Universal stability of magneto-micropolar fluid motions

Criteria for universal stability of the unsteady motion of an incompressible, electrically conducting linear micropolar fluid, i.e. with rigid microinclusions, in the presence of an arbitrary magnetic field, and in an arbitrary bounded time dependent domain are established. The model of the micropolar fluid employed is essentially the one proposed by Eringen. The interaction between the flow field and the magnetic field is manifested through the body force and body couple. Relativistic, Hall, and temperature effects are neglected. The stability method employed is an energy technique due to James Serrin. Certain uniqueness theorems for the unsteady and steady flows of magneto-micropolar fluid are also established.The theorems established for the stability and uniqueness are universal in the sense that they may be applied to any geometry of bounded domains and any distribution of the basic field variables. The present problem finds application in MHD generators with neutral fluid seedings in the form of rigid microinclusions.     

Unsteady Flow of Radiating and Chemically Reacting MHD Micropolar Fluid in Slip-Flow Regime with Heat Generation

An analytical study of the problem of unsteady free convection with thermal radiation and heat generation on MHD micropolar fluid flow through a porous medium bounded by a semi-infinite vertical plate in a slip-flow regime has been presented. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted for in the mass diffusion equation. A uniform magnetic field acts perpendicular on the porous surface absorbing micropolar fluid with a suction velocity varying with time. A perturbation technique is applied to obtain the expressions for the velocity, microrotation, temperature, and concentration distributions. Expressions for the skin-friction, Nusselt number, and Sherwood number are also obtained. The results are discussed graphically for different values of the parameters entered into the equations of the problem.     

Upper bound for the number of degrees of freedom for magneto-micropolar flows and turbulence

We present an explicit bound for the number of degrees of freedom for attractor for an incompressible, electrically conducting micropolar fluid flow in the presence of an arbitrary magnetic field. Then we use it to explain the possible mechanism of maintenance of turbulence that is different from that in N-S flows.     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below

Summary. The effect of a vertical magnetic field on the onset of convective instability in a conducting micropolar fluid (Oldroyd fluid) layer heated from below confined between two horizontal planes under the simultaneous action of the rotation of the system and a vertical temperature gradient is considered. Linear stability theory and normal mode analysis are used to derive an eigenvalue system of order twelve, and an exact eigenvalue equation for natural instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. The effects of magnetic field, the relaxation time and micropolar parameters on the critical Rayleigh number and critical wave number are discussed and presented graphically. The analysis presented in this paper is more general than any previous investigation.     

Some uniqueness and continuous dependence results in the micropolar mixture theory of porous media

The present paper studies the uniqueness and continuous data dependence of solutions of the initial-boundary value problem associated with the micropolar mixture linear theory of porous media. For a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid, an uniqueness result is established. Then we deduce some estimates for describing the continuous dependence of solution with respect to the changes in the body force and body couple and in the initial-boundary given data. Thus, it is shown that the general approach of a binary homogeneous mixture of an isotropic micropolar elastic solid with an incompressible micropolar viscous fluid is well posed.     

MHD boundary-layer flow of a micropolar fluid past a wedge with variable wall temperature

Summary The steady laminar MHD boundary-layer flow past a wedge immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solved numerically using a finite-difference scheme known as the Keller-box method. Numerical results show that the micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

This paper deals with the theoretical investigation of the effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like medium permeability, rotation, non-buoyancy magnetization, coupling parameter, spin diffusion parameter and micropolar heat conduction has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of magnetic parameter M₁ and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.     

Effect of throughflow and magnetic field on Bénard convection in micropolar fluids

Summary The effect of throughflow and magnetic field on the onset of Bénard convection in a horizontal layer of micropolar fluid permeated between two rigid, isothermal and micro-rotation free boundaries is studied. The determination of the critical Rayleigh number entails in solving the eigenvalue problem numerically for which the single-term Galerkin method is employed. It is established that both stabilizing and destabilizing factors can be enhanced by throughflow.     

Effects of suction-injection-combination (SIC) on the onset of Rayleigh-Benard magnetoconvection in a micropolar fluid

The effects of suction-injection-combination (SIC) and magnetic field on the linear stability analysis of Rayleigh-Benard convection in a horizontal layer of an Boussinesq micropolar fluid is studied using a Rayleigh-Ritz techinque. The eigenvalues are obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the micropolar fluid layer heated from below is more stable compared to the classical fluid layer.     

Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical plates

The fully developed electrically conducting micropolar fluid flow between two vertical porous parallel plates is studied in the presence of temperature dependent heat sources including the effect of frictional heating and in the presence of a magnetic field. Profiles for velocity, microrotation and temperature are presented for a wide range of Hartmann numbers and the micropolar parameter. The skin friction, couple stress and Nusselt numbers at the plates are shown in the tables.     

