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.     

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.     

Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

This paper deals with the theoretical investigation of the effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like medium permeability, rotation, non-buoyancy magnetization, coupling parameter, spin diffusion parameter and micropolar heat conduction has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of magnetic parameter M₁ and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.     

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.     

Thermal instability of rotating nanofluid layer

In the present paper we have considered thermal instability of rotating nanofluids heated from below. Linear stability analysis has been made to investigate analytically the effect of rotation. The more important effect of Brownian motion and thermophoresis has been included in the model of nanofluid. Galerkin method is used to obtain the analytical expression for both non-oscillatory and oscillatory cases, when boundaries surfaces are free–free. The influence of various nanofluids parameters and rotation on the onset of convection has been analysed. It has been shown that the rotation has a stabilizing effect depending upon the values of various nanofluid parameters. The critical Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. The necessary and sufficient conditions for the existence of over stability are also obtained.     

Convective instability of a rotating fluid above a rapidly heated surface

Convection in a rotating fluid above a rapidly heated surface is studied experimentally. Pis’ma Zh. Tekh. Fiz. 23, 1–6 (February 12, 1997)     

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.     

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.     

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     

Three-dimensional convection in an inclined layer heated from below

Longitudinal rolls aligned with the component of gravity parallel to the layer represent the preferred mode of convection at onset for a wide range of parameters of the inclined layer problem. As the Rayleigh number is increased beyond the critical value the longitudinal rolls tend to become unstable with respect to the wavy instability. The three-dimensional convection flows evolving from this instability are studied in this paper by numerical computations. The main effect of the wavy distortions of the longitudinal rolls is a decrease of the heat transport. The stability of the steady three-dimensional convection flow with respect to disturbances with the same periodicity interval in the plane of the layer is also investigated. Various instabilities are found in dependence on the Prandtl number P and the angle of inclination and their evolution is studied in a few cases.     

On the possibility of overstable motions of micropolar fluids heated from below

It is shown that the principle of exchange of stability is not always accomplished in the Rayleigh-Bénard problem for fluids with internal structure. The possibility of oscillatory motions and the dependence of the critical Rayleigh number on the wave number is given.     

The effect of microstructure on the thermal convection in a rectangular box of fluid heated from below

The effect of microstructure on the thermal convection in a rectangular box of fluid heated from below has been investigated by applying the micropolar fluid theory. The influence of lateral walls on the convection process in a rectangular box has been determined. The Galerkin method has been employed to get an approximate solution for the eigenvalue problem. The beam functions which satisfy two boundary conditions on each rigid surface have been used to construct the finite roll (cells with two nonzero velocity components depend on all three spatial variables) trial functions for the Galerkin method. The effect of variations of material parameters at the onset of convection has been presented graphically. It is observed that as the distance between the lateral walls increases the effect of one of the material parameters, characterizing the spin-gradient viscosity, at the onset of stability diminishes. A comparison has been made with the corresponding results for a Newtonian fluid.     

Boundary layer growth of a micropolar fluid

The unsteady boundary layer flow of a micropolar fluid along an infinite plate is analysed when the plate undergoes an impulsive motion in its plane. The solution is obtained by Laplace transform technique. The microstructure of the fluid is found to induce a wave-dominated flow pattern, there being two modes of wave-propagation. The characteristics of these waves, during the initial and final stages of the boundary layer growth, are discussed.     

Electromagneto-hydrodynamic instability in a horizontal viscoelastic fluid layer with one relaxation time

Summary The stability of viscoelastic conducting liquid (Walters's liquid B) heated from below in the presence of a magnetic field is considered. Linear stability theory is used to derive an eigenvalue system of sixth order, and an exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh number and wavenumber for the onset of instability are presented graphically as functions of the Chandrasekhar number at a Prandtl numberP _r=100 and for various values of the one relaxation time and the elastic parameters.     

Convective instability of a layer of a ferromagnetic fluid rotating about a vertical axis

Thermal instability in a layer of a ferromagnetic fluid rotating about a vertical axis and permeated by a vertical magnetic field is investigated within the framework of linear theory. Overstability cannot occur if the Prandtl number $\text{P > 1}$. For overstable oscillations, Rayleigh number is plotted as a function of wave number for several values of the magnetization parameter $\text{M}{}_3$. Critical Rayleigh number is found to decrease with increase in $\text{M}{}_3$.     

Thermal boundary layer flow of a micropolar fluid past a wedge with constant wall temperature

Summary The steady laminar flow of micropolar fluids past a wedge has been examined with constant surface temperature. The similarity variables found by Falkner and Skan are employed to reduce the streamwise-dependence in the coupled nonlinear boundary layer equation. Numerical solutions are presented for the heat transfer characteristics with Pr=1 using the fourth-order Runge-Kutta method, and their dependence on the material parameters is discussed. The distributions of dimensionless temperature and Nusselt number across the boundary layer are compared with the corresponding flow problems for a Newtonian fluid over wedges. Numerical results show that for a constant wedge angle with a given Prandtl number Pr=1, the effect of increasing values ofK results in an increasing thermal boundary thickness for a micropolar fluid, as compared with a Newtonian fluid. For the case of the constant material parameterK, however, the heat transfer rate for a micropolar fluid is lower than that of a Newtonian fluid.Nomenclature h Dimensionless microrotation - j Micro-inertia density - K Dimensionless parameter of vortex viscosity - m Falkner-Skan power-law parameter - Re Reynolds number - T Temperature - u, v Fluid velocities in thex andy directions, respectively - U Free stream velocity - x Streamwise coordinate along the body surface - y Coordinate normal to the body surface Greek symbols Thermal diffusivity - Wedge angle parameter - Spin gradient viscosity - Pseudo-similarity variable - Vortex viscosity - Absolute viscosity of the fluid - v Kinematic viscosity - Dimensionless temperature - Density of the micropolar fluid - Angular velocity of micropolar fluid - Stream function