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# Acoustomicrofluidic assembly of oriented and simultaneously activated metal–organic frameworks

## Abstract

The high surface area and porosity, and limitless compound and network combinations between the metal ions and organic ligands making up metal–organic frameworks (MOFs) offer tremendous opportunities for their use in many applications. While numerous methods have been proposed for the synthesis of MOF powders, it is often difficult to obtain oriented crystals with these techniques. Further, the need for additional post-synthesis steps to activate the crystals and release them from the substrate presents a considerable production challenge. Here, we report an acoustically-driven microcentrifugation platform that facilitates fast convective solutal transport, allowing the synthesis of MOF crystals in as short as five minutes. The crystals are not only oriented due to long-range out-of-plane superlattice ordering aided by molecular dipole polarization under the acoustoelectric coupling, but also simultaneously activated during the synthesis process.

## Introduction

Metal–organic frameworks (MOFs)—highly ordered three-dimensional coordination networks comprising inorganic nodal units interconnected by polytopic organic ligands—have recently garnered significant attention because of their exceptionally high Brunauer–Emmett–Teller (BET) surface areas (~104 m2/g) and porosities (up to 90% of its free volume)1,2. Moreover, their structural diversity, arising from the vast number of possible combinations between the metal nodes and organic linkers, facilitates the tailoring of materials with widely different physical, chemical, and geometrical properties for applications that span catalysis, gas separation and sensing, charge transport and storage, and drug delivery, amongst others2,3,4,5,6,7,8.

While MOFs have conventionally been synthesized through a variety of techniques, including hydrothermal, solvothermal, microwave, sonochemical, and electrochemical synthesis9, drawbacks associated with these routes include the random orientation, polycrystallinity and defect-rich nature of the MOFs produced due to inhomogeneities in the diffusion process10,11. Moreover, a significant practical limitation in the large-scale production of these MOF powders is the requirement for subsequent post-synthesis chemical or thermal activation to remove the unreacted solvents trapped within the pores12,13,14. Furthermore, in some instances, thermal and chemical activation via solvent exchange hinder the use of MOFs as green materials, and has often failed to yield the expected internal surface area.

In this work, we report an acoustomicrofluidic platform that facilitates one-step synthesis and activation of MOF powders with high degree of orientation and surface area. The facile, room-temperature operation is rapid—MOFs are produced in as little as 5 min, and are found to already be activated during their synthesis, therefore removing the need for further post-processing activation steps. In particular, we demonstrate the synthesis of HKUST-1 (Cu3(1,3,5-benzenetricarboxylate)n; Cu–BTC) since it is a particularly well-characterized MOF15,16 that facilitates comparison of its geometrical properties. In addition, we show that the technique can also be extended with similar results for the synthesis of Fe-MIL-88B [Fe3O(1,4-benzenedicarboxylate)3Cl]. That the HKUST-1 and Fe-MIL-88B crystals are simultaneous activated during their synthesis in the current process is significant, not just from the standpoint of reducing processing time, complexity and failure, but also from an economic and environmental cost perspective since conventional activation methods under vacuum or via liquid solvent exchange are either energy-intensive or results in waste organic solvents that necessitate facilities for their treatment or disposal, particularly in large-scale manufacture.

## Results

### Synthesis and activation of HKUST-1

The acoustomicrofluidic synthesis platform, which comprises a piezoelectric substrate (lithium niobate; LiNbO3), is schematically shown in Fig. 1a. The pair of interdigital transducers (IDTs) are deliberately patterned off-center on the substrate in order to break the symmetry of the opposing surface acoustic waves (SAWs)—nanometer-amplitude MHz-order electromechanical Rayleigh waves (longitudinal, transversely-polarized, i.e., out-of-plane, surface-propagating compressional waves)—they generate upon application of an oscillating electric field at resonance. The transmission of these asymmetrically opposing waves into a 10 μl sessile liquid drop placed atop the substrate then results in an internal microcentrifugal flow17,18,19,20,21 that has been previously demonstrated for driving extremely efficient micromixing and particle concentration22. Subjecting a drop containing 5 μl of a copper precursor, i.e., copper(II) nitrate hemi(pentahydrate) (Cu(NO3)2 2.5H2O), and 5 μl of trimesic acid (benzene-1,3,5-tricarboxylic acid; H3BTC), both in 1:1 (vol/vol) ethanol–water solutions, to such acoustically-driven microcentrifugation at varying acoustic intensities (1.5, 4.5, 7.5, and 9 Vrms) for 5 min can be seen to induce nucleation and subsequent crystallization of HKUST-1 (Fig. 2), which is amongst the foremost and most common of MOFs reported in the literature given its excellent thermal stability, and superior adsorption and catalytic properties23,24.

Confirmation of the production of stable HKUST-1 MOFs is provided by the Fourier Transform Infrared (FTIR) spectra in Supplementary Fig. 1 in which we verify the asymmetric stretching of the carboxylate groups in the H3BTC molecules at 1508–1623 cm−1 and the symmetric stretching of the COO–Cu2 carboxylate groups at 1384 and 1405 cm−1. Several bands over wavenumbers 1300–600 cm−1 are observed, which can be attributed to the out-of-plane vibration of the H3BTC molecules25,26. Noting the thermal stability of solid HKUST-1 to exceed 300 °C, thermal gravimetric analysis (TGA) of one of the samples (9 Vrms), on the other hand, reveals two major stages in the weight loss behavior of the HKUST-1 crystals, consistent with that observed for bulk HKUST-1 (Supplementary Fig. 2)27. The first weight loss stage occurs at temperatures below 200 °C, which can be attributed to the removal of water and solvent molecules from the surfaces of the HKUST-1 precursors. For octahedral HKUST-1 crystals, the second weight loss stage starts at around 270 °C and ends around 340 °C. The relative weight loss (weight loss/residual weight × 100%) of the HKUST-1 crystals at this stage is approximately 36.618%, roughly corresponding to the theoretical weight loss caused by the combustion of the organic ligands in HKUST-1 in air. With increasing temperature, we observe a slight weight decrease of approximately 8.866% in the temperature range between 330 °C and 450 °C, which can be attributed to the oxidation of Cu2O in air (Cu2O + 1/2O2 → 2CuO).

Further inspection of the orientation of the resultant crystals reveals its strong dependence on the magnitude of the acoustic energy coupled into the drop, which is accompanied by an intensification of the convective microcentrifugation flow (Fig. 2). We first observe from the scanning electron microscopy (SEM) images and size distributions in Fig. 2 (left and center columns) that increasing the flow intensity results in octahedral crystals typical of HKUST-1 that are progressively smaller and more homogeneous in size, decreasing from a mean characteristic dimension of 53.49 ± 4.04 μm at 1.5 Vrms to 34.19 ± 4.95 μm at 4.5 Vrms, 25.83 ± 1.37 μm at 7.5 Vrms and 15.09 ± 1.78 μm at 9 Vrms, as compared to that of bulk HKUST-1 with a mean diameter of 73.15 ± 5.85 μm (Supplementary Table 1). This dimensional reduction is a consequence of the turbulence generated in the drop due to the leakage of the SAW energy into it, given streaming Reynolds numbers28 $${\mathrm{Re}}_{\mathrm{s}} \equiv \hat u{\cal{L}}/\nu \approx 10^3$$ well above the 102 critical value reported for the classical transition to subharmonic turbulence in acoustical flows29,30; here, ν denotes the kinematic viscosity of the fluid, $${\hat{u}}$$ its Lagrangian velocity and $${\cal{L}}$$ the characteristic drop dimension. The turbulent mixing eddies generated in the flow then results in enhanced convective transport, which is known to lead to the formation of smaller crystals31,32, given that the eddy size imposes an upper limitation to the crystal dimension during its growth. This is confirmed not only by the order of magnitude agreement between the Kolmogorov length scale33 $$\eta = (\nu ^3{\cal{L}}/{\cal{U}}^3)^{1/4}\sim {\cal{O}}(10\,{\upmu{\mathrm{m}}})$$, characteristic of the eddy size with the crystal dimension observed, but also the $${\cal{U}}^{ - 3/4}$$ velocity scaling that the crystal dimension obeys, as seen in Supplementary Fig. 3.

More significantly, we note from the powder X-ray diffraction (XRD) spectra in the right column of Fig. 2b–e that the HKUST-1 crystals synthesized from the present technique exhibit a high degree of orientation parallel to the {222} plane, especially at high input voltages. This is in stark contrast to the control experiment in which the crystals that form under slow solvent evaporation of the same drop on identical substrates in the absence of the acoustic forcing show no apparent orientational preference (Fig. 2a, right column). Interestingly, we observe that the increase in the input voltage leads to more prominent vertical, out-of-plane orientation, as can be seen by the appearance of additional peaks parallel to the {222} plane, such as the {333}, {444} and {555} planes at 2θ = 11.7°, 17.7°, 23.7°, and 29.7°, respectively (Fig. 2(b–e), right column).

