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the hele - shaw problem involves two inmiscible newtonian fluids that interact in a narrow gap between two parallel plates .
one of them is of higher viscosity and the other is effectively inviscid .
the model under consideration is valid when surface - tension effects in the plane of the cell are negligible . in the most of the cases
it is known that when a fluid region is contracting , a finite time blow - up can occur , in which a cusp forms in the free surface .
the solution does not exist beyond the time of blow - up .
however , saffman and taylor in 1958 @xcite discovered the long time existence of a continuum set of long bubbles within a receding fluid between two parallel walls in a hele - shaw cell that further have been called the saffman - taylor fingers .
it is worthy to mention that the first non - trivial explicit solution in the circular geometry has been given by polubarinova - kochina and galin in 1945 @xcite .
they also have proposed a complex variable approach , that nowadays is one of the principle tools to treat the hele - shaw problem in the plane geometry ( see , e.g. , @xcite ) .
following these first steps several other non - trivial exact solutions have been obtained ( see , e.g. , @xcite ) . through the similarity in the governing equations ( hele - shaw and darcy )
, these solutions can be used to study the models of saturated flows in porous media .
another typical scenario is given by witten - sander s diffusion - limited - aggregation ( dla ) model ( see , e.g. , @xcite ) .
in both cases the motion takes place in a laplacian field ( pressure for viscous fluid and random walker s probability of visit for dla ) .
one of the ways , in which several new exact solution have been obtained , is to perturb known solutions .
for example , howison @xcite suggested perturbations of the saffman - taylor fingers that led him to new fingering solutions keeping the same asymptotic behavior as time @xmath0 .
recently , hele - shaw flows and saffman - taylor fingering phenomenon have been studied intensively in wedges ( see , e.g. , @xcite nad the references therein ) . in particular , kadanoff @xcite suggested a self - similar interface evolution between two walls in a hele - shaw cell expressed explicitly by a rather simple parametric function with a logarithmic singularity at one of the walls . by this note
we perturb kadanoff s solution and give new explicit solutions with similar asymptotics .
we suppose that the viscous fluid occupies a simply connected domain @xmath1 in the phase @xmath2-plane whose boundary @xmath3 consists of two walls @xmath4 and @xmath5 of the corner and a free interface @xmath6 between them at a moment @xmath7 .
the inviscid fluid ( or air ) fills the complement to @xmath1 .
the simplifying assumption of constant pressure at the interface between the fluids means that we omit the effect of surface tension .
the velocity must be bounded close to the contact point that yields the contact angle between the walls of the wedge and the moving interface to be @xmath8 ( see figure [ fig3 ] ) .
a limiting case corresponds to one finite contact point and the other tends to infinity . by a shift
we can place the point of the intersection of the wall extensions at the origin . to simplify matter , we set the corner of angle @xmath9 between the walls so that the positive real axis @xmath10 contains one of the walls and fix this angle as @xmath11 $ ] .    in the zero - surface - tension model neglecting gravity
, the unique acting force is pressure @xmath12 .
the velocity field averaged across the gap is given by the hele - shaw law ( darcy s law in the multidimensional case ) as @xmath13 .
incompressibility implies that @xmath14 is simply @xmath15 the dynamic condition @xmath16 is imposed on the free boundary @xmath17 .
the kinematic condition implies that the normal velocity @xmath18 of the free boundary @xmath19 outwards from @xmath1 is given as @xmath20 on the walls @xmath21 and @xmath22 the boundary conditions are given as @xmath23 ( impermeability condition ) .
we suppose that the motion is driven by a homogeneous source / sink at infinity .
since the angle between the walls at infinity is also @xmath9 , the pressure behaves about infinity as @xmath24 where @xmath25 corresponds to the constant strength of the source ( @xmath26 ) or sink ( @xmath27 ) .
finally , we assume that @xmath28 is a given analytic curve .
we introduce the complex velocity ( complex analytic potential ) @xmath29 , where @xmath30 is the stream function
. then , @xmath31 by the cauchy - riemann conditions .
