost, campus life and academics.
Using the College Matching Wizard allows you to explore the advantages/disadvantages and definitions of various factors affecting the college selection process.
Click the region below to view colleges by state.


Bravotv.com: What did you think of NeNe's show?
Kandi Burruss: It was a great show. I laughed the whole time.
Bravotv.com: Why did you decide to give Porsha relationship advice?
KB: Porsha and I were just starting to get on the right foot. I didn't want to say too much, but I did feel like I should tell her to take her time. In the past, she's been known to get serious quickly with some guys who didn't necessarily deserve that from her. In this case, I already heard some things about Dennis, but I didn't feel like she and I were in a place where she would receive any negative information well coming from me. So instead of telling her anything bad, I felt that I should at least tell her to take her time and not push the relationship too quickly.
Bravotv.com: Who's the biggest diva of the group when traveling?
KB: Seeing Gregg struggle hurts. I'm so used to him brightening up the room when he walks in. Gregg is normally the one who lifts everybody up, but now it's time for us all to be there for him and lift him up.
"Personally, I can say just from the jump, the situation that NeNe and Gregg are dealing with, that one is the hardest for me," Kandi Burruss told The Daily Dish this week.


text:
Bernstein-Ballentine Counters (early 1950s)
Proportional Counters
These tubes, known as Bernstein-Ballentine counters, were manufactured by the Nuclear Instrument and Chemical Corporation sometime in the early 1950s. They are 7/8" in diameter and 19 to 24" long.
Their purpose was to measure the carbon-14 content of carbon dioxide, often as a means to date organic material (C-14 dating). The samples to be analyzed would be combusted and the resulting carbon dioxide drawn into the tubes for counting. I believe that methane was used as the counting gas (with which the carbon dioxide was mixed). The steel mounting box in the following photograph was convenient for holding and protecting the fragile glass tubes. It also provided some degree of shielding and it covered up the electrical connections.
Quoting the AEC's Radiation Measuring Instruments and Associated Components: "This method involves the rapid combustion of radioactive organic samples by the Van Slyke-Folch technique and assay of the resultant carbon-14 dioxide with the Bernstein-Ballentine proportional counter. Organic samples of from 0.1 to 15 mg. of carbon content can be combusted and the gas measured manometrically and transferred to the proportional tube in 15 minutes or less."
Atomic Energy Commission Radiation Measuring Instruments and Associated Components RIB-8 (Part 2) Supplement, July 1, 1954, page BG-6E.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

It is a very  important issue to simulate multi-component two-phase compressible fluid systems with  a realistic equation of state (e.g. Peng-Robinson equation of state \cite{Peng1976EOS}).  It has a wide range of applications   in  chemical  and reservoir engineering   \cite{kousun2015CMA,qiaosun2014,kousun2015SISC,kousun2015CHE,polivka2014compositional, mikyvska2015General,jindrova2013fast,kousun2018Flash}, especially the pore scale modeling  of subsurface fluid flow including shale gas reservoir.  The classical models  of incompressible two-phase flows  or compositional flows  have been extensively studied and employed   \cite{firoozabadi1999thermodynamics,Moortgat2013compositional,chen2006multiphase,kouandsun2014dgtwophase},  the primal state variables of which are often    pressure, temperature, and chemical composition.  Although the classical models have been widely used, they suffer from a few essential   limitations  as pointed out in \cite{mikyvska2011new, polivka2014compositional};  for example, it is required to construct a pressure  equation since  there is no intrinsic  pressure equation  \cite{polivka2014compositional}.

An alternative  modeling framework, which uses the moles, volume, and temperature  (the so-called  NVT-based framework)  as the primal state variables, has been intensively studied recently \cite{kousun2015CMA,qiaosun2014,kousun2015SISC,kousun2015CHE,kouandsun2016multiscale,polivka2014compositional,jindrova2013fast,mikyvska2015General,kousun2018Flash}. The NVT-based modeling framework  originates from the phase-splitting  calculations  of multi-component fluids at specified  moles, volume and temperature \cite{michelsen1999,mikyvska2011new,Nagarajan1991}.  Very recently, in the NVT-based framework, a general multi-component two-phase compressible flow model is rigorously derived by Kou and Sun in \cite{kouandsun2017modeling} based on   the thermodynamic laws and  realistic equations of state (e.g. Peng-Robinson equation of state).  This model has at least three important features that are distinguished  from the classical models:
 \begin{itemize}
\item  It has thermodynamically-consistent unified formulations for general average velocities and mass diffusion fluxes;
\item It uses diffusive interfaces and  realistic equations of state, and as a result,  it can characterize the fluid compressibility and partial miscibility between different phases;
\item It uses a general  thermodynamic pressure, which  is  a function of the molar density and temperature, and consequently, it is never required to construct the pressure  equation.
\end{itemize}
In addition,  another formulation of the  momentum conservation equation, which is  convenient for numerical simulation,  has been derived in \cite{kouandsun2017modeling} by the relation between the pressure gradient  and  chemical potential gradients. In this paper, we consider how to develop and analyze efficient   numerical methods for this model problem.

 A key challenge   in numerical simulation of diffuse interface models is  to construct  efficient numerical schemes  preserving  the discrete energy-dissipation law \cite{shen2015SIAM,shen2016JCP}.   In constructing  energy-stable numerical schemes for multi-component two-phase compressible flow model,  there are at least two main difficulties: one is   the strong nonlinearity  of bulk Helmholtz free energy density; the other is  the tightly coupling relationship between molar densities and flow velocity through the convection term in the mass balance equations and the stress force arising  from chemical potential gradients in the momentum balance equation.
An energy-dissipation  numerical scheme was developed in \cite{kouandsun2017modeling} based on a convex-concave  splitting of Helmholtz free energy density, but it leads to a nonlinear and coupled system of the mass balance equations and momentum balance equation.  In this paper, we focus on the  linear, decoupled, energy stable numerical schemes.

Recently,  for incompressible two-phase flows, a decoupled approach \cite{Minjeaud2013Uncoupled}  was developed by  introducing an intermediate velocity  in the phase equation to resolve the coupling relation between the velocity and phase function, and this technique was used to construct linear, decoupled, efficient numerical  methods    for phase-field models of  incompressible two-phase flows \cite{shen2015SIAM,shen2016JCP}. When applying  this technique to compressible multi-component two-phase  flow model considered in this paper, we encounter two challenging problems: the first  is how to construct the intermediate velocities since the stress force in the momentum balance equation  is different from phase-field models;  the second is how to treat the  momentum balance equation using intermediate velocities.    The second problem is because at the time-discrete level, the  velocity variable   in the convection term of  the  momentum balance equation shall be consistent with the intermediate  velocities when we  combine the mass balance equation of each component and the  momentum balance equation to derive the variation of the kinetic  energy.  In this work, we will  construct two  intermediate  velocities, both of which can   uncouple the relationship between velocity and molar densities; we will also propose a discrete  formulation of the  momentum balance equation, which involves the intermediate  velocities and consequently consistent with  the mass balance equations.
It is noted that one of the introduced intermediate  velocities is for the first time defined by a component-wise way, and thus, it is very efficient for a special multi-component fluid.

There have been at least four approaches in the literature  dealing with the bulk Helmholtz free energy density derived from Peng-Robinson equation of state for constructing  energy-stable numerical schemes. The  first approach is the convex splitting method \cite{Elliott1993ConvexSpliting,Eyre1998ConvexSplitting}, which has been  popularly used in phase-field models \cite{shen2015SIAM,Wise2009Convex,Eyre1998ConvexSplitting,Baskaran2013convexsplitting}.
 The energy-stable numerical scheme based on the convex splitting method have also been developed and analyzed for the diffuse-interface models with Peng-Robinson equation of state \cite{qiaosun2014,fan2017componentwise,kou2017compositional,kouandsun2017modeling,kousun2018Flash,Peng2017convexsplitting}. The second approach is a modified Newton's method with a relaxation parameter that is dynamically chosen to ensure the energy decay property \cite{kousun2015SISC}.  The third approach is  a fully-implicit unconditionally-stable scheme \cite{kousun2015CHE},  which uses the difference of Helmholtz free energy density to approximate  the chemical potential. The fourth numerical scheme is developed in \cite{Li2017IEQ} based on the invariant energy quadratization (IEQ) approach that is  a novel, efficient method   and  has been  applied to many phase-field models intensively recently \cite{Yang2016IEQ,Yang2017IEQ,Yang2017IEQ2}. Very recently, a scalar auxiliary variable (SAV) approach \cite{Shen2018SAV} is built upon the  IEQ approach. It leads to unconditionally stable numerical schemes, which   only need to solve the linear equations with constant coefficients  at each time step. In this paper, we will apply the SAV approach to treat the mass balance equations and construct linear, unconditionally stable numerical schemes. Moreover, we will develop a component-wise SAV approach for a multi-component flow model, which  uncouples the relationships between multiple components and allows us to solve each component mass balance equation separately.  The schemes for gradient flows of multiple functions in \cite{Shen2018SAV} usually require the computation of eigen-matrix and eigenvalues  to  achieve the decoupled forms, but this computation  cost is free for the proposed component-wise SAV approach.  So the proposed scheme is efficient and easy-to-implement for the case of multiple components.

We must note that the proposed numerical schemes for multi-component two-phase flows are perfect combinations of the above intermediate  velocity approaches and   SAV approaches, which lead to a sequence of linearly decoupled equations.  The proposed schemes  are proved to be unconditionally energy stable.

The rest of this paper is organized as   follows. In Section 2, we will give a brief description  of  the multi-component two-phase flow model.
 In Section 3, we will propose the numerical schemes and prove the unconditional energy stability.  In Section 4,  numerical tests are carried out  to show the   effectiveness of the proposed methods.   Finally,   concluding remarks are provided in Section 5.

\section{Mathematical model of multi-component two-phase flow}

In this section, we briefly introduce  the mathematical model of multi-component two-phase flow with Peng-Robinson equation of state, which is very recently proposed in \cite{kouandsun2017modeling}.

We consider the motion of a mixture fluid   composed of $M$ chemical  components at a constant  temperature.  Let $n_i$ be the molar density of   component $i$, and we denote the molar density vector by ${\bm   n} = [n_1,n_2,\cdots,n_M]^T$.

Mathematical model developed  in \cite{kouandsun2017modeling} can employ  any  realistic   equation of state, for instance, the van der Waals equation of state and Peng-Robinson equation of state (PR-EOS)  \cite{Peng1976EOS}.  PR-EOS  has been widely  applied  in oil reservoir and chemical engineering due to its accuracy.  In this work, we focus on the PR-EOS-based Helmholtz free energy density $f_{b}({\bm   n})$ of a homogeneous bulk  fluid, which has a form as
\begin{eqnarray}\label{eqHelmholtzEnergy_a0}
    f_b({\bm   n})= f_b^{\textnormal{ideal}}({\bm   n}) +f_b^{\textnormal{repulsion}}({\bm   n})+f_b^{\textnormal{attraction}}({\bm   n}),
\end{eqnarray}
where $f_b^{\textnormal{ideal}}$, $f_b^{\textnormal{repulsion}}$ and $f_b^{\textnormal{attraction}}$ are formulated in Appendix \ref{appendixHelmholtz}.

