Viscoelastic fluids in a thin domain

The present paper deals with viscoelastic flows in a thin domain. In particular, we derive and analyse the asymptotic equations of the Stokes-Oldroyd system in thin films (including shear effects). We present a numerical method which solves the corresponding problem and present some related numerical tests which evidence the effects of the elastic contribution on the flow. Introduction. Much literature research has been devoted to non-Newtonian fluids, in a thin film, in both mathematical aspects and applications. It is well known that numerous biological fluids, blood or physiological secretions like tears or synovial fluids, show these non-Newtonian characteristics. In engineering applications people are interested in controling the flows characteristics to suit various requirements such as maintaining the fluid qualities in a wide range of temperatures and stresses. Introduction of additives lead to non-Newtonian behavior of the modern lubricant. Another application domain is linked to polymers, whose non-Newtonian characteristics appear in a wide range of applications such as the molding or injection processes. It is to be noticed that, in most practical applications, the geometry of the flow to be considered is anisotropic. A well-known case deals with the study of boundary layers for complex flows [6, 7, 14]. Another case, which is the subject of the present paper, is the lubrication problem in which the fluid is contained between two close surfaces in relative motion. These two applications lead to two very different mathematical models, essentially since the order of magnitude of the parameters in the approximation process is different. For example, in the boundary layer study, the Reynolds number is large and boundary conditions are prescribed at an infinite distance from the solid phase. In lubrication theory, the Reynolds number cannot be too large and boundary conditions 2000 Mathematics Subject Classification. 76A10, 35B40. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] 1 2 G. BAYADA, L. CHUPIN, AND S. MARTIN are precribed on both surfaces which enclose the fluid. As a consequence, pressure becomes the leading unknown. If such anisotropy can induce some numerical problems in 3D computations, especially as the ratio-aspect of the geometry is sufficiently large, it has however the advantage of allowing some simplification in the equations. So if this approximation process could lead to 2D equations, it could be thought that such simplified equations are easier to solve than the original 3D ones. This explains the amount of work devoted to this topic. Some particular classes of non-Newtonian models have often been considered. This includes the Bingham flow or the quasi-Newtonian fluids (Carreau’s law, the power law or Williamson’s law, in which various stress-velocity relations are chosen, see [16]) and also micropolar ones [3]. For these kind of problems, it has been possible to give, in a rigorous way, some thin film approximations of the 3D equations using a so-called generalized Reynolds equation for the pressure. These models, however, considered the fluid as viscous and elasticity effects were neglected. The introdution of such viscoelastic behavior is primilarly described by the Deborah number, denoted De which can be viewed as a measure of the elasticity of the fluid and is related to its relaxation time. One of the laws which seems the most able to describe viscoelastic flows is the Olroyd-B model. This model is based on a constitutive equation which is an interpolation between purely viscous and purely elastic behaviors, thus introducing a supplementary parameter r which describes the relative proportion of both behaviors (the solvant to solute ratio). Considering the Oldroyd model [15], the momentum, continuity and constitutive equations for an incompressible flow of such a non-Newtonian fluid are, respectively, ρ ( ∂U ∂t⋆ ⋆ +U · ∇U ) − η(1− r)∆U ⋆ +∇p − div σ = 0, (1) div U = 0, (2) λ ( ∂ σ ∂t⋆ ⋆ +U · ∇σ + ga(∇U , σ) ) + f(σ) σ = 2ηrD(U ). (3) In these equations, ρ, η and λ are positive constants which respectively correspond to the fluid density, the fluid viscosity and the relaxation time. Equations (1)–(3) make up a system of 10 scalar equations with 10 unknowns: the lubricant velocity vector U = (u1, u ⋆ 2, w ), the pressure p and the extra-stress symmeric tensor σ = (σ i,j)1≤i,j≤3. The bilinear application ga, −1 ≤ a ≤ 1, is