We note in the XRD spectra a slight shift in the peaks (≈ + 0.4°), which can be attributed to the effect of compressive stresses on the crystal lattice structure arising from the acoustic forcing; a similar observation was recently reported for sodium chloride crystals34. To obtain different deformed models of Cu–BTC, density functional theory (DFT) simulations were carried out using a volume-conserving strain tensor applied to the lattice parameters with strain magnitudes ranging from −0.008 to 0.008 in 0.002 increments. A comparison of the theoretical and experimental powder XRD data using the Pawley fitting method35 then yielded refined cell parameters which accurately resemble that expected for HKUST-1. Matching of the data for the crystals acquired under input voltages of 1.5 and 4.5 Vrms was obtained when a strain of −0.002 that led to slight compression along the c-plane was applied, resulting in cell parameters a = b = 26.30597 Å and c = 26.62462 Å with α = β = 90° and γ = 89.65623°. For the crystals acquired when the input voltage was increased to 7.5 Vrms, matching was obtained with a slight increase in the applied tensor strain of −0.006 that led to more compression along the c-plane to yield cell parameters a = b = 26.30597 Å and c = 26.62483 Å at α = β = 90° and γ = 89.88541°, whereas further compression by applying a strain of −0.008 along the c-plane was required for the crystals obtained when the input voltage was ramped to 9 Vrms, yielding cell parameters of a = b = 26.30597 Å and c = 26.62502 Å at α = β = γ = 90°.

Additionally, we note the full-width at half maximum (FWHM) value of the peak at 2θ = 11.7° was seen to decrease with increasing acoustic intensity, from 0.14° at the lowest input voltage (1.5 Vrms), corresponding to a crystallite size of 59.7 nm, to 0.49° at 9 Vrms, corresponding to a crystallite size of 167.17 nm. This increase in crystallite dimension with increasing electroacoustic coupling into the fluid provides further confirmation of the increasing degree of orientation of the crystals. We also note that the crystallite size at 9 Vrms is larger than the typical 100 nm sizes observed for HKUST-1 surface anchored MOFs (SURMOFs) synthesized using the layer-by-layer technique after 80 cycles—a much longer process requiring several hours36,37.

A possible mechanism by which the out-of-plane crystal orientation arises can be postulated from the solutal dynamics associated with the acoustically-driven microcentrifugation flow. In the absence of the acoustic excitation, the weak convective flow that arises in a sessile drop left to slowly evaporate transports solute molecules to its contact line where the locally singular evaporation rate drives their precipitation to form a particulate ring—the well-known coffee-ring stain effect38,39,40,41, as illustrated in the top row of Fig. 1b. More importantly, the diffusion-dominated transport, with a timescale on the order $${\cal{L}}^2/{\cal{D}}\sim {\cal{O}}\left( {10^3\,{\mathrm{s}}} \right)$$, wherein $${\cal{D}}$$ is the molecular diffusion coefficient, is too slow to enhance the local solutal concentration immediately behind the drying front, i.e., the contact line region, such that the weak intermolecular interactions that arise between the solute molecules are insufficient to result in large long-range vertical ordering of the crystal.

On the other hand, the oscillation of the contact line under the MHz-order SAW vibration42 as it recedes in a stick–slip manner (a consequence of pinning effects in the presence of surface heterogeneities39) is observed to smear out the rings43,44, leading to the successive deposition of monolayers across the entire footprint of the drop (Fig. 1b, bottom row). The turbulent convective flow driven by the acoustics is also sufficiently fast—with time scales on the order $${\cal{L}}/{\cal{U}}\sim {\cal{O}}\left( {1\,{\mathrm{s}}} \right)$$—in transporting the solute to a large region behind the drying front. Together with the locally singular evaporation rate at the contact line, this results in an exponential enrichment of the local solutal concentration in that vicinity, which provides ripe conditions that are known to promote the formation of large, ordered three-dimensional superlattice structures45. The out-of-plane assembly of successive monolayer stacks is further aided by the leakage of the electric field associated with the SAW on the piezoelectric substrate46,47; the ability of an induced electric field in controlling the crystal orientation having been previously reported48. As illustrated in the left and center columns of Fig. 3a, this evanescent electric field in the liquid induces dipoles in the solute molecules, whose polarization results in their vertical stacking along the field gradient orthogonal to the substrate surface, similar to that in ref. 48.

The increase of out-of-plane crystal orientation with the input voltage as seen in the right column of Fig. 2b–e is consistent with such a theory. To test this hypothesis, we repeated the experiments using shear-horizontal SAWs (SH-SAWs), i.e., longitudinal, horizontally-polarized, i.e., in-plane, surface propagating shear waves, generated on a lithium tantalate (LiTaO3) substrate, whose native electric field polarization occurs in the plane of the substrate (Fig. 3b, left and center columns), in contrast to that of the Rayleigh SAW, whose native electric field polarization occurs out of the plane of the substrate (Fig. 3a, left and center columns). Under the same conditions, we observe the crystals produced to possess an in-plane orientation, i.e., parallel to the {200} plane, as shown in the right column of Fig. 3b. We also note the negligible temperature change (from 27.8 to 28.6 °C) in the drop over the acoustic excitation period, even for the highest input voltage, thus eliminating the possibility of heating effects due to either Rayleigh SAW and SH-SAW irradiation on the evaporation dynamics (see Supplementary Fig. 4).

Additionally, the SAW excitation along the substrate results in oscillations in the MOF crystalline structure, which, in turn, squeezes the solvents out of the pores, leading to their simultaneous activation, as observed by the color change in the crystals from a light (solvent-rich) to deep (solvent-poor) purple shade under increasing acoustic field intensities (Fig. 4a). The surface area of the synthesized MOFs are reported in Fig. 4b, in which the N2 adsorption isotherm at 77 K for input voltages of 1.5, 4.5, 7.5 and 9 Vrms yields BET surface areas of approximately 849, 1300, 1434, and 1682 m2 g−1, respectively. The latter values are comparable to that reported in the literature for commercially-available activated HKUST-1 crystals15 (1500–2100 m2 g−1) and more than double the surface area (≈700 m2 g−1) associated with the HKUST-1 crystals produced in the absence of SAW excitation, which matches closely with that reported for the same crystals prior to their activation49.

### Synthesis and activation of Fe-MIL-88B

Similar oriented and simultaneously-activated structures were also observed for the synthesis of Fe-MIL-88B crystals—which comprises a three-dimensional hexagonal structure built up from trimers of FeO6 octahedra linked to benzenedicarboxylate anions forming water-filled tunnels along the c-axis connected by bipyramidal cages, and thus having relatively small pore dimensions. MIL-88B is especially useful particularly for selective adsorption of gases and solvents because its unit cell is able to shrink and swell reversibly under external stimuli or upon solvent removal during activation50,51,52. We, however, note that the oriented and simultaneously-activated structures were only obtained for exposure to the acoustic irradiation at the highest input voltage of 9 Vrms over the same five minute period. The rod-shaped crystals with average lengths of 2–3 μm (Fig. 5a) were seen to be oriented towards the {002} plane, together with the appearance of the accompanying {101} plane (Fig. 5b), characteristic of oriented MIL-88B diffraction reported in the literature52,53,54. BET analysis of these MOFs (Fig. 5c) revealed no accessible porosity for N2 at 77 K due to the large contraction of the pores after dehydration; the BET surface area obtained (≈8.7 m2 g−1) being consistent with that reported previously for closed activated MIL-88B structures55,56. Parenthetically, we note that the activation of mesoporous MOFs with even larger pores sizes than MIL-88B, such as MIL-101, using high boiling point solvents can be particularly challenging due to the tendency of the pores to collapse upon removal of the solvent, or the propensity for pore blockage as a consequence of partial solvent removal; both these challenges can be circumvented in the present case simply by adjusting the input power and duration that the MOFs are exposed to during acoustic activation14. Thermal behavior of the samples, on the other hand, shows free and/or coordinated solvent/water departure occurring up to 200 °C, whereas the MIL-88B exhibits thermal stability up to around 300 °C, typical of these classes of MOFs (Fig. 5d)57.

## Discussion

In summary, we have demonstrated that oriented and simultaneously-activated MOF crystalline powders can be synthesized at room temperature in as little as five minutes (comparable to other synthesis techniques, such as microwave assisted synthesis56,58) using an acoustomicrofluidic platform. In addition to its simplicity and speed, as well as the advantages of simultaneous synthesis and activation in a single step, the process easily affords fine control over the out-of-plane architecture of the crystals through the power intensity delivered to the device. Given the low cost of the device (typically around US\$1 each) and a production rate of 2 g h−1 device−1, we note the potential of the setup to be scaled to achieve industrially-relevant production rates by employing a large number of devices in parallel given an equivalent space-time yield σP of 17,500 kg m−3 day−1, thus making the platform an attractive alternative for environmentally-friendly large-scale MOF production for a vast array of applications59.

## Methods

### Device fabrication

The acoustomicrofluidic device shown in Fig. 1a consists of a piezoelectric substrate that comprises either 127.68° YX lithium niobate (LiNbO3; Roditi Ltd., London, UK) for the Rayleigh SAW experiments, or, 42° YX lithium tantalate (LiTaO3; Fujitsu Laboratories Ltd., Atsugi, Japan) for the SH-SAW experiments, on which an off-centered pair of 300 nm thick straight aluminium interdigitated transducers (IDTs) in a basic full-width interleaved configuration are patterned atop a 20 nm thick chromium layer using sputter deposition and standard UV photolithography. The substrates were optically polished on both sides to render it transparent such that the interior of the fluid drop can be observed from the underside of the device to avoid optical distortion at the liquid–air interface of the drop when visualizing from above. Each IDT consists of 25 finger pairs with an aperture of 12 mm and a gap and width of 110 μm, such that application of a sinusoidal electrical signal through an RF signal generator (N9310A; Agilent Technologies, Santa Clara, CA, USA) and amplifier (10W1000C; Amplifier Research, Souderton, PA, USA) at their resonant frequency of 19.37 MHz gives rise to a Rayleigh SAW (in the case of the LiNbO3 substrate) or a SH-SAW (in the case of the LiTaO3 substrate) with a wavelength of 200 μm. A variety of input voltages (1.5, 4.5, 7.5, and 9 Vrms) to the electrical signal was used in the study, but this was limited to an upper value of 9 Vrms to avoid the liquid being nebulized off the device.