The diffuse interfaces always occurs  between multiple  phases of a realistic fluid.  To  characterize  this feature, a local density gradient  contribution is added to the  free energy density of an inhomogeneous fluid, and consequently, the  general  Helmholtz free energy density (denoted by $f$) is expressed as
\begin{eqnarray}\label{eqHelmholtzfreeenergydensity}
 f({\bm   n})=f_b({\bm   n})+\frac{1}{2}\sum_{i,j=1}^Mc_{ij}\nabla  n_i\cdot\nabla  n_j,
 \end{eqnarray}
 where $c_{ij}(1\leq i,j\leq M)$ are the cross influence parameters depending on temperature but independent of molar densities.  The formulations of $c_{ij}$ can be found  in Appendix \ref{appendixInfluenceParameters}. We assume that the influence parameter matrix $\big(c_{ij}\big)_{i,j=1}^M$ is symmetric and moreover it is positive definite or positive semi-definite.

The chemical potential of component $i$ is defined as
  \begin{eqnarray}\label{eqgeneralchemicalpotential}
   \mu_i({\bm   n})=\frac{\delta  f({\bm   n})}{\delta n_i}=\mu_i^b({\bm   n})-\sum_{j=1}^M\nabla\cdot \(c_{ij}\nabla {n_j}\),~~~\mu_i^b({\bm   n})=\frac{\partial  f_b({\bm   n})}{\partial n_i},
   \end{eqnarray}
  where $\frac{\delta}{\delta n_i}$ denotes the variational  derivative. By the thermodynamical relations, the general  thermodynamical pressure can be formulated as  a function of ${\bm   n}$ at a constant temperature
\begin{eqnarray}\label{eqMultiComponentDefPresGeneralB}
    p({\bm   n}) &=& \sum_{i=1}^Mn_i \mu_i({\bm   n})- f({\bm   n})\nonumber\\
   &=&p_b-  \sum_{i,j=1}^Mn_i\nabla\cdot \(c_{ij}\nabla {n_j}\)-\frac{1}{2} \sum_{i,j=1}^Mc_{ij}\nabla n_i\cdot\nabla n_j,
\end{eqnarray}
where $p_b({\bm   n})=\sum_{i=1}^Mn_i\mu_i^b({\bm   n})-f_b({\bm   n}).$

The overall molar density of a mixture is denoted by $n=\sum_{i=1}^Mn_i$. Let $M_{w,i}$ denote the molar weight of component $i$, and then we denote  the mass density of component $i$ by $\rho_i=n_iM_{w,i}$ and denote the overall mass density of a mixture by $\rho=\sum_{i=1}^M\rho_i$.

 We now describe the governing equations. First, the mass balance equation   for component  $i$    is
\begin{eqnarray}\label{eqGeneralNSEQMass}
\frac{\partial n_i}{\partial t}+\nabla\cdot \({\bm   u} n_i\)+\nabla\cdot  {\bm   J}_i=0,
\end{eqnarray}
where ${\bm   u}$ is a specific or average velocity and ${\bm   J}_i$ is the diffusion flux of component $i$. In general, we can express the diffusion flux  of component $i$  as   \cite{Cogswell2010thesis,kouandsun2016multiscale,kouandsun2017modeling}
\begin{eqnarray}\label{eqMultiCompononentMassConserveDiffusion}
  \bm J_{i}=-\sum_{j=1}^M\mathcal{M}_{ij}\nabla \mu_{j},~~i=1,\cdots, M,
\end{eqnarray}
where   $\bm{\mathcal{M}}=\(\mathcal{M}_{ij}\)_{i,j=1}^M$ is the mobility tensor.   The mobility matrix $\bm{\mathcal{M}}$ shall be symmetric and at least positive semidefinite so that Onsager's reciprocal principle \cite{Groot2015NET} and the second law of thermodynamics are satisfied.

 Three choices of the mobility $\bm{\mathcal{M}}$ in \eqref{eqMultiCompononentMassConserveDiffusion} are provided in \cite{kouandsun2017modeling} as below.   \begin{description}
\item[(A1)] The first mobility choice is to take $\bm{\mathcal{M}}$  as a diagonal positive definite matrix with diagonal elements
\begin{equation}\label{eqMultiCompononentMassConserveDiffusionChoiceA}
\mathcal{M}_{i}=\mathcal{M}_{ii}=\frac{D_in_i}{RT},
\end{equation}
where  $R$ stands for the universal gas constant and  $D_i>0$ is the diffusion coefficient of component $i$. The  diffusion flux  has a form \cite{Cogswell2010thesis,kousun2018Flash} as
${\bm   J}_i  = -\frac{D_in_i}{RT}\nabla \mu_i$.
 In this case, ${\bm   u}$ and ${\bm   J}_i$ is viewed as the mean velocity and general  mixture diffusion fluxes at the constant temperature  and pressure, respectively.
\item[(A2)]     The second choice is to take $\bm{\mathcal{M}}$   as a full matrix
\begin{eqnarray}\label{eqMultiCompononentMassConserveDiffusionChoiceB}
\mathcal{M}_{ii} = \sum_{j=1}^M\frac{\mathcal{D}_{ij} n_in_j}{nRT},~~~~~\mathcal{M}_{ij} = -\frac{\mathcal{D}_{ij} n_in_j}{nRT},~~j\neq i,
\end{eqnarray}
where  the mole diffusion coefficients $\mathcal{D}_{ij}$ satisfy $\mathcal{D}_{ii}=0$ and $\mathcal{D}_{ij}=\mathcal{D}_{ji}>0$ for $i\neq j$.  In this case, ${\bm   u}$ is  the molar-average velocity.
\item[(A3)]  The third mobility $\bm{\mathcal{M}}$ has the following formulation
\begin{eqnarray}\label{eqMultiCompononentMassConserveDiffusionChoiceC}
\mathcal{M}_{ii} = \sum_{j=1}^M\frac{\mathscr{D}_{ij}n_i\rho_j}{M_{w,i}\rho RT},~~~~~\mathcal{M}_{ij} = -\frac{\mathscr{D}_{ij}n_in_j}{\rho RT},~~j\neq i,
\end{eqnarray}
where   the mass diffusion coefficients $\mathscr{D}_{ij}$ satisfy $\mathscr{D}_{ii}=0$ and $\mathscr{D}_{ij}=\mathscr{D}_{ji}>0$ for $i\neq j$.
In this case, ${\bm   u}$ is actually  the mass-average velocity.
\end{description}

We now introduce the thermodynamically-consistent  momentum balance equation, which is expressed as  \cite{kouandsun2017modeling}
\begin{eqnarray}\label{eqGeneralNSEQ}
&&\rho\(\frac{\partial  {\bm   u}}{\partial t}+ {\bm   u}\cdot\nabla { {\bm   u}}\)+\sum_{i=1}^MM_{w,i}{\bm   J}_i\cdot\nabla {\bm   u}=-\nabla p +\nabla\(\lambda\nabla\cdot {\bm   u}\)\nonumber\\
&&~~+\nabla\cdot \eta\(\nabla{\bm   u}+\nabla{\bm   u}^T\)
-\sum_{i,j=1}^M\nabla\cdot \(c_{ij}\nabla n_i\otimes\nabla n_j\),
\end{eqnarray}
where  $\lambda=\xi-\frac{2}{3}\eta$,  and  $\xi$ and   $\eta$ represent the volumetric  viscosity  and   shear viscosity respectively. We assume that $\eta>0$ and $\xi>\frac{2}{3}\eta$, and thus $\lambda>0$.
If ${\bm   u}$ is the mass-average velocity, the term   $\sum_{i=1}^MM_{w,i}{\bm   J}_i\cdot\nabla {\bm   u}$ vanishes, while for the other types of ${\bm   u}$, it  is crucial  to ensure the thermodynamical consistency.
 It is proved in \cite{kouandsun2017modeling} that the gradients of  the pressure and chemical potentials  have the following relation
\begin{eqnarray}\label{eqPresChptlMultiGrad01}
 \sum_{i=1}^Mn_i\nabla  \mu_i
 =\nabla  p+\sum_{i,j=1}^M\nabla\cdot \(c_{ij}\nabla  n_i\otimes\nabla  n_j\),
\end{eqnarray}
and then we   reformulate the momentum conservation equation \eqref{eqGeneralNSEQ}  as
\begin{eqnarray}\label{eqGeneralNSEQb}
&&\rho\(\frac{\partial  {\bm   u}}{\partial t}+ {\bm   u}\cdot\nabla { {\bm   u}}\)+\sum_{i=1}^MM_{w,i}{\bm   J}_i\cdot\nabla {\bm   u}=-\sum_{i=1}^Mn_i\nabla  \mu_i\nonumber\\
   &&~~+\nabla\(\lambda\nabla\cdot {\bm   u}\) +\nabla\cdot \eta\(\nabla{\bm   u}+\nabla{\bm   u}^T\) ,
\end{eqnarray}
which shows that the  fluid motion is  driven by the  chemical potential gradients.

In this work, we consider numerical schemes for the model formulated by  \eqref{eqGeneralNSEQMass} and \eqref{eqGeneralNSEQb} coupling with  the chemical potential \eqref{eqgeneralchemicalpotential} and  the diffusion flux  \eqref{eqMultiCompononentMassConserveDiffusion}. For the boundary conditions, we assume that   all boundary terms in \eqref{eqGeneralNSEQMass} and \eqref{eqGeneralNSEQb} will vanish when integrating by parts is performed; for example, we can use homogeneous Neumann boundary conditions or periodic boundary conditions.

We assume that $\Omega\subset {\mathbb R}^d (1\leq d\leq3)$ is an open, bounded and connected domain with the sufficiently smooth boundary $\partial\Omega$.  The Helmholtz free energy and  kinetic energy within $\Omega$ at a specified time are defined as
\begin{eqnarray}\label{eqPDETotalEnergy}
&F=F_b+F_\nabla ,~~F_b=\int_{\Omega} f_b({\bm   n})d{\bm   x},~~F_\nabla =\frac{1}{2}\sum_{i,j=1}^M\int_{\Omega} c_{ij}\nabla  n_i\cdot\nabla  n_jd{\bm   x},\nonumber\\
&E=\frac{1}{2}\int_{\Omega}\rho|{\bm   u}|^2d{\bm   x}.
\end{eqnarray}
It is proved in \cite{kouandsun2017modeling} that
the total energy, i.e. the sum  of    the Helmholtz free energy and  kinetic energy,    is dissipated with time as
\begin{eqnarray}\label{eqPDETotalEnergyDecay}
\frac{\partial (F+E)}{\partial t} \leq 0.
\end{eqnarray}

In order to use the scalar auxiliary variable (SAV) approach \cite{Shen2018SAV} , we define $H(t)=\sqrt{F_b+\sum_{i=1}^MC_{T,i}N_i^t}$, where  $N_i^t=\int_\Omega n_id{\bm   x}$.  Here,  $C_{T,i}\geq0$   is the thermodynamical coefficient  of component $i$ to ensure  $F_b+\sum_{i=1}^MC_{T,i}N_i^t\geq0$, and the choice of $C_{T,i}\geq0$ may depend on $T$ but independent of molar densities. Then the chemical potential of component $i~(1\leq i\leq M)$ can be reformulated as
\begin{subequations}\label{eqDiscreteChemicalPotential01}
 \begin{equation}\label{eqDiscreteChemicalPotential01A}
\mu_i=\frac{H(t)}{\sqrt{F_b+\sum_{j=1}^MC_{T,j}N_j^t}}\mu_i^{b}-\sum_{j=1}^M\nabla\cdot \(c_{ij}\nabla {n_j}\),
  \end{equation}
  \begin{equation}\label{eqDiscreteChemicalPotential01A}
\frac{\partial H}{\partial t}=\sum_{i=1}^M\int_\Omega\frac{\mu_i^{b}}{2\sqrt{F_b+\sum_{j=1}^MC_{T,j}N_j^t}}\frac{\partial n_i}{\partial t}d{\bm   x}.
  \end{equation}
 \end{subequations}
 The modified Helmholtz free energy  is  defined as
 \begin{equation*}
\mathcal{F}=H^2+F_\nabla -\sum_{i=1}^MC_{T,i}N_i^t.
  \end{equation*}
  In the continuous model, we have $\mathcal{F}=F$, but at the time-discrete level, the modified Helmholtz free energy may be generally different from the original Helmholtz free energy.

