Incompressible micropolar fluid flow over a semi-infinite plate

Micropolar fluid flow over a semi-infinite flat plate has been described by using the parabolic co-ordinates and the method of series truncation in order to study the flow for low to large Reynolds numbers. These co-ordinates permit to study the flow regime at the leading edge. Numerical results have been presented for different Reynolds numbers. Results show a reduction in skin friction.     

Wall slip induced by a micropolar fluid

Lubrication theory is devoted to the study of thin-film flows, More often, the fluid can be considered as a Newtonian one and no-slip boundary conditions can be retained for the velocity at the fluid solid interface. With these assumptions it is possible to deduce from the (Navier) Stokes system a simplified equation describing the flow: the Reynolds equation. It allows to compute the pressure distribution inside the film and to obtain overall performances of a lubricated device such as load and friction coefficient. For very thin films, however, surface effects at the fluid solid interface become very important and no-slip conditions cannot be retained. Solid surfaces exert some influence on the liquid molecules and the effective shear viscosity along the boundary differs from the classical bulk shear viscosity. Moreover, the microstructure of the fluid cannot be ignored, especially the effects of solid-particle additives in the lubricant. Micropolar theory for fluids is often adopted to account of such microstructure and microrotation. In the present study, a thin micropolar fluid model with new boundary conditions at the fluid–solid interface is considered. This condition links velocity and microrotation at the interface by introducing a so-called “boundary viscosity”. By way of asymptotic analysis, a generalized micropolar Reynolds equation is obtained. Numerical results show the influence of the new boundary conditions for the load and friction coefficients. Comparisons are made with other works that retain the no-slip boundary conditions.     

Stokes flow of micropolar fluid past a viscous fluid spheroid with non-zero boundary condition for microrotation

The Stokes axisymmetric flow of an incompressible micropolar fluid past a viscous fluid spheroid whose shape deviates slightly from that of a sphere is studied analytically. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, continuity of shear stresses and spin–vorticity relation at the surface of the spheroid. The hydrodynamic drag force acting on the spheroid is calculated. An exact solution of the problem is obtained to the first order in the small parameter characterizing the deformation. It is observed that due to increased spin parameter value, the drag coefficient decreases. Well-known results are deduced and comparisons are made with classical viscous fluid and micropolar fluids.     

The asymptotic boundary layer on a circular cylinder in axisymmetric micropolar fluid flow

An asymptotic boundary layer analysis is presented using the theory of micropolar fluids due to Eringen. The laminar boundary layer induced on the outside of a long, slender cylinder due to the flow of an incompressible micropolar fluid parallel to the axis of the cylinder is investigated. For reasons of both analytical and practical interests the boundary layer characteristics far down stream from the leading edge are analyzed on the basis of their asymptotic nature. Asymptotic series solutions for the velocity and micro-rotation fields are obtained. An expression for the new micropolar boundary layer thickness is derived. Central to the present investigation is the result that while calculating the skin friction one should take into account the total surface stress effects, not only due to the usual shear stresses but also due to the couple stresses. As a result, it is shown that the micropolar theory does predict a reduction in skin friction as is observed in experiments thus confirming Eringen's well known conjecture.     

Longitudinal oscillations of a sphere in a micropolar fluid

Summary A general expression for the force exerted on a sphere excuting longitudinal oscillations, with small amplitude, in an incompressible micropolar fluid is obtained. This is accomplished by using direct integral consequences of the full field and the constitutive equations written in cartesian coordinates. The results which are independent of any boundary conditions are then applied to calculate the hydrodynamic force experienced by a sphere moving with rectilinear oscillating velocityu(t)=(u _oe^it, 0, 0) in an unbounded micropolar fluid. As a special case, a general expression for the drag in a similar viscous flow is also derived.     

