Stability of a layer of viscoelastic fluid heated from below

The polymerization of methyl methacrylate is accompanied by liberation of heat; this results in overheating of the reaction mass during production of plastics. The temperature distribution in the polymerizing layer is complicated by convection, which disrupts the natural temperature field. Thus, in addition to the stress along the sheet, local internal stresses appear that show up in operation of the product. Product quality and intensification of the polymerization process depend on the critical temperature gradient, which determines the stability threshold of the layer of polymerizing methyl methacrylate. The Rayleigh-Jeffrey problem is considered for a weak viscoelastic fluid described by an integral rheological constitutive relationship. The critical Rayleigh numbers are determined for stationary and oscillatory instabilities with free and ideally heat-conducting rigid boundaries.     

Heat transfer in a vertically-layered porous medium heated from below

Some studies already made have investigated the criterion for onset of convection and heat and mass flow distributions in a porous slab composed of horizontal layers of different materials. This paper reports a study of such criteria for the case where the slab is composed of vertically-aligned strata with different permeabilities and thermal conductivities. This has particular relevance to where blocks of different materials abut in a vertical plane, as well as the case of very narrow highly permeable vertical layers which represent vertical faults in a geological structure. Results indicate that permeability and/or thermal conductivity contrasts between layers can significantly affect the flow pattern and the spatial distribution of the surface heat flux. The concentration of flow in highly permeable faults produces marked irregularities in the heat flow through the surface above them.     

The onset of natural convection in a horizontal air layer heated from below

In this work, the beginning of the instability (onset of convection) of an air layer of infinite width and depth heated from underneath with a constant heat flux is studied. In the theoretical part, the instability is studied using the quasi-static assumption. The functional relationship of Rayleigh number vs. horizontal wave number of the disturbance is obtained in a digital computer using Green's Fonctions for the case of Prandtl number equal to one. Furthermore in order to make a comparison with similar investigations, limiting cases of infinite and very small Prandtl numbers are also taken into consideration.-In the experimental part, Rayleigh numbers corresponding to the onset of manifest convection based on visual observations are investigated by optical methods under various heat fluxes. The average value measured is found to be 145.     

Oscillatory convection in a viscoelastic fluid through a porous layer heated from below

The stability of a viscoelastic fluid in a densely packed horizontal porous layer heated from below is considered using an Oldroyd model. Critical Rayleigh number, wave number, and frequency for overstability are determined by applying the linear stability theory. It is shown that the critical Rayleigh number is invariant under all relevant boundary combinations. Also, it is found that the effect of elasticity of the fluid is to destabilize the system and that of porosity is to stabilize the same. The limiting case of very high Prandtl number and the degenerate case corresponding to the Maxwell model are analyzed in some detail.     

Experimental investigation of convective flows in a plane horizontal fluid layer heated from below and rotating about the vertical axis

The convective coherent structures in a plane horizontal fluid layer, heated from below and capable of rotation about the vertical axis, are experimentally investigated. It is shown that with increase in the supercriticality the time it takes for the convective structures to be formed decreases sharply. Rotation and an increase in the layer thickness-to-diameter ratio lead to an increase in the steady-state attainment time.     

Thermal convection of a viscoelastic fluid in an open-top porous layer heated from below

Buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box is studied based on a modified Darcy's law. The results are compared with those for a Newtonian fluid under the same boundary conditions and those for the viscoelastic fluid under a closed-top boundary. In particular, the critical Darcy–Rayleigh number Ra for onset of convection is determined first by using the linear stability theory. Then the effects of the relaxation time and the retardation time of the viscoelastic fluid on the heat transfer rate and the flow pattern are investigated numerically. The results reveal some interesting properties of thermal convection for the viscoelastic fluid. The relaxation time makes the fluid easier to destabilize while the retardation time tends to stabilize the fluid motion in the porous medium, and larger heat transfer rate can be achieved with larger value of the relaxation time and decreased retardation time. Furthermore, larger relaxation time facilitates earlier bifurcation of the flow pattern as Ra increases, but bifurcation can be postponed with increased retardation time. For larger ratio of relaxation time over retardation time, the flow pattern is more complicated and the frequency of flow oscillation also increases. Finally, large ratio of relaxation time over retardation time can make the open-top boundary impermeable due to the viscoelastic effect on the fluid.     