\section{Energy-stable numerical methods}
In this section,  we aim to develop efficient energy-dissipated semi-implicit time marching scheme for simulating  the above multi-component flow model. The key difficulties  result from the strong nonlinearity  of Helmholtz free energy density and fully coupling relations between molar densities and velocity.   In this work,  our purpose is to uncouple  this tightly coupling relations between molar densities and velocity,  and from this, we will develop linearly decoupled numerical schemes preserving  the feature of energy dissipation.

For a given  time interval $\mathcal{I}=(0,T_{f}]$, where $T_{f}>0$, we divide $\mathcal{I}$ into   $\mathcal{N}$ subintervals $\mathcal{I}_{k}=(t_{k},t_{k+1}]$, where $t_{0}=0$ and $t_{\mathcal{N}}=T_{f}$, and we denote $\delta t_{k}=t_{k+1}-t_{k}$.  For any scalar $v(t)$ or vector $\bm{v}(t)$, we denote by $v^{k}$ or $\bm{v}^{k}$ its approximation at the time $t_{k}$.
The traditional notations $\(\cdot,\cdot\)$ and $\|\cdot\|$ are used to represent the  inner product and norm of $L^2\(\Omega\)$, $\(L^2\(\Omega\)\)^d$ or $\(L^2\(\Omega\)\)^{d\times d}$ respectively.

\subsection{Velocity-density decoupled  semi-implicit  scheme}\label{secVelocityDensityDecoupledScheme}
We try to develop a linear  semi-implicit  scheme that decouples the tight relationship between molar densities and velocity. This scheme allows us to solve the mass balance equations and momentum equation separately.  This scheme can be applied for the model problems with the general diffusion mobility, especially  the cases that the mobility is a full tensor.

We denote ${\bm   n}^{k}= [n_1^k,n_2^k,\cdots,n_M^k]^T$, and define  $\mu_i^{k+1}$  as
 \begin{subequations}\label{eqDiscreteChemicalPotential01}
 \begin{equation}\label{eqDiscreteChemicalPotential01A}
\mu_i^{k+1}=\frac{H^{k+1}+H^k}{2\sqrt{F_b({\bm   n}^k)+\sum_{j=1}^MC_{T,j}N_j^t}}\mu_i^{b}\({\bm   n}^k\)-\sum_{j=1}^M\nabla\cdot \(c_{ij}\nabla {n_j^{k+1}}\),
  \end{equation}
  \begin{equation}\label{eqDiscreteChemicalPotential01B}
\frac{H^{k+1}-H^k}{\delta t_k}=\sum_{i=1}^M\int_\Omega\frac{\mu_i^{b}\({\bm   n}^k\)}{2\sqrt{F_b({\bm   n}^k)+\sum_{j=1}^MC_{T,j}N_j^t}}\frac{n_i^{k+1}-n_i^k}{\delta t_k}d{\bm   x}.
  \end{equation}
 \end{subequations}
  Furthermore, we define an intermediate  velocity ${\bm   u}_\star^{k}$ as
\begin{eqnarray}\label{eqDecoupledDiscreteVelocityStar}
{\bm   u}_\star^{k} = {\bm   u}^k-\frac{\delta t_{k}}{\rho^k}\sum_{i=1}^M n_i^{k}\nabla  \mu_i^{k+1}.
\end{eqnarray}
We note that ${\bm   u}_\star^{k}$ can be viewed as an approximation of ${\bm   u}^{k+1} $ obtained by neglecting the three parts: the convection term, $\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+1}\cdot\nabla {\bm   u}^{k+1}$, and the viscosity terms,  in \eqref{eqDecoupledDiscreteMultiComponentMomentumConserveC01}.

Using the intermediate  velocity ${\bm   u}_\star^{k}$, we construct the semi-implicit time   scheme for the molar density balance equation \eqref{eqGeneralNSEQMass}    as
 \begin{subequations}\label{eqDecoupledDiscreteMultiCompononentMassConserve01}
 \begin{equation}\label{eqDecoupledDiscreteMultiCompononentMassConserve01Main}
 \frac{ n_i^{k+1}-n_i^{k}}{\delta t_{k}}+\nabla\cdot (n_i^{k}{\bm   u}_\star^{k})+\nabla\cdot {\bm   J}_i^{k+1} =0,
 \end{equation}
 \begin{equation}\label{eqDecoupledDiscreteMultiCompononentMassConserve01Diffusion}
{\bm   J}_i^{k+1} = -\sum_{j=1}^M\mathcal{M}^k_{ij}\nabla \mu_{j}^{k+1},
\end{equation}
\end{subequations}
where we denote by $\mathcal{M}^k_{ij}$ the mobility coefficients calculated from molar densities ${\bm   n}^k$ since the mobility coefficients  $\mathcal{M}_{ij}$, generally depending on  molar densities, can be treated explicitly.

 We can see that  only ${\bm   n}^{k+1}$ is  the unknown variable of the equations \eqref{eqDecoupledDiscreteMultiCompononentMassConserve01}. This means that the use of ${\bm   u}_\star^{k}$  eliminates the tight coupling relationship between molar densities and velocity.  We can solve  \eqref{eqDecoupledDiscreteMultiCompononentMassConserve01}   to obtain molar densities ${\bm   n}^{k+1} $. Once  ${\bm   n}^{k+1}$   is calculated,  we can get $\mu_i^{k+1}$,  ${\bm   J}_i^{k+1}$ and ${\bm   u}_\star^{k} $ from \eqref{eqDiscreteChemicalPotential01}, \eqref{eqDecoupledDiscreteMultiCompononentMassConserve01Diffusion} and  \eqref{eqDecoupledDiscreteVelocityStar} respectively.  A semi-implicit scheme for the momentum balance equation \eqref{eqGeneralNSEQb} is  formulated as
\begin{eqnarray}\label{eqDecoupledDiscreteMultiComponentMomentumConserveC01}
&& \rho^{k}\frac{{\bm   u}^{k+1}-{\bm   u}^{k}}{\delta t_{k}}+\rho^{k}{\bm   u}_\star^{k}\cdot\nabla {\bm   u}^{k+1}+\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+1}\cdot\nabla {\bm   u}^{k+1}=-\sum_{i=1}^Mn_i^{k}\nabla  \mu_i^{k+1}\nonumber\\
   &&~~+\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\),
\end{eqnarray}
which is a linear equation of velocity ${\bm   u}^{k+1}$ only.   In the convection term of \eqref{eqDecoupledDiscreteMultiComponentMomentumConserveC01}, the use of ${\bm   u}_\star^{k}$ instead of ${\bm   u}^k$ is consistent with the mass balance equations as shown in the proof of Theorem \ref{thmDecoupledSchemeEnergy}, and moreover, it avoids to use the existing approach in \cite{Minjeaud2013Uncoupled,shen2015SIAM} that needs to impose the overall mass  equation into the momentum  equation for the sake of achieving energy dissipation for phase-field models with the large density ratios. We note that this treatment (i.e., using ${\bm   u}_\star^{k}$  instead of ${\bm   u}^k$ in the convection term of the momentum  equation) can be directly  applied to the phase-field models with different densities.

We now prove that the above linearly  decoupled scheme satisfies the discrete energy dissipation law.  To do this, we define the discrete  kinetic energy and the modified  Helmholtz free energy  as
\begin{subequations}\label{eqDiscreteTotalEnergy}
\begin{eqnarray}\label{eqDiscreteKineticEnergy}
E^k=\frac{1}{2}\int_{\Omega}\rho^k|{\bm   u}^k|^2d{\bm   x},
\end{eqnarray}
\begin{eqnarray}\label{eqDiscreteHelmholtz}
\mathcal{F}^k=|H^k|^2+F^k_\nabla -\sum_{i=1}^MC_{T,i}N_i^t,~~~F_\nabla ^{k}=\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^Mc_{ij}\nabla  n_i^k\cdot\nabla  n_j^kd{\bm   x}.
\end{eqnarray}
\end{subequations}

\begin{thm}\label{thmDecoupledSchemeEnergy}
The modified  total (free) energy, i.e., the sum of    the modified  Helmholtz free energy and   kinetic energy,  determined by  \eqref{eqDecoupledDiscreteMultiCompononentMassConserve01} and \eqref{eqDecoupledDiscreteMultiComponentMomentumConserveC01} associated with \eqref{eqDiscreteChemicalPotential01}  and \eqref{eqDecoupledDiscreteVelocityStar}, is dissipated with time steps, i.e.
\begin{eqnarray}\label{eqDecoupledDiscreteTotalEnergyDecay}
E^{k+1}+\mathcal{F}^{k+1} \leq E^{k}+\mathcal{F}^{k}.
\end{eqnarray}

\end{thm}
\begin{proof}
We first estimate the difference between $|H^{k+1}|^2$ and $|H^k|^2$ using \eqref{eqDiscreteChemicalPotential01B} as
\begin{align}\label{eqCHEConvexSplittingPoreEnergyDecayProof03}
|H^{k+1}|^2-|H^{k}|^2&= (H^{k+1}+H^k)\(H^{k+1}-H^{k}\)\nonumber\\
&= \sum_{i=1}^M\(\frac{\(H^{k+1}+H^k\)\mu_i^{b}\({\bm   n}^k\)}{2\sqrt{F_b({\bm   n}^k)+C_{T,i}N_i^t}},n_i^{k+1}-n_i^k\).
\end{align}
Since the influence parameter matrix $\big(c_{ij}\big)_{i,j=1}^M$ is symmetric and it is positive definite or positive semi-definite,  we have
\begin{align}\label{eqDiscreteMultiCompononentMassConserveProof01}
F_\nabla ^{k+1}-F_\nabla ^{k}&=\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^Mc_{ij}\(\nabla  n_i^{k+1}\cdot\nabla  n_j^{k+1}-\nabla  n_i^k\cdot\nabla  n_j^k\)d{\bm   x}\nonumber\\
&=\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^Mc_{ij}\(\nabla  \(n_i^{k+1}-n_i^k\)\cdot\nabla  n_j^{k+1}+\nabla  n_i^k\cdot\nabla  \(n_j^{k+1}-n_j^k\)\)d{\bm   x}\nonumber\\
&=\sum_{i,j=1}^M\(\nabla  \(n_i^{k+1}-n_i^k\),c_{ij}\nabla  n_j^{k+1}\)- \sum_{i,j=1}^M\(c_{ij}\nabla  \(n_i^{k+1}-n_i^k\),\nabla  \(n_j^{k+1}-n_j^k\)\)\nonumber\\
&\leq\sum_{i,j=1}^M\(\nabla  \(n_i^{k+1}-n_i^k\),c_{ij}\nabla  n_j^{k+1}\)\nonumber\\
&\leq-\sum_{i,j=1}^M\(n_i^{k+1}-n_i^{k},\nabla\cdot \(c_{ij}\nabla {n_j^{k+1}}\)\).
\end{align}
The inequalities \eqref{eqCHEConvexSplittingPoreEnergyDecayProof03} and  \eqref{eqDiscreteMultiCompononentMassConserveProof01} yield
\begin{equation}\label{eqDiscreteMultiCompononentMassConserveProof02}
\mathcal{F}^{k+1}-\mathcal{F}^{k}=|H^{k+1}|^2-|H^{k}|^2+F_\nabla ^{k+1}-F_\nabla ^{k}\leq\sum_{i=1}^M\(\mu_i^{k+1} ,n_i^{k+1}-n_i^k\).
\end{equation}
Substituting \eqref{eqDecoupledDiscreteMultiCompononentMassConserve01} into \eqref{eqDiscreteMultiCompononentMassConserveProof02}, we derive
\begin{align}\label{eqDiscreteMultiCompononentMassConserveProof03}
\frac{\mathcal{F}^{k+1}-\mathcal{F}^{k}}{\delta t_{k}}&\leq-\sum_{i=1}^M\(\nabla\cdot  (n_i^{k}{\bm   u}_\star^{k})+\nabla\cdot {\bm   J}_i^{k+1},\mu_i^{k+1}\)\nonumber\\
&\leq-\sum_{i=1}^M\(\nabla\cdot  (n_i^{k}{\bm   u}_\star^{k}),\mu_i^{k+1}\) -\sum_{i,j=1}^{M}\big(\mathcal{M}^{k}_{ij}\nabla \mu_{i}^{k+1},\nabla \mu_j^{k+1}\big).
\end{align}

