Existence of periodic solutions of the equations of incompressible micropolar fluid flow

The equations of incompressible micropolar fluid flow are a coupled system of vector differential equations involving the two basic vectors, viz. the velocity and the microrotation of the fluid elements. Let D = D (t) be a bounded region in space, and let a flow velocity and a microrotation be prescribed at each point of the boundary of D(t). Assume that D(t) as well as the assigned velocity and microrotation vectors depend periodically on the time t and that the condition (2μ+k)j−4a 0 is satisfied (equation (25) in the text). Further assumptions are that (i) to every continuous initial distribution of the flow fields over D, there corresponds a solution of the field equations for all time t 0 satisfying the prescribed boundary conditions; (ii) there is one solution for which the Reynolds numbers Re, Rm satisfy the condition Re² + Rm² < 80 and this solution is equicontinuous in for all t. Then there exists a unique, stable, periodic solution of the micropolar flow equations in D(t) taking the prescribed values on the boundary. The proof of the theorem rests on a formula describing the rate of decay of the kinetic energy of the difference of two micropolar flows in the domain subject to the same boundary conditions.     

Unsteady mixed convection flow of a thermomicropolar fluid on a long thin vertical cylinder

The unsteady laminar mixed convection boundary layer flow of a thermomicropolar fluid over a long thin vertical cylinder has been studied when the free stream velocity varies with time. The coupled nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite difference scheme in combination with the quasilinearization technique. The results show that the buoyancy, curvature and suction parameters, in general, enhance the skin friction, heat transfer and gradient of microrotation, but the effect of injection is just opposite. The skin friction and heat transfer for the micropolar fluid are considerably less than those for the Newtonian fluids. The effect of microrotation parameter is appreciable only on the microrotation gradient. The effect of the Prandtl number is appreciable on the skin friction, heat transfer and gradient of microtation.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

Heat transfer in boundary layer flow of a micropolar fluid past a curved surface with suction and injection

Van Dyke's singular perturbation technique has been used to study the heat transfer in the flow of a micropolar fluid past a curved surface with suction and injection. The conditions for similar solutions of the thermal boundary layer equations have been obtained. In addition to the usual “no slip” condition for velocity, the two types of boundary conditions used for microrotation are: (i) no relative spin on the boundary; (ii) the anti-symmetric part of the stress tensor vanishes at the boundary. The effect of suction or injection on velocity, microrotation, temperature, skin friction coefficient, wall couple stress coefficient, displacement and momentum thicknesses, rate of heat transfer and adiabatic wall temperature have been studied. It is observed that with the increase of injection velocity, the thickness of the boundary layer is increased and the local drag is reduced. A comparison with the results obtained for a Newtonian fluid reveals that the microelements present in the fluid reduce the velocity and frictional drag, and cool the boundary.     

Effects of fluid layer at micropolar orthotropic boundary surface

Effects of a fluid layer at a micropolar orthotropic elastic solid interface to a moving point load have been studied. After using the Fourier transform an eigen value approach has been employed to solve the problem. The displacement, microrotation and stress components for a micropolar orthotropic elastic solid so obtained in the physical domain are computed numerically by applying the numerical inversion technique. Micropolarity and anisotropy effects along with that of the depth of the fluid layer on various expressions have been depicted graphically for a particular model. Some special cases of interest have been presented     

Unsteady incompressible boundary layer flow of a micropolar fluid at a stagnation point

The flow, heat and mass transfer on the unsteady laminar incompressible boundary layer in micropolar fluid at the stagnation point of a 2-dimensional and an axisymmetric body have been studied when the free stream velocity and the wall temperature vary arbitrarily with time. The partial defferential equations governing the flow have been solved numerically using a quasilinear finite-difference scheme. The skin friction, microrotation gradient and heat transfer parameters are found to be strongly dependent on the coupling parameter, mass transfer and time, whereas the effect of the microrotation parameter on the skin friction and heat transfer is rather weak, but microrotation gradient is strongly affected by it. The Prandtl number and the variation of the wall temperature with time affect the heat-transfer very significantly but the skin friction and micrortation gradient are unaffected by them.     

Lubrication theory for micropolar fluids and its application to a journal bearing

The problem considered is that of a steady laminar flow of an incompressible micropolar fluid in a journal bearing. A two dimensional flow field is considered and order of magnitude arguments are made, which reduce the governing balance equations to a system of coupled, ordinary differential equations. The equations are solved subject to appropriate boundary conditions and the bearing characteristics obtained. The effect of microstructure is elaborated through various graphs. The prominent feature of a micropolar fluid is an increased effective viscosity especially in thin films.     

