Unsteady thermocapillary convection in a nonuniformly heated fluid layer

In many technological processes, thin extended layers of nonuniformly heated fluid are used [1–3]. If they are sufficiently thin, thermocapillary forces have a decisive influence on the occurrence and development of motion of the fluid [4–6]. Investigation of convective motion in such a layer is of great interest for estimating the intensity of heat and mass transfer in technological processes. This paper is a study of unsteady thermocapillary motion in a layer of viscous incompressible fluid with free surface in which a thermal inhomogeneity is created at the initial time. Approximate expressions are obtained for the fields of the velocity, temperature, and pressure in the fluid, and also for the shape of the free surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1991.     

Thermocapillary convection in a thin layer of nonuniformly heated fluid

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–128, March–April, 1989.     

Parametric instability of a nonuniformly heated horizontal layer of liquid dielectric in a variable electric field

The parametric instability of a nonuniformly heated horizontal layer of liquid dielectric with free isothermal boundaries in a transverse electric field is studied analytically. An instability map is obtained. It is shown that instability can develop at some critical electric field strength which depends on the frequency and is several times greater than the critical strength of the constant electric field.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 184–186, September–October, 1993.     

Generation of the mean flow in the fluid layer with a nonuniformly heated wavy boundary

The problem of convection in the horizontal fluid layer with a wavy lower boundary is considered. It is shown that for the periodic temperature distribution with a certain phase shift given on the wavy boundary, in the fluid layer a unidirectional horizontal flow arises. The flow velocity linearly decreases with increase in attitude and depends on the relief distribution wavelength. There is an optimum wavelength (of the order of the layer thickness) at which the velocity reaches its maximum value.     

Localized convective flows in a nonuniformly heated liquid layer

Within the class of exact solutions of the thermal-convection equations in the Oberbeck-Boussinesq approximation, which assumes a linear dependence of the temperature and the vertical velocity component on the height, a non-self-similar behavior of localized disturbances of a special type in a nonuniformly heated liquid layer is studied. It is shown that in an unstably stratified medium these disturbances can evolve to isothermal vortex structures of Burgers type. In the conditions of stable stratification or uniform heating of the layer, the disturbances considered tend to the state of rest in an oscillating or monotonic manner. New solutions describing self-similar convective vortices are found.     

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.     

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.     

Force acting on a body in a nonuniformly heated magnetizable fluid

The force acting on a spherical particle in a nonuniformly heated magnetizable fluid is calculated in the case in which the permeability of the particle material depends arbitrarily on temperature and the strength of the magnetic field, and the permeability of the fluid on temperature. In calculating the force the difference between the thermal conductivities of the particle material and the fluid and, as distinct from [6], the distortion of the applied magnetic field due to the presence of a temperature gradient are taken into account. Accordingly, the expression for the force differs from that obtained in [6]. It is shown that the expression obtained for the force is correct up to terms of the order of a certain power of a small parameter — the ratio of the particle size to the characteristic interval of variation of the parameters (temperature, strength of magnetic field, etc.). A condition determining the value of this power is derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 76–83, March–April, 1989.     

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 the replacement of a non-Newtonian fluid layer in a porous medium

When petroleum is extracted from strata by replacing it with other fluids, the question arises of the stability of the interface. Uniformity in the injection horizons is in practice achieved by such methods as using polymer additives to thicken the replacing fluid or introducing an intermediate layer with non-Newtonian properties between the replacing fluid and the fluid being replaced. In this article the stability of the interface of non-Newtonian fluids being filtered exponentially is investigated in a linear formulation. The condition governing the stability of the interface of two non-Newtonian fluids is found and the influence of the thickness of the intermediate layer on the stability is also demonstrated. The presence of the layer is found to be essential for certain parameters of fluids moving in a porous medium if the replacement is to be stable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 183–186, March–April, 1976.The author wishes to thank V. N. Nikolaevskii for suggesting the subject for this investigation and A. T. Listrov for his constant interest in the work.     

Stability analysis of thermosolutal convection in a horizontal porous layer using a thermal non-equilibrium model

A stability analysis is carried out to investigate the onset of thermosolutal convection in a horizontal porous layer when the solid and fluid phases are not in a local thermal equilibrium, and the solubility of the dissolved component depends on temperature. To study how the reaction and thermal non-equilibrium affect the double-diffusive convection, the effects of scaled inter-phase heat transfer coefficient H and dimensionless reaction rate k on thermosolutal convection are discussed . The critical Rayleigh number and the corresponding wave number for the stability and overstability convections are obtained. Specially, asymptotic analysis for both small and large values of H and k is presented, and the corresponding asymptotic solutions are compared with numerical results. At last, a nonlinear stability analysis is presented to study how H and k affect the Nusselt number.     

