Diffusion in Deforming Porous Media - Department of Mathematics

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

12 R.E. SHOWALTERthe characteristic function of Y m for m = 1, 2, extended Y -periodically to all of IR N .Thus, χ 1 (y) + χ 2 (y) = 1. We shall assume that the set {y ∈ IR N : χ 1 (y) = 1} issmooth and connected. The domain Ω is thus partitioned into the two subdomains{ ( x) }Ω ε m = x ∈ Ω : χ m = 1 , m = 1, 2 .εLet Γ ε 12 ≡ ∂Ωε 1 ∩ ∂Ωε 2 ∩ Ω be that part of the interface of Ωε 1 with Ωε 2 that is interiorto Ω, and let Γ 12 ≡ ∂Y 1 ∩ ∂Y 2 ∩ Y be the corresponding interface in the cell Y .Likewise, let Γ 22 ≡ Y 2 ∩ ∂Y and denote by Γ ε 22 its periodic extension which formsthe interface between those parts of the second component Ω ε 2 which lie withinneighboring εY -cells. These are the local inclusions and we denote them by Y2 ε.The second component Ω ε 2 may be connected, but this is not required.The flow in the porous structure Ω ε 1 is described by a Darcy system that is coupledacross the interface Γ ε 12 to a Biot system for the slow flow and deformation inthe inclusions Ω ε 2 . In the region Ωε 2 we scale the permeability by ε2 . Thus, we shalldenote by c m and ε 2(m−1) κ m the compressibility and the permeability, respectively,in Ω ε m , m = 1, 2. The fluid pressure in the macroporous region Ωε 1 is denoted byp ε 1 (x, t) and the corresponding flux there is given by −κ 1∇p ε 1 . The pressure in the regionΩ ε 2 is pε 2 (x, t) with scaled flux −ε2 κ 2 ∇p ε 2 . It is the resulting very high frequencyspatial variations in the pressure gradients in the second component which lead tolocal storage and corresponding local deformations. These are described by the(small) displacement u(x, t) from the position x ∈ Ω ε 2 , and ε kl(u) ≡ 1(∂ 2 ku l + ∂ l u k )is the (linearized) strain tensor. The stress σ(u) is given by the generalized Hooke’slaw σ ij (u) = a m ijkl ε kl(u) with the positive definite symmetric elasticity tensor a ijklfor a general anisotropic material. The boundary conditions will involve the surfacedensity of forces or traction σ ij n j . The normal will be directed out of Ω ε 2 . Theelastic structure is described by the bilinear form∫∫e(u, v) ≡ a ijkl ε kl (u)∂ j v i dx = a ijkl ε kl (u)ε ij (v) dxΩ ε 2on the space V ε ≡ {v ∈ H 1 (Ω) : v = 0 on Ω ε 1} of admissable displacements. Thelocal operator is obtained by means of Stokes’ theorem, that is,∫e(u, v) = E(u(x)) · v(x) dxΩ ε 2where the formal operator is given by E(u) i = −∂ j a ijkl ε kl (u) , 1 ≤ i ≤ 3 , wheneveru, v ∈ V and E(u) ∈ [L 2 (Ω ε 2 )]3 .The Darcy flow in the first component and the Biot system for the second componentare given by the highly heterogeneous systemc 1 ṗ ε 1 − ∇ · (κ 1 ∇p ε 1) = F 1 in Ω ε 1 ,Ω ε 2p ε 1 = pε 2 , κ ∂p ε 11∂n = ∂p εε2 2κ 2∂n , uε = 0 on Γ ε 12 ,c 2 ṗ ε 2 − ∇ · (ε2 κ 2 ∇p ε 2 ) + b ε∇ · ˙u ε = F 2 andρ 2 ü ε + ε 2 E(u ε ) + b ε ∇p ε 2 = f 2 , in Ω ε 2 ,
DEFORMING POROUS MEDIA 13which is the exact model of the fine-scale structure. It is supplemented with appropriateinitial and boundary conditions, and then it follows that there is a uniquesolution p ε 1 (x, t), pε 2 (x, t), uε (x, t), x ∈ Ω for each ε > 0. Moreover, the threesequences of functions have two-scale limits as ε → 0, and these are given by thetriple of functions p 1 (x, t), P 2 (x, y, t), U(x, y, t), x ∈ Ω, y ∈ Y 2 . Then we showthat this triple is a solution of a Darcy-Biot distributed microstructure systemwhich we now describe.The global flow in the porous structure is determined on the macro-scale by themacroscopic Darcy equation˜c 1 ṗ 1 (x, t) −∂∫A 1 ∂p 1 (x, t) ∂P 2 (x, s, t)ij + κ 2 ds = F 1 (x, t) , x ∈ Ω ,∂x j ∂x i Γ 12∂nwhere ∫ ∂PΓ 12κ 2 (x,s,t)2 ds is the exchange term representing the flow into the local∂ninclusion Y2 ε at the point x ∈ Ω. The flow and deformation within the re-scaledare given by the local Biot systemY ε2c 2 ˙ P 2 (x, y, t) − ∇ y · κ 2 ∇ y P 2 (x, y, t) + b 0 ∇ y · ˙U(x, y, t) = F 2 (x, y, t) , y ∈ Y 2 ,P 2 (x, s, t) = p 1 (x, t) , s ∈ Γ 12 ,P 2 (x, s, t) and κ 2 ∇ y P 2 (x, s, t) are Y -periodic on Γ 22 ,ρÜi(x, y, t) − ∂∂y j(σyij (U(x, y, t)) − δ ijb 0 P 2 (x, y, t) ) = 0 , y ∈ Y 2 ,U(x, s, t) and σ y ij (U(x, s, t))n j − b 0 P 2 (x, s, t)n i are Y -periodic on Γ 22 ,U(x, s, t) = 0 , s ∈ Γ 12 .The subscript y on the gradient indicates that the derivative is taken with respectto the local variable y. The solution P 2 (·, ·), U(·, ·) of the local system depends onthe global pressure p 1 (·) at the point x ∈ Ω. Because of the small size of the cells,this pressure is assumed to be well approximated by the “constant” value p 1 (x, t)on the interface Γ ε 12. Note that if the deformation is suppressed in this system, i.e.,if U(x, y, t) = 0, then this is precisely the model of Arbogast et al [2] for singlephase flow in a doubly-porous medium.This Darcy-Biot model is a very special case, intended only to suggest the structureof the limiting initial-boundary-value problems that arise, and the micromodelsthat come from the particular applications always lead to considerably morecomplicated distributed microstructure models. One can use Biot systems for eachcomponent and scale the parameters for each component in a wide variety of ways,the choice being dependent on the situation. Also, one can start with a Biot systemfor the structure coupled to a fluid flow model either of Stokes type or of slightlycompressible flow type and then investigate the limiting form of the composite forvarious scalings of the parameters. Similar Biot-Biot models have been constructedfor the mechanics of soft biological tissues. These are based on the hypothesis thattissue can be regarded as a composite cellular poroelastic material composed of a
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Page 18: 18 R.E. SHOWALTERDepartment of Math

Diffusion in Deforming Porous Media - Department of Mathematics

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