### MOF synthesis

The HKUST-1 working solution was prepared by dissolving the metal precursor, i.e., 0.875 g (3.62 mmol) copper(II) nitrate hemi(pentahydrate) (Cu(NO3)2 2.5H2O; Sigma Aldrich Pty. Ltd., Castle-Hill, NSW, Australia), and the organic ligand precursor, i.e., 0.42 g (2 mmol) trimesic acid (C6H3(CO2H)3; Sigma Aldrich Pty. Ltd., Castle-Hill, NSW, Australia), in two separate tubes, each containing 12 ml 1:1 (vol/vol) ethanol (Sigma Aldrich Pty. Ltd., Castle-Hill, NSW, Australia) and MilliQ® water (18.2 MΩ.cm, Merck Millipore, Bayswater, VIC, Australia). A 10 μl drop of this solution was then carefully pipetted onto the middle of the device such that one-half of the drop was subjected to the SAW irradiation in one direction from one IDT and the other half was subjected to the SAW irradiation from the opposite direction from the other IDT. Due to this asymmetry, an azimuthal microcentrifugation flow is generated within the drop (Fig. 1a). After 5 min of exposure to the SAW excitation for each input voltage, during which the MOF was observed to nucleate and hence crystallize, the MOF powder was subsequently collected from the device in an Eppendorf® tube (Eppendorf South Pacific Pty. Ltd., North Ryde, NSW, Australia) and reconstituted with 1:1 (vol/vol) ethanol/water to make a 1 ml solution. This suspension was then centrifuged at 5000 rpm for 5 min and thrice washed in 50% ethanol/water, after which it was left to dry at 25 °C in a sealed glass vial prior to further analysis.

Bulk HKUST-1 was prepared using a hydrothermal synthesis method reported in the literature60, in which 0.42 g (2 mmol) BTC was dissolved in 24 ml of 1:1 (vol/vol) ethanol/water. The mixture was stirred for 10 min until a clear solution was obtained. Subsequently, 0.875 g (3.62 mmol) of (Cu(NO3)2 2.5H2O was added to the mixture, followed by agitation for a further 10 min. Once the reactants were completely dissolved in the solvent, the resulting blue solution was left to evaporate to allow the crystallization to occur, through which a blue crystalline powder was obtained. The powder was then washed thrice with 60 ml of a 1:1 (vol/vol) ethanol/water solution and the product left to dry at 25 °C in a sealed glass vial for further analysis.

A similar procedure as that detailed above was repeated for the synthesis of Fe-MIL-88B MOFs, whose working solution was prepared by separately dispersing its precursors, i.e., 31.9 mg iron(III) chloride hexahydrate (FeCl36H2O; Alfa Aesar GmbH & Co KG, Lancashire, United Kingdom) and 19.1 mg 1,4-benzenedicarboxylic acid (C6H4(CO2H)2; Sigma Aldrich Pty. Ltd., Castle-Hill, NSW, Australia), in 2.5 ml dimethylformamide (DMF; Thermofisher Scientific, Waltham, MA, USA).

### MOF characterization

SEM imaging (Philips XL30, FEI, Hillsboro, OR, USA) was employed to characterize the morphology of the MOF crystals. Briefly, the crystals were deposited on a silicon wafer above which a 5 nm gold layer was sputtered over 60 s and imaging was carried out at 10 kV. The size of the MOF crystals was determined through visual inspection of approximately 100 crystals from the SEM digital images using ImageJ (v.1.34, National Institutes of Health, Bethesda, MD, USA). Transmission electron microscopy (TEM; JEM-2100F, JEOL, Frenchs Forest NSW, Australia) images were also obtained by drop casting 3 μl of the MIL-88B crystals onto a holey carbon grid, which was left to dry in air prior to imaging under an acceleration voltage of 80 keV together with a CCD camera (Orius SC1000, Gatan Inc., Pleasanton, CA, USA); post-processing was carried out using the supplied image analysis software (Digital Micrograph® v2.31, Gatan Inc., Pleasanton, CA, USA).

To resolve the crystal structure, powder XRD (D8 Advance, Bruker Pty. Ltd., Preston, VIC, Australia) was conducted with Cu Kα radiation at 40 mA and 40 kV (λ = 1.54 Å), and a 2θ range of 6°–50° with a step size of 0.02°; the scan rate was 2° min−1 for HKUST-1 and 0.0142° min−1 for MIL-88B. The analysis was performed at room temperature and atmospheric pressure. Pawley refinements on the collected powder XRD data were performed using Topas Academic (v4.1.1; Coelho Software, Brisbane, Australia)61; this method is suitable for crystalline powders with preferred orientation since the Pawley fitting treats the peak areas as variables35. A Gaussian function was used to model the background, in combination with a freely refining Chebyshev polynomial using 8 parameters.

FTIR analysis of the samples at room temperature were acquired using a spectrophotometer (Spectrum One; PerkinElmer Inc., Waltham, MA, USA) by placing a 10 μl suspension of the crystals on a diamond substrate, from which transmittance measurements were conducted in the wavenumber range between 500 and 4000 cm−1. The thermal properties of the crystals, on the other hand, were analyzed through TGA (Pyrus 1, PerkinElmer Inc., Waltham, MA, USA). Specifically, 7.5 mg of the crystals were placed in an aluminium stainless steel pan and heated at a rate of 10 °C min−1 under N2 from 35 °C to 800 °C. The BET and Langmuir surface areas were calculated from N2 physisorption measurements by placing approximately 0.5 g of the crystals in a surface area and porosity analyzer (ASAP 2020; Micromeritics Instrument Corp. Norcross, GA, USA) under N2 at 77 K, following degassing of the samples at 25 °C under vacuum overnight prior to measurement. To quantify the overall yield and production rate, the dried MOF powder was weighed using a microbalance (XP56; Mettler Toledo Ltd., Port Melbourne, VIC, Australia). Temperature measurements, on the other hand, were carried out using a handheld thermal camera (Trotec EC060V; Emona Instruments, Pty. Ltd., Melbourne, VIC, Australia).

### DFT modelling

Geometry optimization of perfect and deformed models of Cu–BTC (HKUST-1) were performed using the dispersion corrected density functional theory (DFT-D3) method62, as implemented in the Vienna Ab Initio Simulation Package (VASP, v5.4.4; VASP Software GmbH, Vienna, Austria)63. The initial periodic model for the Cu–BTC structure was taken from previous work23,64. Electron exchange and correlation were described using the generalized gradient approximation Perdew, Burke, and Ernzerhof (PBE)65 form and the projector-augmented wave potentials were used to treat core and valence electrons66. In all cases, we used a plane-wave kinetic energy cut-off of 600 eV and a Gamma-point mesh for sampling the Brillouin zone. To obtain different deformed models of Cu–BTC, a volume-conserving strain tensor was applied to the lattice parameters with a strain magnitude that ranged from −0.008 to 0.008 in increments of 0.002. The models were also DFT-D3 optimized, keeping the volume and the lattice parameters fixed while completely relaxing the ionic positions.

## Data availability

The powder XRD, BET and crystal structure data that support the plots within this paper and other findings of this study are available on the Figshare repository (https://doi.org/10.6084/m9.figshare.8020010).

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## Acknowledgements

The authors acknowledge access to the facilities at the RMIT Microscopy & Microanalysis Research Facility (RMMF), the RMIT Integrated Victorian X-Ray Structural Determination & Materials Characterisation Facility and the helium ion microscope at the Materials Characterisation and Fabrication platform at the University of Melbourne. They also thank Mr. Frank Antolasic and Ms. Nadia Zakhartchouk for technical assistance. A.R.R. and L.L. are grateful for support through RMIT Vice-Chancellor Postdoctoral Research Fellowships. R.B. acknowledges use of the National Computing Infrastructure (NCI) and the Pawsey supercomputing facilities and funding from the Australian Research Council (ARC) for a DECRA fellowship (DE160100987). LYY is funded through an ARC Future Fellowship (FT130100672) and an ARC Discovery Project (DP170101061).

## Author information

Authors

### Contributions

H.A., A.R.R., J.J.R., and L.Y.Y. conceived the original research idea. H.A. carried out the device development, sample synthesis and characterization with assistance in data analysis from A.R.R., J.J.R., L.L., and L.Y.Y. J.J.R., and E.L.H.M. conducted the microscopy. R.B. performed the DFT simulations, while L.K.M. carried out Pawley refinement of the data. All authors contributed, discussed, and wrote the manuscript.

### Corresponding author

Correspondence to Leslie Y. Yeo.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Journal peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Ahmed, H., Rezk, A.R., Richardson, J.J. et al. Acoustomicrofluidic assembly of oriented and simultaneously activated metal–organic frameworks. Nat Commun 10, 2282 (2019). https://doi.org/10.1038/s41467-019-10173-5

• Accepted:

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text:
# mne.inverse_sparse.tf_mixed_norm¶

mne.inverse_sparse.tf_mixed_norm(evoked, forward, noise_cov, loose='auto', depth=0.8, maxit=3000, tol=0.0001, weights=None, weights_min=0.0, pca=True, debias=True, wsize=64, tstep=4, window=0.02, return_residual=False, return_as_dipoles=False, alpha=None, l1_ratio=None, dgap_freq=10, rank=None, pick_ori=None, n_tfmxne_iter=1, verbose=None)[source]

Time-Frequency Mixed-norm estimate (TF-MxNE).

