Advertisement

International Journal of Thermophysics

, Volume 34, Issue 11, pp 2183–2208 | Cite as

Unsteady Flow of Radiating and Chemically Reacting MHD Micropolar Fluid in Slip-Flow Regime with Heat Generation

  • S.  M. Abo-Dahab
  • R.  A. Mohamed
Article

Abstract

An analytical study of the problem of unsteady free convection with thermal radiation and heat generation on MHD micropolar fluid flow through a porous medium bounded by a semi-infinite vertical plate in a slip-flow regime has been presented. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted for in the mass diffusion equation. A uniform magnetic field acts perpendicular on the porous surface absorbing micropolar fluid with a suction velocity varying with time. A perturbation technique is applied to obtain the expressions for the velocity, microrotation, temperature, and concentration distributions. Expressions for the skin-friction, Nusselt number, and Sherwood number are also obtained. The results are discussed graphically for different values of the parameters entered into the equations of the problem.

Keywords

MHD Micropolar fluid Porous medium Thermal radiation  Heat generation Chemical reaction Slip-flow 

List of Symbols

\(B_{0}\)

Magnetic induction (\(T\))

\(C^{*}\)

Species concentration (\(\mathrm {mol}\,{\cdot }\,\mathrm {m}^{-3}\))

\(C_{\mathrm{w }}^{*}\)

Surface concentration (\(\mathrm {mol}\,{\cdot }\,\mathrm{m}^{-3}\))

\(C_{\infty }^{*}\)

Species concentration far from the surface (\(\mathrm {mol}\,{\cdot }\,\mathrm{m}^{-3}\))

\(C_\mathrm{f }\)

Skin-friction coefficient

\(C_\mathrm{m }\)

Couple stress coefficient

\(c_{p}\)

Specific heat at constant pressure (\(\mathrm J \,{\cdot }\, \mathrm{kg}^{-1}\,{\cdot }\, \mathrm K ^{-1}\))

\(D^{*}\)

Chemical molecular diffusivity (\(\mathrm m ^{2}\,{\cdot }\,\mathrm s ^{-1}\))

\(Gc\)

Solutal Grashof number

\(Gr\)

Grashof number

\(g\)

Accelaration due to gravity (\(\mathrm m \,{\cdot }\,\mathrm s ^{-2}\))

\(H\)

Dimensionless heat generation/absorption coefficient

\(j^{*}\)

Microinertia per unit mass (\(\mathrm m ^{2}\))

\(K{^{*}}\)

Permeability of the porous media (\(\mathrm m ^{2}\))

\(K_{1}^{*}\)

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

\(K\)

Diemnsionless permeability

\(k\)

Thermal conductivity of the fluid (\(\mathrm W \,{\cdot }\,\mathrm m ^{-1}\,{\cdot }\,\mathrm K ^{-1}\))

\(M\)

Local magnetic field parameter

\(n\)

Parameter related to microgyration vector and shear stress

\(N\)

Model parameter

\(Nu\)

Nusselt parameter

\(P^{*}\)

Pressure (\(\mathrm{kg}\,{\cdot }\,\mathrm m ^{-1}\,{\cdot }\,\mathrm s ^{-2}\))

\(Pr\)

Fluid Prandtl number

\(Q_{0}\)

Heat generation coefficient (\(\mathrm W \,{\cdot }\,\mathrm m ^{-3}\,{\cdot }\,\mathrm K ^{-1}\))

\(q_\mathrm{r }^{*}\)

Radiative heat flux (\(\mathrm W \,{\cdot }\,\mathrm m ^{-2}\))

\(R^{*}\)

Reaction rate constant (J)

\(R\)

Radiation parameter

\(Re\)

Reynolds number

\(Sc\)

Generalized Schmidt number

\(Sh\)

Sherwood number

\(t^{*}\)

Time (s)

\(t\)

Dimensionless time

\(T^{*}\)

Temperature in the boundary layer (K)

\(T_{\infty }^{*}\)

Temperature far away from the plate (K)

