Onset of stationary Benard-Marangoni convection in a fluid layer with variable surface tension and viscosity

The onset of stationary Benard-Marangoni convective instabilities in a fluid layer with thermally dependent surface tension and viscosity is studied by means of linear stability analysis. The dependence of viscosity and surface tension of the fluid on temperature is assumed to be exponential and linear respectively. The upper surface is free and is subject to a general thermal condition, while the lower boundary is rigid and is fixed at a constant temperature or a constant heat flux. For the latter case, the analytically asymptotic solution of long wavelength is obtained. For pure Benard convection, the system becomes more stable when the Biot number Bi increases. For both the upper and lower boundaries fixed at a constant heat flux, the critical Rayleigh number R-c decreases monotonically with the physical viscosity parameter B, and the corresponding critical wavenumber a(c) vanishes. The Blot number Bi, affecting the system, depends strongly on the parameter Gamma(=M/R). The critical Rayleigh number R-c decreases with Gamma and a jump in the critical wavenumber a(c) for large Gamma and B and small Bi exists. The critical conditions R-c and a(c), for various values of Gamma or Bi, approach constant values as the viscosity parameter B becomes large

Maximum density effects on natural convection in a vertical annulus filled with a non-Darcy porous medium

This article numerically studies the problem of natural convection in a porous medium saturated with cold water, under density inversion, within a vertical annulus. In modeling the flow in the porous medium the non-Darcy effects, which include the Forchheimer inertia and Brinkman viscous effects are taken into account. The governing equations are solved numerically by the finite difference method using the modified strongly implicit procedure. The effects of the inversion parameter Theta(m), radius ratio R*, aspect ratio AR, Forchheimer inertia parameter Fc/Pr, and Darcy number parameter Da on the heat transfer and fluid flow characteristics are discussed in detail. Results show that both the inversion parameter and radius ratio have a significant influence on the flow structure and heat transfer rate in the annulus. It is also found that the mean Nusselt number decreases as the Forchheimer inertia parameter or the Darcy number increases. Moreover, the results obtained here are also compared and favorably agree with numerical results and with experimental data

Influence of viscosity variation on the stationary Benard-Marangoni instability with a boundary slab of finite conductivity

The onset of stationary Bernard-Marangoni instability in a variable-viscosity fluid layer with a boundary slab of finite conductivity is studied. The relations of the viscosity and surface tension of the fluid with the temperature are exponential and linear, respectively. The asymptotic solutions of the long wavelength, for small values of the conductivity and thickness of the solid, are achieved and are very well compared with the numerical results. As the viscosity ratio increases, the validity of the asymptotical solutions is extended for cases of larger values of the thermal conductivity and thickness ratios

Conjugate film condensation and natural convection between two porous media separated by a vertical plate

This article theoretically studies the conjugate film condensation and natural convection along the vertical plate between a saturated vapor porous medium and a fluid-saturated porous medium. The solution takes into consideration the effect of heat conduction along the plate. The governing equations along with their corresponding boundary conditions for film condensation and natural convection are First cast into a dimensionless form by a nonsimilar transformation, and the resulting equations are then solved by the cubic spline collocation method. The primary parameters studied include the thermal resistance ratio of film to plate A, the thermal resistance ratio of natural convection to film B, and the Jakob number Ja of the subcooling degree in the film. The effects of these dimensionless parameters on the plate temperature distribution and the local heat transfer rate on both sides of the plate are discussed in detail. In addition, the interesting engineering results regarding the overall heat transfer rate from the film condensation side to the natural convection side are also illustrated

Comparative analysis of linear and non linear low-Reynolds-number eddy viscosity models to turbulent natural convection in horizontal cylindrical annuli

A computational study is presented of the natural convection flow within horizontal cylindrical annuli at a range of Rayleigh numbers from 8.02 x 10(5) to 1.18 x 10(9) and with a variation of outer and inner diameter ratio Between 2.6 and 4.58. The flow features large recirculation and the coexistence of turbulent and laminar, nearly stagnant regions, signifying the importance of effects arising from streamline curvature and wall viscosity on the evolution of boundary layers over inner and outer cylinder surfaces. This has,first, motivated the application of low-Reynolds-number models, which are able to take wall viscous effects into account. Two such models, a linear and a nonlinear variant, are examined in this study. The nonlinear model can, with the inclusion of cubic-order terms in the nonlinear stress-strain/vorticity relation, further sensitize the model to streamline curvature and is anticipated to give more faithful solutions than the linear model. At low-Rayleigh-number-flow, computational results indicate that both models return laminar and identical solutions, which are in good agreement with the experimental data. However, the nonlinear model is shown to return improved representation of the flow at high Rayleigh numbers, and the results obtained by both models are in reasonable agreement with the data

Maximum density effects on natural convection of micropolar fluids between horizontal eccentric cylinders