Compute L1/L2 + L1 mixed-norm solution on time-frequency dictionary. Works with evoked data [1] [2].

Parameters
evokedinstance of Evoked

Evoked data to invert.

forwarddict

Forward operator.

noise_covinstance of Covariance

Noise covariance to compute whitener.

loosefloat in [0, 1] | ‘auto’

Value that weights the source variances of the dipole components that are parallel (tangential) to the cortical surface. If loose is 0 then the solution is computed with fixed orientation. If loose is 1, it corresponds to free orientations. The default value (‘auto’) is set to 0.2 for surface-oriented source space and set to 1.0 for volumic or discrete source space.

depth

How to weight (or normalize) the forward using a depth prior. If float (default 0.8), it acts as the depth weighting exponent (exp) to use, which must be between 0 and 1. None is equivalent to 0, meaning no depth weighting is performed. It can also be a dict containing keyword arguments to pass to mne.forward.compute_depth_prior() (see docstring for details and defaults). This is effectively ignored when method='eLORETA'.

Changed in version 0.20: Depth bias ignored for method='eLORETA'.

maxitint

Maximum number of iterations.

tolfloat

Tolerance parameter.

weights

Weight for penalty in mixed_norm. Can be None or 1d array of length n_sources or a SourceEstimate e.g. obtained with wMNE or dSPM or fMRI.

weights_minfloat

Do not consider in the estimation sources for which weights is less than weights_min.

pcabool

If True the rank of the data is reduced to true dimension.

debiasbool

Remove coefficient amplitude bias due to L1 penalty.

wsize

Length of the STFT window in samples (must be a multiple of 4). If an array is passed, multiple TF dictionaries are used (each having its own wsize and tstep) and each entry of wsize must be a multiple of 4. See [3].

tstep

Step between successive windows in samples (must be a multiple of 2, a divider of wsize and smaller than wsize/2) (default: wsize/2). If an array is passed, multiple TF dictionaries are used (each having its own wsize and tstep), and each entry of tstep must be a multiple of 2 and divide the corresponding entry of wsize. See [3].

window

Length of time window used to take care of edge artifacts in seconds. It can be one float or float if the values are different for left and right window length.

return_residualbool

If True, the residual is returned as an Evoked instance.

return_as_dipolesbool

If True, the sources are returned as a list of Dipole instances.

alphafloat in [0, 100) or None

Overall regularization parameter. If alpha and l1_ratio are not None, alpha_space and alpha_time are overridden by alpha * alpha_max * (1. - l1_ratio) and alpha * alpha_max * l1_ratio. 0 means no regularization, 100 would give 0 active dipole.

l1_ratiofloat in [0, 1] or None

Proportion of temporal regularization. If l1_ratio and alpha are not None, alpha_space and alpha_time are overridden by alpha * alpha_max * (1. - l1_ratio) and alpha * alpha_max * l1_ratio. 0 means no time regularization a.k.a. MxNE.

dgap_freq

The duality gap is evaluated every dgap_freq iterations.

rankNone | dict | ‘info’ | ‘full’

This controls the rank computation that can be read from the measurement info or estimated from the data. See Notes of mne.compute_rank() for details.The default is None.

New in version 0.18.

pick_oriNone | “normal” | “vector”

Options:

• None

Pooling is performed by taking the norm of loose/free orientations. In case of a fixed source space no norm is computed leading to signed source activity.

• "normal"

Only the normal to the cortical surface is kept. This is only implemented when working with loose orientations.

• "vector"

No pooling of the orientations is done, and the vector result will be returned in the form of a mne.VectorSourceEstimate object. This is only implemented when working with loose orientations.

n_tfmxne_iterint

Number of TF-MxNE iterations. If > 1, iterative reweighting is applied.

verbose

If not None, override default verbose level (see mne.verbose() and Logging documentation for more).

Returns
stcinstance of SourceEstimate

Source time courses.

residualinstance of Evoked

The residual a.k.a. data not explained by the sources. Only returned if return_residual is True.

References

1

A. Gramfort, D. Strohmeier, J. Haueisen, M. Hämäläinen, M. Kowalski “Time-Frequency Mixed-Norm Estimates: Sparse M/EEG imaging with non-stationary source activations”, Neuroimage, Volume 70, pp. 410-422, 15 April 2013. DOI: 10.1016/j.neuroimage.2012.12.051

2

A. Gramfort, D. Strohmeier, J. Haueisen, M. Hämäläinen, M. Kowalski “Functional Brain Imaging with M/EEG Using Structured Sparsity in Time-Frequency Dictionaries”, Proceedings Information Processing in Medical Imaging Lecture Notes in Computer Science, Volume 6801/2011, pp. 600-611, 2011. DOI: 10.1007/978-3-642-22092-0_49

3(1,2)

Y. Bekhti, D. Strohmeier, M. Jas, R. Badeau, A. Gramfort. “M/EEG source localization with multiscale time-frequency dictionaries”, 6th International Workshop on Pattern Recognition in Neuroimaging (PRNI), 2016. DOI: 10.1109/PRNI.2016.7552337

format: plain


text:
## Intermediate Algebra (6th Edition)

Published by Pearson

# Chapter 9 - Section 9.3 - Exponential Functions - Vocabulary, Readiness & Video Check: 4

yes

#### Work Step by Step

Yes, the function $y=2^{x}$ is a function because it passes the horizontal line test. Even though the left side of the graph looks constant, the value of $y$ keeps decreasing ans the value of $x$ decreases.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

Title & Authors
GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES
Kim, Jae-Ryong; Oda, Nobuyuki; Pan, Jianzhong; Woo, Moo-Ha;

Abstract
Let $\small{\varepsilon_#(X)}$ be the subgroups of $\small{\varepsilon(X)}$ consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and $\small{\varepsilon_*(X) }$ be the subgroup of $\small{\varepsilon(X)}$ that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup $\small{\varepsilon_#(X)\cap\varepsilon_*(X)\;of\;\varepsilon(X)}$. We also give some relations between $\small{\pi_n(W)}$, as well as a generalized Gottlieb group $\small{G_n^f(W,X)}$, and a subset $\small{M_{#N}^f(X,W)}$ of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of $\small{\varepsilon(X)}$ consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.
Keywords
self-homotopy equivalences;Gottlieb groups;coGottlieb groups;
Language
English
Cited by
1.
Cocyclic element preserving pair maps and fibrations, Topology and its Applications, 2015, 191, 82
2.
The set of cyclic-element preserving maps, Topology and its Applications, 2013, 160, 6, 794
References
1.
M. Arkowitz, G. Lupton, and A. Murillo, Subgroups of the group of self-homotopy equivalences, Contemporary Mathematics 274 (2001), 21-32

2.
M. Arkowitz and K. Maruyama, Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topology Appl. 87 (1998), no. 2, 133-154

3.
H. J. Baues, Homotopy type and homology, Clarendon Press, Oxford, 1996

4.
E. Dror and A. Zabrodsky, Unipotency and nilpotency in homotopy equivalences, Topology 18 (1979), no. 3, 187-197

5.
D. H. Gottlieb, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-756

6.
P. Hilton, Homotopy theory and duality, Gordon and Breach Science Publishers, New York-London Paris, 1965

7.
P. J. Kahn, Self-equivalences of (n-1)-connected 2n-manifolds, Math. Ann. 180 (1969), 26-47

8.
K. Maruyama, Localization of a certain subgroup of self-homotopy equivalences, Pacific J. Math. 136 (1989), no. 2, 293-301

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K. Maruyama, Localization of self-homotopy equivalences inducing the identity on ho- mology, Math. Proc. Cambridge Philos. Soc. 108 (1990), no. 2, 291-297

10.
K. Maruyama, Stability properties of maps between Hopf spaces, Q. J. Math. 53 (2002), no. 1, 47-57

11.
N. Oda, Pairings and copairings in the category of topological spaces, Publ. Res. Inst. Math. Sci. 28 (1992), no. 1, 83-97

12.
S. Oka, N. Sawashita, and M. Sugawara, On the group of self-equivalences of a mapping cone, Hiroshima Math. J. 4 (1974), 9-28

13.
J. W. Rutter, A homotopy classi¯cation of maps into an induced fibre space, Topology 6 (1967), 379-403

14.
K. Varadarajan, Generalised Gottlieb groups, J. Indian Math. Soc. 33 (1969), 141-164

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G. W. Whitehead, Elements of homotopy theory, Graduate texts in Mathematics 61, Springer-Verlag, New York Heidelberg Berlin, 1978

16.
M. H. Woo and J.-R. Kim, Certain subgroups and homotopy groups, J. Korean. Math. Soc. 21 (1984), no. 2, 109-120


text:
# American Institute of Mathematical Sciences

April 2018, 38(4): 2187-2206. doi: 10.3934/dcds.2018090

## New periodic orbits in the planar equal-mass three-body problem

 School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

* Corresponding author: Duokui Yan

Received  June 2017 Revised  October 2017 Published  January 2018

Fund Project: The authors are supported by NSFC No.11432001 and the Fundamental Research Funds of the Central Universities

It is known that there exist two sets of nontrivial periodic orbits in the planar equal-mass three-body problem: retrograde orbit and prograde orbit. By introducing topological constraints to a two-point free boundary value problem, we show that there exists a new set of periodic orbits for a small interval of rotation angle $\mathit{\theta }$.

