Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

The Onset of Convection in a Strongly Heterogeneous Porous Medium with Transient Temperature Profile

The effect of heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a nonuniform basic temperature gradient resulting from transient heating, is studied analytically using linear stability theory for the case of strong heterogeneity. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained. The case of a strongly nonlinear temperature gradient is studied with the help of a computer package.     

Effects of non-uniform heating on a variable viscosity Rayleigh�CB��nard problem

This study presents a natural convection problem with a temperature-dependent viscosity fluid, driven by buoyancy and influenced by horizontal temperature gradients. A numerical linear stability analysis of the stationary solutions is studied. The horizontal temperature gradients tend to localize motion near the warmer zones and favour pattern formation in the direction perpendicular to the gradient. In fact, the problem is almost 2D in the uniform heating case, but becomes totally 3D in the non-uniform heating case.     

The onset of convection in a horizontal nanofluid layer of finite depth

This paper presents a linear stability analysis for the onset of natural convection in a horizontal nanofluid layer. The employed model incorporates the effects of Brownian motion and thermophoresis. Both monotonic and oscillatory convection for free–free, rigid–rigid, and rigid–free boundaries are investigated. The oscillatory instability is possible when nanoparticles concentrate near the bottom of the layer, so that the density gradient caused by such a bottom-heavy nanoparticle distribution competes with the density variation caused by heating from the bottom. It is established that the instability is almost purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles. It is independent of the contributions of Brownian motion and thermophoresis to the thermal energy equation. Rather, the Brownian motion and thermophoresis enter to produce their effects directly into the equation expressing the conservation of nanoparticles so that the temperature and the particle density are coupled in a particular way, and that results in the thermal and concentration buoyancy effects being coupled in the same way.     

Fluid convection in a rotating porous layer under modulated temperature on the boundaries

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.     

The Onset of Convection in a Heterogeneous Porous Medium with Transient Temperature Profile

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from transient heating or otherwise, is studied analytically using linear stability theory. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained.     

Equilibrium characteristics of the secondary convection in a vertical fluid layer between two flat plates 1

The nonlinear stability of the natural convection in a vertical fluid layer between two flat plates with different temperatures is investigated by a direct method to find the equilibrium states of the secondary convection. We confine ourselves to two-dimensional flows and assume that the aspect ratio of the fluid layer is very large. Since the Prantl number is assumed to be very small, the buoyancy effect caused by temperature disturbances is negligible. As a result we obtained a neutral surface of the energy of the fundamental mode of the secondary convection. It is concluded that there is no finite amplitude instability below the critical Grashof number derived from linear stability theory, and that both the unstable equilibrium solution (threshold amplitude solution) and the stable equilibrium solution (finite amplitude solution) are found outside the neutral curve of the linear stability. Our results are almost consistent with those of Nagata and Busse (1983), but are more accurate and more thorough.     

Onset of Convection in a Porous Rectangular Channel with External Heat Transfer to Upper and Lower Fluid Environments

The conditions for the onset of convection in a horizontal rectangular channel filled with a fluid saturated porous medium are studied. The vertical sidewalls are assumed to be impermeable and adiabatic. The horizontal upper and lower boundary walls are considered as impermeable and subject to external heat transfer, modelled through a third-kind boundary condition on the temperature field. The external fluid environments above and below the channel, kept at different temperatures, provide the heating-from-below mechanism which may lead to the onset of the thermal instability in the porous medium. The linear response of the fluid saturated porous channel, in a basic motionless state, is tested with respect to three-dimensional normal mode disturbances of the temperature field and of the pressure field. The linearised disturbance equations are solved analytically leading to an implicit-form expression of the neutral stability condition, formulated as a functional relationship between the Darcy?CRayleigh number and the continuous longitudinal wave number of the normal modes, for any assigned aspect ratio of the cross-section and for any given Biot number. The analysis of the neutral stability is carried out. The analysis is extended to the case of a channel with a finite length in the longitudinal direction, and with adiabatic and impermeable capped ends.     

多孔介质内黏弹性流体的热对流稳定性研究

基于修正的Darcy模型, 介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展. 通过线性稳定性理论, 分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, Darcy-Brinkman-Oldroyd以及Darcy-Brinkman -Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响. 利用弱非线性分析方法, 揭示失稳临界点附近热对流流动的分叉情况, 以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式. 采用数值模拟方法, 研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果:(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的, 而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞, 最后发展为混沌状态.      

On the delicate state of instability of a vertical riser transporting fluid

The variation in the dynamic characteristics of a flexible riser as the riser transitions from a vertical riser to a catenary-type riser is investigated. It is well known that the straight configuration of a flexible vertical riser conveying fluid destablizes in a divergence-type instability once the velocity of the transporting fluid exceeds a critical speed. As expected, the instability persists if a slight horizontal offset is introduced at the hang-off point. However, as demonstrated in this paper, if a finite horizontal offset is introduced then the instability vanishes and the resulting static configuration of the catenary-type riser is stable regardless of the transport speed of the fluid.     