A domain-independent interaction integral for linear elastic fracture analysis of micropolar materials

This paper establishes a domain-independent interaction integral (DII-integral) for linear elastic fracture mechanics of micropolar elastic solids. The DII-integral has three amazing features that make it effective for solving the fracture parameters of complex micropolar materials. The first one is that the DII-integral can decouple the stress intensity factors (SIFs) and couple stress intensity factors (CSIFs) both of which are the key fracture parameters charactering the crack-tip asymptotic singular fields. In details, the DII-integral is derived from the J-integral by superimposing an actual field and an auxiliary field. By assigning the fracture parameters in the auxiliary field with different values, the SIFs and CSIFs of different crack opening modes can be obtained separately through the DII-integral. The second important feature is that the DII-integral is domain-independent for material nonhomogeneity and discontinuity. Thanks to this feature, the DII-integral becomes extremely effective for the micropolar materials with arbitrary nonhomogeneous properties or complex interfaces. The third feature is that the DII-integral does not contain any derivatives of material properties, which feature facilitate the practical implementation of the DII-integral on complex micropolar materials. Finally, the DII-integral combined with the extended finite element method (XFEM) is employed to solve four representative crack problems and the results show good validity of the DII-integral for complex micropolar materials.     

On uniqueness and continuous dependence for a linear micropolar fluid

In this paper logarithmic convexity arguments are employed to establish the uniqueness and Holder continuous dependence on initial data of a certain class of solutions of an initial boundary value problem for a linear micropolar fluid.     

An additive theory for finite elastic–plastic deformations of the micropolar continuous media

In this paper, the method of additive plasticity at finite deformations is generalized to the micropolar continuous media. It is shown that the non-symmetric rate of deformation tensor and gradient of gyration vector could be decomposed into elastic and plastic parts. For the finite elastic deformation, the micropolar hypo-elastic constitutive equations for isotropic micropolar materials are considered. Concerning the additive decomposition and the micropolar hypo-elasticity as the basic tools, an elastic–plastic formulation consisting of an arbitrary number of internal variables and arbitrary form of plastic flow rule is derived. The localization conditions for the micropolar material obeying the developed elastic–plastic constitutive equations are investigated. It is shown that in the proposed formulation, the rate of skew-symmetric part of the stress tensor does not exhibit any jump across the singular surface. As an example, a generalization of the Drucker–Prager yield criterion to the micropolar continuum through a generalized form of the J ₂-flow theory incorporating isotropic and kinematic hardenings is introduced.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.     

Fundamental solutions for micropolar fluids

New fundamental solutions for micropolar fluids are derived in explicit form for two- and three-dimensional steady unbounded Stokes and Oseen flows due to a point force and a point couple, including the two-dimensional micropolar Stokeslet, the two- and three-dimensional micropolar Stokes couplet, the three-dimensional micropolar Oseenlet, and the three-dimensional micropolar Oseen couplet. These fundamental solutions do not exist in Newtonian flow due to the absence of microrotation velocity field. The flow due to these singularities is useful for understanding and studying microscale flows. As an application, the drag coefficients for a solid sphere or a circular cylinder that translates in a low-Reynolds-number micropolar flow are determined and compared with those corresponding to Newtonian flow. The drag coefficients in a micropolar fluid are greater than those in a Newtonian fluid.     

Boundary layer growth of a micropolar fluid

The unsteady boundary layer flow of a micropolar fluid along an infinite plate is analysed when the plate undergoes an impulsive motion in its plane. The solution is obtained by Laplace transform technique. The microstructure of the fluid is found to induce a wave-dominated flow pattern, there being two modes of wave-propagation. The characteristics of these waves, during the initial and final stages of the boundary layer growth, are discussed.     

Stability of microstretch fluid motions

Criteria of stability of the unsteady motion of incompressible microstretch fluid in an arbitrary time-dependent domain are obtained using a general energy method introduced by Serrin. It is shown that the original motion is stable in the mean if either of the two sets of numbers ($\text{?}{}_1$,$\text{?}{}_2$,$\text{?}{}_3$) or ($\text{σ}{}_1$, $\text{σ}{}_2$,$\text{σ}{}_3$) consists of positive numbers only. These numbers are expressible in terms of the various Reynolds numbers of the original motion. The theorems giving the stability criteria are universal in the sense that they do not depend on the geometry of the domain or the actual distribution of the flow field quantities. The decay of energy of the flow in a rigid and fixed container as well as a theorem on the uniqueness of steady flows are deduced.     

Thermal radiation effects on magnetohydrodynamic mixed convection flow of a micropolar fluid past a continuously moving semi-infinite plate for high temperature differences

Summary An analysis is presented to study the effect of radiation on magnetohydrodynamic mixed convective steady laminar boundary layer flow of an optically thick electrically conducting viscous micropolar fluid past a moving semi-infinite vertical plate for high temperature differences. A uniform magnetic field is applied perpendicular to the moving plate. The density of the micropolar fluid is assumed to reduce exponentially with temperature. The usual Boussinesq approximation is neglected because of the high temperature differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting governing equations are transformed using a similarity transformation and then solved numerically by applying an efficient technique. The effects of radiation parameter R, magnetic parameter M, couple parameter Δ and density/temperature parameter n on the velocity, angular velocity and temperature profiles as well as the local skin friction coefficient, wall couple stress and the local Nusselt number are presented graphically and in tabular form.     

Uniqueness of compressible micropolar fluid flows

The paper examines the uniqueness of compressible micropolar fluid flows over an arbitrary region $\text{R(t)}$ with a smooth boundary $\text{?R(t)}$. It is shown that there is at most one solution of the flow equations and boundary conditions which corresponds to suitably assigned initial values of the density, velocity, microrotation and temperature fields. The analysis rests on the use of differential inequalities involving the time derivatives of certain energy integrals.