We consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem, and the tangential drag force and the effective mass flow are determined up to order O(= 3Â2 ). The value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of O(=). Academic Press

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The vanishing viscosity limit is considered for the incompressible 2D Navier-Stokes equations in a bounded domain. Motivated by studies of turbulent flow we suppose Navier's friction condition in the tangential direction, i.e. the creation of a vorticity proportional to the tangential velocity. We prove the existence of the regular solutions for the Navier-Stokes equations with smooth compatible data and of the solutions with bounded vorticity for initial vorticity being only bounded. Finally, we establish a uniform L ∞ -bound for the vorticity and convergence to the incompressible 2D Euler equations in the inviscid limit.

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International audienceIn this paper we present a phase field model for a fluid-driven fracture in a poroelastic medium. In our previous work, the pressure was assumed given. Here we consider a fully coupled system where the pressure field is determined simultaneously with the displacement and the phase field. To the best of our knowledge, such a model is new in the literature. The mathematical model consists of a linear elasticity system with fading elastic moduli as the crack grows, which is coupled with an elliptic variational inequality for the phase field variable and with the pressure equation containing the phase field variable in its coefficients. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. The phase field variational inequality contains quadratic pressure and strain terms, with coefficients depending on the phase field unknown. We establish existence of a solution to the incremental problem through convergence of a finite dimensional approximation. Furthermore, we construct the corresponding Lyapunov functional that is linked to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation

The recently introduced phase-field approach for pressurized fractures in a porous medium offers various attractive computational features for numerical simulations of cracks such as joining, branching, and nonplanar propagation in possibly heterogeneous media. In this paper, the pressurized phase-field framework is extended to fluid-filled fractures in which the pressure is computed from a generalized parabolic diffraction problem. Here, the phase-field variable is used as an indicator function to combine reservoir and fracture pressure. The resulting three-field framework (elasticity, phase field, pressure) is a multiscale problem that is based on the Biot equations. The proposed numerical solution algorithm iteratively decouples the equations using a fixed-stress splitting. The framework is substantiated with several numerical benchmark tests in two and three dimensions.

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We consider the laminar viscous channel ow over a porous surface. The size of the pores is much smaller than the size of the channel and it is important to determine the e ective boundary conditions at the porous surface. After studying the corresponding boundary layers, we obtain by a rigorous asymptotic expansion Sa man's modi cation of the interface condition observed by Beavers and Joseph. The e ective coe cient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the e ective mass ow are given as powers of the characteristic pore size ". Finally, the interface condition linking the e ective pressure elds, in the porous medium and in the channel, is given and the jump of the e ective pressures is explicitly determined.