Analytical investigation of steady convection of a near-critical Van der Waals gas in a porous thin annular cylinder embedded in a heat-conducting medium

Natural convection of a near-critical Van der Waals gas in a horizontal porous thin annular cylinder embedded in a heat-conducting space with a temperature gradient given at infinity is considered. The two-dimensional problem in a plane orthogonal to the cylinder axis is investigated using analytical methods, accurate to quantities of the order of the annulus thickness. A criterion of the onset of an annulus-section-average flow is obtained. The critical Rayleigh-Darcy number is determined for the general case in which the physical properties of the gas can considerably vary throughout the medium. Several limiting cases are considered and the ranges of their applicability are discussed. It is shown that, as the thermodynamic critical point is approached, the asymptotics of the critical Rayleigh-Darcy number depend on the relation between non-Boussinesq parameters, such as hydrostatic compressibility criteria, the temperature difference, and the nearness to the critical point. In the case of steady convection, an analytical solution is also derived in the case in which the above-mentioned stability threshold is exceeded and the physical properties of the gas vary throughout space only slightly. A comparison with the case of a perfect gas is made.     

Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Investigation of filtration convection in a near-critical Van-der-Waals gas for substantial temperature differences

The qualitative mechanisms of stationary filtration convection in a near-critical Van-der-Waals gas with substantial temperature variations are investigated. The object of the study is a thin porous ring in a vertical heat-conducting plane, saturated with the gas. An adequate mathematical model, based on the averaging of the two-dimensional equations over the ring thickness, is proposed. The dependence of natural-convection characteristics on the non-Boussinesq parameters (compressibility criteria, temperature difference, and proximity to the critical point) is found. The limits of applicability of the previously known regularities related to the conditions of onset and the properties of stationary convection are discussed.     

Free-convection Flow of a Binary Mixture in a Thin Porous Ring

The problem of natural convection of a binary mixture in a thin porous ring is considered. In the simplified formulation steady-state solutions of the problem are obtained. The stability of these solutions is investigated and a stability map is plotted in the plane of the Rayleigh numbers calculated from the temperature and concentration. It is shown that an auto-oscillation convection regime is established in the ring under certain conditions. It is also found that there is a region of variation of the seepage and diffusion-seepage Rayleigh numbers in which three steady-state solutions are stable.     

Influence of anisotropy of the permeability on the convection and heat transfer in a porous annular layer

Investigations into the convective transport of heat in porous materials are of interest for many applications in connection with the problem of increasing the efficiency of thermal insulation. In [1–5], convection in Isotropic porous media was considered. However, in many cases porous materials have an essential anisotropy of their permeability. Convective heat transfer has been inadequately studied for this case. In [6], the linearized equations were used to study the convection between infinite horizontal planes with a filling of an anisotropic material; the value of the critical Rayleigh number was found, and this agreed satisfactorily with experimental data. In the present paper, we investigate numerically convection between two infinite coaxial cylinders with an anisotropic porous filling, using the equations of convection in the Darcy—Boussinesq approximation [1–3]. The permeability tensor in the annular region is constructed from its principal values, which can be found experimentally. A method of calculation is developed and a parametric study made of the structure of the flow and of the local and averaged characteristics of the heat transfer, which are of interest for the design of thermal insulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–64, January–February, 1980.     

Convection in an Obliquely Heated Thin Porous Elliptic Ring

The problem of convection in a thin porous elliptic ring located in an impermeable rock mass is considered. The geothermal gradient is assumed to deviate by a certain angle from the vertical axis of the ellipse. An analytic solution of the problem is obtained in the hydraulic approximation. The stability of the solutions obtained is investigated for a circular ring. The profile of the thermal anomaly in the outer mass due to convection in the contour is given.     

