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1.
Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out. 相似文献
2.
M. S. Malashetty Ioan Pop Rajashekhar Heera 《Continuum Mechanics and Thermodynamics》2009,21(4):317-339
Double diffusive convection in a fluid-saturated rotating porous layer is studied when the fluid and solid phases are not
in local thermal equilibrium, using both linear and nonlinear stability analyses. The Brinkman model that includes the Coriolis
term is employed as the momentum equation. A two-field model that represents the fluid and solid phase temperature fields
separately is used for the energy equation. The onset criterion for stationary, oscillatory, and finite amplitude convection
is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability
of the system. There is a competition between the processes of thermal diffusion, solute diffusion, and rotation that causes
the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute
Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number, and Taylor number on
the stability of the system is investigated. The nonlinear theory based on the truncated representation of Fourier series
method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium
on heat and mass transfer is also brought out. Some of the convection systems previously reported in the literature is shown
to be special cases of the system presented in this study. 相似文献
3.
The triple diffusive convection in an Oldroyd-B fluid-saturated porous layer is investigated by performing linear and weakly nonlinear stability analyses. The condition for the onset of stationary and oscillatory is derived analytically. Contrary to the observed phenomenon in Newtonian fluids, the presence of viscoelasticity of the fluid is to degenerate the quasiperiodic bifurcation from the steady quiescent state. Under certain conditions, it is found that disconnected closed convex oscillatory neutral curves occur, indicating the requirement of three critical values of the thermal Darcy-Rayleigh number to identify the linear instability criteria instead of the usual single value, which is a novel result enunciated from the present study for an Oldroyd-B fluid saturating a porous medium. The similarities and differences of linear instability characteristics of Oldroyd-B, Maxwell, and Newtonian fluids are also highlighted. The stability of oscillatory finite amplitude convection is discussed by deriving a cubic Landau equation, and the convective heat and mass transfer are analyzed for different values of physical parameters. 相似文献
4.
Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and
cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered
to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis
terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory
convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only
two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude
equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the
effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady
streamlines, isotherms, and isohalines for various parameters. 相似文献
5.
The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out. 相似文献
6.
Natalia Strong 《Journal of Mathematical Fluid Mechanics》2008,10(4):488-502
The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal
layer of fluid saturating a porous medium. A constant temperature distribution is assigned on the rigid boundaries, so that
there exists a vertical temperature gradient. The mathematical model is described by equations of filtration convection in
the Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed
using Floquet theory. Employment of the method of continued fractions allows derivation of the dispersion equation for the
Floquet exponent σ in an explicit form. The neutral curves of the Rayleigh number Ra versus horizontal wave number α for the
synchronous and subharmonic resonant modes are constructed for different values of frequency Ω and amplitude A of vibration. Asymptotic formulas for these curves are derived for large values of Ω using the method of averaging, and,
for small values of Ω, using the WKB method. It is shown that, at some finite frequencies of vibration, there exist regions
of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters
of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility
to convection.
相似文献
7.
The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. 相似文献
8.
B. S. Bhadauria 《Transport in Porous Media》2012,92(2):299-320
In the present study, double-diffusive convection in an anisotropic porous layer with an internal heat source, heated and
salted from below, has been investigated. The generalized Darcy model is employed for the momentum equation. The fluid and
solid phases are considered to be in equilibrium. Linear and nonlinear stability analyses have been performed. For linear
theory normal mode technique has been used, while nonlinear analysis is based on a minimal representation of truncated Fourier
series. Heat and mass transfers across the porous layer have been obtained in terms of Nusselt number Nu and Sherwood number Sh, respectively. The effects of internal Rayleigh number, anisotropy parameters, concentration Rayleigh number, and Vadasz
number on stationary, oscillatory, and weak nonlinear convection are shown graphically. The transient behaviors of Nusselt
number and Sherwood number have been investigated by solving the finite amplitude equations using a numerical method. Streamlines,
isotherms, and isohalines are drawn for both steady and unsteady (time-dependent) cases. The results obtained, during the
above analyses, have been presented graphically, and the effects of various parameters on heat and mass transfers have been
discussed. 相似文献
9.
Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer
I. S. Shivakumara S. Sureshkumar N. Devaraju 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(4):513-530
Both linear and weakly nonlinear stability analyses are performed to study thermal convection in a rotating couple-stress
fluid-saturated rigid porous layer. In the case of linear stability analysis, conditions for the occurrence of possible bifurcations are obtained.
It is shown that Hopf bifurcation is possible due to Coriolis force, and it occurs at a lower value of the Rayleigh number
at which the simple bifurcation occurs. In contrast to the nonrotating case, it is found that the couple-stress parameter
plays a dual role in deciding the stability characteristics of the system, depending on the strength of rotation. Nonlinear
stability analysis is carried out by constructing a set of coupled nonlinear ordinary differential equations using truncated
representation of Fourier series. Sub-critical finite amplitude steady motions occur depending on the choice of physical parameters
but at higher rotation rates oscillatory convection is found to be the preferred mode of instability. Besides, the stability
of steady bifurcating equilibrium solution is discussed using modified perturbation theory. Heat transfer is calculated in
terms of Nusselt number. Also, the transient behavior of the Nusselt number is investigated by solving the nonlinear differential
equations numerically using the Runge–Kutta–Gill method. It is noted that increase in the value of Taylor number and the couple-stress
parameter is to dampen the oscillations of Nusselt number and thereby to decrease the heat transfer. 相似文献
10.
