A THIN LAYER OF A NON-NEWTONIAN FLUID FLOWING AROUND A ROUGH SURFACE AND PERCOLATING THROUGH A PERFORATED OBSTACLE

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1 Journal of Mathematical Sciences, Vol. 189, No. 3, March, 213 A THIN LAYER OF A NON-NEWTONIAN FLUID FLOWING AROUND A ROUGH SURFACE AND PERCOLATING THROUGH A PERFORATED OBSTACLE A. Yu. Linkevich Narvik University College, Postboks 385, Narvik 855, Norway anna.linkevitch@gmail.com T. S. Ratiu École polytechnique fédérale de Lausanne, CH-115 Lausanne, Switzerland Moscow Lomonosov State University, Moscow , Russia tudor.ratiu@epfl.ch S. V. Spiridonov Moscow Lomonosov State University, Moscow , Russia spiridonov.s.v@gmail.com G. A. Chechkin Moscow Lomonosov State University, Moscow , Russia chechkin@mech.math.msu.su UDC We study the behavior of a thin layer of a conducting dilatant fluid percolating through a porous obstacle and flowing around a rough surface. The characteristic size of pores in the obstacle (the initial velocity profile and the roughness on the lower boundary play a role of a small parameter of order O(ε. For a family of boundary value problems depending on ε we construct a homogenized (limit problem and prove the homogenization theorem. Thereby we describe the effective behavior of such a non-newtonian fluid. Bibliography: 16titles. Illustrations: 2figures. Introduction This paper continues the investigations started at [1] (cf. also [2] [5], where the behavior of a boundary layer of a microinhomogeneous Newtonian fluid was studied. In particular, a homogenized problem was obtained and the convergence of solutions to the original problem depending on a small parameter to the solution to the homogenized problem was established for a thin layer of a Newtonian fluid percolating through a perforated obstacle and flowing around a rapidly oscillating surface. In this paper, we study a boundary layer of a magnetic dilatant fluid percolating through an obstacle with slit-like pores and flowing around a rough (rapidly oscillating surface. The To whom the correspondence should be addressed. Translated from Problemy Matematicheskogo Analiza 68, January 213, pp /13/1893- c 213 Springer Science+Business Media New York 1

2 Prandtl ideas (expressed at the beginning of the XXth century concerning the behavior of thin boundary layers of a Newtonian fluid can be applied not only to the classical Newtonian fluids, but also to flows of non-newtonian fluids (conducting, magnetic and to some new fluids, for example, to a modified Ladyzhenskaya fluid [6] [9]. Some problems with small parameters in the theory of boundary layers were considered in [1] (an incompressible fluid flowing through a small hole, [11] (perturbations introduced by a harmonic oscillator in a boundary layer on a plate, [12] (homogenization of the Prandtl equations governing a nonmagnetic fluid with rapidly oscillating blowing/suction, [13], (multiscale homogenization of the Prandtl equations [14] (a flow around a rough surface, [15] (estimates for the convergence rate for solutions of the original problems to the solution of the homogenized problem, [16] (homogenization of an inhomogeneous pseudoplastic fluid. In this paper, we study the behavior of a boundary layer of a magnetic dilatant fluid flowing around a rough surface and percolating through a perforated obstacle made of microporous material. We introduce a small parameter ε characterizing the microinhomogeneous structure of the obstacle and the plate roughness. The role of ε can be played by the microinhomogeneity size in the obstacle and the microinhomogeneity size (roughness on the plate. Thus, we study the asymptotic behavior of solutions to a family of the Prandtl equations for the fluid flow in a neighborhood of a rough plate (i.e., in the domain with rapidly oscillating boundary. Moreover, we construct a homogenized (limit as ε problem and prove the convergence of solutions of the original problems to the solution of the homogenized problem. For this purpose, we consider the problem about extending a boundary layer of a fluid percolating through a porous microinhomogeneous obstacle P ε (cf. Figure 1. P ε Fig. 1. The flow of a fluid on a rough surface percolating through an obstacle with slit-like pores. We assume that the obstacle has slit-like pores and the behavior of the fluid remains unchanged along the obstacle. Then the problem under consideration can be reduced to the two-dimensional problem on the vertical cross section perpendicular to the obstacle. 2