We now turn to consider  the difference between $E^{k+1}$ and $E^k$.  We introduce the intermediate  kinetic energy as $$E_\star^{k}=\frac{1}{2}\(\rho^k{\bm   u}_\star^k,{\bm   u}_\star^k\).$$
The difference between $E^{k+1}$ and $E_\star^k$ is estimated as
 \begin{align}\label{eqDecoupledDiscreteMultiComponentMomentumConserveProof01}
E^{k+1}-E_\star^{k}&=\frac{1}{2}\(\rho^{k+1},|{\bm   u}^{k+1}|^2\)-\frac{1}{2}\(\rho^k,|{\bm   u}_\star^{k}|^2\)\nonumber\\
&=\frac{1}{2}\(\rho^{k},|{\bm   u}^{k+1}|^2-|{\bm   u}_\star^{k}|^2\)+\frac{1}{2}\(\rho^{k+1}-\rho^{k},|{\bm   u}^{k+1}|^2\)\nonumber\\
   &= \(\rho^{k}\({\bm   u}^{k+1}-{\bm   u}_\star^{k}\),{\bm   u}^{k+1}\)-\frac{1}{2}\(\rho^{k},|{\bm   u}^{k+1}-{\bm   u}_\star^{k}|^2\)\nonumber\\
   &~~~+\frac{1}{2}\(\rho^{k+1}-\rho^{k},|{\bm   u}^{k+1}|^2\)\nonumber\\
    &\leq \(\rho^{k}\({\bm   u}^{k+1}-{\bm   u}_\star^{k}\),{\bm   u}^{k+1}\)+\frac{1}{2}\(\rho^{k+1}-\rho^{k},|{\bm   u}^{k+1}|^2\).
\end{align}
On the other hand,  we have the overall mass balance equation as
\begin{eqnarray}\label{eqDecoupledDiscreteMultiCompononentMassConserveProof02}
\frac{ \rho^{k+1}-\rho^{k}}{\delta t_{k}}+\nabla\cdot (\rho^{k}{\bm   u}_\star^{k})+\sum_{i=1}^MM_{w,i}\nabla\cdot {\bm   J}_i^{k+1}=0,
\end{eqnarray}
and taking into account the definition of ${\bm   u}_\star^k$, we rewrite \eqref{eqDecoupledDiscreteMultiComponentMomentumConserveC01} as
\begin{align}\label{eqDecoupledDiscreteMultiCompononentMassConserveProof03}
 \rho^{k}\frac{{\bm   u}^{k+1}-{\bm   u}_\star^{k}}{\delta t_{k}}&=-\rho^{k}{\bm   u}_\star^{k}\cdot\nabla {\bm   u}^{k+1}-\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+1}\cdot\nabla {\bm   u}^{k+1}\nonumber\\
   &~~+\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\).
\end{align}
Substituting \eqref{eqDecoupledDiscreteMultiCompononentMassConserveProof02} and \eqref{eqDecoupledDiscreteMultiCompononentMassConserveProof03} into \eqref{eqDecoupledDiscreteMultiComponentMomentumConserveProof01} yields
\begin{align}\label{eqDecoupledDiscreteMultiCompononentMassConserveProof04}
\frac{E^{k+1}-E_\star^{k}}{\delta t^k}&\leq -\(\rho^{k}{\bm   u}_\star^{k}\cdot\nabla {\bm   u}^{k+1}+\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+1}\cdot\nabla {\bm   u}^{k+1},{\bm   u}^{k+1}\)\nonumber\\
   &~~+\(\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\),{\bm   u}^{k+1}\)\nonumber\\
   &~~-\frac{1}{2}\(\nabla\cdot (\rho^{k}{\bm   u}_\star^{k})+\sum_{i=1}^MM_{w,i}\nabla\cdot {\bm   J}_i^{k+1},|{\bm   u}^{k+1}|^2\)\nonumber\\
   &\leq- \left\|\sqrt{\lambda^k}\nabla\cdot {\bm   u}^{k+1}\right\|^2 -\frac{1}{2} \left\|\sqrt{\eta^{k}}\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\)\right\|^2.
\end{align}
We apply  the definition of ${\bm   u}_\star^k$ to derive
\begin{eqnarray}\label{eqDecoupledDiscreteMultiCompononentMassConserveProof05}
E_\star^{k}-E^{k}&=&\(\rho^k\({\bm   u}_\star^{k} - {\bm   u}^k\),{\bm   u}_\star^k\)-\frac{1}{2}\(\rho^k,|{\bm   u}_\star^{k} - {\bm   u}^k|^2\)\nonumber\\
&\leq&\(\rho^k\({\bm   u}_\star^{k} - {\bm   u}^k\),{\bm   u}_\star^k\)\nonumber\\
&\leq&- \delta t_{k}\sum_{i=1}^M \(n_i^{k}\nabla  \mu_i^{k+1},{\bm   u}_\star^k\)\nonumber\\
&\leq&\delta t_{k}\sum_{i=1}^M \(\nabla\cdot \(n_i^{k}{\bm   u}_\star^k\), \mu_i^{k+1}\).
\end{eqnarray}
Combining   \eqref{eqDiscreteMultiCompononentMassConserveProof03}, \eqref{eqDecoupledDiscreteMultiCompononentMassConserveProof04} and \eqref{eqDecoupledDiscreteMultiCompononentMassConserveProof05} yields
 \begin{align}\label{eqDecoupledDiscreteMultiCompononentEnergyDecay}
\frac{E^{k+1}-E^{k}+\mathcal{F}^{k+1}-\mathcal{F}^{k}}{\delta t_{k}}
&\leq-\sum_{i,j=1}^{M}\big(\mathcal{M}^{k}_{ij}\nabla \mu_{i}^{k+1},\nabla \mu_j^{k+1}\big)- \left\|\sqrt{\lambda^k}\nabla\cdot {\bm   u}^{k+1}\right\|^2\nonumber\\
   &~~~~~~~ -\frac{1}{2} \left\|\sqrt{\eta^{k}}\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\)\right\|^2\leq0,
   \end{align}
which yields  the energy dissipation \eqref{eqDecoupledDiscreteTotalEnergyDecay}.
\end{proof}

\subsection{Component-wise, decoupled  semi-implicit  scheme}\label{secComponentwisescheme}
For the case that diffusion fluxes have a diagonal mobility tensor,    we can design  a component-wise, decoupled  semi-implicit  scheme, which  not only uncouples the tight relationship between molar densities and velocity, but also solves the mass balance equations  by a component-wise way.

 We still use ${\bm   n}^{k}= [n_1^k,n_2^k,\cdots,n_M^k]^T$ to denote the molar density vector at the integer time step $k$. Furthermore, we introduce the molar density vector at the fractional time step $\(k+\frac{i}{M}\)$ and denote it by  $ {\bm   n}^{k+\frac{i}{M}}=\[n_1^{k+1},\cdots,n_i^{k+1},n_{i+1}^{k},\cdots,n_M^{k}\]^T$, where $0\leq i\leq M$;  in particular,  $ {\bm   n}^{k+\frac{i}{M}}={\bm   n}^k$ for $i=0$ and $ {\bm   n}^{k+\frac{i}{M}}={\bm   n}^{k+1}$ for $i=M$. The discrete chemical potential  $\mu_i^{k+\frac{i}{M}}(1\leq i\leq M)$ of component $i$ is defined  as
   \begin{subequations}\label{eqFullyDecoupledDiscreteChemicalPotential01}
 \begin{align}\label{eqFullyDecoupledDiscreteChemicalPotential01A}
\mu_i^{k+\frac{i}{M}}&=\frac{H^{k+\frac{i}{M}}+H^{k+\frac{i-1}{M}}}{2\sqrt{F_b({\bm   n}^{k})+\sum_{j=1}^MC_{T,j}N_j^t}}\mu_i^{b}\({\bm   n}^{k+\frac{i-1}{M}}\)\nonumber\\
&~~~-\sum_{j=1}^i\nabla\cdot \(c_{ij}\nabla {n_j^{k+1}}\)-\sum_{j=i+1}^M\nabla\cdot \(c_{ij}\nabla {n_j^{k}}\),
  \end{align}
  \begin{align}\label{eqFullyDecoupledDiscreteChemicalPotential01B}
\frac{H^{k+\frac{i}{M}}-H^{k+\frac{i-1}{M}}}{\delta t_k}=\int_\Omega\frac{\mu_i^{b}\({\bm   n}^{k+\frac{i-1}{M}}\)}{2\sqrt{F_b({\bm   n}^k)+\sum_{j=1}^MC_{T,j}N_j^t}}\frac{n_i^{k+1}-n_i^k}{\delta t_k}d{\bm   x}.
  \end{align}
 \end{subequations}
   A component-wise intermediate  velocity ${\bm   u}_\star^{k+\frac{i}{M}}$ is defined as
\begin{eqnarray}\label{eqFullyDecoupledDiscreteVelocityStar}
{\bm   u}_\star^{k+\frac{i}{M}} = {\bm   u}_\star^{k+\frac{i-1}{M}}-\frac{\delta t_{k}}{\rho^k}n_i^{k}\nabla  \mu_i^{k+\frac{i}{M}},~~~1\leq i\leq M,
\end{eqnarray}
where ${\bm   u}_\star^{k+0}={\bm   u}_\star^{k}={\bm   u}^k$.  Let $\rho_i=M_{w,i}n_i$ be the mass density of component $i$, and then we introduce a mean intermediate  velocity ${\bm   u}_{\star\star}^k$ as
\begin{eqnarray}\label{eqFullyDecoupledDiscreteVelocityStar02}
{\bm   u}_{\star\star}^k = \sum_{i=1}^M\frac{\rho_i^{k}}{\rho^k}{\bm   u}_\star^{k+\frac{i}{M}}.
\end{eqnarray}