Natural convection heat transfer of micropolar fluids near the maximum density in a horizontal eccentric annulus is studied numerically. The governing equations, in terms of vorticity, stream function and temperature, are expressed in a body-fitted coordinate system, and solved numerically by the finite difference method with the modified strongly implicit procedure. The main objective is to investigate the differences of fluid dow and temperature fields for various eccentricities e with Rayleigh numbers Ra ranging from 671 to 5 x 10(4) and inversion parameter Theta(m) variations between 0 and 1. The micropolar parameter Delta is varied from 0 to 1. The results indicate that both inversion parameter and eccentricities have a strong effect on the flow structure and heat transfer rate in the annulus. In comparison to a Newtonian fluid, the mean Nusselt number is reduced for a micropolar fluid. Moreover, good agreement is obtained with the present numerical results and the available experimental data. (C) 1998 Elsevier Science Ltd

Effects of nonuniform temperature gradients on the onset of oscillatory Marangoni convection in a magnetic field

This article theoretically studies the onset of oscillatory Marangoni convection in a horizontal layer of an electrically conducting fluid, to which a nonuniform thermal gradient and a uniform magnetic field are applied. The top surface of a fluid layer is deformably free and the bottom is rigid. By means of the linear stability theory and a normal mode analysis, the eigenvalue equations of the perturbed state are solved by using the fourth-order Runge-Kutta-Gill's method with the shooting technique. The computational results are compared with those known from the literature, and the agreement is found out to be generally good. The results indicate that the critical Marangoni number -Ma(c) increases with increasing the Chandrasekhar number Q, the Prandtl number Pr, or the Biot number Bi of the upper free surface, but decreases with increasing the Crispation number Cr. As compared with the linear temperature profile, the inverted parabolic temperature profile shows higher -Ma(c) values, while the parabolic temperature profile shows lower -Ma(c) values. In addition, for the piecewise linear temperature profiles, the influences of thermal depth on the critical Marangoni number are also obtained

Soret and Dufour effects on free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation

This article numerically studies the combined laminar free convection flow with thermal radiation and mass transfer of non-Newtonian power-law fluids along a vertical plate within a porous medium. The solution takes the diffusion-thermo (Dufour), thermal-diffusion (Soret), thermal radiation and power-law fluid index effects into consideration. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a similarity transformation and the resulting coupled differential equations are then solved by the differential quadrature method (DQM). The effects of the radiation parameter R. the power-law index n, the Dufour number D(f), and the Soret number S(r) on the fluid flow, thermal and concentration fields are discussed in detail. The results indicate that when the buoyancy ratio of concentration to temperature is positive, N>0, the local Nusselt number increases with an increase in the power-law index and the Soret number or a decrease in the radiation parameter and the Dufour number. In addition, the local Sherwood number for different values of the controlling parameters is also obtained. (C) 2010 Elsevier Ltd. All rights reserved

Effect of a non-uniform temperature gradient on the onset of oscillatory Benard-Marangoni convection of an electrically conducting liquid in a magnetic field

This study deals theoretically with the effect of a non-uniform. basic temperature gradient on the linear stability of the oscillatory Benard-Marangoni convection in a horizontal layer of a viscous quiescent, electrically conducting fluid in the presence of a uniform vertical magnetic field. The upper surface of a fluid layer is deformably free and the lower surface is rigid. The eigenvalue equations of the perturbed state obtained from the normal mode analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results show that the critical Rayleigh number Ra-c and the critical Marangoni number -Ma(c) become larger as the Chandrasekhar number Q or the Biot number Bi of the upper free surface increases, and the Crispation number Cr decreases. When compared with the linear temperature profile, the inverted parabolic temperature profile indicates a reinforcement of stability, while the parabolic temperature profile indicates a diminution of stability in both Rayleigh-Benard and Benard-Marangoni convections. In addition, for the piecewise linear temperature profiles, the influences of thermal depth on the critical conditions are also obtained. Comparisons are also made of the critical conditions [-Ma(c), a(c), q(ic)] between the present results and published data sets for the linear basic temperature profile case, and the agreement is found out to be generally good. (C) 2003 Elsevier Science Ltd. All rights reserved

Thermal performance of mixed electroosmotic-pressure driven flows in microtubes with constant wall temperature

A numerical analysis is performed to explore the heat transfer characteristics of mixed electroosmotic and pressure-driven flows in a microtube with constant wall temperature. Thermally fully-developed flow with joule heating is considered. The joule heating is generated by imposed voltage gradient and can be regarded as volumetric heat source. The analysis combines energy equation with overall energy balance equation for a control element to generate a nonclimensional governing equation. Of interest are the effects of the relative duct radius a (ratio of the duct radius to Debye length), the pressure gradient parameter P (ratio of pressure gradient to electroosmotic forces) and the joule number S (ratio of heat generation due to joule heating to heat transfer at the wall) on the temperature distribution and the local heat transfer rate. The results indicate that the Nusselt number increases with an increase in the relative duct radius and the joule number or a decrease in the pressure gradient parameter. The resulting solution due to the conditions of energy unbalance in the flow is also discussed. (C) 2009 Elsevier Ltd. All rights reserved