Mathematical Communications 3998), 97 24 97 Micropolar fluid flow with rapidly variable initial conditions Nermina Mujaković Abstract. In this paper we consider nonstationary D-flow of a micropolar viscous compressible fluid, which is in a thermodynamic sense perfect and polytropic. Assuming that the initial data for the specific volume, velocity, microvorticity and temperature are rapidly variable functions and making use of the method of two-scale asymptotic expansion, we find out the homogenized model of the considered flow. Key words: micropolar fluid, homogenization Sažetak. Homogenizacija D-toka mikropolarnog kompresibilnog fluida. U ovom radu razmatramo nestacionarni D-tok mikropolarnog viskoznog kompresibilnog fluida, koji je u termodinamičkom smislu perfektan i politropan. Pretpostavljajući da su početni podaci za specifični volumen, brzinu, mikrovrtložnost i temperaturu brzo varijabilne funkcije i koristeći metodu dvoskalnog asimptotičkog razvoja, nalazimo homogenizirani model razmatranog toka. Ključne riječi: mikropolarni fluid, homogenizacija AMS subject classifications: 35B27, 35Q35, 76N99 Received July 2, 998 Accepted August 29, 998. Introduction In this paper we consider nonstationary D-flow of a micropolar viscous compressible fluid, which is in a thermodynamic sense perfect and polytropic. Initialboundary value problem for such flow is well posed [9]). Assuming that the initial data for the specific volume, velocity, microvorticity and temperature are locally ɛ - periodic functions, where ɛ > is a small parameter, and making use of the method of two-scale asymptotic expansion [5], []), we find out the homogenized model of the considered flow. It turns out that the homogenized model has the same structure as the original one. The analogous problem for the classical viscous Faculty of Philosophy, University of Rijeka, Omladinska 4, HR-5 Rijeka, Croatia, e-mail: mujakovic@rocketmail.com
98 N. Mujaković compressible fluid was considered in [7]. Homogenization problems for incompressible flows were treated in []-[3] see also [4] and [6], where lubrication problems close to homogenization were considered). Let ρ, ν, ω, θ :], [ R + R ) denote the mass density, velocity, microvorticity and temperature of the fluid in the Lagrangean description, respectively. Governing equations of the flow under consideration are as follows [8]): ρ θ t = Kρ2 θ ν + ρ2 ν t = ρ ω t = A ρ ν + ρ2 =, t 2) ρ ν ) K ρθ), 3) ν [ ρ ρ ω ) 2 + ρ 2 ω ) ω ], 4) ) 2 + ω 2 + Dρ ρ θ ), 5) where K, A and D are positive constants. It is convenient to introduce instead of the mass density ρ) the specific volume V = ρ. Then we obtain the system V V θ t + K θ ν V 2 V 2 V t ν =, 6) ν t )) ν V Kθ =, 7) ) ) ω t A ω ω =, 8) V V ) 2 ν ) 2 ω V 2 ω 2 D ) θ = 9) V V in Q =], [ R +. We take the following homogeneous boundary conditions: ν, t) = ν, t) =, ) for t >. Let the functions ω, t) = ω, t) =, ) θ θ, t) =, t) = 2) V, ν, ω, θ : [, ] R R 3) be -periodic in the second, so-called fast variable y. Then, for sufficiently small ɛ > the functions V ɛ x) = V x, y), 4)
Micropolar fluid flow 99 where νx) ɛ = ν x, y), 5) ωx) ɛ = ω x, y), 6) θx) ɛ = θ x, y), 7) y = x ɛ, 8) are locally ɛ-periodic on ], [. For the system 6) 9) we take the initial conditions V x, ) = V ɛ x), 9) νx, ) = ν ɛ x), 2) ωx, ) = ω ɛ x), 2) θx, ) = θ ɛ x) 22) for x ], [. Let the functions 3) be sufficiently smooth and let the compatibility conditions ν,.) = ν,.) =, 23) on ], [ and the conditions ω,.) = ω,.) = 24) V >, θ > on [, ] 2 25) hold true. Then for each ɛ > the initial-boundary value problem 6) 2), 9) 22) has a unique strong solution in Q, having the properties V ɛ, ν ɛ, ω ɛ, θ ɛ ) 26) V ɛ >, θ ɛ >, on [, ] [, ) 27) [9]). In order to describe a propagation of the initial rapidly variable heterogenities 9) 22), we have to identify a limit of solutions 26), as ɛ tends to zero. 2. Two-scale asymptotic expansion According to the homogenization method [5], []), we shall look for a solution to the problem 6) 2), 9) 22) in the form of two-scale asymptotic expansion: V ɛ, ν ɛ, ω ɛ, θ ɛ )x, t) = V, ν, ω, θ )x, y, t) + ɛv, ν, ω, θ )x, y, t) +, 28) where functions y = x ɛ, V k, ν k, ω k, θ k ) : ], [ R R + R, k =,,... 29)
