Stability of a micropolar fluid layer heated from below

The stability of a layer of micropolar fluid heated from below is studied employing a linear theory as well as an energy method. It is proved that the principle of exchange of stability holds and the critical Rayleigh number is obtained. It is observed that the micropolar fluid layer heated from below is more stable as compared with the classical viscous fluid. The energy method is then used to study the stability under finite disturbances. A variational method is applied to obtain the sharp stability limit. It is found that no subcritical instability region exists and the critical Rayleigh number as derived from the energy method is identical to that of the linear limit.     

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.     

Axisymmetric thermal boundary layer of a micropolar fluid on a cylinder

The laminar micropolar thermal boundary layer in axial flow along a long, thin circular cylinder is investigated using the theory of micropolar fluids formulated by Eringen. The governing energy equation has been solved numerically by the power series approach. Missing values of the thermal functions are tabulated for a wide range of the material parameters, the transverse curvature parameter, and Prandtl number of the fluid.     

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.     

Longitudinal surface curvature effects on boundary layer of a micropolar fluid

Summary The effect of longitudinal surface curvature on steady two-dimensional incompressible laminar boundary layer of a micropolar fluid has been considered. Van Dyke's first oder perturbation analysis is applied to the full equations of motion derived in curvilinear coordinate system which facilitates to carry out boundary layer approximation for flow past a curved surface. This results into two systems of partial differential equations which are called the zeroth order and the first order boundary layer equations. The zeroth order equations are the usual boundary layer equations for a micropolar fluid. The first order equations take into account the longitudinal surface curvature effect explicitly. Similar solution of the governing equations exists if (i) the inviscid flow velocity on the surface varies linearly along the surface and (ii) the longitudinal surface curvature is constant. Numerical results are presented illustrating the dependence of the important flow quantities of both zeroth order and first order boundary layers on the micropolar fluid parameters. The results have been compared with the corresponding results for a Newtonian fluid. It has been found that the skin friction decreases and the wall couple stress increases for convex side of the surface and vice versa for the concave side.

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Einfluß der longitudinalen Oberflächenkrümmung auf die Grenzschicht einer mikropolaren Flüssigkeit

Zusammenfassung Der Einfluß einer longitudinalen Oberflächenkrümmung auf die stationäre, zweidimensionale, inkompressible, laminare Grenzschicht einer mikropolaren Flüssigkeit wird untersucht. Van Dyke's Strömungsanalyse erster Ordnung wird angewendet auf die Bewegungsgleichungen in krummlinigen Koordinaten, die einfach die Grenzschichtnäherung für die Strömung hinter einer gekrümmten Oberfläche aufzeigen. Dies führt zu zwei Systemen partieller Differentialgleichungen, die die Grenzschichtgleichungen nullter und erster Ordnung genannt werden. Die Gleichungen nullter Ordnung sind die üblichen Grenzschichtgleichungen für eine mikropolare Flüssigkeit. Die Gleichungen erster Ordnung berücksichtigen den Einfluß der longitudinalen Oberflächenkrümmung explizit. Ähnliche Lösungen der Grundgleichungen existieren wenn: (i) sich die reibungsfreie Strömungsgeschwindigkeit an der Oberfläche linear entlang der Oberfläche ändert und (ii) die longitudinale Oberflächenkrümmung konstant ist. Numerische Ergebnisse werden angegeben, um die Abhängigkeit der wichtigsten Strömungsgrößen der Grenzschicht nullter und erster Ordnung von den Parametern der mikropolaren Flüssigkeit aufzuzeigen. Die Ergebnisse werden verglichen mit den entsprechenden Ergebnissen für eine Newtonsche Flüssigkeit. Es wurde festgestellt, daß für eine konvexe Fläche die Wandreibung abnimmt und die gekoppelten Spannungen an der Wand zunehmen; das Umgekehrte gilt für eine konkave Fläche.

