On the influence of wavy riblets on the slip behaviour of viscous fluids

In this work, we use the homogenization theory to investigate the capability of wavy riblet patterns to influence the behaviour of a viscous flow near a ribbed boundary. Starting from perfect slip conditions on the wall, we show that periodic oscillations of wavy riblets in the lateral direction may induce a friction effect in the direction of the flow, contrary to what happens with straight riblets. Finally, we illustrate this effect numerically by simulating riblet profiles that are widely used in experimental studies: the V-shape, U-shape, and blade riblets.     

The incompressible limit of the full Navier–Stokes–Fourier system on domains with rough boundaries

We study the incompressible limit of the full Navier–Stokes–Fourier system on condition that the boundary of the spatial domain oscillates with the amplitude and wave length proportional to the Mach number. Assuming the fluid satisfies the complete slip boundary conditions on the oscillating boundary, we identify the asymptotic limit, and, in particular, establish strong (pointwise) convergence of the velocities towards a solenoidal vector field.     

Analytic solutions for unsteady flows over a superhydrophobic surface

A class of unsteady analytic solutions for the evolution of viscous Newtonian fluids over a superhydrophobic surface is derived. The surface is represented by regular rectilinear riblets and the fluid is assumed to flow along the grooves’ direction. A mixed boundary condition is imposed on the surface, consisting in homogeneous Neumann conditions over the riblet voids and homogeneous Dirichlet conditions on the wall intervals. The transition between the above conditions is modelled through a Robin condition with non-constant smooth coefficients in a completely general manner. A global solution is derived and relevant examples, that can be fruitfully adopted as benchmark solutions for testing numerical solvers, are discussed.     

Boundary control of non-well-posed elliptic equations with nonlinear Neumann boundary conditions

In this paper, we consider a boundary control problem governed by a class of non-well-posed elliptic equations with nonlinear Neumann boundary conditions. First, the existence of optimal pairs is proved. Then by considering a well-posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem.     

Lp-solvability of a full superconductive model

In this article the mean-field vortex model arising from the II-type superconductivity is investigated. The vortex model is reduced to a nonlinear hyperbolic–elliptic system of PDEs in a bounded domain. Motivated by experiments, we consider physical boundary conditions, which describe a flux of superconducting vortices through the boundary of the domain. We prove the global solvability for the system. To show the solvability result we take a vanishing “viscosity” limit in an approximated parabolic–elliptic system. Since the approximated solutions do not have a compactness property, we justify this limit transition, using a kinetic formulation of our problem. The main trick is that instead of the nonlinear system, we have to investigate a linear transport equation.     

On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries

We study the asymptotic behavior of solutions to the incompressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are non-degenerate, meaning not oriented in a single direction.     

Limit Behaviour of Solutions to a Class of Equivalued Surface Boundary Value Problems for Parabolic Equations

In this paper, we discuss the limit behaviour of solutions for a class of equivalued surface boundary value problems for parabolic equations. When the equivalued surface boundary \overline{\Gamma}^\varepsilon_1 shrinks to a fixed point on boundary \Gamma_1, only homogeneous Neumann boundary conditions or Neumann boundary conditions with Dirac function appear on \Gamma_1.     

Passing from bulk to bulk-surface evolution in the Allen–Cahn equation

In this paper we formulate a boundary layer approximation for an Allen–Cahn-type equation involving a small parameter ${\varepsilon}$ . Here, ${\varepsilon}$ is related to the thickness of the boundary layer and we are interested in the limit ${\varepsilon \to 0}$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L²-gradient flow with the corresponding Allen–Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Γ- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.     

Global Versus Local Casimir Effect

This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH), we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit, we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.     

Boundary Layers and Incompressible Navier‐Stokes‐Fourier Limit of the Boltzmann Equation in Bounded Domain I

We establish the incompressible Navier‐Stokes‐Fourier limit for solutions to the Boltzmann equation with a general cutoff collision kernel in a bounded domain. Appropriately scaled families of DiPerna‐Lions(‐Mischler) renormalized solutions with Maxwell reflection boundary conditions are shown to have fluctuations that converge as the Knudsen number goes to 0. Every limit point is a weak solution to the Navier‐Stokes‐Fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the Knudsen number. The main new result of the paper is that this convergence is strong in the case of the Dirichlet boundary condition. Indeed, we prove that the acoustic waves are damped immediately; namely, they are damped in a boundary layer in time. This damping is due to the presence of viscous and kinetic boundary layers in space. As a consequence, we also justify the first correction to the infinitesimal Maxwellian that one obtains from the Chapman‐Enskog expansion with Navier‐Stokes scaling. This extends the work of Golse and Saint‐Raymond [20,21] and Levermore and Masmoudi [28] to the case of a bounded domain. The case of a bounded domain was considered by Masmoudi and Saint‐Raymond [34] for the linear Stokes‐Fourier limit and Saint‐Raymond [41] for the Navier‐Stokes limit for hard potential kernels. Neither [34] nor [41] studied the damping of the acoustic waves. This paper extends the result of [34,41] to the nonlinear case and includes soft potential kernels. More importantly, for the Dirichlet boundary condition, this work strengthens the convergence so as to make the boundary layer visible. This answers an open problem proposed by Ukai [46]. © 2016 Wiley Periodicals, Inc.     

Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system

In this paper, we study the stationary flow for a one-dimensional isentropic bipolar Euler-Poisson system (hydrodynamic model) for semiconductor devices. This model consists of the continuous equations for the electron and hole densities, and their current densities, coupled the Poisson equation of the electrostatic potential. In a bounded interval supplemented by the proper boundary conditions, we first show the unique existence of stationary solutions of the one-dimensional isentropic hydrodynamic model, based on the Schauder fixed-point principle and the careful energy estimates. Next, we investigate the zero-electron-mass limit, combined zero-electron mass and zero-hole mass limit, the zero-relaxation-time limit and the Debye-length (quasi-neutral) limit, respectively. We also show the strong convergence of the sequence of solutions and give the associated convergence rates.     

Some Unilateral Boundary-Value Problems for Elastic Bodies with Rugged Boundaries

The system of linear elasticity is considered in a domain whose boundary depends on a small parameter > 0 and has a part with a rugged structure. The rugged part of the boundary may bend sharply and embrace cavities or channels, and as 0, it approaches a limit surface on the boundary of the limit domain. On the rugged part of the boundary, conditions of two types are considered: (I) contact with rigid obstacles (conditions of Signorini type); (II) reaction forces involving the parameter and nonlinearly depending on displacements. We investigate the asymptotic behavior of weak solutions to such boundary-value problems as 0 and construct the limit problem, according to the geometric structure of the rugged part of the boundary and the external surface forces and their dependence on the parameter . In general, the limit problem has the form of a variational inequality over a certain closed convex cone in a Sobolev space. This cone characterizes the boundary conditions of the limit problem and is described in terms of the functions involved in the nonlinear boundary conditions on the rugged boundary. As shown by examples, in the limit, the type of boundary condition may change. To justify these asymptotic results, we give a detailed exposition of some facts about extensions, Korn's inequalities, traces, and nonlinear boundary conditions in partially perforated domains with Lipschitz continuous boundaries. Bibliography: 16 titles.     

The isentropic quantum drift-diffusion model in two or three space dimensions

We investigate the isentropic quantum drift-diffusion model, a fourth order parabolic system, in space dimensions d = 2, 3. First, we establish the global weak solutions with large initial value and periodic boundary conditions. Then we show the semiclassical limit by delicate interpolation estimates and compactness argument.     

Viscous boundary value problems for symmetric systems with variable multiplicities

Extending investigations of Métivier and Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to “real” or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun-Serre-Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.     

Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise

We consider a heat conduction model for solids. Nearest neighbour atoms interact as coupled oscillators exchanging velocities in such a way that the total energy is conserved. The system is considered under periodic boundary conditions. We will show that the system has a hydrodynamic limit given by the solution of the heat equation and we discuss some aspects of the model.     

On a non-isothermal,non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions

We consider a problem describing the motion of an incompressible, non-isothermal, and non-Newtonian fluid in a three-dimensional thin domain. We first establish an existence result for weak solutions of this problem. Then we study the asymptotic analysis when one dimension of the fluid domain tends to zero. A specific weak Reynolds equation, the limit of Tresca fluid–solid boundary conditions, and the limit boundary conditions for the temperature are obtained. The uniqueness result for the limit problem is also proved.     

Asymptotic analysis of singularly perturbed integro-differential equations with zero operator in the differential part

We consider an integro-differential system with identically zero operator in the differential part. We construct a regularized asymptotics of the solution of this system for two cases, one in which the integral operator contains an exponentially varying factor and the other in which it does not. On the basis of the resulting asymptotic expansion, we study the passage to the limit in the system and present conditions under which this passage is uniform on the entire range of the independent variable (including the boundary layer region).     

Spectral stability results for higher-order operators under perturbations of the domain

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.     

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid,assuming that it is in the thermodynamically sense perfect and polytropic.The fluid is between a static solid wall and a free boundary connected to a vacuum state.We take the homogeneous boundary conditions for velocity,microrotation and heat flux on the solid border and that the normal stress,heat flux and microrotation are equal to zero on the free boundary.The proof of the global existence of the solution is based on a limit procedure.We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.