Influence of Inclined Magnetic Field on a Mixed Convective UCM Fluid Flow Through a Porous Medium with Thermophoresis and Brownian Motion

  • Odelu Ojjela
  • K. Pravin Kashyap
  • N. Naresh Kumar
  • Samir Kumar Das
Original Paper
First Online:

Abstract

In this paper, an incompressible electrically conducting mixed convective upper convected Maxwell fluid flow in a porous medium between parallel plates under the influence of inclined magnetic field, thermophoresis and Brownian motion is considered. Assume that there is periodic injection and suction at the lower and upper plates respectively. The temperature and concentration at the lower and upper plates are varying periodically with time. The flow field equations are reduced to nonlinear ordinary differential equations by using similarity transformations and a numerical solution has been obtained by using shooting technique with fourth order Runge–Kutta method. The velocity components, temperature distribution and concentration with respect to different fluid and geometric parameters are discussed in detail and presented in the form of graphs. It is observed that the temperature of the fluid is enhanced with Brownian parameter whereas the concentration decreases with increasing thermophoresis parameter. The present results are compared with the existing literature and are found to be good agreement.

Keywords

UCM fluid Mixed convection Inclined magnetic field Thermophoresis Brownian motion Porous medium 

List of symbols

h

Distance between parallel plates

\(V_{1 } e^{i\omega t}\)

Injection velocity at lower plate

\(V_{2 } e^{i\omega t}\)

Suction velocity at the upper plate

a

Suction injection parameter, \(\frac{V_2 }{V_1 }-1\)

P

Fluid pressure

u(x,y)

Axial velocity

v(x,y)

Radial velocity

Wi

Weissenberg number, \(\frac{\beta V_1 }{h}\)

\(T_{1 } e^{i\omega t}\)

Temperature at the lower plate

\(T_2 e^{i\omega t}\)

Temperature at the upper plate

\(C_{1 } e^{i\omega t}\)

Concentration at the lower plate

\(C_2 e^{i\omega t}\)

Concentration at the upper plate

\(\hbox {T}^{*}\)

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

\(\hbox {C}^{*}\)

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

k

Permeability parameter

Ec

Eckert number,\(\frac{\mu V_1 }{\rho hc(T_2 -T_1 )}\)

Gr

Thermal Grashof number \(\frac{\rho g\beta _T (T_2 -T_1 )h^{2}}{\mu V_1 }\)

Gc

Solutal Grashof number \(\frac{\rho g\beta _c (T_2 -T_1 )h^{2}}{\mu V_1 }\)

Sh

Sherwood number \(\frac{\dot{n}_A }{h\upsilon (C_2 -C_1 )}\)

Nb

Brownian motion parameter \(\frac{D_B (C_2 -C_1 )}{\alpha }\)

Nt

Thermophoresis parameter \(\frac{D_T (T_2 -T_1 )}{\alpha T_2 }\)

Da

Darcy parameter \(\frac{k}{h^{2}}\)

Pr

Prandtl number, \(\frac{\mu c}{k_1 }\)

\(\hbox {B}_{0}\)

Magnetic field strength

Ha

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

g

Acceleration due to gravity

\(\hbox {D}_{\mathrm{B}}\)

Brownian diffusion coefficient

\(\hbox {D}_{\mathrm{T}}\)

Thermophoretic diffusion coefficient

\(\hbox {k}_{1}\)

Thermal conductivity

\(\hbox {T}_{\mathrm{m}}\)

Mean temperature

c

Specific heat

\(\hbox {c}_{\mathrm{p}}\)

Specific heat at constant pressure

\(\hbox {c}_{\mathrm{s}}\)

Concentration susceptibility

Re

Suction Reynolds number, \(\frac{\rho hV_1 }{\mu }\)

\(U_{0}\)

Entrance velocity

Greek Letters

\(\upalpha \)

Thermal diffusivity, \(\frac{k_1 }{\rho c}\)

\(\beta \)

Maxwell parameter

\(\upsigma \)

Electric conductivity

\(\uptheta \)

Inclination of magnetic field with Y-axis

\(\phi \)

Frequency parameter, \(\omega t\)

\(\lambda \)

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

\(\zeta \)

Dimensionless axial variable, \(\left( {\frac{U_0 }{aV_1 }-\frac{x}{h}} \right) \)

\(\uprho \)

Fluid density

\(\upmu \)

Dynamic viscosity

\(\nu \)

Kinematic viscosity

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Notes

Acknowledgments

The authors are thankful to the editor and reviewers for their valuable suggestions to improve the quality of the article. Also one of the authors (K Pravin Kashyap) is grateful to the University Grants Commission, Government of India for providing Senior Research Fellowship (F.2-18/2012(SA-I)).

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Copyright information

© Springer India Pvt. Ltd. 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsDefence Institute of Advanced Technology (Deemed University)PuneIndia

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