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Micropolar fluid flow in a thick domain with multiscale oscillating roughness and friction boundary conditions. (English) Zbl 1489.76004

Statement of the problem: Motivated by applications in lubrication, this paper deals with the periodic homogenisation of a complex PDE-system describing micropolar fluid flow in a thin two-dimensional layer with highly oscillating boundary \[ \Omega^{\epsilon} = \Big\{z = (z_1, \, z_2) \in \mathbb{R}^2, \, 0 < z_1 < L, \, 0 < z_2 < \epsilon^m \, h^{\epsilon}(z_1)\Big\} \, , \] with a parameter \(\epsilon \ll 1\), a natural number \(m \geq 2\), and a function \(h^{\epsilon}(z_1) := h(z_1, \, \frac{z_1}{\epsilon}, \, \frac{z_1}{\epsilon^2}, \ldots, \frac{z_1}{\epsilon^m})\) where \(h \in C^{\infty}(\mathbb{R}^{m+1})\) is given and periodic in all variables. This function is moreover strictly positive and subject to uniform bounds from below and above.
In the cylindrical domain \((0, \, T) \times \Omega^{\epsilon}\) with \(T> 0\), the following PDEs for the unknowns \(u^{\epsilon}\) (velocity field), \(p^{\epsilon}\) (pressure) and \(\omega^{\epsilon}\) (angular micro-rotation field) are studied: \begin{align*} \frac{\partial u^{\epsilon}}{\partial t} - (\nu+\nu_r) \, \Delta u^{\epsilon} + (u^{\epsilon} \cdot \nabla)u^{\epsilon} + \nabla p^{\epsilon} &= 2 \, \nu_r \, \mathrm{rot} \omega^{\epsilon} + f^{\epsilon}\, ,\\ \operatorname{div}v u^{\epsilon} &= 0\, ,\\ \frac{\partial \omega^{\epsilon}}{\partial t} - \alpha \, \triangle \omega^{\epsilon} + u^{\epsilon} \cdot \nabla \omega^{\epsilon} + 4 \, \nu_r \, \omega^{\epsilon} &= 2 \, \nu_r \, \mathrm{rot} u^{\epsilon} + g^{\epsilon} \, . \end{align*} Here \(f^{\epsilon}\) and \(g^{\epsilon}\) are external force/momentum fields, while \(\nu\), \(\nu_r\) and \(\alpha\) are positive viscosity parameters. The \(L\)-periodicity in the \(z_1\)-direction is assumed for all data and variables. On the upper boundary \(\Gamma^{\epsilon}_1 = \{0 < z_1 < L,\, z_2 = \epsilon^m \, h^{\epsilon}(z_1)\}\) of the layer, the variables are prescribed in the form of \(u^{\epsilon} = (U_0, \, 0)\) and \(\omega^{\epsilon} = W_0\) with functions \(U_0\) and \(W_0\) depending only on time. On the lower part \(\Gamma_0 = \{0< z_1 < L, \, z_2 = 0\}\), a non-penetration condition \(u^{\epsilon}_2 = 0\) for the velocity is supplemented by vanishing angular momentum \(\omega^{\epsilon} = 0\), and by the Tresca-type friction condition for the tangential part of the stress \(((\sigma_{ij})) = \big(\big(- p^{\epsilon} \, \delta_{ij} + (\nu+\nu_r) \, (\partial_{z_j} u_i^{\epsilon} + \partial_{z_i} u_j^{\epsilon})\big)\big)\) in the form of \begin{align*} |\sigma^{\epsilon}_{\tau}| < k^{\epsilon} \, \, \Rightarrow \, \, & u_1^{\epsilon} = s_0(t)\, , \\ |\sigma^{\epsilon}_{\tau}| = k^{\epsilon} \, \, \Rightarrow \, \, & \exists \lambda \geq 0 \, : \, u_1^{\epsilon} = s_0(t) - \lambda \, \sigma_{\tau}^{\epsilon} \, , \end{align*} where \(k^{\epsilon}\) is a given function on \((0,T) \times \Gamma_0\) and \(s_0(t)\) is the sliding velocity of the wall.
