Exact solutions of linearized equations of convection of a weakly compressible fluid

A mathematical model of fluid convection under microgravity conditions is considered. The equation of state is used in a form that allows considering the fluid as a weakly compressible medium. Based on the previously proposed mathematical model of convection of a weakly compressible fluid, unsteady convective motion in a vertical band, with a heat flux periodic in time set on the solid boundaries of this band, is considered. This model of convection allows one to study the problem with the boundary thermal model oscillating in an antiphase rather than in-phase mode, while the latter was required for the model of microconvection of an isothermally incompressible fluid. Exact solutions for velocity components and temperature are derived, and the trajectories of fluid particles are constructed. For comparison, the trajectories predicted by the classical Oberbeck-Boussinesq model of convection and by the microconvection model are presented.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 52–63, March–April, 2005.     

Stability of Steady Flow in a Vertical Layer in the Microconvection Model

The stability of steady flow in a vertical gap is analyzed using the perturbation method within the framework of the microconvection model. The resulting spectral problem is not self-conjugate. The stability of the flow to long-wave perturbations is established. It is shown that if the Boussinesq parameter is small, the spectrum of this problem approximates the spectra of the corresponding problems for a viscous heat-conducting fluid and thermogravitational convection with a finite Rayleigh number. The numerical calculations indicate that in the microconvection model the instability develops at smaller wave numbers.     

Stability of the equilibrium of a flat layer in a microconvection model

The stability of the equilibrium state of a flat layer bounded by rigid walls is studied using a microconvection model. The behavior of the complex decrement for longwave perturbations has an asymptotic character. Calculations of the full spectral problem were performed for melted silicon. Unlike in the classical Oberbeck–Boussinesq model, the perturbations in the microconvection model are not monotonic. It is shown that for small Boussinesq parameters, the spectrum of this problem approximates the spectra of the corresponding problems for a heatconducting viscous fluid or thermal gravitational convection when the Rayleigh number is finite.     

Stationary solution of the equations of microconvection in a vertical layer

The stationary problem of convection in liquids is considered using the model of microconvection developed by V. V. Pukhnachev. Velocity profiles for boundary conditions of different classes are constructed. The solutions of the problem under study and the classical problem based on the Oberbeck—Boussinesq model are compared.     

Mixed convection flow over a vertical wedge embedded in a highly porous medium

 The steady mixed convection flow over a vertical wedge with a magnetic field embedded in a porous medium has been investigated. The effects of the permeability of the medium, surface mass transfer and viscous dissipation on the flow and temperature fields have been included in the analysis. The coupled nonlinear partial differential equations governing the flow field have been solved numerically using the Keller box method. The skin friction and heat transfer are found to increase with the parameters characterizing the permeability of the medium, buoyancy force, magnetic field and pressure gradient. However the effect of the permeability and magnetic field on the heat transfer is very small. The heat transfer increases with the Prandtl number, but the skin friction decreases. The buoyancy force which assists the forced convection flow causes an overshoot in the velocity profiles. Both the skin friction and heat transfer increase with suction and the effect of injection is just the reverse. Received on 21 May 1999     

Mixed Convection Adjacent to a Suddenly Heated Horizontal Circular Cylinder Embedded in a Porous Medium

The mixed convection caused when a horizontal circular cylinder is suddenly heated is investigated in the situation when the initial flow past the cylinder is uniform and its direction either upwards or downwards. An analytical series solution, which is valid at small times, is obtained using the matched asymptotic expansions technique. A numerical solution, which is valid at all times and for any values of the Rayleigh and Péclet numbers, is also obtained using a fully implicit finite-difference method. Three different regimes, when either the free or forced convection is dominant or when they have the same order of magnitude, are considered. In the free convection dominated regime, two vortices develop near the sides of the cylinder in both situations of an upward or downward external flow. Comparisons between the analytical and numerical results at small times, as well as a detailed discussion of the evolution of the numerical solution are presented. The numerical results obtained for large Rayleigh, Ra, and Péclet Pe, numbers show that a thermal boundary-layer forms adjacent to the cylinder for any value of the ratio Ra/e. The steady state boundary-layer analysis, similar to that performed by Cheng and Merkin, is analysed in comparison to the numerical solution obtained for large values of Ra and Pe at very large times.     

Convection in a stratified binary mixture near a thermally inhomogeneous vertical surface

The stationary convection in a stratified two-component medium, for example, saline sea water, near a thermally inhomogeneous vertical surface is investigated analytically. Physically different cases of thermal inhomogeneities extended in the vertical or horizontal direction are considered. The solutions obtained can be applied to problems of convection in semibounded horizontal or vertical layers in the presence of thermal inhomogeneities at the “ends” of the layer. The solutions show that in two-component media convection is very specific. In particular, the spatial pattern of the thermal response to inhomogeneous heatingmay significantly differ from the case of an ordinary single-component medium: additional perturbation modes that penetrate deeply into the stably stratified medium appear. For an arbitrarily strong hydrostatic stability of the medium there exists an unexplored mechanism of convective instability related with the difference in the boundary conditions for the two substances. Weak variations of the background stratification of the admixture concentration (salinity) may significantly affect the heat exchange between a vertical surface and the medium. Even a very weak presence of the second component (a small contribution of the admixture stratification to the background density stratification) may lead not only to a significant quantitative change in the thermal response but also to a change in its sign, for example, to a significant decrease in the temperature of the medium in response to a heat influx from the vertical boundary.     