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With 2 Figures     

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     

Thermal instability in a micropolar fluid layer subject to a magnetic field

In the present paper we have considered thermal instability in a heat conducting micropolar fluid layer under the influence of a transverse magnetic field. Assuming the bounding surfaces to be rigid the eigenvalue problem is solved using finite-difference and Wilkinson's iteration techniques. Here it is seen that the instability sets in not only for adverse temperature gradient but also for positive temperature gradient. Both the microtation and the magnetic field are seen to stabilize the fluid layer. However, the stabilizing effect of microrotation becomes less significant when the strength of the magnetic field is large. In the case of heating from below, the critical wave number is seen to be insensitive to increase in the strength of the magnetic field, while it increases significantly when the fluid is heated from above.     

Boundary layer of a microstructural fluid on a plate

A numerical solution of the self-similar problem of the boundary layer of a micropolar model fluid on a semiinfinite plate is obtained.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 36, No. 1, pp. 121–126, January, 1979.     

Convective instability of a micropolar fluid layer by the method of energy

The convective instability of a micropolar incompressible fluid layer heated from below is treated within the framework of Serrin-Joseph's energy method. In presence of coupling between temperature and micro-rotations, a region of subcritical instability is displayed. The influence of the various micropolar parameters on the onset of convection is also analyzed.     

Free convection boundary layer flow of a micropolar fluid past slender bodies

The transverse curvature effects on axisymmetric free convection boundary layer flow of a micropolar fluid past vertical cylinders are investigated using the theory of micropolar fluids formulated by Eringen. The governing equations for momentum, angular momentum and energy have been solved numerically. Missing values of the velocity, angular velocity and thermal functions are tabulated for a wide range of the material parameters, transverse curvature parameter and Prandtl number of the fluid. A comparison has been made with the corresponding results for Newtonian fluids. Micropolar fluids display drag reduction and reduced surface heat transfer rate as compared with Newtonian fluids.     

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.     

Thermal boundary layer of a micropolar fluid jet impinging normally on a flat plate

Summary The heat transfer is studied in the boundary layer formed on a flat plate by the impingement of an incompressible micropolar fluid jet. The thermal boundary layer equations are obtained after writing the governing equations for the steady two-dimensional flow of an incompressible micropolar fluid in cartesian co-ordinate system. The solution for the energy equation inside the boundary layer is obtained as a polynomial in terms of the distance from the stagnation point. The temperature of the plate and the temperature outside the boundary layer are assumed to be constant. The temperature distribution and the dimensionless heat transfer coefficient are presented graphically for various values of the material parameters which arise due to the micropolar property of the fluid. These results have been compared with the corresponding results for a Newtonian fluid.

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Thermische Grenzschicht eines mikropolaren Flüssigkeitsstrahles, der senkrecht auf eine ebene Platte auftrifft

Zusammenfassung Der Wärmeübergang in der Grenzschicht einer ebenen Platte, zufolge des Auftreffens eines inkompressiblen, mikropolaren Flüssigkeitsstrahles wird untersucht. Die thermischen Grenzschichtgleichungen werden aus den Grundgleichungen für die stationäre zweidimensionale Strömung einer inkompressiblen, mikropolaren Flüssigkeit in kartesischen Koordinaten erhalten. Die Lösung der Energiegleichung innerhalb der Grenzschicht wird als Polynom in Termen des Abstandes zum Staupunkt angegeben. Die Temperatur der Platte, sowie die Temperatur außerhalb der Grenzschicht werden als konstant vorausgesetzt. Die Temperaturverteilung und der dimensionslose Wärmeübergangskoeffizient sind graphisch für verschiedene Werte der Materialparameter, zufolge der mikropolaren Eigenschaften der Flüssigkeit, dargestellt. Diese Ergebnisse wurden mit den entsprechenden Ergebnissen für Newtonsche Flüssigkeiten verglichen.

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With 6 Figures     

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.