Oscillating Convection of Nanofluid in Porous Medium

In this paper, oscillatory convection in a horizontal layer of nanofluid in porous medium is studied. For porous medium, Darcy model is applied. A linear stability theory and normal mode analysis method is used to find the solution confined between two free boundaries. The onset criterion for oscillatory convection is derived analytically and graphically. Regimes of oscillatory and non-oscillatory convection for various parameters are derived. The effects of Lewis number, concentration Rayleigh number, Prandtl?CDarcy number (Vadasz Number) and modified diffusivity ratio on the oscillatory convection are investigated graphically. We examine the validity of ??PES?? and concluded that ??PES?? is not valid for the problem.     

Gas-lubricated sector thrust bearing with maximum load capacity

The problem of the optimization of the microgeometry of a gas-lubricated sector thrust bearing with respect to the maximum lift criterion is considered. The gas lubrication Reynolds equation is used for determining the pressure in the lubricating film. The case of low compressibility numbers is studied. The optimum profile is qualitatively determined on the basis of an analysis of the system of necessary conditions for an extremum. The results are presented for various values of the problem parameters. A comparison is made with the data of a number of gas thrust bearing optimization studies.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–42, November–December, 1990.     

Periodic variational problem associated with the linear Reynolds equation

The periodic problem of the optimization of the microgeometry of a gas-lubricated bearing with respect to the maximum lift criterion is considered. The pressure field in the gas film is determined by means of the gas-lubrication Reynolds equation. The case of low compressibility numbers is studied. The system of necessary conditions for an extremum of the variational problem is subjected to an analysis, on the basis of which the existence of an optimum profile with an infinite number of discontinuities is established, and the variational problem is expanded. Finally, the limiting periodic optimum profiles are found.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 3–9, March–April, 1992.     

A Nonlinear Stability Analysis of Convection in a Porous Vertical Channel Including Local Thermal Nonequilibrium

The problem is considered of thermal convection in a saturated porous medium contained in an infinite vertical channel with differentially heated sidewalls. The theory employed allows for different solid and fluid temperatures in the matrix. Nonlinear energy stability theory is used to derive a Rayleigh number threshold below which convection will not occur no matter how large the initial data. A generalized nonlinear analysis is also given which shows convection cannot occur for any Rayleigh number provided the initial data is suitably restricted.     

Hydrothermal convection in a thin porous ring

Natural convection of the fluid in a thin porous ring on whose boundaries steady temperature distributions are maintained is considered. For this problem on the basis of the two-dimensional equations an integrodifferential equation is obtained in the zeroth approximation in terms of a small parameter, namely the relative thickness of convection. A parametric numerical investigation of the flow and temperature fields is carried out.Makhachkala, Kaspiisk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 4–8, November–December, 1994.     

A non-linear analysis of non-isothermal wave propagation in linear-elastic fluid-saturated porous media

The non-isothermal dynamic behaviour of saturated porous media is analysed numerically employing the finite element method and taking energy convection due to large pore fluid displacements into account. A different pore fluid reference temperature is introduced in order to allow properly for heat convection: this concept is usually neglected in the literature and is discussed and analysed herein. The numerical procedure is validated in a simple problem of hot fluid injection in a steady seepage flow and by comparing the numerical results, neglecting energy convection, with those obtained with a novel solution of the linearised equations, presented herein, which is based on the transfer functions and Fourier transforms method. Finally, the effects of energy convection in wave propagation are analysed: in a pervious porous medium the flux of energy due to energy convection is much greater than the one due to heat conduction; in any case, wave propagation can be considered completely adiabatic even when energy convection is taken into account. Thus the validity of the results presented in the literature and based on the linearised theory is demonstrated.     

Coriolis Effect on the Linear Stability of Convection in a Porous Layer Placed Far Away from the Axis of Rotation

The linear stability theory is used to investigate analytically the Coriolis effect on centrifugally driven convection in a rotating porous layer. The problem corresponding to a layer placed far away from the axis of rotation was identified as a distinct case and therefore justifying special attention. The stability of the basic centrifugally driven convection is analysed. The marginal stability criterion is established as a characteristic centrifugal Rayleigh number in terms of the wavenumber and the Taylor number.     

