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Hybrid Numerical Solution of Mixed Convection Boundary Layer Flow of Nanofluid Along an Inclined Plate with Prescribed Surface Fluxes

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Abstract

The present paper contains a numerical study using hybrid approach on mixed convective boundary layer flow of a nanofluid past an inclined plate embedded in a porous medium. The inclined plate is maintained at uniform and constant heat, mass and nanoparticle fluxes, and the behavior of the porous medium is described by the Darcy’s model. The model considered for nanofluids incorporates the effects of Brownian motion and thermophoresis. Using appropriate similarity variables, system of non-linear ordinary differential equations are solved by hybrid approach consisting of finite element method and symbolic computation. The other effect of mixed convection parameter, buoyancy ratio parameter and inclination angle on the temperature, concentration of both salts and nanoparticle volume fraction are shown. It is found that concentration of salts and nanoparticle volume fraction decreases with increasing mixed convection parameter in the presence of nanofluid and both salts.

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Abbreviations

K :

Permeability of the porous medium

\(K_m\) :

Thermal conductivity

Nur :

Reduced Nusselt number

\(Nu_x\) :

Local Nusselt number

\(Ra_x\) :

Local Rayleigh number

\(Shr^1\) :

Reduced solutal sherwood number for salt 1

\(Shr^2\) :

Reduced solutal sherwood number for salt 2

\(Sh_x^1\) :

Local solutal Sherwood number for salt 1

\(Sh_x^2\) :

Local solutal Sherwood number for salt 2

\(\frac{Ra_x}{Pe_x}\) :

Mixed convection parameter (\(Pe_x\ne 0\))

\(Pe_x\) :

Local Peclet number

g :

Acceleration due to gravity

\(D_{TC_1}\) :

Dufour diffusivite

\(D_{TC_2}\) :

Dufour diffusivite

\(D_{C_1T}\) :

Soret diffusivite

\(D_{C_2T}\) :

Soret diffusivite

\(D_{S_1}\) :

Solutal diffusivite of the porous medium

\(D_{S_2}\) :

Solutal diffusivite of the porous medium

\(D_T\) :

Thermophoretic diffusion coefficient

\(D_B\) :

Brownian diffusion coefficient

C :

Nanoparticle volume fraction

\(C_1\) :

Solutal concentration of salt 1

\(C_2\) :

Solutal concentration of salt 2

\(\beta _{C1}\) :

Volumetric solutal expansion coefficient of the fluid

\(\beta _{C2}\) :

Volumetric solutal expansion coefficient of the fluid

\(C_{1\infty }\) :

Ambient solutal concentration of salt 1 attained as y tends to infinity

\(C_{2\infty }\) :

Ambient solutal concentration of salt 2 attained as y tends to infinity

\(C_\infty \) :

Ambient nanoparticle volume fraction

T :

Local fluid temperature

\(T_\infty \) :

Ambient temperature

\(Ld_1\) :

Dufour-solutal Lewis numbers of salt 1

\(Ld_2\) :

Dufour-solutal Lewis numbers of salt 2

Nb :

Brownian motion parameter

\(Nc_1\) :

Buoyancy ratio of salt 1

\(Nc_2\) :

Buoyancy ratio of salt 2

\(Nd_1\) :

Modified Dufour parameters of salt 1

\(Nd_2\) :

Modified Dufour parameters of salt 2

Nr :

Nanofluid buoyancy ratio

\(q_w\) :

Wall heat flux

\(q_{m_1},~q_{m_2}\) :

Solutal mass flux

\(q_{np}\) :

Nanoparticle mass flux

(x, y):

Cartesian co-ordinate

(u, v):

Velocity components along x and y axes

\(\theta \) :

Dimensionless heat transfer

\(\gamma _1\) :

Dimensionless solutal concentration of salt 1

\(\gamma _2\) :

Dimensionless solutal concentration of salt 2

\(\alpha _m\) :

Thermal diffusivity of porous medium

\(\beta _T\) :

Volumetric thermal expansion coefficient of the fluid

\((\rho c)_f\) :

Effective heat capacity of the fluid

\((\rho c)_p\) :

Effective heat capacity of the nanoparticle material

\((\rho c)_m\) :

Effective heat capacity of the porous medium

\(\tau *\) :

Parameter define in the manuscript \(\frac{\varepsilon (\rho c)_p}{(\rho c)_f}\)

\(\upsilon \) :

Kinematic viscosity of the fluid

\(\rho _p\) :

Nanoparticle mass density

\(\rho _f\) :

Fluid density

\(\psi \) :

Stream function

\(\mu \) :

Absolute viscosity of the fluid

\(\eta \) :

Similarity variable

\(\delta '\) :

Acute angle of the plate to the vertical

\(\omega \) :

Condition of the plate

\( \infty \) :

Condition far away from the plate

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Acknowledgements

All authors acknowledge the financial support received through the project supported by SERB. The help rendered by Ms. Rangoli Goyal, Research Scholar, Department of Mathematics is highly acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Rama Bhargava,  Pratibha & Harish Chandra

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  1. Rama Bhargava
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Correspondence to Rama Bhargava.

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Bhargava, R., Pratibha & Chandra, H. Hybrid Numerical Solution of Mixed Convection Boundary Layer Flow of Nanofluid Along an Inclined Plate with Prescribed Surface Fluxes. Int. J. Appl. Comput. Math 3, 2909–2928 (2017). https://doi.org/10.1007/s40819-016-0278-0

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