A general free surface rule for Stokes flow of fluid films over obstacles

Summary A general rule is derived for the free surface profile of a gravity–driven fluid film on an inclined wall as it flows over a local irregularity, or obstacle, in the present case a trench. Starting from an exact analytical solution of Stokes' equations, based on complex function theory, it is proved that the integral of the free surface profile, from its planer asymptotic equilibrium level, vanishes. This general analytical result, which is valid for arbitrary wall irregularities where the spatial extension is small compared to an intrinsic length, is not only an interesting feature in itself but also provides a useful check of the accuracy of numerical schemes that are used to solve such problems.     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

On the motion of a micropolar fluid drop in a viscous fluid

Summary The problems of the flow of a viscous fluid past a micropolar fluid sphere and the flow of a micropolar fluid past a viscous fluid drop are discussed. The expressions for the stream functions, velocities, spins and the drag are obtained in each case and are compared with the classical (viscous fluid past a viscous fluid sphere) results. It is found that the viscosity ratios and the parameters, which arises in connection with the boundary condition, have significant effect upon the drag on the sphere in each case.     

Mixed convection of micropolar fluid about a sphere with blowing and suction

An analysis is performed to study the skin friction characteristics of laminar mixed forced and free convection flow of micropolar fluid about a permeable sphere with various prescribed thermal conditions on the surface. The problem was formulated by applying a suitable variables transformation and solutions were obtained by an efficient difference method. Numerical results were carried out for a wide range of mass transfer parameter as the Prandtl number at 0.7 or 7 with several values of material parameters and buoyancy force parameter of the micropolar fluid. The variations of the local friction factor and local Nusselt number are plotted and discussed. A comparison between the present solution and finite-difference solution from T.S. Chen, etc. is also shown.     

Mixed convection in micropolar fluid flow over a horizontal plate with surface mass transfer

Combined free and forced convection in the boundary layer flow of a micropolar fluid over a horizontal surface is studied. Buoyancy effects on the flow and temperature fields are discussed. The influence of uniform mass transfer from the surface is also considered. Wall friction and heat transfer results are presented for various cases representing the relative effects of blowing or suction as compared to the combined effects of buoyancy and mass transfer.     

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.     

State space approach to unsteady free convection flow of micropolar fluid

Summary The equations of unsteady free convection flow of a micropolar fluid are cast into a matrix form using the state space and Laplace-transform techniques. The results obtained are used to generate solutions in the Laplace-transform domain to a broad class of problems in free convection flow. The technique is applied to a heated vertical plate problem and to a problem pertaining to a plate under uniform heating. The inversion of Laplace-transform is performed using a numerical approach. Numerical results concerning velocity, microrotation and temperature are given and illustrated graphically for both problems.     

The slow stationary flow of a micropolar liquid past a sphere

Summary The paper is concerned with the slow stationary flow of a micropolar incompressible fluid past a sphere. Adopting the Stokesian approach of neglecting the inertial terms in the momentum equation and the bilinear terms in the balance of first stress moments, the equations are integrated and the flow parameters determined. The drag on the sphere is seen to be more in the present case than that in the case of non-polar fluids. It is found that in spite of the couple stress in the fluid, there is no resultant action by it on the sphere. Numerical work shows that the streamlines in the polar case have greater deflection towards the sphere than in the non-polar (or classical) case.     

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.     

Motion of a rigid sphere in a shear field in a micropolar fluid

The motion of a rigid sphere, suspended in a micropolar fluid which is undergoing a shearing motion, is discussed. The expressions for the pressure, velocity and spin in the fluid and those for the force and torque on the sphere are obtained. A compromise boundary condition, relating the spin of the particle with the vorticity vector at the boundary, is employed. The results are compared with the classical values and apart from other interesting observations, it is noted that the torque on the sphere depends upon the various parameters in a complicated manner. By extending the definitions of the effective viscosity for the viscous fluids, an expression for the viscosity of the suspension in the micropolar fluid is derived.     

Flow and heat transfer from a continuous surface in a parallel free stream of micropolar fluid

Boundary layer solutions are presented to investigate the steady now and heat transfer characteristics from a continuous flat surface moving in a parallel free stream of micropolar fluid. Numerical results are presented for the distribution of velocity, micro-rotation and temperature profiles within the boundary layer. The groupings of the material properties of the fluid were allowed to vary over a wide range.     

Analytic solution for flow of a micropolar fluid

Summary The time-independent equations for the two dimensional incompressible micropolar fluid have been considered. Using group method the equations have been reduced to ordinary differential equations and then solved analytically. Finally the boundary value problem has been discussed, and the graphical results are in good agreement with the numerical solution.