Abstract
The linear and nonlinear onset of convection in a viscoelastic nanofluid saturated densely packed horizontal rotating porous layer heated from below and cooled from above is investigated by considering the Oldroyd-B type fluid. The Brinkman model is used to simulate conservation of momentum in the porous medium. The temperature of the three phases—porous matrix, fluid, and nanoparticles— is considered to be under local thermal non-equilibrium, that is, the three phases have there own temperature equations. Linear stability analysis is performed using a one-term Galerkin scheme, while a minimal representation of the truncated Fourier series, involving only two terms, has been utilized for nonlinear analysis. The feasible range of new dimensionless parameters has been predicted using available experimental range of thermo-physical properties of alumina and ethylene-glycol. Obtained results have been presented graphically and discussed in detail.
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Abbreviations
- \(D_{\rm B}\) :
-
Brownian Diffusion coefficient
- \(D_{\rm T}\) :
-
Thermophoretic diffusion coefficient
- \(Da\) :
-
Darcy number
- \(Pr\) :
-
Prandtl number
- \(d\) :
-
Dimensional layer depth
- \(k_{\rm f}\) :
-
Effective thermal conductivity of porous medium
- \(k_{\rm T}\) :
-
Thermal diffusivity of porous medium
- \(Le\) :
-
Lewis number
- \(N_{\rm A}\) :
-
Modified diffusivity ratio
- \(N_{\rm B}\) :
-
Modified particle-density increment
- \(N_{\mathrm{HP}}\) :
-
Nield number for the fluid/particle interface
- \(N_{\mathrm{HS}}\) :
-
Nield number for the fluid/solid-matrix interface
- \(p\) :
-
Pressure
- \(g\) :
-
Gravitational acceleration
- \(Ra\) :
-
Thermal Rayleigh–Darcy number
- \(Rm\) :
-
Basic density Rayleigh number
- \(Rn\) :
-
Concentration Rayleigh number
- \(Ta\) :
-
Taylor number
- \(t\) :
-
Time
- \(T\) :
-
Nanofluid temperature
- \(T_{\rm c}\) :
-
Temperature at the upper wall
- \(T_{\rm h}\) :
-
Temperature at the lower wall
- \({\mathbf{v}}_{\rm D}\) :
-
Darcy velocity \(\varepsilon {\mathbf{v}}\)
- \((x,y,z)\) :
-
Cartesian coordinates
- \(\alpha _{\rm f}\) :
-
Thermal diffusivity of the fluid defined as \(\frac{{k_{\mathrm{f}}}}{(\rho c)}_{\mathrm{f}}\)
- \(\beta\) :
-
Proportionality factor
- \(\gamma _{\rm P}\) :
-
Modified thermal capacity ratio
- \(\gamma _{\rm S}\) :
-
Modified thermal capacity ratio
- \(\varepsilon\) :
-
Porosity
- \(\mu\) :
-
Viscosity of the fluid
- \({\bar{\mu }}\) :
-
Effective viscosity of the porous medium
- \(\varepsilon _{\rm P}\) :
-
Modified thermal diffusivity ratio
- \(\varepsilon _{\rm S}\) :
-
Modified thermal diffusivity ratio
- \(\mu\) :
-
Viscosity of the fluid
- \(\rho _{\rm f}\) :
-
Fluid density
- \(\rho _{\rm p}\) :
-
Nanoparticle mass density
- \((\rho c )_{\rm f}\) :
-
Heat capacity of the fluid
- \((\rho c)_{\rm s}\) :
-
Heat capacity of the solid-matrix material
- \((\rho c)_{\rm p}\) :
-
Heat capacity of the nanoparticle material
- \(\phi\) :
-
Nanoparticle volume fraction
- \(\psi\) :
-
Stream function
- \(b\) :
-
Basic solution
- \(f\) :
-
Fluid phase
- \(p\) :
-
Particle phase
- \(s\) :
-
Solid-matrix phase
- *:
-
Dimensional variable
- \(^{\prime }\) :
-
Perturbation variable
- \(\nabla ^2\) :
-
\(\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} + \frac{\partial ^2}{\partial z^2}\)
- \(\nabla _1^2\) :
-
\(\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial z^2}\)
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Agarwal, S., Rana, P. Thermal stability analysis of rotating porous layer with thermal non-equilibrium approach utilizing Al2O3–EG Oldroyd-B nanofluid. Microfluid Nanofluid 19, 117–131 (2015). https://doi.org/10.1007/s10404-015-1554-8
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DOI: https://doi.org/10.1007/s10404-015-1554-8