Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid

In the analysis of the long-time behavior of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity [T. Gallay, C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier–Stokes equation, Commun. Math. Phys. 255 (1) (2005) 97–129]. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slightly inhomogeneous fluids with respect to localized perturbations.     

On the control of layered flow of a viscous incompressible fluid within MHD

The control of a viscous incompressible flow within MHD is considered. The phenomenon of internal heat release is explored as applied to a layered plane flow in which the kinetic energy dissipates strictly into heat. The influence exerted by the electromagnetic field of the flowing fluid on the surrounding medium is examined, and methods for flow control in the context of digital deposit field technology are considered.     

MHD oblique stagnation-point flow of a micropolar fluid

The steady two-dimensional oblique stagnation-point flow of an electrically conducting micropolar fluid in the presence of a uniform external electromagnetic field (E₀, H₀) is analysed and some physical situations are examined. In particular, if E₀ vanishes, H₀ lies in the plane of the flow, with a direction not parallel to the boundary, and the induced magnetic field is neglected, it is proved that the oblique stagnation-point flow exists if, and only if, the external magnetic field is parallel to the dividing streamline. In all cases it is shown that the governing nonlinear partial differential equations admit similarity solutions and the resulting ordinary differential problems are solved numerically. Finally, the behaviour of the flow near the boundary is analysed; this depends on the three dimensionless material parameters, and also on the Hartmann number if H₀ is parallel to the dividing streamline.     

On fluid limit for the semiconductors Boltzmann equation

This paper is devoted to the derivation of (non-linear) drift-diffusion equations from the semiconductor Boltzmann equation. Collisions are taken into account through the non-linear Pauli operator, but we do not assume relation on the cross section such as the so-called detailed balance principle. In turn, equilibrium states are implicitly defined. This article follows and completes the contribution of Mellet (Monatsh. Math. 134 (4) (2002) 305-329) where the electric field is given and does not depend on time. Here, we treat the self-consistent problem, the electric potential satisfying the Poisson equation. By means of a Hilbert expansion, we shall formally derive the asymptotic model in the general case. We shall then rigorously prove the convergence in the one-dimensional case by using a modified Hilbert expansion.     

Derivation of a quasi‐stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width η_ε. Passing to the limit when ε goes to zero, we find a critical size in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

On pulsatile flow of a viscous fluid in a rotating channel

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK ²=ΩL ²/v which is reciprocal of Ekmann Number and frequency parameter σ=αL ²/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK ² the flow near the plates has a multiple boundary layer character.     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

Solutions with compact support to the Cauchy problem of an equation modeling the motion of viscous droplets

The Cauchy problem to an equation arising in modeling the motion of viscous droplets is studied in the present paper. The authors prove that if the initial data has compact support, then there exists a weak solution which has compact support for all the time.     

Numerical modelling of aeroelastic behaviour of an airfoil in viscous incompressible flow

In this paper, the problem of the numerical approximation of a two-dimensional incompressible viscous fluid flow interacting with a flexible structure is considered. Due to high Reynolds numbers in the range 10⁴ − 10⁶ the turbulent character of the flow is considered and modelled with the aid of Reynolds equations coupled with the k − ω turbulence model. The structure motion is described by a system of ordinary differential equations for three degrees of freedom: vertical displacement, rotation and rotation of the aileron. The problem is discretized in space by the Galerkin Least-Squares stabilized finite element method and the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian method.     

On global motion of a compressible viscous heat-conducting fluid bounded by a free surface

We consider the motion of a viscous compressible heat-conducting fluid in ³ bounded by a free surface which is under constant exterior pressure. We present the global existence theorems in two cases: when the free surface is under the surface tension and without it.     

On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach

The Homotopy Analysis Method of Liao [Liao SJ. Beyond perturbation: introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003] has proven useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the Homotopy Analysis Method, in a fairly general setting. Further, we discuss various convergence requirements on solutions.     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

Two-dimensional problems of stationary flow of a noncompressible viscous fluid in the case of Ossen's linearization

Two-dimensional boundary value problems of flow of a viscous micropolar fluid are investigated in the case of linearization by Oseen's method.     

Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative

This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.     

On the convergence of global and bounded solutions of some evolution equations

It is shown that a special case of the well-known Lojasiewicz gradient inequality is sufficient to give a unified background for many convergence results in gradient or gradient-like systems appearing previously in the Literature. Besides as an illustration we give a direct proof of convergence in the case of 1D wave equations by a suitable adaptation of Zelenyak’s method.     

Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous,incompressible fluids

We introduce a new sharp interface model for the flow of two immiscible, viscous, incompressible fluids. In contrast to classical models for two-phase flows we prescribe an evolution law for the interfaces that takes diffusional effects into account. This leads to a coupled system of Navier–Stokes and Mullins–Sekerka type parts that coincides with the asymptotic limit of a diffuse interface model. We prove the long-time existence of weak solutions, which is an open problem for the classical two-phase model. We show that the phase interfaces have in almost all points a generalized mean curvature.     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.