\(T_\mathrm w ^{*}\)

Temperature at the wall (K)

\(u\)

Dimensionless velocity

\(u^{*}\) and \(v^{*}\)

Velocities along and perpendicular to the plate (\(\mathrm m \,{\cdot }\,\mathrm s ^{-1}\))

\(U_{0}\)

Plate velocity in the direction of flow (\(\mathrm m \,{\cdot }\,\mathrm{s }^{-1}\))

\(U_{\infty }^{*}\)

Free stream velocity (\(\mathrm m \,{\cdot }\,\mathrm s ^{-1}\))

\(K_n\)

Knudsen number

\(h\)

Rarefaction parameter

\(\varrho \)

Characteristic dimension of flow

\(V_{0}\)

Scale of suction velocity

\(x^{*}\) and \(y^{*}\)

Distances along and perpendicular to the plate, respectively (m)

\(y\)

Dimensionless spanwise coordiante normal to the plate

\(\alpha \)

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

\(\beta \)

Ratio of vortex viscosity and dynamic viscosity

\(\beta _{c}\)

Volumetric coefficient of thermal expansion (\(\mathrm K ^{-1}\))

\(\beta _\mathrm{f}\)

Volumetric coefficient of expansion with concentration (\(\mathrm K ^{-1}\))

\(\gamma \)

Spin-gradient velocity (\(\mathrm{kg}\,{\cdot }\,\mathrm m \,{\cdot }\,\mathrm s ^{-1}\))

\(\delta \)

Scalar constant

\(\epsilon \)

Scalar constant

\(\theta \)

Dimensionless temperature

\(K_\mathrm{c}\)

Chemical reaction parameter

\(\eta \)

Scalar constant

\(\mu \)

Fluid dynamic viscosity (\(\mathrm{kg}\,{\cdot }\,\mathrm m ^{-1}\,{\cdot }\,\mathrm s ^{-1}\))

\(\nu \)

Fluid kinematic rotational viscosity (\(\mathrm m ^{2}\,{\cdot }\,\mathrm s ^{-1}\))

\(\nu _\mathrm{r}\)

Kinematic rotational viscosity (\(\mathrm m ^{2}\,{\cdot }\,\mathrm s ^{-1}\))

\(\rho \)

Fluid density (\(\mathrm{kg}\,{\cdot }\,\mathrm m ^{-3}\))

\(\sigma \)

Electrical conductivity of the fluid (\(\mathrm S \,{\cdot }\,\mathrm m ^{-1}\))

\(\sigma ^{*}\)

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

\(\omega \)

Microrotation vector (\(\mathrm m \,{\cdot }\,\mathrm s ^{-1}\))

\(\varGamma \)

Coefficient of gryo-viscosity (or vortex viscosity) (\(\mathrm{kg}\,{\cdot }\,\mathrm m ^{-1}\,{\cdot }\,\mathrm s ^{-1}\))

\(L^{*}\)

Mean free path

\(L_{1}^{*}\)

Slip strength or slip coefficient

\(m_{1}\)

Maxwell’s reflection coefficient

\(A\)