Peristaltic pumping of a micropolar fluid in a tube

Summary.  Peristaltic transport of a micropolar fluid in a circular tube is studied under low Reynolds number and long wavelength approximations. The closed form solutions are obtained for velocity, microrotation components, as well as the stream function and they contain new additional parameters namely, N the coupling number and m the micropolar parameter. In the case of free pumping (pressure difference Δp=0) the difference in pumping flux is observed to be very small for Newtonian and micropolar fluids but in the case of pumping (Δp>0) the characteristics are significantly altered for different N and m. It is observed that the peristalsis in micropolar fluids works as a pump against a greater pressure rise compared with a Newtonian fluid. Streamline patterns which depict trapping phenomena are presented for different parameter ranges. The limit on the trapping of the center streamline is obtained. The effects of N and m on friction force for different Δp are discussed. Received June 20, 2002; revised October 23, 2002 Published online: April 17, 2003 The authors thank the referees for pointing out some mistakes in the governing equations and for the suggestions to improve the presentation of the paper.     

A uniqueness theorem for compressible micropolar flows

Summary Using an energy integral method it is proved that the motion of a non-heat conducting compressible micropolar fluid in a bounded regionV=V(t) is uniquely determined by the initial distributions of velocity, microrotation, density and temperature, together with certain boundary conditions.     

Moving wedge and flat plate in a micropolar fluid

Similarity solutions for a moving wedge and flat plate in a micropolar fluid may be obtained when the fluid and boundary velocities are proportional to the same power-law of the downstream coordinate. The governing partial differential equations are transformed to the ordinary differential equations using similarity variables, and then solve numerically using a finite-difference scheme known as the Keller-box method. Numerical results are given for the dimensionless velocity and microrotation profiles, as well as the skin friction coefficient for several values of the Falkner–Skan power-law parameter (m), the ratio of the boundary velocity to the free stream velocity parameter (λ) and the material parameter (K). Important features of these flow characteristics are plotted and discussed. It is found that multiple solutions exist when the boundary is moving in the opposite direction to the free stream, and the micropolar fluids display a drag reduction compared to Newtonian fluids.     

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Summary The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.Nomenclature h Dimensionless microrotation - j Micro-inertia density - K Dimensionless parameter of vortex viscosity - m Falkner-Skan power-law parameter - Re Reynolds number - T Temperature - u, v Fluid velocities in thex andy directions, respectively - U Free stream velocity - x Streamwise coordinate along the body surface - y Coordinate normal to the body surface Greek symbols Thermal diffusivity - Wedge angle parameter - Spin gradient viscosity - Pseudo-similarity variable - Vortex viscosity - Absolute viscosity of the fluid - v Kinematic viscosity - Dimensionless temperature - Density of the micropolar fluid - Angular velocity of micropolar fluid - Stream function     

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.     

Boundary conditions for micropolar fluids

The effect of different boundary conditions for micropolar fluids is investigated. For steady channel flow it is shown that some commonly applied boundary conditions, such as microrotation rate equals minus one half the vorticity, can only reproduce Navier Stokes results. A general linear relation between microrotation rate and vorticity at rigid boundaries is analyzed and several special cases are examined in some detail. Implications for more general flows are briefly discussed.     

On the slow motion of a cylinder in a micropolar fluid

Slow steady 2-dimensional motion of an incompressible micropolar fluid in the unbounded region exterior to a cylinder of arbitrary cross section is considered. The possibility of a solution in the strict sense of the Stokes' approximation is examined. It is shown that the near and far boundary conditions can be satisfied simultaneously only when the drag on the cylinder is zero and the velocity and microrotation vectors satisfy an integral condition.     

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.     

Natural convection of micropolar fluid in an enclosure with boundary element method

The contribution deals with numerical simulation of natural convection in micropolar fluids, describing flow of suspensions with rigid and underformable particles with own rotation. The micropolar fluid flow theory is incorporated into the framework of a velocity–vorticity formulation of Navier–Stokes equations. The governing equations are derived in differential and integral form, resulting from the application of a boundary element method (BEM). In integral transformations, the diffusion-convection fundamental solution for flow kinetics, including vorticity transport, heat transport and microrotation transport, is implemented. The natural convection test case is the benchmark case of natural convection in a square cavity, and computations are performed for Rayleigh number values up to 10⁷. The results show, which microrotation of particles in suspension in general decreases overall heat transfer from the heated wall and should not therefore be neglected when computing heat and fluid flow of micropolar fluids.     

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

The fully developed free convection 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. The basic equations are solved using quasi-linearization finite difference technique with an error of order 0.5 × 10₋₆. The velocity, microrotation and temperature are displayed in graphs whereas the skin friction, couple stress and Nusselt numbers at the plates are shown in tables. It is noted that the couple stress on either plates increases numerically with increase in micropolar parameter. Also the Nusselt number follows the same pattern for a negative suction velocity.     

Mixed convection boundary layer flow of a micropolar fluid along vertical cylinders and needles

An analysis is presented to investigate the effects of buoyancy and curvature on convection along vertical cylinders and needles placed in a micropolar fluid. The governing equations for momentum, angular momentum and energy are solved numerically by finite difference scheme. The heat transfer results are presented for a range of values of the buoyancy parameters, the curvature parameter and the material parameters of the fluid. The effect of the microrotation boundary conditions on heat transfer is discussed.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.