We construct the semi-implicit time   scheme for the molar density balance equation \eqref{eqGeneralNSEQMass}  of component $i$  as
\begin{subequations}\label{eqFullyDecoupledScheme01}
 \begin{equation}\label{eqFullyDecoupledSchemeComponent}
 \frac{ n_i^{k+1}-n_i^{k}}{\delta t_{k}}+\nabla\cdot (n_i^{k}{\bm   u}_\star^{k+\frac{i}{M}})+\nabla\cdot {\bm   J}_i^{k+\frac{i}{M}} =0,
 \end{equation}
 \begin{equation}\label{eqFullyDecoupledSchemeDiffusion}
{\bm   J}_i^{k+\frac{i}{M}} = -\mathcal{M}^k_{i}\nabla \mu_{i}^{k+\frac{i}{M}},
\end{equation}
\end{subequations}
which is a linear equation of $n_i^{k+1}$ only and can be solved  sequently  from $i=1$ to $M$. The semi-implicit time scheme for the momentum balance equation is
\begin{eqnarray}\label{eqFullyDecoupledSchemeVelocity}
&& \rho^{k}\frac{{\bm   u}^{k+1}-{\bm   u}^{k}}{\delta t_{k}}+\rho^{k}{\bm   u}_{\star\star}^k \cdot\nabla {\bm   u}^{k+1}+\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+\frac{i}{M}}\cdot\nabla {\bm   u}^{k+1}=-\sum_{i=1}^Mn_i^{k}\nabla  \mu_i^{k+\frac{i}{M}}\nonumber\\
   &&~~+\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\).
\end{eqnarray}
Summing \eqref{eqFullyDecoupledDiscreteVelocityStar} from $i=1$ to $M$ yields
\begin{eqnarray}\label{eqFullyDecoupledDiscreteVelocityStar03}
{\bm   u}_\star^{k+1} = {\bm   u}^k-\frac{\delta t_{k}}{\rho^k}\sum_{i=1}^Mn_i^{k}\nabla  \mu_i^{k+\frac{i}{M}}.
\end{eqnarray}
Consequently, the equation \eqref{eqFullyDecoupledSchemeVelocity} can be reformulated as
\begin{eqnarray}\label{eqFullyDecoupledSchemeVelocity02}
&& \rho^{k}\frac{{\bm   u}^{k+1}-{\bm   u}_\star^{k+1}}{\delta t_{k}}+\rho^{k}{\bm   u}_{\star\star}^k \cdot\nabla {\bm   u}^{k+1}+\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+\frac{i}{M}}\cdot\nabla {\bm   u}^{k+1}\nonumber\\
   &&~~=\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\).
\end{eqnarray}
This is a linear equation of velocity ${\bm   u}^{k+1}$ and easy to be solved.
In the convection term of \eqref{eqFullyDecoupledSchemeVelocity02}, we use the mean  intermediate  velocity ${\bm   u}_{\star\star}^{k}$ instead of ${\bm   u}^k$ or  ${\bm   u}_\star^{k+\frac{i}{M}}$ to match  the mass balance equations.

It is apparent that the above component-wise  approach can be directly applied for the IEQ-based component-wise schemes and for the Cahn-Hilliard-type models studied in \cite{kousun2015CHE}.

We now prove that the component-wise, decoupled scheme satisfies the discrete energy dissipation law.

\begin{thm}\label{thmFullyDecoupledSchemeEnergy}
The sum of    the modified  Helmholtz free energy and   kinetic energy determined by  \eqref{eqFullyDecoupledScheme01} and \eqref{eqFullyDecoupledSchemeVelocity02} associated with \eqref{eqFullyDecoupledDiscreteChemicalPotential01},  \eqref{eqFullyDecoupledDiscreteVelocityStar} and \eqref{eqFullyDecoupledDiscreteVelocityStar02} is dissipated with time steps, i.e.
\begin{eqnarray}\label{eqFullyDecoupledDiscreteTotalEnergyDecay}
E^{k+1}+\mathcal{F}^{k+1} \leq E^{k}+\mathcal{F}^{k},
\end{eqnarray}
where $E^k$ and $\mathcal{F}^k$ are still defined as in \eqref{eqDiscreteTotalEnergy}.
\end{thm}
\begin{proof}
Using \eqref{eqFullyDecoupledDiscreteChemicalPotential01B}, we derive the   difference between $|H^{k+\frac{i}{M}}|^2$ and $|H^{k+\frac{i-1}{M}}|^2~ (1\leq i\leq M)$ as
\begin{align}\label{eqFullyDecoupledSchemeEnergyProof01}
|H^{k+\frac{i}{M}}|^2-|H^{k+\frac{i-1}{M}}|^2&=   \(H^{k+\frac{i}{M}}+H^{k+\frac{i-1}{M}}\)\(H^{k+\frac{i}{M}}-H^{k+\frac{i-1}{M}}\)\nonumber\\
&=  \(\frac{\(H^{k+\frac{i}{M}}+H^{k+\frac{i-1}{M}}\)\mu_i^{b}\({\bm   n}^{k+\frac{i-1}{M}}\)}{2\sqrt{F_b({\bm   n}^k)+\sum_{j=1}^MC_{T,j}N_j^t}},n_i^{k+1}-n_i^k\).
\end{align}
The gradient contribution of Helmholtz free energy at the time step $(k+\frac{i}{M})$ can be expressed as
\begin{eqnarray}\label{eqFullyDecoupledSchemeEnergyProof02}
F_\nabla ^{k+\frac{i}{M}}&=&\frac{1}{2}\int_{\Omega}\sum_{j,l=1}^i c_{jl}\nabla  n_j^{k+1}\cdot\nabla  n_l^{k+1}d{\bm   x}+\frac{1}{2}\int_{\Omega}\sum_{j,l=i+1}^Mc_{jl}\nabla  n_j^k\cdot\nabla  n_l^kd{\bm   x}\nonumber\\
&&+\sum_{j=1}^i\sum_{l=i+1}^M\int_{\Omega}c_{jl}\nabla  n_j^{k+1}\cdot\nabla  n_l^kd{\bm   x}.
\end{eqnarray}
Taking into account  $c_{ij}=c_{ji}$ and $c_{ij}>0$,  we derive
\begin{eqnarray}\label{eqFullyDecoupledSchemeEnergyProof03}
F_\nabla ^{k+\frac{i}{M}}-F_\nabla ^{k+\frac{i-1}{M}}&=&\frac{1}{2}\int_{\Omega}c_{ii}\(\nabla  n_i^{k+1}\cdot\nabla  n_i^{k+1}-\nabla  n_i^k\cdot\nabla  n_i^k\)d{\bm   x}\nonumber\\
&&+\int_{\Omega}\sum_{j=1}^{i-1} c_{ij}\nabla  \(n_i^{k+1}-n_i^k\)\cdot\nabla  n_j^{k+1}d{\bm   x}\nonumber\\
&&+\int_{\Omega}\sum_{j=i+1}^{M} c_{ij}\nabla  \(n_i^{k+1}-n_i^k\)\cdot\nabla  n_j^{k}d{\bm   x}\nonumber\\
&\leq&\sum_{j=1}^{i}\(\nabla  \(n_i^{k+1}-n_i^k\),c_{ij}\nabla  n_j^{k+1}\)\nonumber\\
&&+\sum_{j=i+1}^{M}\(\nabla  \(n_i^{k+1}-n_i^k\),c_{ij}\nabla  n_j^{k}\)\nonumber\\
&\leq&-\(n_i^{k+1}-n_i^{k},\sum_{j=1}^{i}\nabla\cdot  c_{ij}\nabla {n_j^{k+1}}+\sum_{j=i+1}^{M}\nabla\cdot  c_{ij}\nabla {n_j^{k}}\).
\end{eqnarray}
By the definition of $\mu_i^{k+\frac{i}{M}}$ given in \eqref{eqFullyDecoupledDiscreteChemicalPotential01}, we obtain from the estimates \eqref{eqFullyDecoupledSchemeEnergyProof01} and  \eqref{eqFullyDecoupledSchemeEnergyProof03} that
\begin{align}\label{eqFDCHEConvexSplittingPoreEnergyDecayProof05}
&|H^{k+\frac{i}{M}}|^2-|H^{k+\frac{i-1}{M}}|^2+F_\nabla ^{k+\frac{i}{M}}-F_\nabla ^{k+\frac{i-1}{M}}\nonumber\\
&~~\leq \(\mu_i^{k+\frac{i}{M}} ,n_i^{k+1}-n_i^k\)\nonumber\\
&~~\leq -\delta t_k\(\mu_i^{k+\frac{i}{M}} ,\nabla\cdot (n_i^{k}{\bm   u}_\star^{k+\frac{i}{M}})-\nabla\cdot \mathcal{M}^k_{i}\nabla \mu_{i}^{k+\frac{i}{M}}\)\nonumber\\
&~~\leq \delta t_k\({\bm   u}_\star^{k+\frac{i}{M}} , n_i^{k}\nabla \mu_i^{k+\frac{i}{M}}\) -\delta t_k\left\|\sqrt{\mathcal{M}^k_{i}}\nabla \mu_{i}^{k+\frac{i}{M}}\right\|^2,
\end{align}
where the second equality is obtained by using \eqref{eqFullyDecoupledScheme01}.
Summing  up \eqref{eqFDCHEConvexSplittingPoreEnergyDecayProof05} from $i=1$ to $M$   yields
\begin{align}\label{eqFDCHEConvexSplittingPoreEnergyDecayProof06}
\mathcal{F}^{k+1}-\mathcal{F}^{k}&=|H^{k+1}|^2-|H^{k}|^2+F_\nabla ^{k+1} -F_\nabla ^{k}\nonumber\\
&=\sum_{i=1}^M\(|H^{k+\frac{i}{M}}|^2-|H^{k+\frac{i-1}{M}}|^2+F_\nabla ^{k+\frac{i}{M}}-F_\nabla ^{k+\frac{i-1}{M}}\)\nonumber\\
&\leq \delta t_k\sum_{i=1}^M\({\bm   u}_\star^{k+\frac{i}{M}} , n_i^{k}\nabla \mu_i^{k+\frac{i}{M}}\) -\delta t_k\sum_{i=1}^M\left\|\sqrt{\mathcal{M}^k_{i}}\nabla \mu_{i}^{k+\frac{i}{M}}\right\|^2.
\end{align}

