An Instability Phenomenon in Hele-Shaw Displacements

We study the linear instability of the displacement of an Oldroyd-B fluid by air in a Hele-Shaw cell, motivated by possible applications in Secondary Oil Recovery (SOR) process. Most numerical methods have usually failed when the Weissenberg numbers Wi (appearing in the constitutive relations) are near 1. We get an approximate formula of the growth rate with a blow-up for some particular Wi = O(1). Therefore the instability at large Weissenberg numbers is due to the model, and not to the computational methods. Our growth rate is quite similar to Saffman-Taylor’s formula (obtained when a Newtonian liquid is displaced by air) if W1 = W2.


Introduction
A Hele-Shaw cell is a technical device consisting of two parallel plates, located at a small distance from each other.With some usual hypothesis -see (Hele-Shaw, 1898) -the averaged (across the plates) velocities of a Stokes flow are verifying an equation quite similar with the Darcy law for the flow in a porous medium, whose permeability is depending on the distance between the plates -see (Bear, 1972;Bird & Stewart, 1960;Lamb, 1933).
We can consider two immiscible fluids with different viscosities in a Hele-Shaw cell.The "contact" zone is a small region, where the viscosity display a (continuous) large variation.In the Hele-Shaw model, this zone is replaced by a sharp interface, where a viscosity jump exists.This model can be used for study the displacement process similar with the Secondary Oil Recovery (SOR) procedure: the oil (at low pressure) from a porous medium is pushed by a second fluid.The linear stability of the interface between two immiscible fluids of Darcy type in a Hele-Shaw cell or porous medium was studied in (Saffman & Taylor, 1958).The obtained growth constant is giving us the well-known Saffman-Taylor instability criterion: the interface is unstable if the displacing fluid is less viscous.The usual displacing fluids (some polymer-solute) used in (SOR) and the oil in a porous reservoir are often non-Newtonian fluids -the constitutive relations between the stress and the strain-rate tensors are non linear.Then seems be useful to study the "Saffman-Taylor" instability in the case of such fluids.
A large number of results in the field of non-Newtonian fluids are given in (Guillope & Saut, 1992;Nittman et all, 1985;Renardy, 2000;Schowlater, 1978;Truesdell & Noll, 1965;Zhao & Maher, 1993).The displacement of Maxwell upperconvected fluids by air in a Hele-Shaw cell are studied numerically in (Mora & Manna, 2009;Mora & Manna, 2010).Some numerical results concerning the instability of the interface between air (as a displacing fluid) and an Oldroyd-B fluid in a Hele-Shaw were obtained in (Wilson, 1990).Here a blow-up of the growth constant for W 2 = 0, W 1 > 2.5 was reported, similar with the fractures observed in the flows of some complex fluids in Hele-Shaw cells -see (Nase et all, 2008), (Nittman et all, 1985;Zhao & Maher, 1993).
In this paper we study the modal linear stability of the interface appearing when an Oldroyd-B fluid is displaced by air in a Hele-Shaw cell.The Weissenberg numbers W i appear in the constitutive relations (3) and we consider W i = O(1).The basic flow (the same as in (Wilson, 1990)) is described in section 2. The perturbations system is derived in section 3. We use the particular perturbations (26), depending on the arbitrary parameter α.If α checks the condition (43), then we can neglect some terms in the constitutive relations and we get approximate expressions of some components of the extra-stress tensor.An approximate formula of the growth rate of perturbations is given in section 4. We get dispersion curves which are similar with numerical results given in (Wilson, 1990).Our growth constant is similar with the Saffman-Taylor's formula if W 1 = W 2 .The main point is following: the particular perturbations (26) lead us to the blow-up of the growth constant (48) for W 1 − W 2 ≈ 0.4.Then the large Weissenberg number instability is due to the model, and not to the computational methods.We conclude in section 5, where we further deomostrate that our basic flow becomes Newtonian if W 1 = W 2 (that means the relaxation and retardation time constants are equal in the constitutive relations (3)).