Modeling of microconvection in a fluid between heat conducting solids

Unsteady convection in a fluid under weak gravity is modeled. Convection in a rectangular domain elongated in the direction of gravity and enclosed between two heat-conducting solids is investigated in the case of heat insulation of the ends of the rectangle and the periodic heat flow through the outer boundaries of the solids. In this case, the condition of zero total heat flux is satisfied. Convective fluid motions are described using two mathematical models: the classical Oberbeck-Boussinesq model and the microconvection model for an isothermally incompressible fluid. Results of the numerical studies confirm the quantitative and qualitative differences between the flow characteristics calculated using the two convection models. Fluid particle trajectories are presented. Effects due to various physical characteristics of the problem are studied.     

Diffusion–Convection in a Porous Medium with Impervious Inclusions at Low Flow Rates

The aim of this paper is to develop a macroscopic model for the transport of a passive solute, by diffusion and convection, in a heterogeneous medium consisting of impervious solids periodically distributed in a porous matrix. In the porous part, the flow is described by Darcy's law. Attempt is made to derive the macroscopic equation governing the average concentration field in the equivalent macroscopic medium and the macroscopic transport parameters. The analysis is conducted in the case when convection and diffusion are of the same order of magnitude at the macroscopic level, that is, when the Péclet number is of order 1. The proposed macroscopic model is obtained using the homogenization method for periodic structures with a double scale asymptotic expansion, in which the small parameter is the ratio between the two characteristic lengths l (the period scale of the impervious bodies distribution) and L (the scale of the macroscopic sample). The macroscopic parameters, which characterize the multiporous medium, depend solely on the transport parameters in the porous matrix and on the geometry of the impervious inclusions without any phenomenological assumption. Numerical computations are performed using a finite element method for several geometries of the solid inclusions, in two- and three-dimensional cases.     

Nonboussinesq Thermal Convection in Microgravity under Nonuniform Heating

The model of subsonic flows is used to numerically the effect of thermal expansion of a fluid on the formation of naturally convective flows for small Rayleigh numbers (microconvection) and spatially periodic distribution of heat flows on the boundaries of the domain occupied by the fluid.     

Study of nonlinear convection in a sparsely packed porous medium using spectral analysis

Nonlinear study cellular convection in a sparsely packed fluid saturated porous medium is investigated, considering the Brinkman model, using the technique of spectral analysis. It is established how the Brinkman model with free-free boundaries generalizes the study of convection in a porous medium in the sense that it yields the results tending to those of viscous and Darcy flows respectively for very small and very large values of the permeability parameter σ². It also provides results for the transition zone (i.e. 10²<σ²<10³). The cross-interaction of the linear modes caused by non-linear effects are considered through the modal Rayleigh number R_γ. The possibility of the existence of steady solution with two self-excited modes in certain regions is predicted. The similarities of present analysis with and advantages over that of the power integral technique are brought out. A detailed discussion of the heat transport, with the effect of permeability thereon, is made. The theoretical values of the Nusselt number are found to be in good agreement with experimental results.     

Stability of Thermal Convection in a Fluid-Porous System Saturated with an Oldroyd-B Fluid Heated from Below

A linear stability analysis is conducted for thermal convection in a two-layer system composed of a fluid layer overlying a porous medium saturated with an Oldroyd-B fluid heated from below. It is found that the convection pattern in the system is controlled by the porous medium when the ratio of the depth of the fluid layer to that of the porous medium is small. However, the fluid layer takes a predominant role if the depth ratio exceeds a critical value. Compared with stationary convection, the switching point from a porous-dominated mode to a fluid-dominated mode for oscillatory convection is located at a lower depth ratio. The effects of different parameters on stationary convection and oscillatory convection are also investigated in detail.     

Boundary layers in free convection

The problem of free convection and mass transfer near a vertical wall is studied for the cases where the motion is described by the classical Oberbeck-Boussinesq model and the model of microconvection. In both cases, boundary layers are developed at high Schmidt numbers. Formulas for Nusselt (local and overall) numbers are obtained by solving the relevant problems for these layers. Initial asymptotic forms are also considered. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 92–100, May–June, 2000.     

Review of nucleate pool boiling bubble heat transfer mechanisms

Enhanced convection, transient conduction, microlayer evaporation, and contact line heat transfer have all been proposed as mechanisms by which bubbles transfer energy during boiling. Models based on these mechanisms contain fitting parameters that are used to fit them to the data, resulting a proliferation of “validated” models. A review of the recent experimental, analytical, and numerical work into single bubble heat transfer is presented to determine the contribution of each of the above mechanisms to the overall heat transfer. Transient conduction and microconvection are found to be the dominant heat transfer mechanisms. Heat transfer through the microlayer and at the three-phase contact line do not contribute more than about 25% of the overall heat transfer.     

