Symmetries and solution of a micropolar fluid flow through a cylinder

Summary The system of equations of motion for a micropolar fluid inside a circular cylinder subjected to longitudinal and rotational motion is considered. Classical Lie symmetries of the system of equations are studied. Classes of invariant solutions corresponding to different symmetry subgroups are obtained.     

Fundamental Oseen solution for the 2-dimensional flow of a micropolar fluid

The steady, incompressible flow of a micropolar fluid in 2 dimensions is considered. The Oseen linearization of the convective operator is introduced, and the associated problem for the fundamental solution is formulated. Solution of the fundamental problem is obtained in explicit form under a certain restriction on the physical parameters of the problem. Utilization of the fundamental solution in the investigation of general flow problems is discussed.     

Numerical solution of micropolar fluid flow in a channel with porous walls

In this paper, we have studied the effect of suction on the laminar flow of a micropolar fluid in the entrance region of a porous channel. The velocity and the microrotation at the entry of the channel are taken to be those of Poiseuille flow of a micropolar fluid through a non porous channel. The similarity variables introduced into the partial differential system, governing the flow, lead to a set of ordinary nonlinear differential equations. The boundary value problem governing the flow is solved by a numerical method based upon quasilinearisation, parametric differentiation and extrapolation. The velocity profiles and the stream lines, presented graphically, reveal several interesting conclusions.     

Numerical solution of a micropolar fluid flow between two rotating coaxial disks

Summary Steady flow of a micropolar fluid between two rotating disks of infinite radius rotating at different/same speeds has been investigated. The lower and upper disks rotate with angular velocities and S respectively. By using similarity transformation method, the equations of motion are reduced to a set of ordinary non-linear coupled differential equations. The resulting non-linear equations are linearized by quasilinearization technique and integrated with the help of fourth order Runge-Kutta method via orthonormalization. Effects of micropolarity parameters on velocity components for different values of Reynolds numbers Re (200, 1000 etc.) andS (–1.0, 0.0, 0.5, 1.0) have been studied. Multiplicity of the solutions have been obtained for different values ofS and micropolar parameters at Re=1000. It is found that only one of the four micropolar parameters (₁) influences translation velocity components significantly. At high Re, flows similar to flows observed by Batchelor Stewartson and Holodniok etc., have been obtained. A new type of multiple solution has been obtained forS=1.0 which needs further investigation.     

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.     

Numerical solution of micropolar fluid flow between a rotating and a porous stationary disc

The steady flow of an incompressible micropolar fluid between a rotating and a stationary disc is studied when a uniform suction is applied on the stationary disc. The nonlinear coupled equations involving velocity and microrotation components have been solved numerically using quasilinearisation technique. The pressure coefficient at any radius r of the rotating disc varies linearly with micropolar parameter R. The three velocities and microrotation components have been displayed graphically. It is observed that for low suction and high rotation, both radial and axial flows are of cellular type for small R but turn unidirectional for large values of R. The flow regimes are completely reversed in the case of blowing.     

Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid

Summary The present work examines the flow of a third grade fluid and heat transfer analysis between two stationary porous plates. The governing non-linear flow problem is solved analytically using homotopy analysis method (HAM). After combining the solution for the velocity, the temperature profile is determined for the constant surface temperature case. Graphs for the velocity and temperature profiles are presented and discussed for various values of parameters entering the problem.     

Stability of periodic flow in a micropolar fluid

The stability of unidirectional periodic flow in a micropolar fluid is treated. An analytic expression is found for the critical Reynolds number of stability loss.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 60, No. 4, pp. 670–679, April, 1991.     

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.     

Numerical solution of heat transfer in micropolar fluid flow in a channel with porous walls

Using the basic equations of heat conducting micropolar fluid, heat transfer through a channel of porous walls has been studied. The temperature distribution is obtained for different values of suction parameter $\text{S}$, convective parameter $\text{E}$ and micropolar parameter $\text{R}$.     

A class of exact solutions for the flow of a micropolar fluid

The flow of a micropolar fluid in an orthogonal rheometer is considered. It is shown that an infinite number of exact solutions characterizing asymmetric motions are possible. The expressions for pressure in the fluid, the components of the forces and couples acting on the plates are obtained. The effect of microrotation on the flow is brought out by considering numerical results for the case of coaxially rotating disks.     