3 1 Statement of the Problem and the Main Result 1.1 Description of the domain and boundary conditions Let a function U ε (y describe the fluid velocity after passing the obstacle, i.e., this function is the initial velocity profile. We assume that the function U ε (y satisfies the following natural conditions: 1 U ε > fory>, 2 U ε U strongly in L 2 (R, 3 the family of functions U ε is uniformly bounded on R. We also assume that the surface roughness is small in comparison with the boundary layer thickness. Let the plate surface be described by the equality y = F ε (x, where F ε uniformly converges to zero as ε andf ε ( =. For an example of such a function one can consider F ε = εf (x/ε, where F (s is a 1-periodic function. We set d(x, y = σ(x, yb2 (x ρ > and introduce the following notation (cf. Figure 2: σ is the magnetoconductivity of the fluid, B is the magnetic induction component, orthogonal to the streamlined plate, ρ = 1 is the fluid density, u ε (x, y (orv ε (x, y is the flow velocity field parallel (or orthogonal to the plate, (U ε (y, is the initial flow velocity, (,V(x is the velocity on the lower boundary, and (U (x, is the velocity on the upper boundary of the domain (the velocity of the main fluid flow. y P ε U (x U U ε d(x, y v u x O X V Fig. 2. The boundary layer of a non-newtonian magnetic fluid percolating through a porous obstacle and flowing around a rough surface. We consider the family of problems ν ( u ε n 1 u ε u ε u ε y y y v u ε ε y = d(x, y(u (x u ε U du, u ε + v ε y = 1 <n<, (1.1 3

4 in D ε = { <x<x,f ε (x <y< } with the boundary conditions u ε (,y=u ε (y, u ε (x, F ε (x =, u ε (x, y U (x, v ε (x, F ε (x = V (x, y. (1.2 Definition 1.1. The following problem about the boundary layer extension is referred to as a homogenized problem for the family of problems (1.1 (1.2: ν ( u n 1 u u u y y y v u y = d(x, y(u (x u U du, u + v y =. in D = { <x<x, <y< } with the boundary conditions 1 <n<, (1.3 u(,y=u(y, u(x, =, v(x, = V (x, u(x, y U (x, y. (1.4 Definition 1.1 is justified below. 1.2 Statement of the problem in the von Mises variables From now on, we assume that 1 <n<+. We consider the auxiliary problem ν ( ũ ε y y ũ ε + ṽε y = n 1 ũ ε y ũ ε ũε ṽε ũε y = dε (x, y(u (x ũ ε U du, (1.5 in D = { <x<x, <y< } with the boundary conditions ũ ε (,y=u ε (y, ũ ε (x, =, ṽ ε (x, = V (x, ũ ε (x, y U (x, y. (1.6 We make the change of the von Mises variables (x, y (x, ψ for the problems (1.5, (1.6 and (1.3, (1.4: x = x, ψ = ψ(x, y, (1.7 where u = y, v V =, ψ(x, =, (1.8 u, v are solutions to the corresponding problem, V is the vertical velocity component on the lower boundary of the domain. For a new unknown we take w(x, ψ =u 2 (x, y. The domain D is transformed into the domain Ω = { <x<x, <ψ< }. 4