2 N. Mujaković are -periodic in the fast variable y R. The idea of the method is to insert the expansion 28) into the system 6) 9) and to identify the powers of ɛ. In this way we obtain a sequence of equations for the functions 29). Our purpose is to identify the function V, ν, ω, θ ) that is, at least in a formal sense, a limit of the solutions 26), as ɛ tends to zero. In order to present computations in a simple form, it is useful to consider first x and y as independent variables and next to replace y by x ɛ. Note that the operator, applied to the function ϕx, x ɛ ) becomes In that what follows we use the notation ϕx) = + ɛ y. 3) ϕx, y)dy. 3) For simplicity we shall assume that the functions ν, ω and θ do not depend on the fast variable: ν = ν x, t), ω = ω x, t), θ = θ x, t); 32) in fact one can easily see that 32) holds true. The equation 6). Identifying the terms of the order ɛ, we get V t ν ν =. 33) y Integrating this equation over ], [ with respect to variable y and taking into account 32) and the periodicity of the function ν, we obtain V t ν =. 34) The equation 7). Identifying the terms of the order ɛ, one gets )) ν y V + ν y Kθ =. 35) Therefore, there exists a function αx, t), such that After integration with respect to y we have ν + ν y Kθ = αv. 36) ν Kθ = α V, 37)
Micropolar fluid flow 2 or From 36) and 38), it follows α = V ν Kθ ν y = V V V ν Kθ ). 38) ). 39) Identifying the terms of the order ɛ and taking into account 39), we obtain ν t )) ν V Kθ )) ν y V + ν2 y Kθ + + )) V ν y V V Kθ =. 4) Integrating with respect to y, we get ν t )) ν V Kθ =. 4) The equation 8). Identifying the terms of the order ɛ, we have )) ω y V + ω =. 42) y Therefore, ther exists a function βx, t), such that Integrating with respect to y, we obtain or From 43) and 45), it follows ω + ω y = βv. 43) ω = β V, 44) β = V ω. 45) ω y = V V V ω. 46) The terms of the order ɛ together with 46)) lead to the equation ω = ω A t V ) ω V + ω y V + ω2 y V ω )). 47) V After integration with respect to y, we obtain ω V t V A ω V ) ) ω =. 48)
22 N. Mujaković The equation 9). The terms of the order ɛ give )) θ y V + θ =. 49) y Therefore, ther exists a function γx, t), such that Integrating with respect to y, we have θ + θ y = γv. 5) or Inserting it in 5), we obtain θ = γ V, 5) γ = V θ. 52) θ y = V V V θ. 53) The terms of the order ɛ together with 53) give θ t = K θ ν V K2 θ V ) 2 + V + K2 V V ) 2 θ ) 2 + V V ) 2 +D y V ω V ) 2 θ + θ2 y V θ V ) 2 ν 2KV V ν ) 2 θ ) 2 + V ω ) 2 + D )). θ V ) 54) After integration with respect to y, we have θ V t + K ) ν 2 V ) ω 2 ) 2 V ) 2 ω ) 2 D V θ V ) =. 55) θ ν V ) 2 3. The homogenized problem For the functions V, ν, ω and θ, describing a macroscopic behaviour of the flow, we get the system 34), 4), 48), 55). From ) 2) it follows ν, t) = ν, t) =, 56) ω, t) = ω, t) =, 57) θ θ, t) =, t) = 58)
Micropolar fluid flow 23 for t >. Taking into account that the functions V ɛ, ν ɛ, ω ɛ and θ ɛ tend in some sense to the functions V, ν, ω and θ, respectively [5]), from 9) 22) we conclude that the following initial conditions hold true: for x ], [. Because of 23) 25) the conditions V x, ) = V x), 59) v x, ) = ν x), 6) ω x, ) = ω x), 6) θ x, ) = θ x) 62) ν ) = ν ) =, 63) ω ) = ω ) =, 64) V >, θ > on [, ] 65) are satisfied. Therefore, we have the following result: Theorem. The homogenized initial-boundary value problem 34), 4), 48), 55), 56) 62) has a unique strong solution, having the properties V >, θ > on [, ] [, ). 66) Remark. If the functions 3) do not depend on the first variable x, then the functions 4) 7) are ɛ-periodic and, consequently, V, ν, ω and θ are constants; it means that the initial state of the macroscopic flow in this case is a homogeneous one. Remark 2. A rigorous proof of a convergence V ɛ, ν ɛ, ω ɛ, θ ɛ ) V, ν, ω, θ ) as ɛ is an open problem. Acknowledgement. The author wishes to thank Professor I. Aganović University of Zagreb) for a stimulating discussion on the topic. References [] I. Aganović, Z. Tutek, Homogenization of micropolar fluid flow through a porous medium, in: Mathematical Modelling of Flow through Porous Media, A. Bourgeot et al, Eds.), World Sci., 995., 3 3. [2] I. Aganović, Z. Tutek, Homogenization of unsteady Stokes flow of micropolar fluid through a porous medium, ZAMM 77997), S2, 55 56. [3] I. Aganović, Z. Tutek, Nonstationary micropolar fluid flow through porous medium, Nonlinear Analysis 3997), 37 378. [4] G. Bayada, G. Lukaszewicz, On micropolar fluids in the theory of lubrication, Internat. J. Engrg. Sci. 34996), 447 49.
24 N. Mujaković [5] A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Nort-Holland, 978. [6] M. Boukrauche, M. S. Mostefai, On micropolar fluids in the theory of lubrification, Equipe d Analyse Numerique Lyon-Saint-Etienne, Preprint N 4, 997. [7] E.W, Propagation of oscillations in the solutions of -D compressible fluid equations, Commun. in Partial Differential Equations 7992), 347 37. [8] N. Mujaković, One-dimensional flow of compressible viscous micropolar fluid: a local existence theorem, Glasnik matematički, to appear. [9] N. Mujaković, One-dimensional flow of compressible viscous micropolar fluid: a global existence theorem, Glasnik matematički, to appear. [] E. Sancez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics 27, Springer-Verlag, 98.