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.     

Thermal relaxation effect on magnetohydrodynamic instability in a rotating micropolar fluid layer heated from below

Summary. The effect of a vertical magnetic field on the onset of convective instability in a conducting micropolar fluid (Oldroyd fluid) layer heated from below confined between two horizontal planes under the simultaneous action of the rotation of the system and a vertical temperature gradient is considered. Linear stability theory and normal mode analysis are used to derive an eigenvalue system of order twelve, and an exact eigenvalue equation for natural 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. The effects of magnetic field, the relaxation time and micropolar parameters on the critical Rayleigh number and critical wave number are discussed and presented graphically. The analysis presented in this paper is more general than any previous investigation.     

Numerical simulation of unsteady MHD natural convection of CNT-water nanofluid in square cavity heated sinusoidally from below

ABSTRACT

Numerical unsteady natural convection flow in a square cavity in the presence of uniform magnetic field is investigated. The cavity is filled with CNT-water nanofluid and heated from below with sinusoidal temperature distribution. This study has been conducted for the certain pertinent parameters of Rayleigh number Ra, Hartmann number Ha and the CNT volume fraction from Φ?=?0 to 0.12. The results indicate, that for large values of Ha, increasing Φ results in an increase of the normalized average Nusselt number. The rate corresponding to CNT- volume fraction nanoparticles of 10% ensures an enhancement about 50% of the heat transfer rate compared to the standard case. The flow undergoes a period bifurcation at a Rayleigh number beyond the value 10⁶. A critical Rayleigh number Ra_c of 1.077?×?10⁶ is then computed. Finally, a correlation giving the average Nusselt number as a function of the frequency of the periodic regime is established.     

High-Rayleigh-number turbulent natural convection on an inclined surface

High-Rayleigh-number natural-convection heat transfer on inclined surfaces is treated. The analysis of the effect of thermogravitational forces on turbulent transfer of momentum and heat in the wall region is based on the balance equations for the second moments of pulsations of velocity and temperature. Analytical dependences are derived for the coefficient of turbulent thermal conductivity and for the Nusselt number in a wide range of values of the Prandtl number.     

Free convection in h horizontal porous layer with a partly heated or cooled wall

The natural convective flow in a fluid-saturated porous medium is considered for an infinite horizontal channel with the bottom wall being partially heated or cooled. The flow and heat transfer are analysed for a range of values of the two non-dimensional parameters which define the problem, namely the Rayleigh number Ra and aspect ratio . Numerical solutions are obtained for = 1 and = 0.1 for both heated and cooled cases and for a range of values of Ra. In the heated case, the nature of the flow is seen to change from unicellular for smaller values of Ra to multicellular for large Ra, with the value of Ra at this changeover being decreased as is decreased. Also, for this case, a range of values of Ra is found over which both unicellular and multicellular flows are possible. For the cooled case, a boundary layer is seen to develop on the bottom wall as Ra is increased for both the values of taken. Finally, a solution for 1 is obtained and is compared with the numerical solutions for = 0.1.     

Effect of a non-constant magnetic field on natural convection in a horizontal porous layer heated from the bottom

An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting fluid. The magnetic field is assumed to be induced by two long wires, carrying current, parallel to the horizontal boundaries of the system. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The governing parameters of the problem under study are the thermal Rayleigh number, Ra, Hartmann number, Ha, position of the electrical wires, d, current intensity ratio, r, and aspect ratio of cavity, A. An analytical solution, valid for a shallow layer (A ? 1), is derived on the basis of the parallel flow approximation. The critical Rayleigh number, Ra _c , for the onset of motion is derived in closed form in terms of the parameters of the problem. For finite-amplitude convection the heat and flow characteristics predicted by the analytical model are found to agree well with a numerical study of the full governing equations.     

Steady and unsteady forced convection past an inclined elliptic cylinder

Summary Discussed in this paper is the two dimensional problem of forced convection past an inclined elliptic cylinder. Both the steady state and unsteady cases have been considered for moderate Reynolds numbers, R, in the range 40R70 with a Prandtl number Pr=1. The numerical results show that there is good agreement between the steady and limiting unsteady heat transfer quantities and Nu which are the average and local Nusselt numbers respectively. The isotherm plots show marked differences as R is increased due to vortex shedding which is known to take place. Lastly, the early development for high Reynolds number flow past a harmonically oscillating and translating thin elliptic cylinder is addressed to illustrate the effect of rotation on the convective heat transfer process. The main role of rotation is to cause fluctuations in the time variation of and also to distort the thermal tail region emanating from the rear tip of the ellipse.     