let us consider an auxiliary parametric complex @xmath32-plane , @xmath33 .
we set @xmath34 , @xmath35 , @xmath36 , @xmath37 , @xmath38 , and construct a conformal univalent time - dependent map @xmath39 , @xmath40 , so that being continued onto @xmath41 , @xmath42 , and the circular arc @xmath43 of @xmath44 is mapped onto @xmath19 ( see figure [ fig4 ] ) .
this map has the expansion @xmath45 about infinity and @xmath46 .
the function @xmath47 parameterizes the boundary of the domain @xmath1 by @xmath48 , @xmath49 .
we will use the notations @xmath50 , @xmath51 .
the normal unit vector in the outward direction is given by @xmath52 therefore , the normal velocity is obtained as @xmath53 the superposition @xmath54 is the solution to the mixed boundary problem ( [ 1 ] ) , ( [ 2 ] ) , ( [ 4 ] ) in @xmath55 , therefore , it is the robin function given by @xmath56 .
on the other hand , @xmath57 the first lines of ( [ 6 ] ) , ( [ 7 ] ) give us that @xmath58 the resting lines of ( [ 6 ] ) , ( [ 7 ] ) imply @xmath59
we are looking for a solution in the form @xmath60 where @xmath61 is regular in @xmath55 with the expansion @xmath62 about infinity .
the branch is chosen so that @xmath63 , being continued symmetrically into the reflection of @xmath55 is real at real points .
the equation ( [ 8 ] ) implies that on @xmath43 the function @xmath63 satisfies the equation @xmath64 taking into account the expansion of @xmath63 we are looking for a solution satisfying the equation @xmath65 changing the right - hand side of the above equation one would obtain other solutions . the general solution to ( [ eq ] )
can be given in terms of the gauss hypergeometric function @xmath66 as @xmath67    ht    we note that @xmath68 vanishes for @xmath69 , therefore , the function @xmath47 is locally univalent , the cusp problem is degenerating and appears only at the initial time @xmath70 and the solution exists during infinite time .
the resulting function is homeomorphic on the boundary @xmath41 , hence it is univalent in @xmath55 .
this presents a case ( apart from the trivial one ) of the long existence of the solution in the problem with suction ( ill - posed problem ) . to complete our solution we need to determine the constant @xmath71 .
first of all we choose the branch of the function @xmath72 so that the points of the ray @xmath73 have real images .
this implies that @xmath74 .
we continue verifying the asymptotic properties of the function @xmath75 as @xmath76 .
the slope is @xmath77= \alpha+\frac{\pi}{2}+\lim\limits_{\theta\to\alpha-0}\arg\left(\sqrt{\frac{2qt}{\alpha}}+ \frac{e^{i\frac{\pi\theta}{\alpha}}-1}{e^{i\frac{\pi\theta}{\alpha}}+1}\right)=\alpha+\pi.\ ] ] to calculate shift we choose @xmath71 such that @xmath78=0.\ ] ] using the properties of hypergeometric functions we have @xmath79 therefore , @xmath80 .
we present numerical simulation in figure  [ fig14 ] .
the special case of angle @xmath81 has been considered by kadanoff @xcite .
the hypergeometric function is reduced to arctangent and we obtain @xmath82 this function maps the domain @xmath83 onto an infinite domain bounded by the imaginary axis @xmath84 , the ray @xmath85 of the real axis and an analytic curve @xmath19 which is the image of the circular arc , see figure [ fig8 ] .
kadanoff s solution ( [ kadanoff ] ) can be thought of as a logarithmic perturbation of a circular evolution with the trivial solution @xmath86 .
a simple way to generalize the solution ( [ kadanoff ] ) is to perturb another function .
for example , one may choose @xmath87 we find the solution @xmath88 in the form @xmath89 similarly to the preceding section .
then the equation ( [ 8 ] ) is satisfied when @xmath90 where @xmath91 as @xmath92 in the unit circumference .
