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.     

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.     

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.     

Steady circulatory flow of a second grade fluid about a rotating porous cylinder

An exact solution for the circulatory flow of an incompressible second grade fluid about a rotating porous cylinder is given. The solution is expressed in terms of the confluent hypergeometric functions and it is valid for all values of the cross-Reynolds number, the elastic number and the ratio of the circulation at infinity to that on the surface of the cylinder. The velocity, the vorticity and the torque exerted by the fluid on the cylinder are calculated. It is shown that there are some discrepancies between the results obtained by the exact solution and those obtained by the perturbation solution which is valid for small values of the elastic number.     

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 flow near the edge of a disc at rest in a rotating fluid

Summary The Reynolds number Re being based on the angular velocity of the fluid and the radius of the disc, it is shown that within a distance O(Re ^-2/3) from the edge of the disc, the flow is determined by the Navier-Stokes equations. The boundary-value problem describing this flow is formulated. The asymptotic behaviour of its solution is investigated analytically and its complete numerical solution is evaluated. Results for various physical quantities, among them the additional torque due to the Navier-Stokes flow, are presented.     

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.     

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.     

On the stability of a hot rotating layer of micropolar fluid

The convective stability of a horizontal layer of incompressible micropolar fluid heated from below and rotating about a vertical axis has been investigated on the basis of linear theory, using normal mode analysis. The boundaries are assumed to be free. After introducing the corrections to the basic equations considered by Sastry and Rao [1], it has been found that the rotation has a destabilizing effect which contradicts the earlier assertion presented in [1].Moreover, microinertia, which does not affect the stability of a hot horizontal layer of incompressible micropolar fluid in the absence of rotation [2], is found to have destabilizing effect.     

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.     

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.     

Off-centered stagnation flow towards a rotating disc

Stagnation flow towards an off-centered rotating disc is studied. A three dimensional similarity transform reduces the Navier-Stokes equations to a set of nonlinear ordinary differential equations which are integrated numerically. It is found that the non-alignment complicates the flow field and surface shear, but does not affect the torque. Similarity equations for convective heat transfer with dissipation are also obtained.     

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.     

Flow of a micropolar fluid between two parallel plates rotating about non-coincident axes

In this paper, we have investigated the flow of a micropolar fluid between two parallel plates rotating about two non-coincident axes. An exact solution of the differential system governing the velocity and microrotation is obtained. The effects of the rotation of the plates and the presence of the micropolar elements are discussed in detail.     

Steady flow of a power-law fluid past a cylinder

Summary Considered in this paper is the two-dimensional steady flow of a power-law fluid past a stationary circular cylinder. The governing nonlinear equations, expressed in terms of a stream function and vorticity, were solved by finite differences for Reynolds numbers (based on the radius of the cylinder)R=5,20, 40 for various power-law indices,n. Parameters such as the drag coefficient, separation angle, wake length and critical Reynolds number are presented and contrasted with those of a Newtonian fluid (n=1) to illustrate the non-Newtonian effects. For a given-Reynolds number a consistent behaviour withn was observed in the parameters for the ranges considered. The results obtained for the Newtonian case agree well with documented results.     

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.     

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     

The slow stationary flow of incompressible micropolar fluid past a spheroid

The paper examines the slow stationary flow of incompressible micropolar fluid past a spheroid (prolate and oblate) adopting the Stokesian approximation, so that the inertial terms in the momentum equation and the bilinear terms in the balance of first stress moments are neglected. The flow over the space outside the body is analyzed and the velocity, microrotation, stress and couple stress are obtained analytically in infinite series form. The drag on the body is determined and it is observed that there is no couple exerted on the body. Numerical studies are undertaken to see the variation of the drag with respect to the geometric as well as the physical flow parameters. These have been presented in the form of figures. Micropolarity of the fluid has an augmenting effect on the drag. In an Appendix, an alternative method of determining the drag is indicated.