5 In the von Mises variables, the problem (1.5, (1.6 is written as L ε (w ε ν wε 2 n 1 x ( w ε n 1 w ε w ε V w ε = 2d vm (x, ψ(u w ε 2U du in the domain Ω with the boundary conditions (1.9 w ε (x, =, w ε (,ψ=w ε (ψ, w ε (x, ψ (U 2 (x, ψ, (1.1 where W ε ( y U ε (ηdη Uε 2 (y, d vm (x, ψ(x, y d ε (x, y. (1.11 Moreover, by the properties of the replacement, there exists a function b(x such that d vm (x, ψ = d vm (,ψb(x. Thus, the function d vm written in the von Mises variables is independent of ε. The following compatibility condition holds as ψ : ν Wε 2 n 1 W ε n 1 W ε +2U (Ux ( V (W ε +2d vm (,ψ(u ( W ε =O(ψ. (1.12 Definition 1.2. By a generalized solution to the problem (1.9 (1.12 we mean a function w ε (x, ψ such that w ε is continuous, bounded, and positive for ψ>, w ε is bounded in Ω, w ε kψ for ψ ψ 1, k = const >, and w satisfies the boundary condition (1.1 for ψ = and ψ, and the following integral identity holds: Ω ( ν w ε 2 n 1 n 1 w ε φ + 1 w wε φ + V w wε φ +2dvM φ 2U ( du wε + dvm φ dψ (1.13 for any function φ(x, ψ that is continuous in Ω, has bounded derivative φ in Ω, and vanishes for ψ = and sufficiently large ψ. The equivalence of these statements is shown in [12, Chapter 1]. The von Mises form of the problem (1.3 (1.4 is as follows: L (w ν ( w w 2 n 1 n 1 w w V w = 2d vm (x, ψ(u w 2U du in Ω with the boundary conditions (1.14 w(x, =, w(,ψ=w (ψ, w(x, ψ (U 2 (x, ψ, (1.15 5

6 where W ( y U(ηdη U 2 (y, (1.16 and the compatibility condition ν W W 2 n 1 n 1 W +2U (Ux ( V (W +2d vm (,ψ(u ( W =O(ψ (1.17 as ψ. Here, d vm (x, ψ(x, y d(x, y. 1.3 The main results Theorem 1.1 (existence. Let F ε be a continuously differentiable function on R such that F ε ( =. A solution u ε (x, y, v ε (x, y to the problem (1.1, (1.2 exists if and only if there exist functions ũ ε (x, y and ṽ ε (x, y that solve the Prandtl equations (1.5 in D with the conditions (1.6. In this case, d ε (x, y =d(x, y + F ε (x, wherethed ε and d are taken from Equations (1.5 and (1.3 respectively. In this case, u ε (x, y =ũ ε (x, y F ε (x, v ε (x, y =ṽ ε (x, y F ε (x F ε(x ũ ε (x, y F ε (x. Theorem 1.2 (convergence. We assume that 1 U(y > for y>, U( =, U ( >, andu(y U ( as y, 2 U ε (y > for y>, U ε ( =, U ε( >, andu ε (y U ( as y, 3 du, V (x, andd(x, y are infinitely differentiable on [,X ], 4 U(y, U (y, andu (y are bounded for y< and satisfy the Hölder condition, 5 U ε (y, U ε(y, andu ε (y are bounded for y< and satisfy the Hölder condition, 6 Ux (x > for x X, 7 U ε (y <U ( and U(y <U ( for any y>, ε>. 8 V (x, 9 U(x U ( for ψ Ψ, 1 U ε (x U ( for ψ Ψ, Then there exist X X and ε > such that 1 for any ε<ε there exists a unique generalized solution to the problem (1.9 (1.12 and a unique generalized solution to the problem (1.14 (1.17 in Ω, 2 the family of solutions to the problems (1.9 (1.12 converges to the solution to the problem (1.14 (1.17 as ε so that w ε w uniformly in Ω. 2 Proof of the Main Results Proof of Theorem 1.1. We introduce new independent variables ξ = x and η = y F ε (x. It is easy to see that = ξ η F ε(x, y = η. (2.1 6