[1ex]
The main results: Several aspects of the problem, as for instance the existence analysis for weak solutions in Hilbert spaces, or also the two-scale homogenisation, have been considered by the same team of authors in previous publications. The present paper focusses on the derivation of the limit effective problem for \(\epsilon \rightarrow 0\) and \(m\geq 2\) independent scales. By means of the rescaling \(y_1 = z_1\) and \(y_2 = z_2/\epsilon^m \, h^{\epsilon}(z_1)\) of the position-variables, the problem is first reformulated on the fixed domain \(\Omega = (0, \, L) \times (0,1)\). Then, \(\epsilon\)-independent estimates are established, and the multiscale (weak) compactness techniques developed by G. Allaire [SIAM J. Math. Anal. 23, No. 6, 1482–1518 (1992; Zbl 0770.35005)], and G. Allaire and M. Briane [Proc. R. Soc. Edinb., Sect. A, Math. 126, No. 2, 297–342 (1996; Zbl 0866.35017)], help extracting converging subsequences.
If the data are subject to certain natural regularity assumptions, and they satisfy the following scaling conditions: \begin{gather*} \epsilon^{2m} \, f^{\epsilon}(t, \, y) = f(t, \, y, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m}), \quad \epsilon^{2m} \, g^{\epsilon}(t, \, y) = g(t, \, y, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m})\, ,\\ \epsilon \, k(t, \, y_1) = k(t, \, y_1, \, \frac{y_1}{\epsilon}, \ldots, \frac{y_1}{\epsilon^m}) \, , \end{gather*} where \(f\), \(g\) and \(k\) are fixed functions, then the solutions converge to limits \[u^{\epsilon} \rightarrow \rightarrow^{m+1} u^0, \quad p^{\epsilon} \rightarrow \rightarrow^{m+1} p^0, \quad \omega^{\epsilon}\rightarrow \rightarrow^{m+1} \omega^0 \, ,\] in the sense of \(m+1\)-scale convergence. The convergence analysis for the pressure, which is introduced only as a distribution, requires a more delicate treatment.
For the limits \(u^0\), \(p^0\) and \(\omega^0\) several structural properties are established. An interesting feature is for instance the asymptotic condition affecting the divergence of the velocity field, in which the geometry of the problem is encoded via occurrence of the function \(h\).
Finally, it is shown how the limits can be computed by means of solving a few purely linear second-order elliptic problems with the time as a parameter. An elliptic variational inequality accounts for the Tresca-type friction condition.

MSC:

76A05 Non-Newtonian fluids
76M50 Homogenization applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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References:

[1] Allaire, G., Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 1482-1518 (1992) · Zbl 0770.35005
[2] Allaire, G.; Briane, M., Multiscale convergence and reiterated homogenisation, Proc. R. Soc. Edinb., 126, 297-342 (1996) · Zbl 0866.35017
[3] Barnes, H. A., A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure, J. Non-Newton. Fluid Mech., 56, 221-251 (1995)
[4] Boukrouche, M.; Paoli, L., Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary, SIAM J. Math. Anal., 44, 1211-1256 (2012) · Zbl 1259.35164
[5] Boukrouche, M.; Boussetouan, I.; Paoli, L., Non-isothermal Navier-Stokes system with mixed boundary conditions and friction law: uniqueness and regularity properties, Nonlinear Anal., Theory Methods Appl., 102, 168-185 (2014) · Zbl 1452.76042
[6] Boukrouche, M.; Paoli, L.; Ziane, F., Unsteady micropolar fluid flow in a thin domain with Tresca fluid-solid interface law, Comput. Math. Appl., 77, 11, 2917-2932 (2019) · Zbl 1442.76124
[7] Chen, Q.; Miao, C., Global well-posedness for the micropolar fluid system in critical Besov spaces, J. Differ. Equ., 252, 2698-2724 (2012) · Zbl 1234.35193
[8] Dong, B.; Chen, Z., Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows, Discrete Contin. Dyn. Syst., 23, 765-784 (2009) · Zbl 1170.35336
[9] Dong, B.; Li, J.; Wu, J., Global well-posedness and large-time decay for the 2D micropolar equations, J. Differ. Equ., 262, 3488-3523 (2017) · Zbl 1361.35143