Stability of a nonuniformly heated fluid in a porous horizontal layer

We investigate the stability of a nonuniformly heated fluid in the gravitational field in a plane horizontal porous layer through which vertical forced motion is effected. A similar system was studied in [1, 2]. In the present paper, the nonuniformity of the permeability of the porous layer with respect to the depth and the dependence of the viscosity of the saturating fluid on the temperature are taken into account in addition. As a result of the application of the linear stability theory, an eigenvalue problem arises, which is solved numerically. A family of curves representing the dependence of the critical modified Rayleigh number Ra _k ^⋆ on the injection parameter (the Péclet number Pe) for different degrees of inhomogeneity of the permeability and the viscosity is obtained. It is found that although Pe=0 corresponds to Ra _k ^⋆ for uniform permeability and viscosity and the stability increases monotonically as Pe increases, the presence of nonuniformity of the permeability or the viscosity leads to the appearance of a stability minimum in the region Pe≈1, while under the simultaneous influence of these two factors, the minimum is shifted into the region Pe≈2. The results of the paper can be used, for example, in the investigation of heat transfer in the case of forced fluid motion in the fissures of a permeable rock mass, when, in the case of pumping through a horizontal fissure, the fluid penetrates vertically across its permeable walls into the stratum. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1986.     

On the nonlinear stability of a fluid layer of a mixture heated and salted from below

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, is studied, in the stress-free boundary case. A stabilizing effect of gradient of solute on thermal convection is shown and a globally nonlinear exponential stability theorem is proved. In particular, when the ratio r of the Schmidt and the Prandtl number is less than 1, a region of coincidence of linear and nonlinear critical parameters is found. The stability of plane parallel convective flows (plane Couette and Poiseuille flows with linear temperature and concentration profiles) is also studied. Stability conditions independent of Reynolds number are found.     

Secondary flow in horizontal channels heated from below

Experiments have been performed to investigate onset and development of the buoyancy driven secondary flow in a horizontal parallel plate channel with uniform bottom heating. Flow visualization in water (Pr ≈ 7) was performed by injecting a continuous sheet of dye into the bottom boundary layer just up-stream of the heated surface, and operating conditions in the ranges 125 < Re < 1,000 and4.7 x 10⁴ < Gr ^* < 8.0 x 10⁶ were considered. Top, side, and end views revealed onset of the secondary flow as thermal plumes, which rise from the heated surface and form pairs of counter-rotating vortices. Subsequent development of the flow is characterized by a breakdown in the regular plume structure and transition to buoyancy driven turbulence. Onset of the secondary flow is advanced by increasing the heat flux and/or decreasing the flow rate, and results may be correlated in terms of a critical Grashof number and a dimensionless longitudinal distance. Liquid crystal sheets applied to the heated surface reveal significant spanwise temperature variations due to the secondary flow. The unsteadiness of the flow is discussed and comparisons are made to previous experimental and numerical work.     

Heat transfer in an unevenly heated porous layer

For a uniform saturated porous layer heated from below, the dependence of the quantity of heat transferred on the distribution of the heat source is investigated. It is found, using perturbation methods and numerical techniques, that very small nonuniformities in the heat source having the same wavelength as the preferred convection mode significantly reinforce natural convection.     

Steady convective motions in a plane horizontal fluid layer with permeable boundaries

In a plane horizontal fluid layer bounded by permeable plane surfaces which are heated to different temperatures and between which transverse flow takes place with uniform velocity, convection occurs at a definite critical Rayieigh number. The study of the disturbance spectrum and the convective stability, made within the framework of linear theory in [1], showed that convective instability in the layer with permeable boundaries, just as in the case of the Rayieigh problem, is associated with the development of monotonie disturbances. It turns out that the transverse motion in the layer leads to a considerable increase of the Rayieigh number. Linear theory does not permit analysis of the development of the disturbances in the supercritical region. Analysis of the developed nonlinear motion can be made only on the basis of the complete nonlinear convection equations.In this investigation we made a numerical study of nonlinear motions in the supercritical region. Calculations were made on a computer via the grid method. Solutions are obtained for the nonlinear equations of motion over a wide range of Rayieigh numbers for different values of the Peclet number, whichdefines the intensity of the transverse motion in the layer.The author wishes to thank E. M. Zhukovitskii for his guidance, and G. Z. Gershuni and E. L. Tarunin for their interest and assistance in the study.     