The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ-QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls (LRs) only depend on the depth ratio. With the existence of the shear flow, the effects of the depth ratio, the Reynolds number, the Prandtl number, the stress relaxation, and strain retardation times on the transverse rolls (TRs) are also studied. Additionally, the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system. In contrast to the case for Newtonian fluids, the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases. 相似文献
11.
《European Journal of Mechanics - B/Fluids》2000,19(2):213-227
Two- and three-dimensional convection flows in a horizontal layer of a low Prandtl number fluid heated from below and rotating about a vertical axis are studied numerically with a Galerkin method. Solutions for subcritical steady finite amplitude convection and convection in the form of standing oscillations are obtained. Parameter regimes that appear to be attainable in laboratory experiments have been emphasized. The stability of subcritical two-dimensional steady convection has been investigated and three-dimensional chaotic states of convection have been found. 相似文献
12.
The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances.
If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value Rc steady-state motions of the liquid are not possible. With R>Rc a steady convection arises, whose amplitude near the threshold is small and proportional to (R–Rc)1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to Rc, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value Rc. The numerical solution permits following the development of the steady motion which arises with R>Rc in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method. 相似文献
13.
The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer
The double diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer, which is heated and
salted from below, is studied analytically. The modified Darcy equation that includes the time derivative term is used to
model the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes, and frequency
of oscillations are obtained analytically using linear theory. The effect of anisotropy parameter, solute Rayleigh number,
Lewis number, couple stress parameter, and Vadasz number on the stationary, oscillatory, and finite amplitude convection is
shown graphically. It is found that the thermal anisotropy parameter, couple stress parameter, and solute Rayleigh number
have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The mechanical anisotropy parameter
has destabilizing effect on stationary, oscillatory, and finite amplitude convection. The Lewis number has stabilizing effect
in the case of stationary and finite amplitude modes, with dual effect in the case of oscillatory convection. Vadasz number
advances the onset of oscillatory convection. The heat and mass transfer decrease with an increase in the values of couple
stress parameter, while both increase with an increase in the value of solute Rayleigh number and mechanical anisotropy parameter.
The thermal anisotropy parameter and Lewis number have contrasting effect on the heat mass transfer. 相似文献
14.
We investigate Rayleigh–Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the
fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and
anisotropic permeability) is used to describe the flow, whilst the two-equation model is used for the energy equation (for
the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the
onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed. 相似文献
15.
Rachid Khiri 《International Journal of Non》2004,39(4):593-604
The problem of finite-amplitude thermal convection in a horizontal layer of a low Prandtl number heated from below and rotating about a vertical axis is studied. Linear stability and weak non-linear theories are used to investigate analytically the Coriolis effect on gravity-driven convection. The non-linear steady problem is solved by perturbation techniques, and the preferred mode of convection is determined by a stability analysis. Finite-amplitude results, obtained by using a weak amplitude, correspond to both stationary and oscillatory convections. These amplitude equations permit to identify from the post-transient conditions that the fluid is subject to Pitchfork bifurcation in the stationary convection and Hopf bifurcation in the oscillatory convection. Thereafter, in the small perturbations hypothesis, an amplitude solution is evaluated and drawn in time and space scales. 相似文献
16.
The problem of the convection of a weakly compressible fluid is considered. In the free convection equations a heat source function is taken into account. The stability of the equilibrium state of a horizontal layer relative to small perturbations is studied using the linearization method. On the basis of numerical calculations it is shown that the mechanical equilibrium state of the fluid is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found. In the calculations values of the physical parameters typical of Lake Baikal were used. 相似文献
17.
A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail. 相似文献
18.
The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases. 相似文献
19.
The effect of thermal modulation on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic
fluid is investigated by a linear stability analysis. Darcy’s law with viscoelastic correction is used to describe the fluid
motion. The perturbation method is used to find the critical Rayleigh number and the corresponding wavenumber for small amplitude
thermal modulation. The stability of the system characterized by a correction Rayleigh number is calculated as a function
of the thermal and mechanical anisotropy parameters, the viscoelastic parameters and the frequency of modulation. It is found
that the onset of convection can be delayed or advanced by the factors represented by these parameters. The results of the
problem have possible implications in mantle convection. 相似文献
20.
B. Straughan 《Journal of Mathematical Fluid Mechanics》2014,16(4):727-736
We show that for many classes of convection problem involving a porous layer, or layers, interleaved with finite but non-deformable solid layers, the global nonlinear stability threshold is exactly the same as the linear instability one. The layer(s) of porous material may be of Darcy type, Brinkman type, possess an anisotropic permeability, or even be such that they are of local thermal non-equilibrium type where the fluid and solid matrix constituting the porous material may have different temperatures. The key to the global stability result lies in proving the linear operator attached to the convection problem is a symmetric operator while the nonlinear terms must satisfy appropriate conditions. 相似文献