7 Taking into account (2.1, we write the system (1.5 in the form u ε (u εξ F εu εη +u εη v ε = ν η ( u εη n 1 u εη +U U + d(ξ,η + F ε (ξ(u u ε, u εξ F εu εη + v εη =. (2.2 We set ṽ ε := v ε F εu ε, ũ ε := u ε, d ε (ξ,η :=d(ξ,η + F ε (ξ. (2.3 Then (2.2 is written as ũ ε ũ ε ξ + ũε ηṽ ε = ν η ( ũ εη n 1 ũ εη +U U + d ε (ξ,η(u ũ ε, ũ ε ξ + ṽε η =. (2.4 On the boundary, we have ũ ε ξ= = u ε ξ= = U ε (η + F ε ( = U ε (η, ũ ε η= = u ε (ξ,f ε (ξ =, ṽ ε η= = v ε η= + F ε(ξu ε η= = V (ξ, ũ ε (ξ,η U (ξ, η. Remark 2.1. The existence and uniqueness of a solution is proved in the same way as Theorem in [12, Chapter 8, Section 8.4]. Proof of Theorem 1.2. We begin with the following auxiliary assertion. Lemma 2.1. Let w 1 and w 2 be two solutions to the Prandtl equations L(w 1 ν ( w 1 w1 2 n 1 n 1 w 1 w 1 V w 1 +2dvM (x, ψ(u w 1 +2U du in Ω={ <x<x, <ψ< } with the conditions = (2.5 w 1 (,ψ=w 1 (ψ, w 1 (x, =, w 1 (x, ψ (U 2 (x, ψ, (2.6 where and W 1 ( y U 1 (ηdη U1 2 (y, L(w 2 ν ( w 2 w2 2 n 1 n 1 w 2 w 2 V w 2 +2dvM (x, ψ(u w 2 +2U du = (2.7 7

8 in Ω={ <x<x, <ψ< } with the conditions where w 2 (,ψ=w 2 (ψ, w 2 (x, =, w 2 (x, ψ (U 2 (x, ψ, (2.8 W 2 ( y U 2 (ηdη U2 2 (y. Let the assumptions of Theorem 1.1 be satisfied. Then the solution monotonically depends on the initial conditions: w 1 (x,ψ w 2 (x,ψ dψ Proof. By [12, Theorem ], there exists Ψ Ψ such that By the definition of a generalized solution, Ψ X [ ν ( w 1 2 n 1 w 1 (,ψ w 2 (,ψ dψ x [,X]. (2.9 w 1 (x, ψ (U 2 (x, ψ Ψ, (2.1 w 2 (x, ψ (U 2 (x, ψ Ψ. (2.11 n 1 w 1 w 2 n 1 w 2 +2V (x ( w 1 w 2 φ +2U ( du φ +2 ( + dvm w1 w1 w 2 φ w2 ] φ dψ = (2.12 for any φ from the definition of a generalized solution. We introduce functions Θ(λ andθ ε (λ by the formula {, λ,, λ, Θ(λ = Θ ε (λ = λε 1, λ ε, 1, λ >, 1, λ ε. Substituting φ(x, ψ =Θ ε (w 1 (x, ψ w 2 (x, ψθ ε (x x into (2.12, we find Ψ X [ ν ( w 1 2 n 1 n 1 w 1 w 2 n 1 w 2 dθε (w 1 w 2 dλ +2V (x ( w 1 w 2 φ ++2U ( du which implies 8 Ψ X [ ( w1 w 2 + dvm w1 φ+v (x ( w 1 w 2 φ + U ( du ( w1 w 2 Θ ε (x x w2 ] φ dψ =, w2 + dvm w1 ] φ dψ.