Citation: Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
##### References:
 [1] R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38. doi: 10.1007/BF01228732. [2] K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325. [3] K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5. [4] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357. [5] A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294. [6] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993. [7] M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285. doi: 10.1007/BF01228647. [8] W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. [9] C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353. doi: 10.1023/A:1020128408706. [10] T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015. [11] B. Shi, R. Liu, D. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835. doi: 10.1515/ans-2017-6028. [12] G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935. doi: 10.1007/s00205-017-1116-1. [13] G. Yu, Spatial double choreographies of the Newtonian $2n$-body problem, arXiv: 1608.07956.

show all references

##### References:
 [1] R. Broucke and D. Boggs, Periodic orbits in the planar general three-body problem, Celestial Mech., 11 (1975), 13-38. doi: 10.1007/BF01228732. [2] K. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses, Annals of Math., 167 (2008), 325-348. doi: 10.4007/annals.2008.167.325. [3] K. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Comm. Math. Phys., 291 (2009), 403-441. doi: 10.1007/s00220-009-0769-5. [4] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357. [5] A. Chenciner, Action minimizing solutions in the Newtonian n-body problem: From homology to symmetry, Proceedings of the International Congress of Mathematicians (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 279-294. [6] W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993. [7] M. Hénon, A family of periodic solutions of the planar three-body problem, and their stability, Celestial Mech., 13 (1976), 267-285. doi: 10.1007/BF01228647. [8] W. Kuang, T. Ouyang, Z. Xie and D. Yan, The Broucke-Hénon orbit and the Schubart orbit in the planar three-body problem with equal masses, preprint, arXiv: 1607.00580. [9] C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dyn. Astro., 83 (2002), 325-353. doi: 10.1023/A:1020128408706. [10] T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76. doi: 10.1016/j.physd.2015.05.015. [11] B. Shi, R. Liu, D. Yan and T. Ouyang, Multiple periodic orbits connecting a collinear configuration and a double isosceles configuration in the planar equal-mass four-body problem, Adv. Nonlinear Stud., 17 (2017), 819-835. doi: 10.1515/ans-2017-6028. [12] G. Yu, Simple choreography solutions of the Newtonian N-body problem, Arch. Rational Mech. Anal., 225 (2017), 901-935. doi: 10.1007/s00205-017-1116-1. [13] G. Yu, Spatial double choreographies of the Newtonian $2n$-body problem, arXiv: 1608.07956.
A picture of the periodic orbit extended by the action minimizer $\mathcal{P}_0$ with $\theta = \pi/10$. The three dots represent the starting configuration $Q_S$, and the three crosses represent the ending configuration $Q_E$ of $\mathcal{P}_0$. In the graph, the red curve is the trajectory of body 2, the blue curve is for body 1 and the black curve is for body 3.
The configurations $Q_S$ and $Q_E$ are shown, where blue dots represent $q_1$, red dots represent $q_2$ and black dots represent $q_3$. In $Q_S$, three masses are on the $x-$axis with an order $q_{2x} \leq q_{1x} \leq q_{3x}$. In $Q_E$, three masses form an isosceles triangle, whose symmetry axis is a counterclockwise $\theta$ rotation of the $x-$axis. $q_1$ is on the symmetry axis and $q_2$ is above the symmetry axis in $Q_E$.
In each figure, the horizontal axis is $\theta/\pi$, and the vertical axis is the action value $\mathcal{A}$. In each subfigure, the black curve is the graph of the test path's action $\mathcal{A}_{test}$; the purple curve is the graph of $\mathcal{A}_{Euler} = g(\theta)$ in (31) and the red curve is the graph of $f_2(\theta)$ in (31), which is the lower bound of $\mathcal{A}_{col}$.
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.181\pi, \, 0.183 \pi]$.
 $\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$ $0.1$ $( 0.11378134, \, 0.035120592)$ $( -1.1144897, \, 0.089194621)$ $0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$ $0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$ $0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 )$ $0.5$ $( 0.16980434, \, 0.14251378 )$ $( -1.0092177, \, 0.43553762)$ $0.6$ $( 0.18860029, \, 0.15603011)$ $(-0.96132044, \, 0.51676178)$ $0.7$ $(0.20644197, \, 0.16368275 )$ $( -0.90499096, \, 0.59464214)$ $0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889)$ $0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832)$ $1$ $(0.28822237, \, 0 ) R(\theta)$ $(-0.144111185, \, 1.0455201)R(\theta)$
 $\mathbf{\theta_0= 0.182 \pi, \, \, \, \, \, \theta \in [0.181 \pi, \, 0.183 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1109, \, 0 )$ $( -1.1189,\, 0 )$ $0.1$ $( 0.11378134, \, 0.035120592)$ $( -1.1144897, \, 0.089194621)$ $0.2$ $(0.12212758, \, 0.068373015 )$ $( -1.1012708, \,0.17787543 )$ $0.3$ $(0.13510199, \, 0.098115021 )$ $(-1.0792782, \, 0.26552094 )$ $0.4$ $( 0.15147483, \, 0.12309328 )$ $( -1.0485685, \, 0.35159462 )$ $0.5$ $( 0.16980434, \, 0.14251378 )$ $( -1.0092177, \, 0.43553762)$ $0.6$ $( 0.18860029, \, 0.15603011)$ $(-0.96132044, \, 0.51676178)$ $0.7$ $(0.20644197, \, 0.16368275 )$ $( -0.90499096, \, 0.59464214)$ $0.8$ $( 0.22204784, \, 0.16582253 )$ $(-0.84036636, \, 0.66850889)$ $0.9$ $( 0.23430842, \, 0.16304077 )$ $( -0.76761388, \, 0.73763832)$ $1$ $(0.28822237, \, 0 ) R(\theta)$ $(-0.144111185, \, 1.0455201)R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.176\pi, \, 0.181 \pi]$.
 $\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$ $0.1$ $( 0.12300003, \, 0.038069081 )$ $(-1.1122108, \, 0.087838457 )$ $0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$ $0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$ $0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 )$ $0.5$ $(0.18460176, \, 0.15344287 )$ $(-1.0074550, \, 0.42920169 )$ $0.6$ $( 0.20503978, \, 0.16765957 )$ $( -0.95979063, \, 0.50939364 )$ $0.7$ $(0.22435124, \, 0.17556129 )$ $( -0.90372458, \, 0.58636242 )$ $0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 )$ $0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 )$ $1$ $( 0.31146657, \, 0 ) R(\theta)$ $(-0.155733285, \, 1.0357709)R(\theta)$
 $\mathbf{\theta_0= 0.18 \pi, \, \, \, \, \, \theta \in [0.176 \pi, \, 0.181 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1198, \, 0 )$ $(-1.1166, \, 0 )$ $0.1$ $( 0.12300003, \, 0.038069081 )$ $(-1.1122108, \, 0.087838457 )$ $0.2$ $(0.13225041, \, 0.074032720 )$ $( -1.0990555, \, 0.17518597 )$ $0.3$ $(0.14657647, \, 0.10606411 )$ $( -1.0771702, \, 0.26154328 )$ $0.4$ $(0.16456742, \, 0.13280675 )$ $( -1.0466117, \, 0.34639491 )$ $0.5$ $(0.18460176, \, 0.15344287 )$ $(-1.0074550, \, 0.42920169 )$ $0.6$ $( 0.20503978, \, 0.16765957 )$ $( -0.95979063, \, 0.50939364 )$ $0.7$ $(0.22435124, \, 0.17556129 )$ $( -0.90372458, \, 0.58636242 )$ $0.8$ $( 0.24118113, \, 0.17757208 )$ $(-0.83938055, \, 0.65945294 )$ $0.9$ $( 0.25437250, \, 0.17435316 )$ $(-0.76690612, \, 0.72795349 )$ $1$ $( 0.31146657, \, 0 ) R(\theta)$ $(-0.155733285, \, 1.0357709)R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.165\pi, \, 0.176 \pi]$.
 $\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$ $0.1$ $( 0.14967657,\, 0.046886901 )$ $(-1.1020407,\, 0.083918190 )$ $0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$ $0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$ $0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 )$ $0.5$ $( 0.22901187,\, 0.18445175 )$ $( -0.99813266,\, 0.41070550 )$ $0.6$ $(0.25429011,\, 0.20016641 )$ $( -0.95089814,\, 0.48777071 )$ $0.7$ $(0.27778726,\, 0.20832504 )$ $( -0.89534096,\, 0.56190183 )$ $0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 )$ $0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 )$ $1$ $( 0.37739476,\, 0 ) R(\theta)$ $( -0.18869738,\, 1.0021646)R(\theta)$
