Effect of Coriolis force on the thermosolutal convection threshold in a rotating annular Hele-Shaw cell

In this study, the linear stability analysis is used to determine the onset of thermosolutal convection in fluids confined in rotating annular Hele-Shaw cell. The fluid layer is submitted to radial gradients of temperature and concentration. The effects of both Coriolis force and curvature parameter on the stationary and oscillatory convection are investigated when the Prandtl number is of the order of unity or larger than unity.     

The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. Attention is focused on the case of constant flux upper and lower boundaries, a case for which the critical horizontal wavenumber is zero, and attention is also concentrated on the case of a shallow layer. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The combination of permeability heterogeneity and conductivity heterogeneity can be either stabilizing or destabilizing for the present case.     

A note on the free convection boundary layer on a horizontal circular cylinder with constant heat flux

The equations governing the free convection boundary-layer flow on a horizontal circular cylinder on which there is a prescribed surface heat flux are solved using a finite-difference scheme. This numerical solution is then used to compare the accuracy of two proposed series expansions, a Blasius expansion and a Görtler-type expansion. It is shown that the former method is better at estimating temperature profiles while the latter is better at estimating velocity profiles.

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Bemerkung über die freie Konvektionsgrenzschicht an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom

Zusammenfassung Die Grenzschichtgleichungen für die freie Konvektion an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom durch die Oberfläche wird mit Hilfe eines finiten Differenzverfahrens gelöst. Die numerisch ermittelten Ergebnisse werden nachher für den Vergleich der Genauigkeit von zwei Reihendarstellungen der Lösung der Grenzschichtsgleichungen benützt. Diese Reihen sind vom Blasiusbzw. Görtier-Typ. Es wird bemerkt, daß die Reihendarstellung von Blasius die Temperaturprofile besser beschreibt, während die Reihenentwicklung der Görtlerschen Art für die Geschwindigkeitsprofile eine gute Übereinstimmung mit der exakten Lösung zeigt.

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Nomenclature a radius of the cylinder - g acceleration of gravity - P _r Prandtl number - Q prescribed (constant) heat flux - T temperature of the fluid - t ₀ temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r Grashof number=g Q a ⁴/v² - coefficient of thermal expansion - x thermal conductivity - v kinematic viscosity - _w skin friction     

Effect of surface tension on the onset of convection in a layer of liquid with a free surface

The problem of stability of a plane horizontal layer of liquid heated from below is considered with surface tension at the upper surface taken into account. The problem is stated in section 1, proof of the existence of stability threshold is given in section 2, while section 3 concerns the construction of neutral curves by numerical methods and with the stabilizing effect of surface tension on the state of equilibrium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 89–92, May–June, 1969.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000     

Effect of modulation on onset of thermal convection of a second-grade fluid layer

This study examines the stability of a horizontally extended second-grade fluid layer heated from below, when a steady temperature difference between the walls is superimposed on sinusoidal temperature perturbations. A linear stability analysis proposed by Venezian (J. Fluid Mech. 35 (1969) 243) is employed to obtain the critical Rayleigh numbers for different types of temperature modulation. The free–free and isothermal boundary conditions are considered so as to allow analytic solutions. The stability characterized by the shift in critical Rayleigh number R_2c is calculated as a function of the modulation frequency ω, the Prandtl number Pr, and the viscoelastic parameter Q. It is found that the onset of convection can be delayed or advanced by these parameters.     

Constant heat flux solution for mixed convection boundary layer viscoelastic fluid

The mixed convection boundary layer of a viscoelastic fluid past a circular cylinder with constant heat flux is discussed. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. The governing non-similar partial differential equations are transformed into dimensionless forms and then solved numerically using the Keller-box method by augmenting an extra boundary condition at infinity. Numerical results obtained in the form of velocity distributions and temperature profiles are presented for a range of values of the dimensionless viscoelastic fluid parameter. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the momentum boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point.     

Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer

Both linear and weakly nonlinear stability analyses are performed to study thermal convection in a rotating couple-stress fluid-saturated rigid porous layer. In the case of linear stability analysis, conditions for the occurrence of possible bifurcations are obtained. It is shown that Hopf bifurcation is possible due to Coriolis force, and it occurs at a lower value of the Rayleigh number at which the simple bifurcation occurs. In contrast to the nonrotating case, it is found that the couple-stress parameter plays a dual role in deciding the stability characteristics of the system, depending on the strength of rotation. Nonlinear stability analysis is carried out by constructing a set of coupled nonlinear ordinary differential equations using truncated representation of Fourier series. Sub-critical finite amplitude steady motions occur depending on the choice of physical parameters but at higher rotation rates oscillatory convection is found to be the preferred mode of instability. Besides, the stability of steady bifurcating equilibrium solution is discussed using modified perturbation theory. Heat transfer is calculated in terms of Nusselt number. Also, the transient behavior of the Nusselt number is investigated by solving the nonlinear differential equations numerically using the Runge–Kutta–Gill method. It is noted that increase in the value of Taylor number and the couple-stress parameter is to dampen the oscillations of Nusselt number and thereby to decrease the heat transfer.     