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

Carbon dioxide injected into saline aquifers dissolves in the resident brines increasing their density, which might lead to convective mixing. Understanding the factors that drive convection in aquifers is important for assessing geological CO₂ storage sites. A hydrodynamic stability analysis is performed for non-linear, transient concentration fields in a saturated, homogenous, porous medium under various boundary conditions. The onset of convection is predicted using linear stability analysis based on the amplification of the initial perturbations. The difficulty with such stability analysis is the choice of the initial conditions used to define the imposed perturbations. We use different noises to find the fastest growing noise as initial conditions for the stability analysis. The stability equations are solved using a Galerkin technique. The resulting coupled ordinary differential equations are integrated numerically using a fourth-order Runge–Kutta method. The upper and lower bounds of convection instabilities are obtained. We find that at high Rayleigh numbers, based on the fastest growing noise for all boundary conditions, both the instability time and the initial wavelength of the convective instabilities are independent of the porous layer thickness. The current analysis provides approximations that help in screening suitable candidates for homogenous geological CO₂ sequestration sites.     

Heat transfer in a horizontal fluid layer heated from below upon rotation of one of the boundaries

The mixed convection in a horizontal fluid layer which is generated by uniform heating from below and by rotation of one of the boundaries of the layer was studied experimentally. The region occupied by the fluid is a cylinder of radius320 mm and height45 mm. Either the upper or the lower boundary together with the side wall rotates. For Rayleigh numbersRa≃2·10 ⁷, in a broad range of Reynolds numbers, based on experimental data we constructed mean-temperature profiles along the normal to the upper boundary and with a uniform step over the radius. In addition, we obtained data on the radial thermal stratification of the fluid, the integral flow through the fluid layer, and information on temperature fluctuations. The complicated character of the dependence of the heat transfer on the Reynolds number was shown. The obtained dependences of the heat transfer and temperature inhomogeneity on Reynolds numbers was explained qualitatively. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 126–133, May–June, 1998.     

Stability of plane-parallel convective fluid flow in a horizontal layer relative to spatial perturbations

The stability of stationary plane-parallel convective flow between horizontal planes along which a constant temperature gradient is given, is investigated relative to spatial perturbations. It is shown that the flow crisis is caused by spiral perturbations in a broad range of Prandtl number values (P > 0.24). Spiral perturbations are developed in unstably stratified fluid layers adjoining the upper and lower layer boundaries, and are of Rayleigh nature.     

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.     

Natural convection of non-Newtonian fluids in a horizontal porous layer

Heat and Mass Transfer - The buoyancy-induced flows of non-Newtonian fluids in a horizontal fluid saturated porous layer is studied analytically and numerically using the power-law model to...     

Free-forced convection from a heated longitudinal horizontal cylinder embedded in a porous medium

The flow and heat transfer from a heated semi-infinite horizontal circular cylinder which is moving with a constant speed into a porous medium is considered. It is assumed that the Grashof and Reynolds numbers are large so that the governing equations are the three dimensional boundary-layer equations. A numerical procedure for solving these equations is described and the asymptotic solutions which are valid both near and distant from the leading edge of the cylinder are presented. The range of validity of these asymptotic solutions is discussed and the results are compared in detail with the full numerical solution. The problem is of practical importance, for example in the drilling of pipes into a geothermal reservoir.

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Freie erzwungene Konvektion von einem beheizten schlanken horizontalen Zylinder, eingebettet in ein poröses Medium

Zusammenfassung Es wird die Strömung und der Wärmeübergang an einem beheizten, halbunendlichen horizontalen Kreiszylinder betrachtet, der mit konstanter Geschwindigkeit sich in ein poröses Medium bewegt. Dabei wird angenommen, daß die Grashof- und Reynolds-Zahlen groß sind, so daß die Bestimmungsgleichungen von den dreidimensionalen Grenzschichtgleichungen gebildet werden. Es wird ein numerisches Verfahren zur Lösung dieser Gleichungen beschrieben und eine asymptotische Lösung präsentiert, die sowohl in der Nähe als auch in großem Abstand von dem vorderen Ende des Zylinders gültig ist. Der Gültigkeitsbereich dieser asymptotischen Lösungen wird diskutiert und die Ergebnisse werden im Detail mit vollständigen numerischen Lösungen verglichen. Das Problem ist z.B. beim Eindringen von Rohrleitungen in geothermische Reservoire von praktischer Wichtigkeit.