defined by ga(∇U , σ) = σ ·W (U)−W (U) · σ − a(σ ·D(U) +D(U) · σ) where D(U) and W (U) are respectively the symmetric and skew-symmetric parts of the velocity gradient ∇U. Usually, D(U) is called the rate of strain tensor and W (U ) is called the vorticity tensor. Notice that the parameter a is considered to interpolate between upper convected (a = 1) and lower convective derivatives (a = −1), the case a = 0 being the corotational case [9]. Note that taking r = 1 allows us to recover various forms of the generalized Maxwell model. Then, by choosing f as the identity, this model is the classical Maxwell one. By introducing a linearized form of f (see in particular [17]), the Phan Tien-Tanner laws [18] are obtained. Conversely, a Newtonian flow is described by choosing r = 0. From the mathematical point of view, few results exist concerning the existence or VISCOELASTIC FLUIDS IN A THIN DOMAIN 3 uniqueness of a solution for true 3D or 2D viscoelastic models [5, 8, 13], also the way to obtain the related thin film approximation is mainly heuristic. A primary approach, which is often used in engineering literature, is to take the parameter defining the (relative) thickness of the flow as the leading small parameter and to use the Deborah number as a pertubation parameter. This has been carried out in the lubrication field by Tichy [19] starting from the upper convected Maxwell model (r = 1, f = Id, a = 1). The case of a Deborah number of the same order of magnitude as the relative thickness has been studied by Tichy and Huang for the UCM Maxwell model and by Bellout [17] for the Phan Tien-Tanner model. In all these researchs, a nonlinear Reynolds equation is gained, allowing the pressure in the thin film to be directly computed. The same procedures can also include the free boundary upper surface of the flow (thin coating problem) or the inertia [10, 20, 21]. However, the goal of these last studies is different, as the primary unknown is not an equation for the pressure but an equation describing the evolution of the free boundary (a generalized shallow water equation). The present paper addresses the mathematical and numerical study of a large class of viscoelastic thin film flows described by an Olroyd-B model in which the Deborah number has the same order of magnitude as the thickness of the fluid. This assumption allows the order of Newtonian and non-Newtonian contributions (see [17] for mechanical comments) to be balanced. Boundary conditions are chosen to be applied to the usual lubrication problems. After scaling both equations and the stress tensor in an adequate way, we are able to obtain an asymptotic 2D problem. This problem generalizes the work of Bellout and Tichy, and concerns not only the rheological model but also can take the 2D dimension (instead of 1D for the pressure asymptotic problem) into account. Obtaining the asymptotic problem is partly an heuristic process, so we have to rigorously prove the solvability of this problem. This is the goal of Section 2 which is divided in two parts for sake of clarity. The newtonian case (r = 0) is studied first and a new way to obtain an existence and uniqueness result for the problem is proposed using velocity as a leading unknown. This type of approach can be easily generalized to the viscoelastic case by using a monotonicity property of the nonlinear term. Interestingly, an existence and uniqueness result is obtained exactly for the same range of the r parameters as in the initial 3D problem. In numerous problems in thin fields, it is possible to eliminate the velocity in the limit problem, thus only retaining a Reynolds equation with respect to the pressure. It is different in our case and we have to solve a nonlinear coupled problem in which a degenerate Stokes equation is still present. A new algorithm related to the Uzawa one is presented and the convergence theorems are given. Lastly, numerical comparisons between various models are given and the importance of obtained 2D and not only a 1D approximation is emphasized. 1. Mathematical formulation. The space coordinates are denoted by (x1, x ⋆ 2, z ) or more simply by (x, z) with x = (x1, x ⋆ 2). Let ω be a fixed bounded domain of the plane z = 0. We suppose that ω has a Lipschitz continuous boundary ∂ω. The upper surface of the gap is defined by z = H(x) with H ∈ C(ω). Let Ω be the following set (see Fig.1): Ω = {(x, z) ∈ R, x ∈ ω and 0 < z < H(x)}. 4 G. BAYADA, L. CHUPIN, AND S. MARTIN