Finite-amplitude regimes of mixed convection in a vertical layer with undulating boundaries

The method of finite differences is used to construct convective motions in a vertical layer with sinusoidally curved boundaries, fluid being pumped through longitudinally. Apart from steady and oscillation regimes, found earlier by analytical means for small amplitudes of undulation and slow pumping through [1, 2], new, essentially nonlinear, types of motion are discovered in the form of two-stroke cycles, and also of complex multi-revolution cycles which are two-dimensional resonance tori. The regions are determined in which regimes of various types exist.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–20, January–February, 1987.The author is grateful to E. M. Zhukhovitskii for constant interest in the study, and also to V. S. Anishchenko and A. A. Nepomnyashchii for useful discussions.     

Effect of non-uniform temperature gradient and magnetic field on onset of Marangoni convection heated from below by a constant heat flux

This paper investigates the effect of non-uniform temperature gradient and magnetic field on Marangoni convection in a horizontal fluid layer heated from below and cooled from above with a constant heat flux. A linear stability analysis is performed. The influence of various parameters on the convection onset is analyzed. Six non-uniform basic temperature profiles are considered, and some general conclusions about their desta- bilizing effects are presented.     

The onset of double-diffusive convection in a nanofluid layer

The onset of double-diffusive convection in a horizontal layer of a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition the thermal energy equations include regular diffusion and cross-diffusion terms. The stability boundaries for both non-oscillatory and oscillatory cases have been approximated by simple analytical expressions. Physical significance of the obtained results is discussed.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Fluid convection in a rotating porous layer under modulated temperature on the boundaries

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.     

Investigation of low-intensity convection regimes in a rectangular cavity with a heat flux on the boundary

The characteristics of heat transfer during natural thermogravitational fluid convection of low intensity in a rectangular cavity heated from below (cooled from above) are investigated. Local convection effects are studied. The dependence of local superheating (supercooling) on the Grashof number and the cavity side ratio is found for single-, two-and three-vortex steady motions. The limits of the convection regimes are estimated.     

Hydrodynamics in weak force fields onset of steady thermocapillary convection in a spherical fluid layer under zero-g conditions

In the study of cellular convection in an infinite plane fluid layer with a free surface, both the Archimedes and thermocapillary forces [1–3] have been cited as reasons for the onset of convection. This has also been confirmed experimentally [4], When mass forces are absent or negligibly small it is natural to pose the question of the onset of pure thermocapillary convection or convection caused only by the surface tension gradients (see [2–3]). In the present paper, this problem is examined for a spherical fluid layer under zero-g conditions.     

Nonstationary laminar heat convection in a closed domain for a given heat flux

Characteristic modes of the time development of nonstationary heat convection in a closed planar domain upon a sudden supply of heat from the lateral surface are considered for Rayleigh numbers 10³–10⁷. Estimates of the boundaries of the beginning of the influence of convection on the temperature field and the buildup of a quasistationary convection mode in the range of Rayleigh and Fourier numbers are given. Characteristics of the circulation flow, the singularities of the temperature-field configuration and of the heat transfer from the wall to the fluid, are investigated. The mechanism for the origination and disappearance of vertical temperature differences, caused by convection, and the dependence of the vertical temperature differences on the Rayleigh and Fourier numbers, on the thermal mode of the boundary, and the domain geometry, are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 109–117, July–August, 1970.The author is grateful to T. D. Pirumov and T. V. Volokitin for assistance in performing the computations.     

Thermocapillary convection in a plane liquid layer with concentration heat sources

Thermocapillary instability of a plane liquid binary-mixture layer with time-dependent surface tension is studied under weightlessness conditions. The liquid is heated (or cooled) due to heat release by an active admixture. The heat release rate is proportional to the active-component concentration. The admixture is transported by convection and diffusion. The active component “burns up” with time. The neutral curves for monotonous and oscillating disturbances are found for different values of the nondimensional parameters. Some nonlinear convection regimes are studied numerically by a finite-difference method. The dependence of the convective flow intensity on the Marangoni number is determined. The phase portraits of unsteady regimes are found.     

Unsteady free convection on a vertical cylinder with variable heat and mass flux

The unsteady natural convection boundary layer flow over a semi-infinite vertical cylinder is considered with combined buoyancy force effects, for the situation in which the surface temperature T ^′ _w(x) and C ^′ _w(x) are subjected to the power-law surface heat and mass flux as K(T ^′/r) = −ax ⁿ and D(C ^′/r) = −bx ^m . The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson method. Numerical results are obtained for different values of Prandtl number, Schmidt number ‘n’ and ‘m’. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt and Sherwood numbers are shown graphically. The local Nusselt and Sherwood number of the present study are compared with the available result and a good agreement is found to exist. Received on 7 July 1998     

Thermocapillary convection in a liquid layer with a quasi-point heat source in the substrate

By the method of tracer particles the velocity field of thermocapillary convection in a thin layer of silicone oil, excited by a quasi-point heat source in the rigid substrate, is investigated as a function of the layer thickness, the temperature of the heater, and the liquid viscosity. The vertical velocity distributions are plotted in several cross-sections at different distances from the vortex axis. A novel method of measuring the profile of the thermocapillary depression, based on mirror reflection from the free liquid surface of radiation scattered by a tracer particle, is proposed. The central segment of the profile of the thermocapillary depression is obtained for different values of the layer thickness, the liquid viscosity and the heater temperature.