Abstract
The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed.
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Abbreviations
- A h :
-
Ratio of heat capacities
- a :
-
Horizontal wavenumber
- c :
-
Specific heat
- c p :
-
Specific heat at constant pressure
- Da :
-
Darcy number, k/d 2
- d :
-
Thickness of the fluid layer
- f :
-
Modulation temperature gradient
- \({\vec{g}}\) :
-
Gravitational acceleration
- l, m:
-
Wave numbers in the x- and y-directions
- M :
-
Modified thermal capacity ratio, \({A_{\rm h}/\varepsilon}\)
- p :
-
Pressure
- Pr :
-
Modified Prandtl number, \({\nu \varepsilon^{2}/\kappa}\)
- R :
-
Rayleigh number, αgΔTd 3/νκ
- \({\vec{q}}\) :
-
Velocity
- T :
-
Temperature
- t :
-
Time
- (x, y, z):
-
Space co-ordinates
- α :
-
Volumetric expansion coefficient
- \({\varepsilon}\) :
-
Porosity of the medium
- \({\bar{{\varepsilon}}}\) :
-
Small amplitude of the thermal modulation
- ρ :
-
Density
- ω :
-
Frequency
- \({\varphi}\) :
-
Phase angle
- κ :
-
Effective thermal diffusivity
- μ :
-
Viscosity
- μ v :
-
Viscoelastic constant of Walters B liquid
- \({\nabla_{\rm h}^2}\) :
-
Horizontal Laplacian operator
- Γ P :
-
Elastic parameter, \({\mu_{\rm v}\varepsilon/\rho_0 d^{2}}\)
- b :
-
Basic state
- c :
-
Critical
- 0:
-
Reference value
- *:
-
Dimensionless quantity
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Shivakumara, I.S., Lee, J., Malashetty, M.S. et al. Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium. Transp Porous Med 87, 291–307 (2011). https://doi.org/10.1007/s11242-010-9682-9
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DOI: https://doi.org/10.1007/s11242-010-9682-9