 $\mathbf{\theta_0= 0.173 \pi, \, \, \, \, \, \theta \in [0.165 \pi, \, 0.176 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $( 0.1454,\, 0 )$ $( -1.1064,\, 0 )$ $0.1$ $( 0.14967657,\, 0.046886901 )$ $(-1.1020407,\, 0.083918190 )$ $0.2$ $(0.16194195,\, 0.090810510 )$ $(-1.0889788,\, 0.16740323 )$ $0.3$ $(0.18067059,\, 0.12932570 )$ $( -1.0672593,\, 0.25001086 )$ $0.4$ $(0.20377305,\, 0.16080303 )$ $( -1.0369499,\, 0.33127610 )$ $0.5$ $( 0.22901187,\, 0.18445175 )$ $( -0.99813266,\, 0.41070550 )$ $0.6$ $(0.25429011,\, 0.20016641 )$ $( -0.95089814,\, 0.48777071 )$ $0.7$ $(0.27778726,\, 0.20832504 )$ $( -0.89534096,\, 0.56190183 )$ $0.8$ $(0.29798679,\, 0.20961764 )$ $( -0.83155952,\, 0.63247939 )$ $0.9$ $(0.31364961,\, 0.20493038 )$ $(-0.75965878,\, 0.69882326 )$ $1$ $( 0.37739476,\, 0 ) R(\theta)$ $( -0.18869738,\, 1.0021646)R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.146\pi, \, 0.165 \pi]$.
 $\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$ $0.1$ $( 0.18658246,\, 0.059576674 )$ $(-1.0816238,\, 0.078768710)$ $0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$ $0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$ $0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003)$ $0.5$ $( 0.29489886,\, 0.22338233 )$ $( -0.97774453,\, 0.38605495)$ $0.6$ $(0.32695826,\, 0.23945199 )$ $( -0.93070180,\, 0.45871449)$ $0.7$ $(0.35597786,\, 0.24655415 )$ $( -0.87545128,\, 0.52868620)$ $0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118)$ $0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 )$ $1$ $(0.46894463,\, 0) R(\theta)$ $( -0.234472315,\, 0.94630060)R(\theta)$
 $\mathbf{\theta_0= 0.16 \pi, \, \, \, \, \, \theta \in [0.146 \pi, \, 0.165 \pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.1803, \, 0 )$ $( -1.0860,\, 0 )$ $0.1$ $( 0.18658246,\, 0.059576674 )$ $(-1.0816238,\, 0.078768710)$ $0.2$ $( 0.20429715,\, 0.11438956 )$ $(-1.0685224,\, 0.15716562)$ $0.3$ $( 0.23056861,\, 0.16092393 )$ $( -1.0467705,\, 0.23480016)$ $0.4$ $( 0.26186968,\, 0.19737903 )$ $( -1.0164725,\, 0.31125003)$ $0.5$ $( 0.29489886,\, 0.22338233 )$ $( -0.97774453,\, 0.38605495)$ $0.6$ $(0.32695826,\, 0.23945199 )$ $( -0.93070180,\, 0.45871449)$ $0.7$ $(0.35597786,\, 0.24655415 )$ $( -0.87545128,\, 0.52868620)$ $0.8$ $(0.38039590,\, 0.24583791 )$ $( -0.81208876,\, 0.59538118)$ $0.9$ $(0.39901814,\, 0.23851003 )$ $( -0.74069823,\, 0.65815496 )$ $1$ $(0.46894463,\, 0) R(\theta)$ $( -0.234472315,\, 0.94630060)R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.12\pi, \, 0.146 \pi]$.
 $\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$ $0.1$ $( 0.24197875, \, 0.079876150 )$ $(-1.0375636, \, 0.072168012 )$ $0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$ $0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$ $0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 )$ $0.5$ $(0.40588817, \, 0.26512985 )$ $( -0.93107630, \, 0.35354922 )$ $0.6$ $(0.44748116, \, 0.27663491 )$ $( -0.88330444, \, 0.41974435 )$ $0.7$ $(0.48345138, \, 0.27815124 )$ $( -0.82742450, \, 0.48317696 )$ $0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 )$ $0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 )$ $1$ $(0.59692629, \, 0) R(\theta)$ $( -0.298463145, \, 0.84227284 )R(\theta)$
 $\mathbf{\theta_0= 0.132 \pi, \, \, \, \, \, \theta \in [0.12 \pi, \, 0.146\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2306, \, 0)$ $( -1.0421, \, 0)$ $0.1$ $( 0.24197875, \, 0.079876150 )$ $(-1.0375636, \, 0.072168012 )$ $0.2$ $( 0.27249601, \, 0.14948693 )$ $( -1.0240177, \, 0.14401235 )$ $0.3$ $( 0.31440456, \, 0.20354167 )$ $( -1.0016223, \, 0.21516044 )$ $0.4$ $(0.36051601, \, 0.24151617 )$ $( -0.97057606, \, 0.28517262 )$ $0.5$ $(0.40588817, \, 0.26512985 )$ $( -0.93107630, \, 0.35354922 )$ $0.6$ $(0.44748116, \, 0.27663491 )$ $( -0.88330444, \, 0.41974435 )$ $0.7$ $(0.48345138, \, 0.27815124 )$ $( -0.82742450, \, 0.48317696 )$ $0.8$ $(0.51262863, \, 0.27151295 )$ $( -0.76358450, \, 0.54323643 )$ $0.9$ $(0.53419879, \, 0.25829537 )$ $( -0.69191492, \, 0.59928148 )$ $1$ $(0.59692629, \, 0) R(\theta)$ $( -0.298463145, \, 0.84227284 )R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.089\pi, \, 0.12 \pi]$.
 $\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$ $0.1$ $(0.28699887, \, 0.097087533 )$ $( -0.99513360, \, 0.068868347)$ $0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$ $0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$ $0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 )$ $0.5$ $(0.50439276, \, 0.27354220 )$ $(-0.88455797, \, 0.33601715 )$ $0.6$ $(0.55150942, \, 0.27662332 )$ $( -0.83534366, \, 0.39783468 )$ $0.7$ $(0.59074871, \, 0.27023456 )$ $( -0.77798406, \, 0.45632275 )$ $0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 )$ $0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 )$ $1$ $(0.69032914, \, 0) R(\theta)$ $( -0.34516457, \, 0.74809598 )R(\theta)$
 $\mathbf{\theta_0= 0.1 \pi, \, \, \, \, \, \theta \in [0.089 \pi, \, 0.12\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2680, \, 0)$ $( -0.9999, \, 0 )$ $0.1$ $(0.28699887, \, 0.097087533 )$ $( -0.99513360, \, 0.068868347)$ $0.2$ $(0.33390510, \, 0.17420817 )$ $( -0.98094971, \, 0.13739837 )$ $0.3$ $(0.39193700, \, 0.22664276 )$ $( -0.95760880, \, 0.20514818)$ $0.4$ $(0.45044849, \, 0.25815778 )$ $( -0.92539244, \, 0.27156414 )$ $0.5$ $(0.50439276, \, 0.27354220 )$ $(-0.88455797, \, 0.33601715 )$ $0.6$ $(0.55150942, \, 0.27662332 )$ $( -0.83534366, \, 0.39783468 )$ $0.7$ $(0.59074871, \, 0.27023456 )$ $( -0.77798406, \, 0.45632275 )$ $0.8$ $(0.62156413, \, 0.25649856 )$ $( -0.71271941, \, 0.51078103 )$ $0.9$ $(0.64359767, \, 0.23707560 )$ $( -0.63979436, \, 0.56051180 )$ $1$ $(0.69032914, \, 0) R(\theta)$ $( -0.34516457, \, 0.74809598 )R(\theta)$
The positions of $\tilde{q}_{i, j} = \tilde{q}_{i}(\frac{j}{10}) \, (i = 1,2, \, j = 0,1, \dots, 10)$ in $\mathcal{P}_{test, \, \theta}$ corresponding to $\theta \in [0.084\pi, \, 0.089 \pi]$.
 $\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$ $0.1$ $( 0.30528479, \, 0.10386817 )$ $( -0.97782312, \, 0.068296014 )$ $0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$ $0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$ $0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 )$ $0.5$ $(0.54454099, \, 0.26800959 )$ $(-0.86520333, \, 0.33225532 )$ $0.6$ $(0.59300170, \, 0.26663595 )$ $(-0.81520260, \, 0.39268889 )$ $0.7$ $(0.63277000, \, 0.25631044 )$ $( -0.75700467, \, 0.44941510 )$ $0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 )$ $0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 )$ $1$ $(0.72320338, \, 0) R(\theta)$ $( -0.36160169, \, 0.71048767 )R(\theta)$
 $\mathbf{\theta_0= 0.085 \pi, \, \, \, \, \, \theta \in [0.084 \pi, \, 0.089\pi ]}$ $t$ $\tilde{q}_1$ $\tilde{q}_2$ $0$ $(0.2818, \, 0)$ $( -0.9827, \, 0 )$ $0.1$ $( 0.30528479, \, 0.10386817 )$ $( -0.97782312, \, 0.068296014 )$ $0.2$ $( 0.36044290, \, 0.18168802 )$ $( -0.96333464, \, 0.13622832 )$ $0.3$ $(0.42517243, \, 0.23070675 )$ $( -0.93953669, \, 0.20329795 )$ $0.4$ $( 0.48803512, \, 0.25746053 )$ $( -0.90673504, \, 0.26887596 )$ $0.5$ $(0.54454099, \, 0.26800959 )$ $(-0.86520333, \, 0.33225532 )$ $0.6$ $(0.59300170, \, 0.26663595 )$ $(-0.81520260, \, 0.39268889 )$ $0.7$ $(0.63277000, \, 0.25631044 )$ $( -0.75700467, \, 0.44941510 )$ $0.8$ $(0.66356766, \, 0.23917705 )$ $( -0.69090435, \, 0.50167763 )$ $0.9$ $(0.68521757, \, 0.21687569 )$ $( -0.61721738, \, 0.54873962 )$ $1$ $(0.72320338, \, 0) R(\theta)$ $( -0.36160169, \, 0.71048767 )R(\theta)$
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# How do you find the percent change given the original number is 90 and new number is 95?