[10] Dong, B.; Wu, J.; Xu, X.; Ye, Z., Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst., 38, 8, 4133-4162 (2018) · Zbl 1397.35204
[11] Dong, B.; Zhang, Z., Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differ. Equ., 249, 200-213 (2010) · Zbl 1402.35220
[12] Duvaut, G.; Lions, J. L., Les inéquations en mécanique et physique (1972), Dunod: Dunod Paris · Zbl 0298.73001
[13] El Kissi, N.; Piau, J. M., Slip and friction of polymer melt flows, Rheol. Ser., 5, 357-388 (1996)
[14] Eringen, A. C., Simple microfluids, J. Math. Mech., 2, 205-217 (1964) · Zbl 0136.45003
[15] Eringen, A. C., Theory of micropolar fluids, J. Math. Mech., 16, 1, 1-18 (1966)
[16] Fujita, H., Flow Problems with Unilateral Boundary Conditions (1993), Leçons au Collège de France
[17] Fujita, H., A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, Math. Fluid Mech. Model., 888, 199-216 (1994) · Zbl 0939.76527
[18] Fujita, H.; Kawarada, H.; Sasamoto, A., Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions, Adv. Numer. Math., 14, 17-31 (1994) · Zbl 0844.76018
[19] Fujita, H.; Kawarada, H., Variational inequalities for the Stokes equation with boundary conditions of friction type, (Recent Developments in Domain Decomposition Methods and Flow Problems, vol. 11 (1998)), 15-33 · Zbl 0926.76092
[20] Fujita, H., Remarks on the Stokes flows under slip and leak boundary conditions of friction type, (Topics in Mathematical Fluid Mechanics, vol. 10 (2002)), 73-94 · Zbl 1189.76135
[21] Fujita, H., A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math., 149, 57-69 (2002) · Zbl 1058.76023
[22] Jiu, Q.; Liu, J.; Wu, J.; Yu, H., On the initial- and boundary-value problem for 2D micropolar equations with only angular velocity dissipation, Z. Angew. Math. Phys., 68, 68, 107 (2017) · Zbl 1386.35342
[23] Kashiwabara, T., On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type, J. Differ. Equ., 254, 756-778 (2013) · Zbl 1253.35102
[24] Le Roux, C.; Tani, A., Steady flows of incompressible Newtonian fluids with threshold slip boundary conditions, (Mathematical Analysis in Fluid and Gas Dynamics, vol. 1353 (2004)), 21-34
[25] Le Roux, C.; Tani, A., Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30, 595-624 (2007) · Zbl 1251.76008
[26] Liu, J.; Wang, S., Initial-boundary value problem for 2D micropolar equations without angular viscosity, Commun. Math. Sci., 16, 2147-2165 (2018) · Zbl 1414.35170
[27] Łukaszewicz, G., Micropolar Fluids. Theory and Applications (1999), Birkhauser: Birkhauser Boston · Zbl 0923.76003
[28] Łukkassen, G.; Nguetseng, G.; Wall, P., Two-scale convergence, Int. J. Pure Appl. Math., 35, 86 (2002) · Zbl 1061.35015
[29] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20, 608-623 (1989) · Zbl 0688.35007
[30] Pahlavan, A. A.; Freund, J. B., Effect of solid properties on slip at a fluid-solid interface, Phys. Rev. E, 83, Article 021602 pp. (2011)
[31] Pit, R.; Hervet, H.; Léger, L., Friction and slip of a simple liquid at solid surface, Tribol. Lett., 7, 147-152 (1999)
[32] Rao, I. J.; Rajagopal, K. R., The effect of the slip boundary condition on the flow of fluids in a channel, Acta Mech., 135, 113-126 (1999) · Zbl 0936.76013
[33] Saito, N.; Fujita, H., Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition, (The Navier-Stokes Equations. The Navier-Stokes Equations, Lecture Note Pure Appl. Math., vol. 223 (2001)), 73-86 · Zbl 0995.35048
[34] Saito, N., On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. RIMS Kyoto Univ., 40, 345-383 (2004) · Zbl 1050.35029
[35] Sochi, T., Slip at fluid-solid interface, Polym. Rev., 51, 309-340 (2011)
[36] Wang, D.; Wu, J.; Ye, Z., Global regularity of the three-dimensional fractional micropolar equations, J. Math. Fluid Mech., 22, 2, Article 28 pp. (2020) · Zbl 1434.35122
[37] Xue, L., Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations, Math. Methods Appl. Sci., 34, 1760-1777 (2011) · Zbl 1222.76027
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