Abstract
The effect of time-periodic temperature modulation at the onset of convection in a Boussinesq porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion. 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. The perturbation method is applied for computing the critical Rayleigh and wave numbers for small amplitude temperature modulation. The shift in the critical Rayleigh number is calculated as a function of frequency of modulation, concentration Rayleigh number, porosity, Lewis number, and thermal capacity ratio. It has been shown that it is possible to advance or delay the onset of convection by time-periodic modulation of the wall temperature. The nanofluid is found to have more stabilizing effect when compared to regular fluid. Low frequency is destabilizing, while high frequency is always stabilizing for symmetric modulation. Asymmetric modulation and only lower wall temperature modulation is stabilizing for all frequencies when concentration Rayleigh number is greater than one.
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Abbreviations
- \(c\) :
-
Nanofluid specific heat at constant pressure
- \(c_p \) :
-
Specific heat of the nanoparticle material
- \(\left( {\rho c} \right)_\mathrm{m} \) :
-
Effective heat capacity of the porous medium
- \(d_\mathrm{p} \) :
-
Nanoparticle diameter
- \(g\) :
-
Gravitational acceleration
- \(D_\mathrm{B} \) :
-
Brownian diffusion coefficient (\({\mathrm{m}^{2}}/\mathrm{s}\))
- \(D_\mathrm{T} \) :
-
Thermophoretic diffusion coefficient (\({\mathrm{m}^{2}}/\mathrm{s}\))
- \(h_\mathrm{p} \) :
-
Specific enthalpy of the nanoparticle material
- \(H\) :
-
Dimensional layer depth (\(m\))
- \(j_\mathrm{p} \) :
-
Diffusion mass flux for the nanoparticles
- \(j_\mathrm{p,T} \) :
-
Thermophoretic diffusion
- \(k\) :
-
Thermal conductivity of the nanofluid
- \(k_\mathrm{B} \) :
-
Boltzman’s constant
- \(k_\mathrm{m} \) :
-
Effective thermal conductivity of the porous medium
- \(k_\mathrm{p} \) :
-
Thermal conductivity of the particle material
- \(Le\) :
-
Lewis number
- \(N_\mathrm{A} \) :
-
Modified diffusivity ratio
- \(N_\mathrm{B} \) :
-
Modified particle-density increment
- \(p^{*}\) :
-
Pressure
- \(p\) :
-
Dimensionless pressure, \({p^{*}K}/{\mu \alpha _\mathrm{m}}\)
- \(q\) :
-
Energy flux relative to a frame moving with the nanofluid velocity v
- \(Ra\) :
-
Thermal Rayleigh–Darcy number
- \(Rm\) :
-
Basic-density Rayleigh number
- \(Rn\) :
-
Concentration Rayleigh number
- \(t^{*}\) :
-
Time
- \(t\) :
-
Dimensionless time, \({t^{*}\alpha _\mathrm{m} }/{\sigma H^{2}}\)
- \(T^{*}\) :
-
Nanofluid temperature
- \(T\) :
-
Dimensionless temperature, \(\frac{T^{*}-T_\mathrm{c}^*}{T_\mathrm{h}^*-T_\mathrm{c}^*}\)
- \(T_\mathrm{c}^*\) :
-
Temperature at the upper wall
- \(T_\mathrm{h}^*\) :
-
Temperature at the lower wall
- \(T_\mathrm{R} \) :
-
Reference temperature
- \(\left({u,v,w} \right)\) :
-
Dimensionless Darcy velocity components \(\left( {u^{*},v^{*},w^{*}} \right)H/{\alpha _\mathrm{m} }\)
- \(v\) :
-
Nanofluid velocity
- \(v_\mathrm{D} \) :
-
Darcy velocity \(\varepsilon \text{ v}\)
- \(v_\mathrm{D}^{*} \) :
-
Dimensionless Darcy velocity \(\left( {u^{*},v^{*},w^{*}} \right)\)
- \(V_\mathrm{T} \) :
-
Thermophoretic velocity
- \(\left( {x,y,z} \right)\) :
-
Dimensionless Cartesian coordinate \({\left( {x^{*},y^{*},z^{*}} \right)}/H\); \(z\) is the vertically upward coordinate
- \(\left({x^{*},y^{*},z^{*}} \right)\) :
-
Cartesian coordinates
- \(\alpha _\mathrm{m} \) :
-
Thermal diffusivity of the porous medium, \(\frac{k_\mathrm{m} }{\left( {\rho c_p } \right)_\mathrm{f} }\)
- \(\tilde{\beta }\) :
-
Proportionality factor
- \(\varepsilon \) :
-
Porosity of the medium
- \(\varepsilon _\mathrm{t} \) :
-
Amplitude of the modulation
- \(\mu \) :
-
Viscosity of the fluid
- \(\tilde{\mu }\) :
-
Effective viscosity of the porous medium
- \(\rho \) :
-
Fluid density
- \(\rho _\mathrm{p} \) :
-
Nanoparticle mass density
- \(\sigma \) :
-
Heat capacity ratio
- \(\phi ^{*}\) :
-
Nanoparticle volume fraction
- \(\phi \) :
-
Relative nanoparticle volume fraction, \(\frac{\phi ^{*}-\phi _\mathrm{c}^*}{\phi _\mathrm{h}^*-\phi _\mathrm{c}^*}\)
- \(\varOmega \) :
-
Dimensional frequency
- \(\omega \) :
-
Dimensionless frequency \(\left({=}{\varOmega H^{2}/k} \right)\)
- \(\psi \) :
-
Phase angle: \(\psi =0\), symmetric modulation; \(\psi =\pi \), antisymmetric modulation; \(\psi =-i\infty ,\) only lower wall temperature modulation
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Umavathi, J.C. Effect of Thermal Modulation on the Onset of Convection in a Porous Medium Layer Saturated by a Nanofluid. Transp Porous Med 98, 59–79 (2013). https://doi.org/10.1007/s11242-013-0133-2
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DOI: https://doi.org/10.1007/s11242-013-0133-2