9 Passing to the limit as ε, we find Ψ x = Ψ x [ ( w1 w 2 [ ( w1 w 2 Θ(w 1 w 2 +V(x ( w 1 w 2 Θ(w 1 w 2 + U ( du + dvm ( w1 w 2 + ]dψ Θ(w 1 w 2 +V(x ( w 1 w 2 Θ(w 1 w U ( du + dvm w1 w2 ] Θ(w 1 w 2 dψ, where (a + =max(a,. Since f + Θ(f = (f, f + Θ(f = (f, we find Ψ x [ ( w1 w 2 + +V (x ( w 1 w U ( du +dvm ( w1 w 2 + ]dψ and further Ψ ( w 1 w 2 + x=x dψ which implies Ψ + Ψ ( w 1 w 2 + x= dψ Ψ x ( w 1 w 2 + x=x dψ U ( du + dvm ( w1 w 2 + dψ, Ψ ( w 1 w 2 + x= dψ. We argue in a similar way in the case ( w 1 w 2. By (2.1 and (2.11, we arrive at the required assertion. Arguing in the same way as in [12, Chapter 8, Section 4], one can show that for the family w ε (x, ψ we have the uniform boundedness of the derivative with respect to ψ, i.e., w ε (x, ψ M 1. (2.13 Lemma 2.1 and the estimate (2.13 imply the required convergence. Remark 2.2. Under the replacement (2.3, the original problem goes to the problem (1.5, (1.6 whose solutions tend to the solution of the homogenized problem (1.3, (1.4 as ε. Thus, we have proved the convergence of solutions of the original problem (1.1, (1.2 to the solution of the homogenized problem (1.3, (1.4. Simultaneously, we justified Definition

10 Acknowledgments The work was financially supported by the Government grant of the Russian Federation under the Resolution No. 22 On measures designed to attract leading scientists to Russian institutions of higher education according to the Agreement No. 11.G , signed by the Ministry of education and science of the Russian Federation, the leading scientist, and Lomonosov Moscow State University (on the basis of which the present research is organized. T. S. Ratiu was also supported by Swiss NSF grant No G. A. Chechkin was also supported by the Russian Foundation for Basic Research (grant No References 1. A. Yu. Linkevich, S. V. Spiridonov, and G. A. Chechkin, On boundary layer of Newtonian fluid, flowing around a rough surface and percolating through a perforated obstacle [in Russian], Ufim. Mat. Zh. 3, No. 3, ( S. V. Spiridonov, Homogenization of a stratified magnetic fluid problem with microinhomogeneous magnetic field and boundary data [in Russian], Probl. Mat. Anal. 44, (21; English transl.: J. Math. Sci., New York 165, No 1, ( S. V. Spiridonov and G. A. Chechkin, Percolation of the boundary layer of a Newtonian fluid through a perforated obstacle [in Russian], Probl. Mat. Anal. 45, (21. English transl.: J. Math. Sci., New York 166, No 3, ( A. Yu. Linkevich, S. V. Spiridonov, and G. A. Chechkin, On the asymptotic behavior of solutions to the Prandtl equations for a stratified magnetic fluid, In: Book of Abstracts of the International Conference Differential Equations and Related Topics dedicated to the 11-th Anniversary of Prominent Mathematician Ivan G. Petrovskii, p. 424, Moscow State Univ. Press, Moscow ( A. Yu. Linkevich, S. V. Spiridonov, and G. A. Chechkin, On estimates of solutions to the Prandtl equations for a stratified microinhomogeneous fluid, In: Abstracts of the Sixth International Conference on Differential and Functional Differential Equations (August 14-21, 211, Moscow, Russia, pp , Moscow ( V. N. Samokhin, G. M. Fadeeva, and G. A. Chechkin, Ladyzhenskaya s modification of the Navier Stokes equations and the boundary layer theory [in Russian], Vestn. Mosk. Univ. Pechat. No. 5, ( V. N. Samokhin, G. M. Fadeeva, and G. A. Chechkin, The continuous dependence of a solution to the boundary layer equation on the profile of initial velocities [in Russian], Vestn. Mosk. Univ. Pechat. No. 4, ( V. N. Samokhin, G. M. Fadeeva, and G. A. Chechkin, Asymptotics of solutions to boundary layer equations for a generalized Newtonian medium under a close to symmetric external flow [in Russian], Probl. Mat. Anal. 59, (211; English transl.: J. Math. Sci., New York 177, No 1, ( V. N. Samokhin, G. M. Fadeeva, and G. A. Chechkin, Equations of a boundary layer for a modified Navier Stokes equations [in Russian], Tr.Semin.im.I.G.Petrovskogo28, (211. 1

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