We define the intermediate  kinetic energy as $$E_\star^{k+\frac{i}{M}}=\frac{1}{2}\(\rho^k{\bm   u}_\star^{k+\frac{i}{M}},{\bm   u}_\star^{k+\frac{i}{M}}\).$$
Using the definition \eqref{eqFullyDecoupledDiscreteVelocityStar} of intermediate velocities, we derive
\begin{eqnarray}\label{eqFullyDecoupledDecoupledExplicitDiscreteVelocityStarProof01}
E_\star^{k+\frac{i}{M}}-E_\star^{k+\frac{i-1}{M}}&=&\frac{1}{2}\(\rho^k{\bm   u}_\star^{k+\frac{i}{M}},{\bm   u}_\star^{k+\frac{i}{M}}\)-\frac{1}{2}\(\rho^k{\bm   u}_\star^{k+\frac{i-1}{M}},{\bm   u}_\star^{k+\frac{i-1}{M}}\)\nonumber\\
&=&\(\rho^k\({\bm   u}_\star^{k+\frac{i}{M}} - {\bm   u}_\star^{k+\frac{i-1}{M}}\),{\bm   u}_\star^{k+\frac{i}{M}}\)-\frac{1}{2}\(\rho^k,\left|{\bm   u}_\star^{k+\frac{i}{M}} - {\bm   u}_\star^{k+\frac{i-1}{M}}\right|^2\)\nonumber\\
&\leq&\(\rho^k\({\bm   u}_\star^{k+\frac{i}{M}} - {\bm   u}_\star^{k+\frac{i-1}{M}}\),{\bm   u}_\star^{k+\frac{i}{M}}\)\nonumber\\
&=&- \delta t_{k} \(n_i^{k}\nabla  \mu_i^{k+\frac{i}{M}},{\bm   u}_\star^{k+\frac{i}{M}}\).
\end{eqnarray}
The sum of \eqref{eqFullyDecoupledScheme01} multiplied by $M_{w,i}$ leads to the overall  mass balance equation
\begin{eqnarray}\label{eqFullyDecoupledDiscreteMultiCompononentMassConserveProof02}
\frac{ \rho^{k+1}-\rho^{k}}{\delta t_{k}}=-\nabla\cdot \(\rho^{k}{\bm   u}_{\star\star}^k\) -\sum_{i=1}^MM_{w,i}\nabla\cdot {\bm   J}_i^{k+\frac{i}{M}},
\end{eqnarray}
where \eqref{eqFullyDecoupledDiscreteVelocityStar02} is also used to get the first term on the right-hand side.
We  estimate the difference between $E^{k+1}$ and $E_\star^k$ as
\begin{align}\label{eqFullyDecoupledDiscreteMultiComponentMomentumConserveProof01}
\frac{E^{k+1}-E_\star^{k+1}}{\delta t_k}&=\frac{1}{2\delta t_k}\(\rho^{k+1},|{\bm   u}^{k+1}|^2\)-\frac{1}{2\delta t_k}\(\rho^k,|{\bm   u}_\star^{k+1}|^2\)\nonumber\\
    &\leq \(\rho^{k}\frac{{\bm   u}^{k+1}-{\bm   u}_\star^{k+1}}{\delta t_k},{\bm   u}^{k+1}\)+\frac{1}{2}\(\frac{\rho^{k+1}-\rho^{k}}{\delta t_k},|{\bm   u}^{k+1}|^2\)\nonumber\\
&\leq -\(\rho^{k}{\bm   u}_{\star\star}^k \cdot\nabla {\bm   u}^{k+1}+\sum_{i=1}^MM_{w,i}{\bm   J}_i^{k+\frac{i}{M}}\cdot\nabla {\bm   u}^{k+1},{\bm   u}^{k+1}\)\nonumber\\
    &~~~+\(\nabla\(\lambda^k\nabla\cdot {\bm   u}^{k+1}\)+\nabla\cdot \eta^k\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\),{\bm   u}^{k+1}\)\nonumber\\
    &~~~-\frac{1}{2}\(\nabla\cdot \(\rho^{k}{\bm   u}_{\star\star}^k\)+\sum_{i=1}^MM_{w,i}\nabla\cdot {\bm   J}_i^{k+\frac{i}{M}},|{\bm   u}^{k+1}|^2\)\nonumber\\
    &\leq- \left\|\sqrt{\lambda^k}\nabla\cdot {\bm   u}^{k+1}\right\|^2 -\frac{1}{2} \left\|\sqrt{\eta^{k}}\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\)\right\|^2,
\end{align}
where the third equality is obtained by using \eqref{eqFullyDecoupledSchemeVelocity02} and \eqref{eqFullyDecoupledDiscreteMultiCompononentMassConserveProof02}.
Combining  \eqref{eqFullyDecoupledDecoupledExplicitDiscreteVelocityStarProof01} and \eqref{eqFullyDecoupledDiscreteMultiComponentMomentumConserveProof01} yields
 \begin{align}\label{eqFullyDecoupledDiscreteMultiComponentMomentumConserveProof02}
E^{k+1}-E^{k}&=E^{k+1}-E_\star^{k+1}+\sum_{i=1}^M\(E_\star^{k+\frac{i}{M}}-E_\star^{k+\frac{i-1}{M}}\)\nonumber\\
    &\leq - \delta t_k\left\|\sqrt{\lambda^k}\nabla\cdot {\bm   u}^{k+1}\right\|^2 -\frac{1}{2}\delta t_k \left\|\sqrt{\eta^{k}}\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\)\right\|^2\nonumber\\
    &~~~- \delta t_{k}\sum_{i=1}^M \(n_i^{k}\nabla  \mu_i^{k+\frac{i}{M}},{\bm   u}_\star^{k+\frac{i}{M}}\).
\end{align}

Finally,   it is derived from \eqref{eqFDCHEConvexSplittingPoreEnergyDecayProof06} and  \eqref{eqFullyDecoupledDiscreteMultiComponentMomentumConserveProof02} that
 \begin{align}\label{eqDecoupledDiscreteMultiCompononentEnergyDecay}
\frac{E^{k+1}-E^{k}+\mathcal{F}^{k+1}-\mathcal{F}^{k}}{\delta t_{k}}&\leq-\left\|\sqrt{\mathcal{M}^k_{i}}\nabla \mu_{i}^{k+\frac{i}{M}}\right\|^2- \left\|\sqrt{\lambda^k}\nabla\cdot {\bm   u}^{k+1}\right\|^2\nonumber\\
   & ~~~-\frac{1}{2} \left\|\sqrt{\eta^{k}}\(\nabla{\bm   u}^{k+1}+\(\nabla{\bm   u}^{k+1}\)^T\)\right\|^2,
   \end{align}
which yields  the energy dissipation \eqref{eqFullyDecoupledDiscreteTotalEnergyDecay}.
\end{proof}

\section{Numerical tests}

In this section,  the proposed methods are applied to simulate  multi-component two-phase   flow problems.  We consider a binary   mixture and a ternary mixture in   a square  domain $\Omega$ with the length $20$~nm. The  boundary conditions are taken as ${\bm   u}=0,~{\bm   J}_i\cdot\bm{\gamma}_{\partial\Omega}=0$ and $\nabla  n_i\cdot\bm{\gamma}_{\partial\Omega}=0$ on the boundary $\partial\Omega$, where $\bm\gamma_{\partial\Omega}$ is the normal unit outward vector  to  $\partial \Omega$.
For spatial discretization,     a  uniform  rectangular mesh with $40\times40$ elements is used;  the cell-centered finite difference method and the upwind scheme are employed  to discretize  the mass balance equation;  the finite volume method  on the staggered mesh \cite{Tryggvason2011book}  is applied for the momentum balance equation. We note that  the above spatial discretization methods  have  equivalent relationships with special mixed finite element methods under specified  quadrature rules \cite{arbogast1997mixed,Girault1996Mac}.  The energy parameters are chosen as $C_{T,i}=0$.

\subsection{Binary mixture}
In this example, we consider a binary   mixture composed of methane (C$_1$) and pentane (C$_5$) at a constant temperature 310 K.  At the initial time, a square shape  droplet is located in   the center of the domain.  The initial gas molar densities of C$_1$ and C$_5$     are $7.4302$ kmol/m$^3$ and $0.6736$ kmol/m$^3$ respectively, while the initial liquid molar densities of C$_1$ and C$_5$  are $6.8663$ kmol/m$^3$ and $ 4.7915$ kmol/m$^3$  respectively. The initial molar density distributions for  C$_1$ and C$_5$ are illustrated in  Figure \ref{SquareC1andC5MolarDensityOfC5Temperature310K}(a) and (d) respectively.  We use the diffusion mobility formulation given by  \eqref{eqMultiCompononentMassConserveDiffusionChoiceB}  with the coefficients $\mathcal{D}_{12}=\mathcal{D}_{21}=10^{-8}$ m$^2$/s. The volumetric  viscosity and    the shear viscosity are taken  as $\xi=\eta=10^{-4}$ Pa$\cdot$s.  The time step size is taken as $10^{-12}$ s, and 200  time steps are simulated.

The  velocity-density decoupled  method proposed in Sub-section \ref{secVelocityDensityDecoupledScheme} is applied to simulate the dynamical evolution of the square-shaped droplet. In Figure \ref{SquareC1andC5EnergyTemperature310}, we show the evolution profiles of the modified  total energy (i.e., the sum of    the modified  Helmholtz free energy and   kinetic energy) with time steps;  we also depict the original total energy (i.e., the sum of    the original  Helmholtz free energy and   kinetic energy) for the sake of comparison.
It is observed from Figure \ref{SquareC1andC5EnergyTemperature310}(a) that although the modified total (free) energy is slightly less than the original energy, both  total (free) energies are   strictly dissipated with time steps, and moreover,   Figure \ref{SquareC1andC5EnergyTemperature310}(b), which is a zoom-in plot of Figure \ref{SquareC1andC5EnergyTemperature310}(a)  in the later time steps, demonstrates  that both total (free) energies remain to decrease.   As a result, the proposed method can preserve the energy-dissipation  feature.

  Figure \ref{SquareC1andC5MolarDensityOfC5Temperature310K} depicts the evolution process  of each component molar density, and it   is clearly observed that the droplet is gradually reshaping  to a circle from its initial square shape due to   chemical potential gradients.  In Figures \ref{SquareC1andC5VelocityTemperature310K}, we show  the fluid motion  driven by chemical potential gradients, including  the velocity field and  magnitudes of both velocity components.

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.45\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=2in]{figs/SquareC1andC5TotalEnergyTemperature310.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.45\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=2in]{figs/SquareC1andC5TotalEnergyPartTemperature310.eps}
            \end{minipage}
            }
           \caption{Binary mixture:  total energy dissipation with time steps.}
            \label{SquareC1andC5EnergyTemperature310}
 \end{figure}

\begin{figure}
            \centering \subfigure[C$_1$ at the initial time]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC1Temperature310KiT0.eps}
            \end{minipage}
            }
            \centering \subfigure[C$_1$ at the 80th time step]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC1Temperature310KiT80.eps}
            \end{minipage}
            }
           \centering \subfigure[C$_1$ at the 200th time step]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC1Temperature310KiT200.eps}
            \end{minipage}
            }
            \centering \subfigure[C$_5$ at the initial time]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC5Temperature310KiT0.eps}
            \end{minipage}
            }
            \centering \subfigure[C$_5$ at the 80th time step]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC5Temperature310KiT80.eps}
            \end{minipage}
            }
           \centering \subfigure[C$_5$ at the 200th time step]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/SquareC1andC5MolarDensityOfC5Temperature310KiT200.eps}
             \end{minipage}
            }
            \caption{Binary mixture:  molar densities of C$_1$ and C$_5$  at  different time steps.}
            \label{SquareC1andC5MolarDensityOfC5Temperature310K}
 \end{figure}

\begin{figure}
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/SquareC1andC5VelocityQuiverTemperature310KiT80.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/SquareC1andC5VelocityXTemperature310KiT80.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/SquareC1andC5VelocityYTemperature310KiT80.eps}
            \end{minipage}
            }
           \caption{Binary mixture:   (a) flow quiver, (b) magnitude contour of $x$-direction velocity component, and (c) magnitude contour of $y$-direction velocity component   at the  80th  time step.}
            \label{SquareC1andC5VelocityTemperature310K}
 \end{figure}

\subsection{Ternary mixture}
In this example, we consider a ternary   mixture composed of methane (C$_1$) pentane (C$_5$) and  decane (C$_{10}$) at a constant temperature 323 K.   The initial gas molar densities of C$_1$, C$_5$  and C$_{10}$   are $10.516$ kmol/m$^3$,  $0.77$ kmol/m$^3$ and  $0.184$ kmol/m$^3$ respectively, while the initial liquid molar densities of C$_1$, C$_5$  and C$_{10}$  are $7.8412$ kmol/m$^3$,  $1.9925$ kmol/m$^3$ and $1.433$ kmol/m$^3$  respectively.  At the initial time, there are two square-shaped  droplets in   the  domain, as shown in the first figures of Figures \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323K}, \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323K} and \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323K} respectively.  The diffusion fluxes are  formulated by  \eqref{eqMultiCompononentMassConserveDiffusionChoiceA}  with  the diffusion coefficients $D_i=3\times10^{-8}$ m$^2$/s $(1\leq i\leq3)$. The volumetric  viscosity and    the shear viscosity are set  as $\xi=\eta=10^{-4}$ Pa$\cdot$s.  We take the time step size equal to $10^{-12}$ s, and we simulate the evolution process for 1000  time steps.

