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Lie Group Transformation on MHD Double-Diffusion Convection of a Casson Nanofluid over a Vertical Stretching/Shrinking Surface with Thermal Radiation and Chemical Reaction

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Abstract

This paper is concern with the study of steady magnetohydrodynamic radiative dissipative boundary layer flow of a reactive Casson nanofluid over a permeable vertical nonlinear stretching/shrinking surface. We have incorporated the combined effects of viscous and Ohmic dissipation, thermophoresis and Brownian motion on heat and mass transfer in Casson nanofluid in the presence of chemical reaction. The governing equations are reduced to a system of nonlinear ordinary differential equations with associated boundary conditions using scaling group transformations. The reduced nonlinear ordinary differential equations are then solved numerically by Runge–Kutta–Fehlberg fifth-order method with shooting technique. The effects of magnetic field, suction/injection parameter, Prandtl number, Eckert number, Brownian motion parameter, thermophoresis parameter and Lewis number on local Nusselt number and local Sherwood number are analyzed. The results show that the physical parameters have significant influence on the flow velocity, surface shear stress, local Nusselt number and local Sherwood number.

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Abbreviations

b :

Constant related to stretching sheet velocity

C :

Casson nanoparticle volume fraction

\(Cf_x\) :

Skin fraction coefficient

\(C_{w}\) :

Casson nanoparticle volume fraction at the wall

\(C_{\infty }\) :

Free stream Casson nanoparticle volume fraction

\(D_B\) :

Coefficient of Brownian diffusion, m\(^{2}\) s\(^{-1}\)

\(D_T\) :

Coefficient of thermophoretic diffusion, m\(^{2}\) s\(^{-1}\)

f :

Dimensionless stream function

\(p_y\) :

Yield stress of the non-Newtonian fluid

g :

Acceleration due to gravity

R :

Rate of chemical reaction

Le :

Lewis number

Nb :

Brownian motion number

Nr :

Buoyancy ratio number

Nt :

Thermophoresis parameter

\({Nu_x}\) :

Local Nusselt number

Pr :

Prandtl number

\(p_y\) :

Yield stress of the fluid

\({q_r}\) :

Thermal radiative heat flux, W m\(^{-2}\)

\({R_a}\) :

Local Rayleigh number

S :

Suction parameter

T :

Temperature of the fluid, K

\(T_{\infty }\) :

Free stream temperature, K

\(T_w\) :

Temperature at the wall K

u :

Velocity component in x-direction, m s\(^{-1}\)

\({u_w}\) :

Stretching sheet velocity, m s\(^{-1}\)

U :

Free stream velocity of the fluid, m s\(^{-1}\)

v :

Velocity component in y-direction, m s\(^{-1}\)

x, y :

Direction along and perpendicular to the plate, respectively

\( {\alpha }\) :

Fluid thermal diffusivity, m\(^{2}\) s\(^{-1}\)

\({\beta }\) :

Coefficient of thermal expansion

\({\gamma }\) :

Casson fluid parameter

\({\gamma _{1}}\) :

Chemical reaction number

\({\eta }\) :

Similarity variable

\(\theta \) :

Dimensionless temperature of Casson nanofluid

\(\kappa \) :

Coefficient of thermal conductivity, W m\(^{-1}\) K\(^{-1}\)

\(\kappa ^{*}\) :

Mean absorption coefficient, m\(^{-1}\)

\({\mu _B}\) :

Plastic dynamic viscosity of Casson nanofluid

\({\nu }\) :

Kinematic viscosity of Casson nanofluid, m\(^{2}\) s\(^{-1}\)

\({\pi _{c}}\) :

Critical value of the product based on the non-Newtonian fluid

\({\rho _{f}}\) :

Density of Casson nanofluid, kg m\(^{-3}\)

\({\rho _{p}}\) :

Casson nanoparticle mass density, kg m\(^{-3}\)

\((\rho c)_f\) :

Heat capacity of Casson nanofluid

\((\rho c)_p\) :

Effective heat capacity of the nanoparticle material

\(\sigma ^*\) :

Stefan–Boltzmann constant, W m\(^{-2}\) K\(^{-4}\)

\(\tau \) :

Parameter defined by ratio between \((\rho c)_p\) and \((\rho c)_f\)

\({\phi }\) :

Dimensionless nanoparticle volume fraction

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Authors and Affiliations

  1. Department of Mathematics, Siksha Bhavana, Visva-Bharati University, Santiniketan, West Bengal, 731 235, India

    Dulal Pal

  2. Department of Mathematics, Surendranath College, Kolkata, West Bengal, 700009, India

    Netai Roy

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Pal, D., Roy, N. Lie Group Transformation on MHD Double-Diffusion Convection of a Casson Nanofluid over a Vertical Stretching/Shrinking Surface with Thermal Radiation and Chemical Reaction. Int. J. Appl. Comput. Math 4, 13 (2018). https://doi.org/10.1007/s40819-017-0449-7

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