Investigation of the effect of the Prandtl number on the local properties of low-intensity convection in a rectangular region heated from below

Local heat transfer and temperature separation in the presence of steady-state natural thermal convection in a rectangular region heated from below have been numerically studied for various Prandtl numbers. The local features of the low-intensity convection have been analyzed. The dependences of the local effects on the Grashof and Prandtl numbers have been obtained. The effect of the Prandtl number on the boundaries of the convection regimes has been estimated.     

Similarity flow in interaction of a shock wave with an inclined heated channel

A study is made of gasdynamic flow that initiates when a shock wave propagates along a thin heated channel. Analytical conditions of the onset of an unsteady flow precursor are obtained. The flow similarity is proved experimentally; precursor characteristics vs shock wave and heated channel parameters are analyzed.Institute of Geosphere Dynamics, Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 54, No. 5, pp. 543–547, May, 1993.     

Thermocapillary convection in a shallow annular pool heated from inner wall

In order to discover the effect of heating direction on the thermocapillary convection, three-dimensional numerical simulations have been conducted in a shallow annular pool (depth d=1 mm) of silicone oil (0.65 cSt, Pr=6.7) differentially heated from the inner wall. The linear stability analysis predicts a critical temperature difference (ΔT_c) for the incipience of hydrothermal wave (HTW) is 4.58K, which is less than ΔT_c=5.0K for a pool heated at the outer wall. Numerical simulations predict that two groups of HTWs, propagating in azimuthal directions opposite to each other, always coexist in the pool. Simulations with pool rotation around the axis elucidate that pool rotation up to 1.05 rad/s gives no distinguishable effect on the behavior of HTW in an inner heated pool, whereas the HTW patterns in an outer heated pool are significantly modified by rotation.     

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.     

Free convection from a heated vertical cylinder embedded in a fluid-saturated porous medium

Summary We consider the free convection boundary layer flow induced by a heated vertical cylinder which is embedded in a fluid-saturated porous medium. The surface of the cylinder is maintained at a temperature whose value above the ambient temperature of the surrounding fluid varies as then ^th power of the distance from the leading edge. Asymptotic analyses and numerical calculations are presented for the governing nonsimilar boundary layer equations and it is shown that, whenn<1, the asymptotic flowfield far from the leading edge of the cylinder takes on a multiple-layer structure. However, forn>1, only a simple single layer is present far downstream, but a multiple layer structure exists close to the cylinder leading edge. We have shown that the fully numerical and asymptotic calculations are in stisfactory agreement, especially for exponentsn close to zero. Comparisons of the present numerical solutions obtained using the Keller-box method with previous numerical solutions using local methods are also given.List of symbols a radius - scaled streamfunctions - f ₀,f ₁,f ₂ inner zone streamfunctions whenn<1 - leading order streamfunctions inn>1, 1 asymptotic solution - F ₀,F ₁ outer zone streamfunctions whenn<1 - G large parameter satisfyingG=X ² lnG - g gravitational acceleration - K permeability of the porous medium - n exponent in prescribed temperature law - r radial co-ordinate - r rescaled radial co-ordinate - R Darcy-Rayleigh number - T temperature of convective fluid - T _w temperature of cylinder at leading edge - T ambient temperature of fluid - u velocity in axial direction - v velocity in azimuthal direction - w velocity in radial direction - x axial co-ordinate - x escaled axial co-ordinate - X dimensionless axial co-ordinate - thermal diffusivity of the saturated medium - coefficient of thermal expansion - constant in the boundary conditions forF ₀ - dimensionless radial co-ordinate - co-ordinate for the outer zone in then<1 solution - scaled radial co-ordinates - scaled fluid temperature - similarity variable for then=1 problem - nondimensionalisation constant (Eq. (9)) - viscosity of fluid - scaled axial co-ordinates - density of fluid - co-ordinate for the inner zone in then<1 solution - azimuthal co-ordinate - similarity variables for then>1 problem - streamfunction     

Investigation of motion of transversely blown layer on an inclined surface

On the basis of an experimental investigation recommendations are given for the computation of nonuniformities of the motion of a layer along an inclined surface in the presence of transverse blowing.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 3, pp. 403–408, March, 1970.     

Stability of a layer of liquid flowing down an inclined plane

Summary The stability of the flow down an inclined plane is studied for small angles of inclination β. The same problem has been studied by S. P. Lin, however using an incorrect boundary condition. The correctly formulated eigenvalueproblem is solved by a numerical integration of the Orr-Sommerfeld equation employing the orthonormalization technique. It is shown that in the range 3′<β<1° a decrease in β means a decrease in the critical Reynolds number for the hard mode, which is a shear wave modified by the presence of the free surface. In that range the stability is still more or less governed by the stability of the soft waves, which are essentially surface waves modified by the presence of shear. For values of β<1′ the stability is governed by the hard mode, contradictory to Lin's statements. In that case instability occurs at high Reynolds numbers.