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Nomenclature a radius of cylinder - Gr Grashof number (=g(T_w-Ta/²) - g acceleration due to gravity - permeability in the porous medium - Nu local Nusselt number - total heat flux from cylinder - q _w heat flux from cylinder - r radial co ordinate - Ra Rayleigh number (=g (T_w - T_t8) a/ ) - Re Reynolds number (=U _t8 a/) - T temperature - u, v, w speeds inx, , r directions - x axial co ordinate - equivalent thermal diffusivity - thermal expansion coefficient - ratioGr/Re - similarity variable - dimensionless temperature (=(T- T)/(T _w- T) - kinematic viscosity - azimuthal co ordinate - w cylinder surface - free stream     

Motion of gas bubbles in a nonuniformly heated liquid

The thermocapillary drift of air bubbles in water was studied experimentally. The linear connection between the drift velocity and the temperature gradient predicted by theory was confirmed. The experimental and theoretical results are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–57, September–October, 1979.We thank M. T. Sharov for assistance in setting up the experiment.     

Convection in a horizontal fluid layer with specific-volume inversion

The effect of the position of the inversion point within the layer on the critical values of the Rayleigh number and the amplitudes of the rectangular-cell convective flows is numerically investigated. The monotonic instability of the mechanical equilibrium of the fluid with respect to small perturbations periodic along the layer is studied by the linearization method. The Lyapunov-Schmidt method is used to construct the secondary steady convective flows. The applicability of these methods in incompressible fluid stability problems was demonstrated in [8–10]. The calculations show that, starting from a certain value of the parameter , the branching is subcritical for any cell side ratio and a fixed wave vector modulus. For smaller values of the nature of the branching depends on the cell side ratio. This points to subcritical branching and hysteresis effects in those cases in which the periodicity of the perturbations is determined by external factors (corrugation of the boundary, spatially periodic temperature modulation, etc.). It is noted that the rectangular convection amplitude tends to zero when the cell side ratio tends to 3, the value at which hexagonal cellular convection is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1989.The author wishes to thank V. I. Yudovich for his interest and useful advice and the participants in the Rostov State University Computational Mathematics Department's Scientific Seminar for discussing the results.     

Steady laminar flow past a heated horizontal plate embedded in a saturated porous medium

In this paper we study the boundary layer equations for steady laminar flow past a heated horizontal plate embedded in a saturated porous medium by adopting the formulation of Chandrasekhara [3], Kolar and Sastri [7]. The velocity distribution and temperature distribution are determined by using the implicit Crank-Nicolson-Predictor-Corrector method of finite difference scheme [7] and [1]. With the help of a compute the distributions are estimated at both (i+1/2)th and (i+1)th levels and they are presented in tabular form. The curves for these distributions are plotted. We calculate the shear stress and skin friction at the wall and observe that the skin-friction directly depends upon the dimensions of the plate and inversely depends upon the Reynolds numberRe. The heat flux and the Nusselt number are evaluated. Further we observe that the Nusselt number depends upon the length of the porous plate.

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Stetige laminare Strömung über eine in einem gesättigten porösen Medium eingebettete horizontale Platte

Zusammenfassung Diese Untersuchung befaßt sich mit den Grenzschichtgleichungen für eine stetige laminare Strömung über eine beheizte Platte, die in ein gesättigtes poröses Medium eingebettet ist, mittels des Formalismus von Chandrasekhara [3], Kolar und Sastri [7]. Die Geschwindigkeits- und Temperaturverteilung wurden unter Benutzung der impliziten Crank-Nicolson-Korrekturmethode des Finiten-Elemente-Schemas bestimmt. Die (i+1/2). und die (i+1). Ebene der Verteilungen wurden mit Computer-Hilfe berechnet und in Tabellenform dargestellt. Die Graphen der Temperatur- und Geschwindigkeitsverteilung wurden ausgeplottet. Die Schubspannungen und die Oberflächenreibung an der Wand wurden berechnet und es konnte festgestellt werden, daß die Oberflächenreibung direkt von der Größe der Platte abhängt und umgekehrt proportional der Reynolds-ZahlRe ist. Der Wärmestrom und die Nusselt-Zahl wurden bestimmt. Weiterhin konnte festgestellt werden, daß die Nusselt-Zahl von der Länge der porösen Platte abhängt.

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Nomenclature C _p specific heat of the convective fluid - D skin friction - k permeability of the porous medium - k _f thermal conductivity of the fluid - k _m the coefficient of thermal conductivity of the porous medium - k _s the conductivity of the solid matrix - N(x) Nusselt number - q(x) specific heat flux - Re local Reynolds number - T temperature - T ₀ temperature of the free stream - T _w temperature of the plate - u velocity in thex-direction - u ₀ velocity of the free stream - V velocity iny-direction - x coordinate axis along the plate - y coordinate axis normal to the plate Greek symbols thermal diffusivity - thickness of the velocity boundary layers in thex direction - thickness of the velocity boundary layer in they-direction - the porosity of the medium - dimensionless variable - kinematic viscosity of the fluid - density of the fluid - shear stress