Displacement of oil by water in a single elastic capillary

In this paper we consider the evolution of a free boundary separating two immiscible viscous fluids with different constant densities. The joint motion of liquids in the solid skeleton is described by Stokes equations coupled with Lame equations, driven by the input pressure and the force of gravity. We prove the existence and uniqueness of classical solutions global in time, and we emphasize the study of the properties of the moving boundary separating the two fluids.


Introduction
This paper follows our previous paper [1], where we considered the evolution of a free boundary separating two immiscible viscous fluids in a single capillary of an absolutely rigid solid body. Here we consider the same flow in a single capillary Qf c Q of an elastic solid body.
In dimensionless variables the evolution of a flow is driven by the input pressure and the force of gravity. More precisely, in this problem we have to find the velocity U (x, t), the pressure pf (x, t) and the density pf (x, t) of the non-homogeneous liquid in Qf, and the displacements us(x, t) and the pressure ps(x, t) of the elastic skeleton in Qs = Q\Qf from the following system of differential equations: V -Pf + pf e = 0, V -U = 0, x e Qf, 0 < t < T, V -Ps + pse = 0, V -us = 0, x e Qs, 0 < t < T, Ps = 2kD(us) -p sI, p, = const is the viscosity of the liquids, к = const is the Lame coefficient, e is the given vector, ps is the density of the solid body, and I is the unit tensor.
The mass and momentum conservation laws dictate the coincidence of velocities and normal tensions in the liquid and solid components, d us u = -, Pf • n = Ps • n, To simplify our considerations we pass to the homogeneous boundary conditions at S±, P; • e1 = 0, x e S±, i = f , s, 0 < t < T, by introducing a new pressure pf ^ pf -p 0(x), ps ^ ps -p 0(x), p 0(x) = 2p+(x)(1 -X i).
So, we may expect that the free boundary Г(^ will not touch the given boundaries S± at least for some time interval 0 < t < T.
At the boundaries S± for 0 < t < T and at initial moment t = 0, the density pf is piecewise constant and assumes two positive values characterizing the distinct phases of the flow Pf (x, 0) = p0 (x), x e Qf, ( ) where p0(x) = p± for x e Q±(0).
If the velocity u (x , t) of the liquid is sufficiently smooth, then the Cauchy problem dt determines a mapping In particular, the free boundary Г (t) is determined as a set The problem treated here is that of finding the velocity u (x , t) and pressure pf (x, t) of the liquid in pores, the displacement us(x, t) and pressure ps(x, t) of the solid skeleton, and the density pf (x, t) of the liquid from the above equations and the initial and boundary data.
Note that it is nonlinear because of the coupling term u • V pf in (2).
It is shown below that the evolution described by the above equations preserves the existence of two subdomains Q±(t), each occupied by one of the fluids, that are separated at time t > 0 by a regular free boundary r(t). Thus, the problem studied is equivalent to finding {U ,p f, us,p s}, and the moving boundary r(t).
Theorems on the existence of generalized solutions to the Navier-Stokes system for non homogeneous incompressible fluids were obtained in, e.g., [2][3][4][5][6][7][8][9] (without a detailed anal ysis of the set where the density is discontinuous). The existence and uniqueness of the classical solution to the Stokes equations for a non-homogeneous liquid with Dirichlet data have been proved in [10], and with Neumann data in [1]. The Muskat problem at the microscopic level with corresponding homogenization has been considered in [11].
Finally, we explain our motivation to study this particular problem. It is well known [12] that the Darcy system of filtration, describing the macroscopic flow of a homogeneous incompressible liquid in some bounded domain Q, is a result of homogenization of the Stokes system for an incompressible viscous liquid occupying a periodic pore space Qs c Q in an absolutely rigid solid body.
The more complicated macroscopic motion of two immiscible incompressible viscous liquids is governed by the Muskat problem. In this model one looks for the free boundary r(t) c Q, which separates two different domains Q+(t) c Q and Q-(t) c Q, Q+(t) U r(t) U Q-(t) = Q, occupied by different fluids. In each of the domains Q±(t) the liquid motion is described by its own Darcy system of filtration, and at the free boundary the normal velocities of the liquids coincide with the normal velocity of the free boundary.
Thus, we may expect that, as in the case of the filtration of a single liquid, the Muskat problem should be a homogenization of the initial boundary value problem for the Stokes system with a non-homogeneous liquid, pA us + gps e = 0, V -ue =0, ^ = 0, (17) at in a periodic pore space Qs of an absolutely rigid solid body Q with the following boundary and initial conditions: where p°(x) = p + = const, x e Q+ (0), pe 0(x) = p = const, x e Qs (0), Q+(0) U Q-(0) = Qs, p is the viscosity andge is the acceleration due to gravity.
But until now this fact has not been proven and it may not be true for an absolutely rigid solid body.
Some indirect arguments confirm this guess.
As a first argument, note that the problem (17)-(19) possesses the evident a priori esti mate max p 2 / |vue(x, t)|2 dx < C0, where C0 is independent of e.
In order to pass to the limit in the transport equation (2) for the density ps, as the size e of pores goes to zero, one needs at least uniform boundedness of the gradient of the velocity ue: Due to the boundary condition (18), the contact points of the free boundary and the solid skeleton will be permanently fixed at the initial position. Numerical implementa tions predict the appearance of a water tongue, which grows with time (see Figure 1). The  free boundary and solid body begin to move ( Figure 6), and homogenization conserves the free boundary which separates the two liquids.
The aim of this paper is to show that the problem (2)-(6), (8), (10) (13) and (14) for an elastic solid body has a solution with a smooth free boundary, which divides the two liq uids.
Next we introduce the space of functions with non-integer derivatives. To do this straight forwardly we consider the half-spaces Rf = jx = (x1,x2) e R 2 : |x1| < ж ,x2 > 1 j, whereMis a Fourier transformation of v: According to [13] (Chapter 2, Theorem 2.3, p.71) For smooth functions we define the following norms: We say that the function u(x) belongs to the space Ca (П) if and it belongs to the space We say that the surface Г e П is Ck+a regular if in local coordinates it is presented by Ck+a regular functions.
Our principal result is the following.
Namely, let M be the set of all continuous functions P e C(G), G = Qf x (0, T), such that For fixed e > 0 we define the following nonlinear operator:  More precisely, we first derive

