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PHYSICAL CONFIGURATION FOR A RAYLEIGH – B Ѐ NARD SITUATION
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The objective of the paper is to study the Rayleigh-Bѐnard convection in second order fluid by replacing the classical Fourier heat law by non-classical Maxwell-Cattaneo law using Galerkin technique. The eigen value of the problem is obtained using the general boundary conditions on velocity and third type of boundary conditions on temperature. A l...
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... This is due to the absence of Cattaneo number. From Figures 6 and 7, we witness the dual nature of the Prandtl number Pr depending on the Cattaneo number G. If the Cattaneo number G is above the threshold value, then on increasing Pr there is a decrease in osc c R as noticed in the work of Nagouda and Pranesh [25] and if the Cattaneo number G is below the threshold value, then on increasing Pr there is an increase in osc c R as noticed in the work of Swamy et al. [18]. In Fig. 8 In Fig. 9 critical Da − will decrease the porous medium permeability and hence the convective instability is impeded. ...
The classical stability analysis is used to examine the combined effect of viscoelasticity and the second sound on the onset of porous medium ferroconvection. The fluid and solid matrix are assumed to be in local thermal equilibrium. Considering the boundary conditions appropriate for an analytical approach, the critical values pertaining to both stationary and oscillatory instabilities are obtained by means of the normal mode analysis. It is observed that the oscillatory mode of instability is preferred to the stationary mode of instability. It is shown that the oscillatory porous medium ferroconvection is advanced through the magnetic forces, nonlinearity in magnetization, stress relaxation due to viscoelasticity, and the second sound. On the other hand, it is observed that the presence of strain retardation and porous medium delays the onset of oscillatory porous medium ferroconvection. The dual nature of the Prandtl number on the Rayleigh number with respect to the Cattaneo number is also delineated. The effect of various parameters on the size of the convection cell and the frequency of oscillations is also discussed. This problem may have possible implications for technological applications wherein viscoelastic magnetic fluids are involved.
... The readers may find more details on the viscoelastic fluids in the papers of Larson [6,7] and the textbook of Phan-Thien [8]. For thermal convective instabilities problems the constitutive equations for the viscoelastic Maxwell [9,10], Jeffreys [11,12] and second order [13,14] fluids has some popularity. Several investigations has used these models to study the hydrodynamics of viscoelastic fluids heated from below. ...
... Sekhar and Jaylatha [21] have studied the Rayleigh-Bènard convection in Second order fluid. Smita and Pranesh [25] investigated on the Rayleigh-Bènard convection in second order fluid with non-classical Maxwell-Cattaneo law. The eigenvalues of the problem were found using general boundary conditions on the velocity and temperature under Galerkin technique. ...
A Linear stability analysis is performed to study the effect of non-uniform temperature
gradient on the onset of Rayleigh-Bénard convection in a second order fluid with Maxwell-Cattaneo law. The eigenvalue is obtained for general boundary condition on velocity and
temperature using Galerkin technique. The influence of various parameters on the onset of
convection has been analyzed. One linear and five non-linear temperature profiles are
considered and their comparative influence on the onset of convection is discussed. It is
observed that the Cattaneo number and Prandtl number advances the onset of convection but
the Viscoelastic parameter delays the onset of convection.
... A well known consequence of this law is that heat perturbations propagate with an infinite velocity. This drawback of the classical law motivated Maxwell [1], Cattaneo [2], Lebon and Cloot [3], Dauby et al. [4], Straughan [5], Siddheshwar [6], Pranesh [7], Pranesh and Kiran [8,9,10] and Pranesh and Smita [11] to adopt a non-classical Maxwell-Cattaneo heat flux law in studying Rayleigh-Bénard/Marangoni convection to get rid of this unphysical results. This Maxwell-Cattaneo heat flux law equation contains an extra inertial term with respect to the Fourier law. ...
The effect of imposed time-periodic boundary
temperature (ITBT, also called temperature modulation) and
magnetic field at the onset of convection is investigated by
making a linear analysis. The classical Fourier heat law is
replaced by the non-classical Maxwell-Cattaneo law. The
classical approach predicts an infinite speed for the propagation
of heat. The adopted non-classical theory involves wave type of
heat transport and does not suffer from the physically
unacceptable drawback of infinite heat propagation speed. The
Venezian approach is adopted in arriving at the critical Rayleigh
number, correction Rayleigh number and wave number for
small amplitude of ITBT. Three cases of oscillating temperature
field are examined: (a) symmetric, so that the wall temperatures
are modulated in phase, (b) asymmetric, corresponding to out-of
phase modulation and (c) only the lower wall is modulated. The
temperature modulation is shown to give rise to sub-critical
motion. The shift in the critical Rayleigh number is calculated as
a function of frequency and it is found that it is possible to
advance or delay the onset of convection by time modulation of
the wall temperatures. It is shown that the system is most stable
when the boundary temperatures are modulated out-of-phase. It
is also found that the results are noteworthy at short times and
the critical eigenvalues are less than the classical ones.
... A well known consequence of classical Fourier heat conduction law is that heat perturbations propagate with an infinite velocity.This drawback of the classical law motivated Maxwell [22], Cattaneo [23], Lindsay and Stranghan [24], Straughan and Franchi [25], Siddheshwar [26], Pranesh [27], Pranesh and Kiran [28,29] and Pranesh and Smita [30] to adopt a non-classical heat flux Maxwell-Cattaneo law in studyingRayleigh-Bénard/Marangoni convection to get rid of this unphysical results. This Maxwell-Cattaneo equation contains an extra inertial term with respect to the Fourier law: ...
The effect of suction-injection-combination (SIC) on the onset of Rayleigh-Bénard magnetoconvection in a micropolar fluid with
Maxwell-Cattaneo law is studied using the Rayleigh-Ritz technique. The eigenvalue is obtained for free-free, rigid-rigid and
rigid-free velocity boundary combinations with isothermal and adiabatic on the spin-vanishing boundaries. A linear stability
analysis is performed. The influence of various micropolar fluid parameters on the onset of convection has been analyzed.It is
found that by adjusting SIC it is possible to control the convection in micropolar fluid with Maxwell–Cattaneo law. It is also
observed that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or antigravity.
The classical approach predicts an infinite speed for the propagation of heat. The present non-classical theory involves a
wave type heat transport and does not suffer from the physically unacceptable drawback of infinite heat propagation speed.It is
found that the results are noteworthy at short times and the critical eigenvalues are less than the classical ones.
... Smita and Pranesh [23] studied the problem of the onset of Rayleigh-Bénard convection in a second order Colemann-Noll fluid by replacing the classical Fourier heat flux law with non-classical Maxwell-Cattaneo law. The Galerkin method is employed to determine the critical values. ...
In this chapter we focus on the equation that expresses the first law of thermodynamics in a porous medium. We start with a simple situation in which the medium is isotropic and where radiative effects, viscous dissipation, and the work done by pressure changes are negligible. Very shortly we shall assume that there is local thermal equilibrium so that T
s = T
f = T, where T
s and T
f are the temperatures of the solid and fluid phases, respectively. Here we also assume that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. More complex situations will be considered in Sect. 6. 5. The fundamentals of heat transfer in porous media also are presented in Bejan et al. (2004) and Bejan (2004a).