Plane-parallel advective flow in a horizontal incompressible fluid layer with rigid boundaries

The exact solution of the Navier-Stokes equations that describes the plane-parallel advective flow in a plane incompressible fluid layer with rigid boundaries at which either the linear distribution of temperature of different signs or the linear horizontal temperature gradient is given is presented.     

Investigation of the onset of thermo-bioconvection in a suspension of oxytactic microorganisms in a shallow fluid layer heated from below

This paper investigates the combined effect of density stratification due to oxytactic upswimming and heating from below on the stability of a suspension of motile oxytactic microorganisms in a shallow fluid layer. Different from traditional bioconvection, thermo-bioconvection has two destabilizing mechanisms that contribute to creating the unstable density stratification. This problem may be relevant to a number of geophysical applications, such as the investigation of the dynamics of some species of thermophiles (heat loving microorganisms) living in hot springs. By performing a linear stability analysis, we obtained a correlation between the critical value of the bioconvection Rayleigh number and the traditional, “thermal” Rayleigh number. It is established that heating from below makes the system more unstable and helps the development of bioconvection.     

Stability of a radially bounded rotating fluid heated from below

The stability of a bounded rotating cylinder of fluid heated from below is treated mathematically under the assumptions of stationary onset and axisymmetry. Critical Rayleigh numbers are computed by Galerkin's method as a function of the Taylor number and cylinder aspect ratio for Taylor numbers,10⁶. The constraining effect of the side walls is shown to decrease with either increasing or increasing radius/height ratios. For>10⁶, most cylinders, excluding extremely tall ones, will appear infinite in horizontal extent as far as stability characteristics are concerned. The form of the motion at onset is discussed in relation to previous work.     

Convective heat transfer in a locally heated plane incompressible fluid layer

The problem of convection in a plane horizontal layer of incompressible fluid with rigid boundaries when the temperature is constant on the lower boundary and has a parabolic profile on the upper boundary can be reduced to solution of a system of time-dependent one-dimensional equations. An analytic solution of the problem is obtained directly at the extremum point. Together with the wellknown solutions which describe heat transfer for the linear temperature distribution on the boundaries, the results obtained make it possible to calculate the heat flux through a thin slit for an arbitrary given heating of a thin fluid layer between heat-conducting bodies.     

Convective heat transfer in a rotating horizontal cylindrical fluid layer

This paper describes the thermal convection and heat transfer in a cylindrical fluid layer rotating around a horizontal axis, with various constant temperatures set at the layer boundaries. The influence of the rotational speed of the cylindrical fluid layer on the convective heat transfer in this layer is studied. The study results are presented as functions of dimensionless parameters that characterize the action of two convective mechanisms: centrifugal and thermal-oscillatory. It is shown that, with low rotational speed, the heat transfer is determined by quasistationary gravitational convection.     

Control of the convective equilibrium stability of a fluid heated from below

Potentially unstable equilibrium is stabilized by means of a method of control based on the single-channel feedback principle using the characteristics of the solution of the complete nonlinear thermal convection equations. The convection equations are solved by a finite-difference method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 6–11, July–August, 1990.     

The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms

The aim of this paper is to present a continuum model for thermo-bioconvection of oxytactic bacteria in a porous medium and investigate the combined effects of microorganisms' upswimming and heating from below on the stability of bioconvection in a horizontal layer filled with a fluid saturated porous medium. Different from traditional bioconvection, thermo-bioconvection has two destabilizing mechanisms that contribute to creating the unstable density stratification. This problem may be relevant to a number of geophysical applications, such as the investigation of the dynamics of oxytactic species of thermophiles (heat loving microorganisms) living in hot springs, microbial-enhanced oil recovery, and modeling oil- and gas-bearing sedimentary basins. The utilization of the Galerkin method to solve a linear stability problem leads to a correlation between the critical value of the bioconvection Rayleigh number and the traditional “thermal” Rayleigh number.     

Control of the formation of Benard cells in a horizontal rectangular duct heated from below

Flows in horizontal chemical vapor deposition (CVD) ducts heated from below play an important role for manufacturing thin semiconductor layers. However, the formation of Benard cells prevents the growth of layers of uniform thickness. The present paper aims at finding conditions for time-mean uniform growth of the layer on the bottom of a rectangular CVD duct of aspect ratio 2 and heated from below. Experimental studies of various side wall temperature distributions were performed to identify conditions appropriate for controlling the Benard cells. It was found that flows in the chaotic state can provide the uniform horizontal temperature distribution preferable for the uniform growth of semiconductor layers.