Conjugate natural convection between horizontal concentric cylinders filled with a porous medium

A conjugate problem of natural convection between two horizontal concentric, isothermal cylinders filled with a fluid-saturated porous medium is studied. The flow field and energy equations are solved under the conditions of equality in temperature but with jump in heat flux at the fluid-solid interface. Numerical results by a finite-difference technique are presented for a large variation in the parameters entering the problem. A comparison of the solution with non-conjugate problem is also given.Das konjugierte Problem der freien Konvektion zwischen zwei horizontalen, konzentrischen, isothermen Zylindern, die mit einem fluid-gesättigten porösen Medium gefüllt sind, wurde hier untersucht.Das Strömungsfeld und die Energiegleichung sind unter den Bedingungen gelöst worden, daß die Temperatur gleich ist, aber ein Wärmeflußsprung am Übergang von der Flüssigkeits- zur Festphase stattfindet. Die numerischen Ergebnisse werden mit dem Finite-Differenzen-Verfahren für viele Parametervariationen dargestellt. Zudem wurde ein Lösungsvergleich mit einem nicht konjugierten Problem gegeben.     

Weakly Nonlinear Convection in a Porous Layer with Multiple Horizontal Partitions

We consider convection in a horizontally uniform fluid-saturated porous layer which is heated from below and which is split into a number of identical sublayers by impermeable and infinitesimally thin horizontal partitions. Rees and Genç (Int J Heat Mass Transfer 54:3081–3089, 2010) determined the onset criterion by means of a detailed analytical and numerical study of the corresponding dispersion relation and showed that this layered system behaves like the single-sublayer constant-heat-flux Darcy–Bénard problem when the number of sublayers becomes large. The aim of the present work is to use a weakly nonlinear analysis to determine whether the layered system also shares the property of the single-sublayer constant-heat-flux Darcy–Bénard problem by having square cells, as opposed to rolls, as the preferred planform for convection.     

An investigation of convection effects in complete and almost complete contact problems

The problem of a rigid, square-ended and an almost square-ended rigid punch sliding with both plane and anti-plane velocity components is studied. It is shown that, for a truly complete contact, if the contacting body is incompressible, convection effects are absent. Introducing either: (a) local rounding or (b) finite compressibility of the contacting body into the problem introduces convection, and hence an inconsistency into the bulk/pointwise application of the orthogonal friction law.     

Transient Free Convection Fluid Flow in Domains Partially Filled with Porous Media

The problem of transient free convection in domains partly filled with porous substrates is investigated analytically using Laplace transformation technique. Four configurations are considered which are subject to an isothermal heating boundary condition. The Brinkman-extended Darcy model is adopted to describe the hydrodynamics behavior of the porous domain.     

AN ALGEBRAIC MULTIGRID METHOD FOR COUPLED THERMO-HYDRO-MECHANICAL PROBLEMS

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Effect of Internal Heat Generation on the Onset of Marangoni Convection in a Fluid Layer Overlying a Layer of an Anisotropic Porous Medium

Linear stability analysis has been performed to investigate the effect of internal heat generation on the criterion for the onset of Marangoni convection in a two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid. The upper non-deformable free surface and the lower rigid surface are assumed to be insulated to temperature perturbations. The fluid flow in the porous layer is governed by the modified Darcy equation and the Beavers–Joseph empirical slip condition is employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly. Besides, analytical expression for the critical Marangoni number is also obtained by using regular perturbation technique with wave number as a perturbation parameter. The effect of internal heating in the porous layer alone exhibits more stabilizing effect on the system compared to its presence in both fluid and porous layers and the system is least stable if the internal heating is in fluid layer alone. It is found that an increase in the value of mechanical anisotropy parameter is to hasten the onset of Marangoni convection while an opposite trend is noticed with increasing thermal anisotropy parameter. Besides, the possibilities of controlling (suppress or augment) Marangoni convection is discussed in detail.     

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.