Mar 21, 2017

5.56% increase

#### Explanation:

"Change" means "difference" in something. Difference in math is found by subtracting.
$95 - 90 = 5$ increase

"Per cent" means "out of 100" (cent as in a century has 100 years)
Set up the problem with (part)/(whole)=(part)/(whole). 90 was the original whole.

$\frac{x}{100} = \frac{5}{90}$

$x = 5 \cdot \frac{100}{90} = 5.5555$ increase


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My Coogam Moyu 3x3 for a 2x2

Sub1Hour

Member
I dont think they do anything to the cube they just resell cubes for like \$10 more expensive.
I know, that's why I asked for a picture so we could find out the actual model of the cube.

asdfghjklohhnhn

Member
Sorry guys, I don’t think there’s a reason for me to post a picture because I’m already trading with @PingPongCuber

PingPongCuber

Member
Letting you all know that we did not do a trade

asdfghjklohhnhn

Member
Sorry, yeah, I just forgot to post a picture and a comment saying that!


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# Fundamental properties of subatomic particles

1. Jul 5, 2011

### Amir Livne

A physics hobbyist such as myself, trying to understand high-energy experiments from the recent decades, often hears about symmetries in the model, conservation laws, Feynman diagrams and so on. These are all intuitive properties but very far from a basic world view of "what happens microscopically".

What I would like to ask, is what elementary quantities completely define the state of a single particle. In Newtonian gravitation these would be mass, position, and momentum. That's 7 real numbers, 6 of them unlimited and one restricted to positive values.

In QM things seem to be more difficult, and I understand much less. For a start, there is not isolated particle in vacuum, and even if there was, there would be no values for these properties. So instead - and my first question is whether this is an interesting property - I suggest counting operators whose eigenspaces are of dimension 1. These would include position and momentum, 6 operators with continuous spectrum. That's all there is for a free particle, meaning $$i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi$$

What do you get for QED? As I understand, you have mass and electric charge, which are operators with discrete spectrum but an infinite set of eigenvalues. If you consider a state with a single electron it's in an eigenspace with one specific value for mass and electric charge. How does spin relate to this? I read everywhere that spin can be measured along any direction and give 2 values. But how many linearly independent spin operators are there? What is the spectrum for a system with n electrons (my common sense tells me n+1, but I don't know really know the physics)? And is there any other observable independent of the above?

The same question goes for the theory of electro-weak interactions and for the Standard Model. Do these models simple tack on more and more degrees of freedom on each particle, or does something more basic change? What are those parameters? Can extensions to the SM be state along the same lines? Like - "instead of being 3 independent neutrino operators, we now believe in stealth neutrinos so there are 4"

2. Jul 7, 2011

### Drakkith

Staff Emeritus
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# Singularities of $\ \frac{z-1}{z^2 \sin z} \$

Find all singularities of $\ \frac{z-1}{z^2 \sin z} \$

Determine if they are isolated or nonisolated.

This is not hard, it is z = 0 and z = k*pi.

But how do I:

For isolated singularities, determine if they are removable or nonremovable and, if nonremovable, determine their order.

Do I need to expand this to a power series? If so, I have no idea where to attack...

-

Since for any $\,k\in\Bbb Z-\{0\}\,$ we have that

$$\lim_{z\to k\pi}(z-k\pi)\frac{z-1}{z^2\sin z}\stackrel{\text{L'Hospital}}=\lim_{z\to k\pi}\frac{z-1}{2z\sin z+z^2\cos z}=$$

$$=\frac{k\pi -1}{2k\pi \sin k\pi+k^2\pi^2\cos k\pi}=\frac{(-1)^k(k\pi -1)}{k^2\pi^2}$$

all these singularities are isolated (in fact, simple poles).

About the case $\,k=0\,$:

$$\lim_{z\to 0}z^3\frac{z-1}{z^2\sin z}=\lim_{z\to 0}\frac{z}{\sin z}(z-1)=-1$$

Thus we have here a pole of order $\,3\,$

-


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# Numerical investigation on purposeful limitations of the anti-stall fin for an axial fan: one-factor analyses

### Impact of anti-stall fin

#### Experimental validation

The experimental course of and facility of this research totally complied with the worldwide customary22. As proven in Fig. 6 (or Fig. 2), the outlet chamber setup was adopted, and a straight duct that has twice the axial size for the fan diameter was linked between the fan outlet and chamber inlet. The circulate settling means contained in the chamber had secured the correct porosity23. Relative humidity, barometric stress, and dry-bulb temperature had been measured to calculate the density. The density and rotational velocity had been respectively transformed as the identical values with the computational setup. The circulate price was adjusted with nozzles and was calculated from the differential stress ((Delta {P}_{s})); the circulate price vary that might not be measured with a mix of nozzles was dealt with with a servo blower behind the chamber, and this was essential to compensate for the stress loss contained by the nozzles within the system. The stress and rotational velocity had been measured with stress manometers and a laser tachometer (or a stroboscope). The uncertainty for stress manometers, stroboscope, and dry-bulb temperature detectors was 0.001–0.005 kilo-pascals for the vary of 0–1.33 kilo-pascals, 0.1–1 revolutions-per-minute for the vary of 40–35,000 revolutions-per-minute, and 0.07 °C for the vary of 0–60 °C, respectively.

Determine 7 represents the (Q)(P) curve for every case of none and ASF connected (left), and the rise or lower price of whole stress rise for the case of ASF connected (proper); right here, ASF akin to the reference set in Desk 2 was in contrast preferentially. Within the case of none, the optimistic gradients had been included within the stalling circulate charges lower than 0.8 ({varPhi }_{d}). Nevertheless, within the case of ASF connected, the optimistic gradients contained within the case of none had been fully reversed to grow to be unfavorable. The ASF-attached axial fan stably recovered efficiency degradation within the stalling circulate charges and allowed to type unfavorable gradients to 0.5 ({varPhi }_{d}). Though it could be a strict declaration, the ASF’s purposeful limitation needs to be evaluated by figuring out whether or not the (Q)(P) curve varieties unfavorable gradients within the circulate price vary over 0.5 ({varPhi }_{d}).

#### Anti-stalling mechanism

Every contour for circumferential velocity (({v}_{theta })), axial velocity (({v}_{a})), and static stress (({varPsi }_{s})) on the sectional aircraft close to the blade LE is proven in Figs. 9, 10, and 11, because the entrance view: the sectional aircraft relies on the sectional line in Fig. 8; Fig. 8 is an enlarged view for the viewing window in Fig. 4; Figs. 9, 10, and 11 have every legend on the left, and the legend on Fig. 8 is for subsequent dialogue concerning (Q)-criterion; the circumferential velocity contour (Fig. 9) signifies the next worth because it turns into stronger in opposition to the blade’s rotational course, and the axial velocity contour (Fig. 10) signifies the next worth because the part towards the downstream turns into stronger; concentric circles in every sectional aircraft mark each 0.1 from the hub (0) to the shroud (1). Within the case of none, every unfavorable portion of ({v}_{theta }) and ({v}_{a}) (Figs. 9a, 10a) had been steadily developed over the thicker span because the circulate price decreased. Right here, the portion of unfavorable ({v}_{theta }) was thicker than that of unfavorable ({v}_{a}) at every circulate price level, i.e., ({v}_{theta }) contained a backward (backflow) and ahead parts from the shroud to the deeper spans; this implies recirculation within the axial course was fashioned with rotation within the circumferential course. Nevertheless, these circulate patterns had been largely managed by ASF (see Figs. 9b, 10b). This might be confirmed as the primary trigger that ASF may suppress the optimistic gradients within the stalling circulate charges. In Fig. 11a for the case of none, the static stress decreased because the circulate price decreased. From a theoretical perspective, a lower in circulate price means a rise in inlet static stress; nevertheless, as proven in Fig. 10a, the backflow may act as a blockage within the circulate passage, inflicting a rise in ({v}_{a}) of major stream. Alternatively, in Fig. 11b for the case of ASF connected, the static stress elevated because the circulate price decreased. In the meantime, every stress and suction facet of ASF (Fig. 8) might be recognized with Fig. 11b.

As additional particulars for the interior circulate discipline, Figs. 12 and 13 had been illustrated: it was primarily based on the information in Fig. 8; the red-circumferential traces denote the axial coordinates each 0.1 ({D}_{2}), and the blade LE line corresponds to 0.3 on the (x)-axis in Fig. 3; the limiting streamlines, which don’t have any legends (white), had been plotted on the imaginary planes passing by the axis; the imaginary planes handled as clear in Fig. 8, however opaque (black) in Figs. 12 and 13; a vortex identification technique ((Q)-criterion24), was employed with iso-surface that was coated with a circumferential velocity contour with the legend on Fig. 8, and the circumferential velocity contour signifies the next worth because it turns into stronger in opposition to the blade’s rotational course; the figures could be centered on the shroud as a result of it’s a three-dimensional view with superimposed imaginary planes. Within the case of none (Fig. 12), the backflow (limiting streamline) and rotating parts ((Q)-criterion) from the blade LE had been strongly developed towards the upstream because the circulate price decreased. As anticipated, the backflow and rotating parts had been largely suppressed within the case of ASF connected (Fig. 13). The vortexes remoted on the ASF’s stress facet nearly misplaced their velocity and couldn’t go to the suction facet in order that it was troublesome to trigger any instability contained in the circulate passage. From the mixture with Figs. 9 and 10, the residual backflow and rotating parts within the fin-to-fin pitch didn’t penetrate 0.9 spans or much less. From the outcomes on this part, the ASF’s mechanism might be said as comply with: prevening the event of backflow and re-directing the circumferential velocity parts to the axial course. In the meantime, the above footnotes for Figs. 12 and 13 could be equally utilized to the figures within the part under.