we choose a consistent form of @xmath93 as @xmath94 integration yields @xmath95 where @xmath71 is a constant of integration . satisfying the conditions on the walls we deduce that @xmath96 , and finally , we get a logarithmic perturbation of the elliptic evolution as @xmath97 see the interface evolution in figure  [ fig19 ] .
the next goal is to obtain perturbations of the logarithmic term of kadanoff s solution that with the same asymptotic as @xmath98 , such that the interface has finite contact points at a finite moment .
let us consider the function @xmath99 the functions @xmath100 , @xmath101 are to be chosen such that equation ( [ 8 ] ) is satisfied for the moving interface as well as the conditions of impermeability and univalence hold .
the local univalence is followed from the first restriction @xmath102 . substituting @xmath103 into equation ( [ 8 ] ) and comparing the fourier coefficients ,
we derive the following system of equations for the functions @xmath100 and @xmath101 : @xmath104 this system can be easily solved and the first integrals are @xmath105 @xmath106 where @xmath107 are the constants of integration .
let us assume the initial condition @xmath108 .
making use of the system ( [ u1],[u2 ] ) we arrive at the explicit function @xmath109 inverse to @xmath100 @xmath110 that exists , is continuous , and increases in the interval @xmath111 .
therefore , the function @xmath100 increases from @xmath112 to @xmath113 as @xmath114 . by ( [ u1 ] )
we conclude that @xmath115 as @xmath98 .
the rotation of @xmath103 is exactly kadanoff s solution when @xmath116 , and @xmath101 is appropriately chosen as in ( [ kadanoff ] ) . to make a numerical simulation
one may use the newton method of the solution of a non - linear system ( howison @xcite presented the numerical approximation of an analogous solution in a narrow channel ) , see figure  [ fig20 ] .    choosing @xmath112 rather close to 1
, one may give an explicit analytic approximation by , e.g. , introducing two functions @xmath117 @xmath118 the initial conditions @xmath112 and @xmath119 are to satisfy the inequalities @xmath120 , @xmath121 . to proceed , we simplify putting @xmath122 .
then these inequalities are satisfied for @xmath123 .
it is easily seen from ( [ u2 ] ) that @xmath124 .
then @xmath125 too . similarly , @xmath126 and @xmath127
both @xmath128 and @xmath100 tend to 1 rapidly and the error @xmath129 is of the same order for @xmath130 .
now we evaluate the error @xmath129 for @xmath131 , and claim that @xmath132 to prove this we estimate the distance @xmath133 between the inverse function   and @xmath134 as @xmath135 the derivative of @xmath136 is @xmath137 since the function @xmath138 increases , the function @xmath133 may have a critical point @xmath139 , @xmath140 , which is the maximal solution to the equation @xmath141 in the interval @xmath142 .
the latter equation implies @xmath143 that proves  .
moreover , @xmath144 is decreasing and non - negative as a function of the initial condition @xmath112 , that vanishes as @xmath145 .
therefore , given a small positive number @xmath146 , we may choose @xmath122 close to 1 such that @xmath128 approximates @xmath100 with the precision @xmath146 during the whole time @xmath131 .
desired quantity @xmath112 satisfies the equation @xmath147 a similar conclusion may be made for the function @xmath101 and its approximation @xmath148 ( note that @xmath149 ) .
moreover , the mapping @xmath150 converges to kadanoff s solution as @xmath98 .
a.  arnodo , y.  couder , g.  grasseau , v.  hakim , m.  rabaud , _ uncovering the analytical saffman - taylor finger in unstable viscows fingering and diffusion - limited aggregation _ , phys .
* 63 * ( 1989 ) , no .
9 . 984987 .
we consider a zero - surface - tension two - dimensional hele - shaw flow in an infinite wedge . 
 there exists a self - similar interface evolution in this wedge , an analogue of the famous saffman - taylor finger in a channel , exact shape of which has been given by kadanoff . 
 one of the main features of this evolution is its infinite time of existence and stability for the hadamard ill - posed problem . 
 we derive several exact solutions existing infinitely by generalizing and perturbing the one by kadanoff .