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 in vertical internal flow of a micropolar fluid

The theory of micropolar fluids due to Eringen is used to formulate a set of equations for the flow and heat transfer characteristics of the combined convection micropolar flow in vertical channels. It is found that the microstructure and substructure parameters have significant effects on the flow and thermal fields. By making the Newtonian solvent more and more micropolar, it is possible to obtain drag reduction as well as reduced heat transfer characteristics.     

Steady flow of a micropolar fluid due to a rotating disc

Summary Three-dimensional, axially-symmetric, steady flow of a micropolar fluid, due to a rotating disc, is considered. The resulting equations of motion are solved numerically, for four different combinations of the six parameters involved, using the Gauss-Seidel iterative procedure and Simpson's rule. Results are presented both in tabular and graphical form.     

Steady forced flow of a micropolar fluid against a rotating disc

The steady forced flow of an incompressible micropolar fluid against a rotating disc is considered. The flow due to a rotating disc in an infinite fluid which is at rest and the axisymmetric stagnation flow on a flat plate are particular cases of this present problem. The equations of motion are solved numerically using the Gauss-Seidel iterative procedure and the Simpson's rule. The results are given in tabular form and compared with the known results for a Newtonian fluid.     

Numerical solution of thermal instability of a rotating micropolar fluid layer

In this paper, we have considered the thermal instability of a rotating, heat conducting, micropolar fluid layer heated from below and confined between two rigid boundaries. The onset of thermal instability is governed by a linear eigenvalue problem. The solution of the eigenvalue problem is obtained by using finite difference method and Wilkinson's iteration technique. The effects of rotation and micropolar parameters on the critical Rayleigh number and the wave number at the threshold of instability are discussed in detail.     

Faxen's laws for a micropolar fluid

Summary Faxen's formulas for the drag and torque on a rigid spherical particle immersed in a Stokes flow of a viscous incompressible fluid are extended for the case of an incompressible micropolar fluid.     

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.     

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.     

Heat transfer in the stagnation point flow of a micropolar fluid

Summary Similar solutions of the equations describing the thermal boundary layer of a micropolar fluid on a plane wall are found to exist for the stagnation point flow when the wall temperature variation is parabolic. The two types of boundary conditions used for microrotation are: (a) the relative spin of the particles on the boundary is related to the skew symmetric part of the stress on the boundary by a parameter which is a measure of the concentration of microelements, and (b) the couple stress on the boundary is related to the relative spin of the particles on the boundary by a friction factor which accounts for the rotational slip of the fluid along the boundary. The velocity, microrotation and temperature fields have been presented graphically for various values of the boundary condition parameters. The skin friction coefficient, wall couple stress coefficient, displacement and momentum thicknesses and rate of heat transfer have been tabulated. A comparison with the corresponding results for a Newtonian fluid has been made.

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Wärmeübergang in der Staupunktsströmung eines mikropolaren Fluids

Zusammenfassung Es werden ähnliche Lösungen der Gleichungen, die die thermische Randschicht eines mikropolaren Fluids längs einer ebenen Wand beschreiben, für die Staupunktsströmung bei parabolischer Änderung der Wandtemperatur gefunden. Zwei Typen von Randbedingungen werden für die Mikrorotation verwendet: (a) Der relative Spin der Teilchen am Rand ist verknüpft mit dem schiefsymmetrischen Anteil der Spannungen am Rand über einen Parameter, der ein Maß für die Konzentration der Mikroelemente darstellt. (b) Die Momentenspannung an der Berandung ist mit dem relativen Spin der Teilchen am Rand mit einem Reibungsfaktor verknüpft, der den Drehslip des Fluids längs der Berandung beschreibt. Die Geschwindigkeits-, Mikrorotations- und Temperaturfelder werden graphisch für verschiedene Werte des Parameters für die Randbedingungen dargestellt. Der Wandreibungskoeffizient, der Koeffizient der Wandmomentenspannung, Verschiebung und Impulsflußdicke und die Wärmeübergangsrate werden tabelliert. Ein Vergleich mit den entsprechenden Ergebnissen der Newtonschen Flüssigkeit wird angestellt.

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With 4 Figures