The Oldroyd-B Fluid and the Basic Flow
We consider a horizontal Hele-Shaw cell, parallel with the xOy plane and neglect the gravity.An Oldroyd-B fluid is displaced by air in the positive direction of Ox.The distance between plates is denoted by b and the cell length cell is l such that ϵ = b/l << 1.We use the following notations: the extra-stress and strain-rate tensors: τ, S; the velocity, pressure and the fluid viscosity: u = (u, v, w), p, µ; the relaxation and retardation (time) constants: λ 1 , λ 2 ; the matrix containing the velocity derivatives: V; the strain-rate tensor: S := (V + V T )/2; (V i j ) T := V ji .
The flow equations, the free-divergence condition and the constitutive relations are The last two relations are not containing the partial time derivatives τ t and S t , because we consider a steady flow.p x , p y , p z , τ i j,x , τ i j,y , τ i j,z are denoting the x, y, z partial derivatives of τ i j and p.The following boundary conditions are used: a) No-slip condition for the velocity components on the plates.b) Laplace's law in the neighborhood of the basic air-fluid interface.
We study the linear stability of the following basic flow, denoted by the super index 0 : The basic extra-stress tensor is given by the following equation where Therefore the components of the basic extra-stress are and we get the basic flow equations: The equations ( 5) -(10) (used also in (Wilson, 1990)) give us We conclude that a negative constant G exists such that p 0 x (x) = µu 0 zz = G, therefore the basic velocity u 0 can be obtained in terms of G: The following characteristic velocity U is introduced then we have the relations The basic air-fluid interface is As in (Wilson, 1990), the basic pressure can depend on the time t: In the following we consider the moving coordinate system x = x − Ut, then the basic interface is x = 0.But (with no confusion) we still use the notation x = 0 for the basic interface.

The Perturbations System
The small perturbations of the basic solution are denoted by u, v, w, p, τ, The perturbation of the basic interface is denoted by η.We consider that a fluid element that was originally on the interface remains here, then it follows (in other words, the interface is material).
In the frame of the linear stability, the free-divergence relation is also verified by the components of the velocity perturbation, then u x + v y + w z = 0. We integrate across the plates, we use the condition w = 0 on z = 0, z = b, then we get In this paper we consider the particular perturbations such that (which verify the above condition) and we get w z = 0. Then the boundary conditions are giving us w = 0, which is obtained in (Wilson, 1990) by using a numerical method .
We introduce the small perturbations in the constitutive equations and in the expressions of the upper convected derivatives and get In the frame of the linear stability (that means by neglecting the second order terms in perturbations) it follows We have and by using ( 6), ( 9) and ( 17) we get

The Sability Analysis
We consider the following perturbations of the basic solution: We obtain a formula for the growth constant σ corresponding to the perturbations ( 26), which is displaying a blow-up (in terms of the wavenumbers n) for some particular values of the Weissenberg numbers.
Near the basic interface x = 0 we use the following Laplace's law as in (Renardy, 2000;Wilson, 1990) (see ( 13) and ( 18) for definitions of G, η): Here (η yy + η zz ) is the approximate expression of the total curvature of the perturbed interface and γ is the surface tension on the interface.The above relation gives the growth constant expression: The problem is to compute < p − τ 11 > in terms of the basic and perturbed velocities.For this, in the following we search a particular solution τ 33 , τ 13 , τ 23 , τ 12 , p x , p z , τ 11 in terms of u, v, u 0 , v 0 .
We define the dimensionless quantities where W i are the Weissenberg numbers.From ( 12), ( 13) and ( 26) we get In the following we use only dimensionless quantities, then we omit the ′ .
The flow equations and the dimensionless quantities (28) give us Proposition 1.We prove that are possible solutions of the perturbations system (22).
)/e, then τ 12,y can be approximated by the formula Proof.The constitutive relations ( 22) and the dimensionless quantities (28) give us The condition gives us As ϵ << 1, from ( 42) -( 44) we get We conclude that (W 2 − W 1 )u 0 (u y + v x ) xy ϵ can be neglected in front of (u y + v x ) y and (41) gives the relation (38).
Remark 2. The above result is an important improvement compared with (Wilson, 1990), where instead of (26) was used the expansion u ∝ exp(−nx + σt cos(ny), v ∝ exp(−nx + σt) sin(ny).( 45) By using (45), the second and third partial derivatives of (u, v) with respect to x, y contain the factors n 2 exp(−nx + σt) and n 3 exp(−nx + σt), which are not bounded in terms of n when x → 0, even if exp(σt) < 1.Moreover, the expansion (45) is not giving the explicit expression of the perturbations amplitude, which in our paper is 6β|z(1 − z)|.Therefore the expansion (26) and the condition (43) allow us to avoid the singularity near x = 0 and to neglect, in a rigorous way, some terms in the constitutive relations.(47) First we get a 22 = a 23 = a 33 = 0.The third row gives a 31 = 0, from the second row it follows a 21 = 0 and the first row is giving a 11 = 0. Thus A = 0 and τ 0 − µ2S 0 = 0 -that means the basic flow (54) is Newtonian.
When W 1 −W 2 ≈ 0.408, the growth constant formula (48) gives us a strong destabilization effect, compared with (Saffman & Taylor, 1958) -see Remark 3 and Figure 2. Our dispersion curves are similar with the numerical results of (Wilson, 1990).We conclude that the instability for W i = O( 1) is a property specific to the used flow pattern -in our case a Hele-Shaw displacement..3, 0.35, 0.37, 0.38 (upper).