We employ the  component-wise,  decoupled numerical  scheme proposed in Sub-section \ref{secComponentwisescheme}. The original and modified  total energies and their zoom-in plots are shown in Figure \ref{MCTwoSquareCh4andTwoHydrocarbonsTotalEnergyTemperature323}. We still see that both of total energies are dissipated  with time steps. We also note that in practical computations, the component-wise method is really effective for the mixtures composed of multiple components since it only needs to solve one more mass-balance equation as a new component is added.

Figures \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323K}, \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323K} and \ref{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323K} illustrate  the  molar density configurations of  three components.  In Figure \ref{MCTwoSquareCh4andTwoHydrocarbonsVelocityTemperature323K}, we depict the fluid motion, including   the velocity fields and  magnitudes of  velocity components,  at different time steps. The simulation results show that due to   chemical potential gradients,   two droplets      are first  emerging  with each other, and at the later time,  the merged  droplets are gradually  reshaping  into a circle.

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.45\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=2in]{figs/MCTwoSquareCh4andTwoHydrocarbonsTotalEnergyTemperature323.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.45\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=2in]{figs/MCTwoSquareCh4andTwoHydrocarbonsTotalEnergyPartTemperature323.eps}
            \end{minipage}
            }
           \caption{Ternary mixture: energy dissipation with time steps.}
            \label{MCTwoSquareCh4andTwoHydrocarbonsTotalEnergyTemperature323}
 \end{figure}

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT0.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT30.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT100.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT200.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT500.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323KiT1000.eps}
            \end{minipage}
            }
           \caption{Ternary mixture: C$_1$ molar densities at  the the initial(a), 30th(b), 100th(c), 200th(d), 500th(e) and 1000th(f)   time step  respectively.}
            \label{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfCH4Temperature323K}
 \end{figure}

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT0.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT30.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT100.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT200.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT500.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323KiT1000.eps}
            \end{minipage}
            }
           \caption{Ternary mixture: C$_{5}$ molar densities at  the initial(a),  30th(b), 100th(c), 200th(d), 500th(e) and 1000th(f)   time step  respectively.}
            \label{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC5Temperature323K}
 \end{figure}

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT0.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT30.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT100.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT200.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT500.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323KiT1000.eps}
            \end{minipage}
            }
           \caption{Ternary mixture: C$_{10}$ molar densities at  the initial(a),   30th(b), 100th(c), 200th(d), 500th(e) and 1000th(f)     time step  respectively.}
            \label{MCTwoSquareCh4andTwoHydrocarbonsMolarDensityOfnC10Temperature323K}
 \end{figure}

\begin{figure}
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityQuiverTemperature323KiT100.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityXTemperature323KiT100.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityYTemperature323KiT100.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityQuiverTemperature323KiT500.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityXTemperature323KiT500.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityYTemperature323KiT500.eps}
            \end{minipage}
            }
           \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityQuiverTemperature323KiT1000.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
               \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityXTemperature323KiT1000.eps}
            \end{minipage}
            }
            \centering \subfigure[]{
            \begin{minipage}[b]{0.3\textwidth}
            \centering
             \includegraphics[width=0.95\textwidth,height=0.9\textwidth]{figs/MCTwoSquareCh4andTwoHydrocarbonsVelocityYTemperature323KiT1000.eps}
            \end{minipage}
            }
           \caption{Ternary mixture:   flow quivers (left column), magnitude contours of $x$-direction velocity component (center column), and magnitude contours of $y$-direction velocity component (right column)   at the 100th(top row), 500th(center row), and 1000th(bottom row) time step  respectively.}
            \label{MCTwoSquareCh4andTwoHydrocarbonsVelocityTemperature323K}
 \end{figure}

\section{Conclusions}

Two decoupled  energy-stable numerical  schemes are developed for    multi-component two-phase compressible flows with a  realistic equation of state (e.g. Peng-Robinson equation of state). In these methods,  the scalar auxiliary variable (SAV) approach is applied to deal with the bulk Helmholtz free energy, and moreover, we propose a component-wise SAV approach, which is extremely  efficient and easy-to-implement for multi-component flows. In order to uncouple the tight  relationship between velocity and molar densities, we introduce two intermediate  velocities, one  of which has a component-wise form matching the component-wise SAV approach.
The intermediate  velocities are involved in the discrete  formulation of the  momentum balance equation, which establishes  the consistent relationships with  the mass balance equations.
The proposed numerical  schemes  only need to solve  a sequence of linear equations at each time step.  The discrete unconditional energy dissipation laws of the proposed methods are proved rigorously.  Numerical results validate  the effectiveness of the proposed methods.

\begin{appendix}\label{appendix}

\section{Helmholtz free energy density}\label{appendixHelmholtz}
Let $R$ be the universal gas constant and let $T$ be the specified temperature. The three contributions of  Helmholtz free energy density $f_{b}({\bm   n})$ based on Peng-Robinson equation of state are formulated as
 \begin{eqnarray}\label{eqHelmholtzEnergy_a0_01}
    f_b^{\textnormal{ideal}}({\bm   n})= RT\sum_{i=1}^{M}n_i\(\ln n_i-1\),
\end{eqnarray}
\begin{eqnarray}\label{eqHelmholtzEnergy_a0_02}
    f_b^{\textnormal{repulsion}}({\bm   n})=-nRT\ln\(1-bn\),
\end{eqnarray}
\begin{eqnarray}\label{eqHelmholtzEnergy_a0_03}
    f_b^{\textnormal{attraction}}({\bm   n})= \frac{a(T)n}{2\sqrt{2}b}\ln\(\frac{1+(1-\sqrt{2})b n}{1+(1+\sqrt{2})b n}\),
\end{eqnarray}
where  $n=\sum_{i=1}^Mn_i$ is the overall molar density. Here,  $a$ and $b$ are the energy parameter and  the covolume respectively, which depend on  the mixture composition and  temperature.
Let us denote by $T_{c_i}$ and $P_{c_i}$  the   critical temperature and critical pressure of component $i$ respectively.  For the $i$th component, we let the reduced temperature be  $T_{r_i}=T/T_{c_i}$.
The parameters $a_{i}$ and $b_{i}$ are calculated as
\begin{eqnarray}\label{eqHelmholtzEnergy_ab}
   a_{i}= 0.45724\frac{R^2T_{c_i}^2}{P_{c_i}}\[1+m_i(1-\sqrt{T_{r_i}})\]^2,~~~~b_{i}= 0.07780\frac{RT_{c_i}}{P_{c_i}}.
\end{eqnarray}
We denote by $\omega_i$  the acentric factor of component $i$.
 The coefficients $m_i$ are calculated  by the following relations
\begin{eqnarray*}
 m_i=0.37464 + 1.54226\omega_i-  0.26992\omega_i^2 ,~~\omega_i\leq0.49,
\end{eqnarray*}
\begin{eqnarray*}
 m_i=0.379642+1.485030\omega_i-0.164423\omega_i^2 +0.016666 \omega_i^3,~~\omega_i>0.49.
\end{eqnarray*}
  We denote by $y_i=n_i/n$ the mole fraction of component $i$ and let $k_{ij}$ be the  binary interaction coefficients for the energy parameters.  Then we calculate  $a(T)$ and $b$   as
\begin{eqnarray*}
   a =\sum_{i=1}^M\sum_{j=1}^M y_i y_j \(a_ia_j\)^{1/2}(1-k_{ij}),~~~~b =\sum_{i=1}^M  y_i b_{i}.
\end{eqnarray*}

We list some physical parameters of three substances in Table \ref{tabParametersPREOS}.
 \begin{table}[htp]
\caption{Physical parameters}
\begin{center}
\begin{tabular}{cccccc}
\hline
Substance & $P_c$(bar) & $T_c$(K) & Acentric factor & $M_w$(g/mole)\\
\hline
methane     &  45.99                          & 190.56                            &0.011              &16.04 \\
pentane      &  33.70                          & 469.7                              &0.251              &72.15\\
decane       &  21.1                            & 617.7                              &0.489              &142.28\\
\hline
\end{tabular}
\end{center}
\label{tabParametersPREOS}
\end{table

\section{Influence parameters}\label{appendixInfluenceParameters}
 The  influence parameters are generally  assumed to rely  on the temperature but  independent of    molar densities.  We  now provide  the formulations of the  influence parameters. First, we formulate the  influence parameter $c_{i}$ of  component $i$ as \cite{miqueu2004modelling}
\begin{eqnarray*}
 c_i=a_ib_i^{2/3}\[\gamma_i(1-T_{r_i})+\phi_i\],
\end{eqnarray*}
where  $a_i$ and $b_i$ are given in \eqref{eqHelmholtzEnergy_ab}, and  $\gamma_i$ and $\phi_i$ are the coefficients
 correlated merely with the acentric factor $\omega_i$ of   component $i$
 by the following relations
\begin{eqnarray*}
 \gamma_i=-\frac{10^{-16}}{1.2326+1.3757\omega_i},~~~~\phi_i=\frac{10^{-16}}{0.9051+1.5410\omega_i}.
\end{eqnarray*}

The cross influence parameter between binary components   is generally  calculated  as a modified geometric mean of the pure component
influence parameters $c_i$ and $c_j$
\begin{eqnarray*}\label{eqCij}
 c_{ij}= (1-\beta_{ij})\sqrt{c_ic_j},
\end{eqnarray*}
where  $\beta_{ij}$ are the binary interaction coefficients   satisfying the symmetry $c_{ij}=c_{ji}$ and $\beta_{ii}=0,~0\leq\beta_{ij}<1$.   In  numerical tests, we take $\beta_{ij}=0.5$ for $i\neq j$.