L2-estimates for the solutions of the modified problem (29), (30),
and using the Fourier transform's techniques we find

uniform estimates for the solutions of the problem (29), (30) in Holder' s spaces.
This part of the proof contains a lot of technical details and is very difficult to follow. Unfortunately, the linear problem that arises is completely novel and requires special consideration. The standard method for classical differential equations consists of: (1) a Fourier transform of the original problem, In what follows, by C we denote constants depending only on C0, p±, and ps.
Proof First of all, note that due to linearity of the problem it suffices to find corresponding a priori estimates.
To prove the first estimate in (36) To estimate the right-hand side in (38) we introduce a new function u(x, t): d us u(x, t) = uf (x, t) for x e Qf, and u(x, t) = -(x, t) for x e Qs. d t It is easy to see that u e Wj(Q), u(x, t) = 0 for x e S0 (see the boundary condition in (31)), Therefore, we may apply the Friedrichs-Poincare inequality [15] f | u(x, t)|2 dx < C f D(u(x, t ) ) : D(u(x, t)) dx, Q Q which together with (38) and (39) imply j D(uf (x, t0) ) : D(uf (x, t0)) dx Q f < C ( f D(uf (x, t ) ) : D(uf (x, t)) dxdt + C0(e).

Jo J Qf
In turn, if we put J0 J Qf we arrive at the differential inequality Proof For given p we may find the solution U = Ф1(р) of the problem (33), then solve the initial boundary value problem (34) and find p = Ф2(и^) = Ф(р) such that The first estimate follows from the maximum principle and shows that Ф transforms M into itself, and the smoothness of p follows from existence theorems for parabolic equa tions with smooth coefficients ( [13], p.320).
So, if we prove the continuity of the operator Ф, then Ф would be completely continuous due to (41). Finally, the Schauder fixed point theorem [14] permits us to find a fixed point The nonlinear operator Ф2 is also continuous.
Coming back to (47) we conclude that Pf (uf,s,pf ) e LTO ((0, T); Wl(Qf )), and (47) is equivalent to the Stokes equation The right-hand side F = f + pse of the differential equation belongs to LTO (G). Therefore we may use the same arguments as in [1] and conclude that for any Щ c Qf u[ ,s e LTO((0, T); W^ft)) for any m >2.
Now we apply the same arguments for the solid component: and Ps(us,s,p f) e LTO ((0, T);L2(Qs)). □
Note that wk( f ,0,0), j and are known functions, and the functions j and are infinitely smooth in t.