Suction parameter

References

  1. 1.
    G. Lukaszewicz, Micropolar Fluids: Theory and Applications (Birkhäuser, Boston, MA, 1966)Google Scholar
  2. 2.
    A.C. Eringen, Int. J. Eng. Sci. 2, 205 (1964)Google Scholar
  3. 3.
    C. Eringen, J. Math. Mech. 16, 1 (1966)Google Scholar
  4. 4.
    C. Eringen, J. Math. Anal. Appl. 38, 480 (1972)Google Scholar
  5. 5.
    K.A. Helmy, ZAMM 78, 255 (1998)Google Scholar
  6. 6.
    Y.J. Kim, Acta Mech. 148, 106 (2001)CrossRefGoogle Scholar
  7. 7.
    F.S. Ibrahim, I.A. Hassanien, A.A. Baker, Canad. J. Phys. 82, 775 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    P.S. Hiremath, P.M. Patil, Acta Mech. 98, 143 (1993)CrossRefMATHGoogle Scholar
  9. 9.
    Y.J. Kim, Int. J. Heat Mass Transf. 44, 2791 (2001)CrossRefMATHGoogle Scholar
  10. 10.
    Y.J. Kim, Acta Mech. 156, 17 (2004)Google Scholar
  11. 11.
    R. Muthucumaraswamy, P. Ganesan, J. Appl. Mech. Tech. Phys. 42, 665 (2001)Google Scholar
  12. 12.
    A.J. Chamkha, Int. Commun. Heat Mass Transf. 30, 413 (2003)Google Scholar
  13. 13.
    R. Muthucumaraswamy, P. Ganesan, Forschung Ingenieurwesen 66, 17 (2000)CrossRefGoogle Scholar
  14. 14.
    R. Muthucumaraswamy, P. Ganesan, Acta Mech. 147, 1 (2001)CrossRefGoogle Scholar
  15. 15.
    R. Muthucumaraswamy, P. Ganesan, Forschung Ingenieurwesen 67, 129 (2002)CrossRefGoogle Scholar
  16. 16.
    A. Raptis, C. Perdikis, Int. J. Non-Linear Mech. 41, 527 (2006)ADSCrossRefMATHGoogle Scholar
  17. 17.
    M.A. Seddeek, A.A. Darwish, M.S. Abdelmeguid, Commun. Nonlinear Sci. Numer. Simulat. 12, 195 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  18. 18.
    F.S. Ibrahim, A.M. Elaiw, A.A. Bakr, Commun. Nonlinear Sci. Numer. Simulat. 13, 1056 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  19. 19.
    R. A. Mohamed, Ibrahim. A. Abbas, S. M. Abo-Dahab, Commun. Nonlinear Sci. Numer. Simulat. 14, 1385 (2009)Google Scholar
  20. 20.
    P.K. Sharma, R.C. Chaudhary, Emir. J. Eng. Res. 8, 33 (2003)Google Scholar
  21. 21.
    P. K. Sharma, Mathematicas XIII, 51 (2005)Google Scholar
  22. 22.
    H. Kumar, S.S. Tak, Acta Cienc. Indica 33, 1043 (2007)MATHGoogle Scholar
  23. 23.
    K. Khandelwal, A. Gupta, N.C. Poonam, Ganita 54, 203 (2003)Google Scholar
  24. 24.
    R.C. Chaudhary, A.K. Jha, Appl. Math. Mech. 29, 1179 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    M.A. Mansour, R.A. Mohamed, M.M. Abd-Elaziz, S.E. Ahmed, Int. J. Appl. Math. Mech 3, 99 (2007)Google Scholar
  26. 26.
    R.S.R. Gorla, A.A. Mohamadien, M.A. Mansour, I.A. Hassanien, Numer. Heat Transfer A28, 253 (1995)ADSCrossRefGoogle Scholar
  27. 27.
    E.M. Abo-Eldahab, M.A. El-Aziz, Appl. Math. Comput. 162, 881 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    M.M. Rahman, M.A. Sattar, ASME J. H. Transfer 128, 142 (2006)CrossRefGoogle Scholar
  29. 29.
    M.M. Rahman, I.A. Eltayb, S.M.M. Rahman, Therm. Sci. 13, 23 (2009)CrossRefGoogle Scholar
  30. 30.
    V.M. Soundalgekar, ASME J. Heat Transfer 99, 499 (1977)CrossRefGoogle Scholar
  31. 31.
    E.M. Abo-Eldahab, A.F. Ghonaim, Appl. Maths. & Comp. 169, 500 (2005)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    M.A. Rahman, T. Sultan, Nonlinear Analysis Modeling and Control 13, 71 (2008)MATHGoogle Scholar
  33. 33.
    R.A. Mohamed, S.M. Abo-Dahab, IJTS 48, 1800 (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceSVUQena Egypt
  2. 2.Department of Mathematics, Faculty of ScienceTaif UniversityTaif Saudi Arabia

Personalised recommendations