\end{appendix}

\small

redpajama_set_name: RedPajamaArXiv


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text:
Keanu Reeves is making a new Constantine movie
17 September 2022 von admin
Keanu Reeves circa 2005
Photo: Franco Origlia (Getty Images)
Warner Bros. revealed today that it's bringing Keanu Reeves back to play John Constantine, the famed DC Comics warlock, for a sequel to Reeves' 2005 "superhero" film.
The sequel would re-team Reeves with original director Francis Lawrence, who's all over the news this week—having recently also been tapped to direct a biopic about the band Sublime. The proposed film also comes as Warner Bros., uh… Well, how do we put this delicately? Let's go with "Attempts to solidify its comic book movie strategy into something that looks like it might have come from an actual corporate machine, and not a guy who couldn't find his ass with two hands and a map to the ass store."
(The studio, recently merged with Discovery, is apparently struggling to find someone who wants the unenviable job of running their overall superhero movie division, i.e., spending the next two years of their life getting compared unfavorably to Marvel's Kevin Feige while Warner/Discovery CEO David "Movie Murderer" Zaslav breathes directly down their neck.)
It's not entirely hard to see why Constantine might appeal for a sequel treatment, though. Reeves is, if anything, even more popular now than he was back in 2005, when the original movie—based off of DC's Hellblazer comics, and a character originally co-created by Alan Moore—clocked a respectable $230 million at the box office. The character has also been kept nicely alive in the public consciousness: Matt Ryan played him (with significantly more comics-faithful swagger than Reeves, we feel moved to note) for several years across multiple D.C.-based TV shows, and the Constantine family name is currently alive and well in Netflix's Sandman. Plus, the sequel is being written by Akiva Goldsman and produced by J.J. Abrams' Bad Robot—and while Goldsman's presence isn't necessarily a hallmark of quality, it does promise that a decent chunk of change is likely to get tossed at this thing.
It'll be interesting to see how much of the original Constantine cast returns for the film; Rachel Weisz co-starred in the first movie, and Tilda Swinton and Peter Stormare both had major roles—Stormare's turn as Satan at the film's climax is still one of his best, most scenery-chewing performances in a career that's never lacked for them.
Nach dem Erfolg von Cobra Kai ist ein neuer Karate-Kinderfilm auf dem Weg
Eheim Vivaline 180 Led Beleuchtung


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You will have a walk along the famous Albert Street, where Art Nouveau buildings range from richly decorated to modest, from Decorative Art Nouveau to National Romanticism. The buildings are highly decorative yet deeply functional.
Decorated with various symbols and mythological characters, each building tells us a different story. Scary lion and screaming men masks side by side with contemplative women's faces, mysterious sphinxes, delicately carved flowers, birds and animals - it is easy to get lost in such an abundance of images. But don't worry - our guides will help you interpret the symbolic content of the decorations, and tell you a lot of myths and legends.
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text:
Q: Where is the data directory in Redis? After writing some data to a redis server, I could read the data from a client.
However, how can I find the data directory on the file system?

A: Find your Redis configuration directory, probably /etc/redis. Then look in the config file called redis.conf and find the line that starts dir.
It will look similar to this:
dir /etc/redis/database

This will do the job slowly but surely if you can't be bothered to look :-)
sudo find / -name "redis.conf" -exec grep "^dir" {} \; 2> /dev/null
dir /etc/redis

or if you want the config filename as well:
sudo find / -name "redis.conf" -exec grep -H "^dir" {} \; 2> /dev/null
/private/etc/redis/redis.conf:dir /etc/redis

Other possibilities you can check are whether Redis was started with a custom config file as its first parameter like this:
redis-server /path/to.custom/config-file

or with the dir option set on the commandline like this:
redis-server dir /path/to/data

Use
ps -aef | grep redis

to look for these options.

A: Because Redis config file can be located in several possible places (depending on the system or container, such as /opt/redis/ in my case), a general solution to find the currently configured location of the RDB file (as set using dir in redis.conf - if config file is used at all*) is:
$ cat $(cd / && find | grep redis.conf) | grep dir

*Note that true to its simplicity Redis by default ships without any config files (using built-in configuration which depends on the version, see docs), and this is indeed the case for the official redis:latest container.

A: Quickest method: use redis-cli.
redis-cli config get dir

If you have authentication configured, you will need to pass that in using -a password Replacing "password" with your password.

redpajama_set_name: RedPajamaStackExchange


The level of swamp water in the swimming pool has been dropping slowly but steadily as the sump pump grinds on. Eventually the pump becomes too gunked up with congealed algae and leaves in the shallow water, and we must empty the rest of the pool by hand. Time for a bucket brigade!
The bucket brigade starts with me down in the bottom of the pool, trying not to slip on the sloped surface, scooping up buckets full of rank liquid. Allysen and Jayce are positioned topside, taking the pails from me. There's no really handy place to dump the slop, so they decide to hurl the water through a wrought iron fence into the empty lot beyond. Memo for the future: It's really important how you aim the bucketful of water. If you don't, it has a tendency to come right back at you.
The iguana is back in the pool!
We have had a variety of wildlife visitations to the pool, starting with the tarantula, and followed by a South American cane toad (with toxic skin). They may all be looking for water, but the pool is now empty. Today (January 21), we sight our first iguana, lounging at the pool's edge. (Jayce just missed it, as she took the 3 a.m. flight home, and in fact has just texted to report wheels down in Boston.) Ugly sucker, very prehistoric in appearance.
Everyone stops work to gawk at the critter, which soon goes into the empty pool, keeping Estevan company as he power washes the sides of the pool.
Eventually he drives the iguana out, and it disappears over the hill. An hour later, it reappears. I chase it away with a broom, and again it disappears over the hill.
Half an hour later it's back. In the pool. Or no, this is a different one—larger and greener in color, perhaps four feet long from nose to tip of tail, and a foot and a half long in main body. What it hopes to gain in a dry pool, I do not know. It seems to be thinking the same thing.
What we're going to do about it, I do not know. Leaving it to poop in the newly cleaned pool is not an option. Apparently this is a common problem in the area, for anyone who maintains a pool. In fact, iguanas have undergone a population explosion on the island (to which they are an invasive species, imported as pets and released), and now pose a serious threat to fruit crops and other commerce. The government actively encourages the practice of killing them and selling the meat. This one eventually comes to its end at the hands of Michael the tree man, who grabs it, balls it up, and dispatches it out of our sight.


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text:
Rockets at NBA trade deadline and their future projection: Hollinger and Iko
MLB analysts think the New York Yankees are in real trouble
Naval Sports of the Week Released by Navy Federal Credit Union
Top Seeded, QB History at Super Bowl 57
East Haven Girls Soccer Foursome Unite at Albertus Magnus
Home » Hockey: Clarkson-St. Lawrence Rivalry Back on the Ice
Hockey: Clarkson-St. Lawrence Rivalry Back on the Ice
Having not yet visited Sector 10, follow these steps for a free upgrade.
By Mohamad Fiverr January 5, 2021 Updated: December 29, 2022 No Comments5 Mins Read
To understand the new smart watched and other pro devices of recent focus, we should look to Silicon Valley and the quantified movement of the latest generation.
Apple's Watch records exercise, tracks our moves throughout the day, assesses the amount of time we are stood up and reminds us to get up and move around if we have been sat for too long – let's not forget Tim Cook's "sitting is the new factor" line.
As you're tapping, scrolling, and swiping on your phone, you probably don't give much thought to the fact that your apps are consuming electricity – just that they're chipping away at your battery life.
It's engaging. It's hopeful. It pushes the plot forward.
I cannot wait till we can get to that point where he's like, 'Boys, I got the call-up. I am going to The Show.'
THOMAS DOE
Before you click away, disappointed that I brought up something as practical and boring as time management, hear me out. My intent is not to crush your spirits.
Greatness Awaits You
The average U.S. iPhone owner uses 1-2 gigabytes of LTE data per month, according to some studies. That data comes at an energy cost that's opaque, because we can't see it directly.
You can't look to your peers in the industry, because all they're going to tell you is that they are killing it, crushing it, changing the world.
Also Read: Journey towards Design Perfection with Google Studio
Still rocking in Euro 2022.
To its detractors, love at first sight must be an illusion – the wrong term for what is simply infatuation, or a way to sugarcoat lust.
Take a social media hiatus. According to a recent survey from Everyday Health, 60 to 70 percent of young adults say they check their social media platforms at least daily. Why not dial that back for the holidays?
Stick to your new routine. It's tempting to shake up your typical schedule when the holidays come around, but women's health expert and BINTO founder Suzie Welsh emphasizes the importance of routine amid the revelry.
Be intentional about fitness. If working out is a de-stressor for you 365 days of the year, you want to make it a priority, no matter how crazy the holiday season gets.
The truth, of course, is that these people have been lying to you all along.
Sixty percent of Africans are under the age of 25. To see Africa through their eyes is to see a continent brimming with potential and opportunity: https://t.co/Sgxobv2iga pic.twitter.com/3NvQOwwTb5
— Bill Gates (@BillGates) September 10, 2018
A federal government initiated report conducted by the Allen Consulting Group released in July 2011 proposed, amongst other detail, various standards of reporting criteria ranging from voluntary to a comprehensive evaluation conducted by qualified energy rating assessors.
Vanity is the Healthiest Thing in Life
There were a lot of cut outs in the waists of gowns at the Critics' Choice Awards and there were mostly chic and fun with a little peak of skin. This is not a little peak.
Has he really done it!
They will all come together to decide the winner of the prize, which was won last year by British menswear designer Grace Wales Bonner. The successful candidate will be given a $300,000 grant and year-long business mentorship, helping them to develop every area of their brand.
Read More: Years of Empathy, Delusion, Joy.
We like the idea of a long top over pants – it's nervy – but the combo of no straps, very long layers of ruffling and her cleavage looking smashed the latest records.
This is reflected in the basic idea to Kate Ballis' photo series Beaches Above. By shooting the world from a bird's eye. Success isn't about the end result, it's about what you learn.
Facebook has been less active than Google on the public relations front, but has joined its lobbying efforts – going as far as hiring experienced lobbyists who know Prime Minister Scott Morrison personally.
Here's what we know – and don't – about Tech's Novel Use.
Joe Doe tested negative for Tech Literacy.
Shane's diagnosis could spell disaster for his campaign.
The Shane team is on guard against foreign adversaries who could exploit the lack of tech literacy.
1. There is always the New Project, the New Opportunity.
The Ovals: You Beauty.
Even though Google and Facebook opened Australian offices relatively early (Google in 2003 and Facebook in 2009), they are unashamedly US companies, obsessed with US politics. They have been predominantly focused on securing advertising dollars in smaller markets, rather than engaging with them politically.
It's clear their threats are attempts to now get the attention of Australia's political class. And if the platforms follow through.
2. To be successful, follow your passion and see where it leads you.
Onwards to victory.
Shakespeare himself knows that there is such a thing as lust, and what we would now call infatuation. He's no fool. People who exhibit the perfectionism are fearful of failure.
Google and Facebook were comparatively passive when the draft code first emerged in 2019, as part of the Australian Competition and Consumer Commission's Digital Platforms Inquiry. Providing advance notice of any changes.
Ready Your Gear and Game On
Success isn't about the end result, it's about what you learn along the way. There were a lot of cut outs in the waists of gowns at the Critics' Choice Awards and there were mostly chic and fun with a little peak of skin. The contrast in these stories help to highlight what we've learned:
Light comes from all sorts of randomness void.
It's a blessing, but also a terrible defect sensational.
Smart phones are a massive energy drain.
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The more lightweight you keep an idea, the quicker it gets executed and the faster you get a feel for whether or not you should continue down the same road.
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