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Unsteady MHD Mixed Convective Flow of Chemically Reacting and Radiating Couple Stress Fluid in a Porous Medium Between Parallel Plates with Soret and Dufour Effects

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Abstract

This article attempts to analyze the chemical reaction, Soret and Dufour effects on an incompressible magnetohydrodynamic mixed convection flow of radiating couple stress fluid between parallel porous plates. It is assumed that the flow is generated due to periodic suction or injection at the plates, and the non-uniform temperature and concentration of the plates are assumed to be varying periodically with time. The governing partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations; then, the resulting equations are solved numerically by the quasilinearization technique. The effects of various fluid and geometric parameters on non-dimensional velocity components, temperature distribution, concentration, skin friction, heat and mass transfer rates are discussed in detail through graphs and tables. It is observed that the temperature of the fluid is enhanced, whereas the concentration is decreased with the increasing of Soret and Dufour parameters. Also, it is noticed that the magnetic field and porous medium exhibit the similar effect on velocity components, temperature distribution, and concentration. It has been noticed that the present results have the better agreement with the existing literature.

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Abbreviations

\({k_2}\) :

Chemical reaction rate

\({C}\) :

Concentration

\({C_1}\) :

Concentration of the lower plate

\({C_2}\) :

Concentration of the upper plate

\({c_{{{\rm s}}}}\) :

Concentration susceptibility

\({\bar{J}}\) :

Current density

\({T^{\ast}}\) :

Dimensionless temperature, \({\frac{T-T_1 e^{i\omega t}}{\left( {T_2 -T_1 } \right)e^{i\omega t}}}\)

\({C^{\ast}}\) :

Dimensionless concentration, \({\frac{C-C_1 e^{i\omega t}}{\left( {C_2 -C_1 } \right)e^{i\omega t}}}\)

h :

Distance between parallel plates

Du:

Dufour number, \({\frac{D_1 k_{{\rm T}} \dot{n}_{{\rm A}} \rho c}{\upsilon^{2}V_2 c_{{s}} k}}\)

E :

Eckert number, \({\frac{\mu V_2 }{\rho hc(T_2 -T_1 )}}\)

\({\bar{E}}\) :

Electric field

p :

Fluid pressure

Ha:

Hartmann number, \({B_0 h\sqrt{\frac{\sigma }{\mu }}}\)

\({\bar{b}}\) :

Induced magnetic field

\({D^{-1}}\) :

Inverse Darcy parameter, \({\frac{h^{2}}{k_1 }}\)

\({B_0}\) :

Magnetic flux density

\({D_{1}}\) :

Mass diffusivity

\({\dot{n}_{{{A}}}}\) :

Mass transfer rate

\({k_{3}}\) :

Mean absorption coefficient

\({T_{{m}}}\) :

Mean temperature

Kr:

Non-dimensional chemical reaction parameter, \({\frac{k_2 h^{2}}{D_1 }}\)

\({k_1}\) :

Permeability parameter

Pr:

Prandtl number, \({\frac{\mu c}{k}}\)

Rd:

Radiation parameter, \({\frac{16 \sigma_1 T_1^3 }{3 k_3 k}}\)

D :

Rate of deformation tensor

Re:

Reynolds number, \({\frac{\rho V_2 h}{\mu }}\)

Sc:

Schmidt number, \({\frac{\upsilon}{D_1 }}\)

Sh:

Sherwood number, \({\frac{\dot{n}_{{{\rm A}}}}{h\upsilon(C_2 -C_1 )}}\)

Gm:

Solutal Grashof number, \({\frac{\rho g\beta_{{\rm C}} (C_2 -C_1 )h^{2}}{\mu V_2 }}\)

Sr:

Soret number, \({\frac{D_1 k_{{\rm T}} \upsilon V_2 }{cT_{{m}} \dot{n}_{{{\rm A}}}}}\)

c :

Specific heat at constant temperature

\({\sigma_1}\) :

Stefan–Boltzmann constant

\({V_1}\) :

Injection velocity at the lower plate

\({V_2}\) :

Suction velocity at the upper plate

a:

Suction–injection ratio (for mixed suction case), \({1-\frac{V_1 }{V_2 }}\)

\({a_{1}}\) :

Suction–injection ratio (for mixed injection case), \({\frac{V_2 }{V_1 }-1}\)

\({T}\) :

Temperature

\({T_1}\) :

Temperature of the lower plate

\({T_2}\) :

Temperature of the upper plate

k :

Thermal conductivity

\({k_{{T}}}\) :

Thermal–diffusion ratio

Gr:

Thermal Grashof number, \({\frac{\rho g\beta_{{T}} (T_2 -T_1 )h^{2}}{\mu V_2 }}\)

t :

Time

\({\bar{b}}\) :

Total magnetic field

\({\hat{i}, \hat{j}, \hat{k}}\) :

Unit vectors

\({\bar{Q}}\) :

Velocity vector

u :

Velocity component in X-direction

v :

Velocity component in Y-direction

\({\beta_{{T}}}\) :

Coefficient of thermal expansion

\({\beta_{{C}}}\) :

Coefficient of solutal expansion

\({\alpha}\) :

Couple stress parameter, \({\sqrt{\frac{\eta }{\mu h^{2}}}}\)

\({\lambda}\) :

Dimensionless y coordinate, \({\frac{y}{h}}\)

\({\zeta}\) :

Dimensionless axial variable, \({\left( {\frac{U_0 }{aV_2 }-\frac{x}{h}} \right)}\)

\({\sigma}\) :

Electric conductivity

\({\rho}\) :

Fluid density

\({\mu}\) :

Fluid viscosity

\({\mu^{\prime}}\) :

Magnetic permeability

\({\beta}\) :

Non-dimensional frequency parameter, \({\omega t}\)

\({\bar{\omega}}\) :

Rotation vector

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Authors and Affiliations

  1. Department of Applied Mathematics, Defence Institute of Advanced Technology (Deemed University), Pune, 411025, India

    Odelu Ojjela & N. Naresh Kumar

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  1. Odelu Ojjela
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  2. N. Naresh Kumar
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Correspondence to Odelu Ojjela.

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Ojjela, O., Naresh Kumar, N. Unsteady MHD Mixed Convective Flow of Chemically Reacting and Radiating Couple Stress Fluid in a Porous Medium Between Parallel Plates with Soret and Dufour Effects. Arab J Sci Eng 41, 1941–1953 (2016). https://doi.org/10.1007/s13369-016-2045-2

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