### One-factor analyses

From the reference set in Desk 2, ({l}_{r}) was evaluated inside the variable vary. Determine 14a exhibits the (Q)(P) curve (left) and the gradient (({a}_{{x}_{1}leftrightarrow {x}_{2}}); proper) in every circulate price vary:

$${a}_{{x}_{1}leftrightarrow {x}_{2}}=frac{{varPsi }_{x2}-{varPsi }_{x1}}{{varPhi }_{x2}-{varPhi }_{x1}} , left({x}_{2}={x}_{1}+{10}^{-1}proper)$$

(5)

the place subscripts ({x}_{1}) and ({x}_{2}) denote the a number of of normalized circulate price primarily based on the design circulate price. The ASF with ({l}_{r}/{D}_{2}=) 0.01625 misplaced its operate at 0.5 ({varPhi }_{d}). It’s because the rotating circulate parts, which needs to be blocked by the ASF, handed over the ASF’s backside and invaded the suction facet (see Fig. 14b). Accordingly, separated vortex cores had been twisted collectively within the fin-to-fin pitch, and its circumferential velocity was increased than the case of reference set at 0.5 ({varPhi }_{d}) (Fig. 13). The backflow sample between the blades and information vanes was moderately just like the case of none at 0.5 ({varPhi }_{d}) (Fig. 12); the backflow grew to become stronger towards downstream. From an analogous mechanism (see Fig. 14c), the ASF with ({l}_{r}/{D}_{2}=) 0.01 misplaced its operate within the circulate price vary lower than 0.6 ({varPhi }_{d}). As a reminder of Desk 1, ({delta }_{t}/{D}_{2}=) 0.0028, the place ({delta }_{t}) denotes the tip clearance between a blade tip and casing. In the appropriate graph of Fig. 14a, ({l}_{r}) grew to become progressively delicate at decrease circulate charges. Alternatively, a reasonably radical and abrupt tendency was confirmed on the level the place the ASF misplaced its operate for ({l}_{r}). Which means if the ASF has an acceptable ({l}_{r}), its operate will be retained nearly unchanged. From the outcomes, the purposeful limitation for ({l}_{r}) might be proposed as ({l}_{r}/{D}_{2}=) 0.0225 or extra.

#### Axial size

The analysis for ({l}_{a}) was carried out from Fig. 15. In Fig. 15a, the ASF with ({l}_{a}/{D}_{2}=) 0.05 misplaced its operate at 0.5 ({varPhi }_{d}). Particularly, the tendency was extra dramatic than ({l}_{r}), i.e., ({l}_{a}) was troublesome to be analyzed as being delicate to circulate price, and the ASF misplaced its operate extra dramatically. In Fig. 15b, the backflow and rotating parts weren’t sufficiently blocked as ({l}_{a}) was shortened in order that the unfavorable circulate patterns may go the ASF’s LE. Right here, a backflow akin to the axial size of roughly 0.2–0.3 ({D}_{2}) was developed inside the fin-to-fin pitch, akin to the case of none at 0.5 ({varPhi }_{d}) (Fig. 12). This was accompanied by the rotating parts, as anticipated. Though considerably filtered vortexes had been confirmed on the ASF’s suction facet, it was unimaginable to satisfy the operate of ASF. The purposeful limitation for ({l}_{a}) might be proposed as ({l}_{a}/{D}_{2}=) 0.075 or extra.

#### Axial hole

Determine 16 depicts a one-factor evaluation for (delta); (delta) could be essentially the most vital parameter, as talked about in Sect. 2. In Fig. 16a, the ASF exhibiting (delta /{D}_{2}=) 0.05 misplaced its operate at 0.7 ({varPhi }_{d}). The tendency to lose operate was gradual, in contrast to that of ({l}_{r}) and ({l}_{a}); (delta) can also acquire a radical or dramatic tendency like ({l}_{r}) and ({l}_{a}) when it turns into wider past the variable vary, however it’s emphasised that the outline was premised on the purposeful limitation (reversal of gradient) outlined on this research. From the empirical background, a rise in (delta) means that the likelihood for suppressing the backflow from blade LE disappears steadily. As (delta) will increase, the annular vortex core confirmed in Fig. 12 can steadily get better its authentic depth for every circulate price. Because of this, the formation nearly equivalent to Fig. 12 was confirmed in Fig. 16b, for a similar circulate price (0.7 ({varPhi }_{d})). The rotating parts interfered with the ASF’s TE; nevertheless, it was inadequate to type a unfavorable gradient on the (Q)(P) curve. Even Fig. 16b exhibits that the entire patterns of the interior circulate discipline seemed just like the case of none; the ASF didn’t act any efficiency. Thus, the purposeful limitation on (delta) needs to be introduced as (delta /{D}_{2}=) 0.04 or much less. As a warning earlier than making use of the ASF, a earlier evaluation on the meridional aircraft is strongly advisable to stop contact between the ASF’s TE and rotor’s LE; the identical is true within the case of high-pressure followers (or fluid equipment) that will have issues with thrust.

#### Variety of fins

From Fig. 17, a one-factor evaluation was carried out for (Z). Primarily based on the 2 graphs in Fig. 17a, the ASF within the case of (Z=) 4 misplaced its operate at 0.5 ({varPhi }_{d}), and the tendency might be analyzed as radical. In Fig. 17b, the viewing window was prolonged as indicated in Fig. 4 to indicate one full fin-to-fin pitch. The annular vortex core inside one pitch appeared extra restrained than that seen on the similar circulate price (0.5 ({varPhi }_{d})) in Fig. 12; nevertheless, the longer size of 1 pitch as a result of smaller (Z) brought on that the backflow and rotating parts couldn’t be sufficiently suppressed. Though the ASF had the identical dimensions because the reference set solely aside from (Z), strongly developed unfavorable circulate patterns had been noticed within the ASF’s stress facet in addition to the ASF’s suction facet. Due to this fact, the purposeful limitation on (Z) was recognized as at the least 7 or extra. For making use of the ASF, if an ASF-attached casing duct is individually manufactured to get replaced with the non-ASF casing duct (current one), i.e., if a technique apart from on-site welding or fastening is utilized, it could be advisable to pick (Z) equal to the variety of fan’s blades; by positioning the fins into every blade-to-blade pitch, the ASF-attached casing duct will be pushed in a course parallel to the axis, with out disassembling the rotor.

#### Tangential angle

The analysis for (theta) was carried out for every course (+ and (-)) as indicated in Fig. 2; ‘ + ’ denotes opposition to the blade’s rotational course, and ‘(-)’ denotes the blade’s rotational course. First, as proven in Fig. 18a, the ASF with (theta =) + 60° misplaced its operate at 0.5 ({varPhi }_{d}) when (theta) was given to be reverse to the blade’s rotational course, and the tendency was radical. The project of +(theta) to the ASF implies that the backflow and rotating parts, which needs to be remoted on the ASF’s stress facet, can obtain a squeezing impact within the recessed area. Accordingly, though the interior circulate sample within the fin-to-fin pitch (Fig. 18b) carefully resembled the situation of the reference set (Fig. 13) at 0.5 ({varPhi }_{d}), the backflow and rotating parts close to the ASF’s stress facet brought on an extra backflow towards upstream amounting to roughly 0.5 ({D}_{2}); this backflow was even stronger than that of none at 0.5 ({varPhi }_{d}). Determine 19 exhibits the outcomes during which (theta) was given in the identical course because the blade’s rotational course. The ASF with (theta =) (-) 60° misplaced its operate at 0.5 ({varPhi }_{d}), whereas the tendency was not radical (see Fig. 19a). On this case, it appeared that the non-isolated circulate on the ASF’s stress facet prolonged its drive alongside the circumferential course moderately than upstream and overflowed to the suction facet (see Fig. 19b). From the outcomes, the purposeful limitation for (theta) might be introduced inside (pm) 45°. Right here, the insensitivity of ASF to (theta) will be thought of as an extra benefit as a result of it doesn’t require a excessive focus throughout the welding or fastening course of. In the meantime, Fig. 20 depicts the re-presentation of every proper graph in Figs. 14a, 18a, and 19a when it comes to ({l}_{r}) to be able to take into account a correlation between ({l}_{r}) and (theta); the size from the casing to the meanline of ASF’s backside was estimated primarily based on the traditional to the casing-fin intersection. The ASF with (pm theta) misplaced its operate at 0.5 ({varPhi }_{d}) when ({D}_{2})-normalized ({l}_{r}) (({l}_{r}/{D}_{2})) was estimated as roughly 0.029. Due to this fact, it’s extra advantageous to connect the ASF exhibiting correspondingly shorter ({l}_{r}) with out (theta) moderately than intentional (theta).


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# Basic Math MCQ 12 (Work and days)

MCQ 12:  P and Q can complete a work in 15 days and 10 days respectively. They started the work together and then Q left after 2 days. P alone completed the remaining work. The work was finished in__________days.

a) 12               b) 15                  c) 22                d) 20

Solution:

Let the amount of wok = x

For P

work in 15 days = x

work in 1 day =x/15

Or we simply write

Wp = x/15

Similarly for Q

Wq= x/10

Worl done by both P & Q in one day together = x/15+x/10= 5x/30= x/6

or

simply

Wp+Wq = x/6

Work done in two days = $$2\times \frac{x}{6} =\frac{x}{3}$$

Remaining work = $$x -\frac{x}{3}=\frac{2x}{3}$$

This remaining work is to be done by P alone !

We arrange data for P as under;

 Work Days x 15 2x/3 Z

Here it is direct proportion in work and days i-e more wok more days , so

$\frac{\frac{2x}{3}}{x}=\frac{z}{15}$

$\frac{2}{3}=\frac{z}{15}$

$z= \frac {2\times15}{3}=10$

So the remaining work will be done by P in 10 days

Total day in which work was done by P = work done with Q + work done as alone = 2+10 = 12 days !

So option  a is correct!

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