On the stability of the natural convection flow in a square cavity heated from the side

The stability of the steady laminar natural-convection flow of air (Prandtl number 0.71) and water (Prandtl number 7.0) in a square cavity is calculated by numerically solving the unsteady, two-dimensional Navier-Stokes equations. The cavity has a hot and cold vertical wall and either conducting or adiabatic horizontal walls. The flow looses its stability at a lower Rayleigh number in the case of conducting horizontal walls than in the case of adiabatic horizontal walls. The flow of water is more stable than the flow of air. Directly above the critical Rayleigh number the unsteady flow shows a single-frequency oscillation. Air in the case of adiabatic horizontal walls is an exception and shows two frequencies. The instabilities in the cavity seem to be related to well-known elementary instability mechanisms. In the case of conducting and adiabatic horizontal walls the instability seems to be related to a Rayleigh/Bénard and a Tollmien-Schlichting instability respectively. The second instability for air in the case of adiabatic horizontal walls seems to be related to an instability after a hydraulic jump.     

Instability and Viscous Dissipation in the Horizontal Brinkman Flow through a Porous Medium

The convective instability activated by the sole effect of viscous dissipation in a fluid saturated porous layer is studied. The basic parallel flow in a highly permeable porous medium is analysed by considering the viscous heating contribution in the local energy balance, by assuming a thermally insulated lower boundary and an isothermal upper boundary. The Brinkman model of momentum transfer is adopted. Arbitrarily oriented oblique roll disturbances are considered in the linear stability analysis. Among them, the longitudinal rolls, having axis parallel to the basic flow direction, are shown to be the preferred mode of instability. Some considerations on the reliability of the Brinkman model, when the viscous dissipation contribution is not negligible and when the flow conditions are close to the limiting case of a clear fluid, are finally expressed.     

Influence of spatial modulation of the temperature field on the stability of two-dimensional steady flow in a horizontal layer of fluid

A study is made of the stability of the steady periodic regime that arises in a horizontal layer of fluid in the presence of spatial modulation of of the temperature on the solid bottom boundary. The upper free boundary of the layer is in contact with the atmosphere. The fundamental resonance values of the wave number of the modulation are found; there are five of them. If the temperature of the lower boundary of the layer is constant, and the temperature gradient is not too large, the fluid is in equilibrium. When the temperature gradient passes through the critical value, the equilibrium ceases to be stable, and steady convection develops in the fluid [1]. In the presence of spatial modulation of the temperature on the lower boundary of the layer the fluid cannot be in equilibrium, and a spatially periodic steady regime is established in it. The aim of the present paper is to find the critical values of the temperature gradient at which this fundamental steady regime becomes unstable and a secondary steady regime develops in the fluid. An analogous problem for the case when both boundaries of the layer are free surfaces and without allowance for the influence of the atmosphere has been solved by Vozovoi and Nepomnyashchii [2].     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions

The stability of thermocapillary flow developed in a slowly rotating fluid layer under microgravity conditions is investigated. Both boundaries of the layer are free and assumed to be plane. The tangential thermocapillary Marangoni force exerts on the boundaries, where heat transfer takes place in accordance with the Newton law, the temperature of the medium in the neighborhood of the boundaries being a linear function of the coordinates. The axis of rotation is perpendicular to the liquid layer, rotation is weak so that the centrifugal force can be neglected. Being the solution of the Navier-Stokes equations, the thermocapillary flow in question can be described analytically. The neutral curves which describe the wavenumber dependence of the critical Marangoni number for various Taylor numbers and various directions of the horizontal temperature gradient on the layer boundaries are obtained within the framework of the linear stability theory. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.     

Effect of rotation on ferromagnetic porous convection with a thermal non-equilibrium model

The effect of rotation on the onset of thermal convection in a horizontal layer of ferrofluid saturated Brinkman porous medium is investigated in the presence of a uniform vertical magnetic field using a local thermal non-equilibrium (LTNE) model. A two-field model for temperature representing the solid and fluid phases separately is used for energy equation. The condition for the occurrence of stationary and oscillatory convection is obtained analytically. The stability of the system has been analyzed when the magnetic and buoyancy forces are acting together as well as in isolation and the similarities as well as differences between the two are highlighted. In contrast to the non-rotating case, it is shown that decrease in the Darcy number Da and an increase in the ratio of effective viscosity to fluid viscosity Λ is to hasten the onset of stationary convection at high rotation rates and a coupling between these two parameters is identified in destabilizing the system. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. Besides, the influence of magnetic parameters and also parameters representing LTNE on the stability of the system is discussed and the veracity of LTNE model over the LTE model is also analyzed.     

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.     

Thermal Instability in a Horizontal Porous Channel with Horizontal Through Flow and Symmetric Wall Heat Fluxes

The linear thermoconvective instability of the basic parallel flow in a plane and horizontal porous channel is investigated. The boundary walls are assumed to be impermeable and subject to symmetric and uniform heat fluxes. The wall heat fluxes produce either a net heating or a net cooling of the fluid saturated porous medium. A horizontal mass flow rate is externally impressed leading to a stationary basic state with a temperature gradient inclined to the vertical. A region of possibly unstable thermal stratification exists either in the lower half-channel (boundary heating), or in the upper half-channel (boundary cooling). The convective instability of the basic flow is governed by the Rayleigh number and by the Péclet number. In the case of boundary heating, the thermal instability arises when the Rayleigh number exceeds its critical value, that depends on the Péclet number. The change of the critical Rayleigh number as a function of the Péclet number is determined numerically for arbitrary normal modes oblique to the basic flow direction. The most dangerous modes are the longitudinal rolls, with a wave vector perpendicular to the basic velocity. There exists a minimum value of the Péclet number, 19.1971, below which no linear instability is detected.     

Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.