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Nonlinear Two-Dimensional Convection in a Nanofluid Saturated Porous Medium

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Abstract

In this article, we study the linear and nonlinear thermal instability in a horizontal porous medium saturated by a nanofluid. For this, the momentum equation with Brinkman model has been used. Also, it incorporates the effect of Brownian motion along with thermophoresis. The linear stability is based on normal mode technique, and for nonlinear analysis, the truncated Fourier series involving only two terms has been used. The expression of Rayleigh number for linear theory has been derived, and the effects of various parameters on Rayleigh number have been presented graphically. Weak nonlinear theory is used to find the concentration and the thermal Nusselt numbers. The behavior of the concentration and thermal Nusselt numbers is investigated and depicted graphically, by solving the finite amplitude equations using a numerical method.

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Abbreviations

D B :

Brownian diffusion coefficient

D T :

Thermophoretic diffusion coefficient

Da :

Darcy number

Pr :

Prandtl number

d :

Dimensional layer depth

k T :

Effective thermal conductivity of porous medium

k m :

Thermal diffusivity of porous medium

Le :

Lewis number

N A :

Modified diffusivity ratio

N B :

Modified particle-density increment

p :

Pressure

g :

Gravitational acceleration

Ra :

Thermal Rayleigh–Darcy number

Rm :

Basic density Rayleigh number

Rn :

Concentration Rayleigh number

t :

Time

T :

Temperature

T c :

Temperature at the upper wall

T h :

Temperature t the lower wall

v :

Nanofluid velocity

v D :

Darcy velocity \({\varepsilon \bf{v}}\)

(x*, y*, z*):

Cartesian coordinates

α :

Horizontal wave number

β :

Proportionality factor

\({\varepsilon }\) :

Porosity

μ :

Viscosity of the fluid

\({\bar\mu}\) :

Effective viscosity of the porous medium

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρ c)f :

Heat capacity of the fluid

(ρ c)m :

Effective heat capacity of the porous medium

(ρ c)p :

Effective heat capacity of the nanoparticle material

γ :

Parameter defined as \({\frac{(\rho c)_{\rm m}}{(\rho c)_{\rm f}}}\)

\({\phi}\) :

Nanoparticle volume fraction

ν :

Kinematic viscosity μ/ρ f

ψ :

Stream function

α :

Wave number

ω :

Frequency of oscillation

b :

Basic solution

*:

Dimensional variable

’:

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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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

    B. S. Bhadauria

  2. DST-Centre for Interdisciplinary Mathematical Sciences, Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi, 221005, India

    Shilpi Agarwal & Anoj Kumar

Authors
  1. B. S. Bhadauria
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  2. Shilpi Agarwal
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  3. Anoj Kumar
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Correspondence to B. S. Bhadauria.

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Bhadauria, B.S., Agarwal, S. & Kumar, A. Nonlinear Two-Dimensional Convection in a Nanofluid Saturated Porous Medium. Transp Porous Med 90, 605–625 (2011). https://doi.org/10.1007/s11242-011-9806-x

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