The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion

Linear and nonlinear stability analysis for the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The critical Rayleigh number, wave number for stationary and oscillatory mode and frequency of oscillations are obtained analytically using linear theory and the non-linear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convection is shown pictorially. We also study the effect of time on transient Nusselt number and Sherwood number which is found to be oscillatory when time is small. However, when time becomes very large both the transient Nusselt value and Sherwood value approaches to their steady state values.     

Heat Transport in an Anisotropic Porous Medium Saturated with Variable Viscosity Liquid Under Temperature Modulation

Thermal instability in a horizontal porous medium saturated with temperature-dependant viscous fluid has been considered, and the effect of time-periodic temperature modulation has been investigated. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small and the disturbances are expanded in terms of power series of amplitude of convection. A weak non-linear stability analysis has been performed for the stationary mode of convection, and heat transport in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg–Landau equation, is calculated. The effects of thermo-rheological parameter, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. It is found that an increment in the value of thermo-rheological parameter results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.     

Gravity Affected Lateral Dispersion and Diffusion in a Stationary Horizontal Porous Medium Shear Flow

Transport of dissolved species by a carrier fluid in a porous medium comprises advection and diffusion/dispersion processes. Hydrodynamic dispersion is commonly characterized by an empirical relationship, in which the dispersion mechanism is described by contributions of molecular diffusion and mechanical dispersion expressed as a function of the molecular Peclét number. Mathematically these two phenomena are modeled by a constant diffusion coefficient and by velocity dependent dispersion coefficients, respectively. Here, the commonly utilized Bear--Scheidegger dispersion model of linear proportionality between mechanical dispersion and velocity, and the more complicated Bear--Bachmat model derived on a streamtube array model porous medium and better describing observed dispersion coefficients in the moderate molecular Peclét number range, will be considered. Analyzing the mixing flow of two parallelly flowing confluent fluids with different concentrations of a dissolved species within the frames of boundary layer theory one has to deal with transverse mixing only. With the Boussinesq approximation being adopted approximate analytical solutions of the corresponding boundary layer system of equations show that there is no effect of density coupling on concentration distributions across the mixing layer in the pure molecular diffusion regime case. With the Peclét number of the oncoming flow growing beyond unity, density coupling has an increasing influence on the mixing zone. When the Peclét number grows further this influence is successively reduced until its disappearance in the pure mechanical dispersion regime.     

g-Jitter free convection flow in the stagnation-point region of a three-dimensional body

A numerical solution of the effect of a small fluctuating gravitational field characteristic of g-jitter is presented. Specifically the problem of free convection boundary layer flow near a three-dimensional stagnation point of attachment resulting from a step change in its constant surface temperature is considered. The transformed non-similar boundary layer equations are solved using the Keller-box method, which is essentially an implicit finite-difference scheme. Numerical results are given for a value of the Prandtl number, Pr = 0.72 with the forcing amplitude, ε, and the forcing frequency, Ω. It is shown that g-jitter affects considerably the flow characteristics, namely the skin friction and the rate of heat transfer. Comparison with earlier results for the case of constant gravity field show very good agreement.     

Unsteady natural convection in a liquid-saturated porous enclosure with local thermal non-equilibrium effect

Stability analysis of free convection in a liquid-saturated sparsely-packed porous medium with local-thermal-non-equilibrium (LTNE) effect is presented. For the vertical boundaries free–free, adiabatic and rigid–rigid, adiabatic are considered while for horizontal boundaries it is the stress-free, isothermal and rigid–rigid, isothermal boundary combinations we consider. From the linear theory, it is apparent that there is advanced onset of convection in a shallow enclosure followed by that in square and tall enclosures. Asymptotic analysis of the thermal Rayleigh number for small and large values of the inter-phase heat transfer coefficient is reported. Results of Darcy–Bénard convection (DBC) and Rayleigh–Bénard convection can be obtained as limiting cases of the study. LTNE effect is prominent in the case of Brinkman–Bénard convection compared to that in DBC. Using a multi-scale method and by performing a non-linear stability analysis the Ginzburg–Landau equation is derived from the five-mode Lorenz modal. Heat transport is estimated at the lower plate of the channel. The effect of the Brinkman number, the porous parameter and the inter-phase heat transfer coefficient is to favour delayed onset of convection and thereby enhanced heat transport while the porosity-modified ratio of thermal conductivities shows the opposite effect.

     

Steady thermocapillary motion in a horizontal layer of liquid metal locally heated from above

The article considers stationary thermocapillary convection in a thin horizontal layer of fluid with Prandtl number Pr < 1 when it is being locally heated from above in conditions in which the curvature of the free surface is small. It is shown that the motion has a cellular structure. The size of the convective cell is determined from the solution to the spectral problem to which the integration of the free convection system of equations reduces. If the Maragoni (Péclet) number is sufficiently high, the length of the convective cell turns out to be large in comparison with the thickness of the layer. The convection picture is considered and an expression obtained for the velocity of the developing flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 146–152, November–December, 1984.