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 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 $\text{R}$.     

Flow of a micropolar fluid in a diverging channel

The steady flow of an incompressible micropolar fluid in a diverging channel is studied. An approximate solution for the equations of motion is obtained in the presence of the inertia terms. It is observed that in the case of a micropolar fluid, the radial flow pattern of a Newtonian fluid is accompanied by a cross flow which is more pronounced near the apex of the channel. The variation of the streamlines and the micro-rotation are plotted and discussed.     

Flow of micropolar fluid through a constricted channel

Micropolar fluid flow through a non-symmetric channel with continuous constriction is studied. Perturbation method of solution is obtained in terms of slope parameter. The expressions for velocity, micro rotation, shear stress, axial pressure gradient and point of separation are derived. The effect of some parameters on shear stress and point of separation are studied.     

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.     

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.     

The dispersion of a chemically reacting solute in a micropolar fluid

The paper presents solution for the dispersion of a solute in a micropolar fluid flow in a circular pipe in the presence of an irreversible first order chemical reaction. It is shown that the solute is dispersed relative to a plane moving with the mean speed of the flow with an effective Taylor dispersion coefficient, dependent on two parameters, κ/μ and λ characteristic of the micropolar fluid and also on the reaction rate parameter a for a homogeneous reaction in the bulk of the fluid in which the solute is dispersed. It is found that for a given the effective Taylor dispersion coefficient decreases with an increase in κ/μ for fixed λ while it increases with increase in λ for fixed κ/μ. It decreases with increase in a for assumed values of κ/μ and λ.     

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.     

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.     

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     

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.     

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.     

Unsteady heat transfer in a micropolar fluid flowing in a plane channel

Heat transfer in a micropolar fluid flowing in a plane channel following an abrupt change in the wall temperature is investigated. The obtained results indicate that in several cases the fluid microstructure has a considerable effect on the main heat-transfer characteristics.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 6, pp. 994–999, December, 1979.     

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.     

On the stability of a hot layer of micropolar fluid

A critical study on the stability of a hot layer of micropolar fluid heated from below with free boundaries has been investigated. The analysis shows that the method by which the previous investigators (Datta and Sastry, and Pérez-García and Rubí) obtained the critical Rayleigh number is not justified and the final result obtained thereby is erroneous. The correct solution to the problem has been presented. Moreover, it is found that the possibility of having an overstable marginal state which was shown by one of the previous investigators (Pérez-García and Rubí) is not justified. The correct approach proves the validity of the principle of exchange of stabilities for this problem. The results show that the criteria of micropolar stability have some interesting features having no classical analogue.     

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.     

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.