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Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using Cattaneo–Christov heat flux with convective heating

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Abstract

The present article investigates the effect of second-order slip, chemical reaction and Soret and Dufour effects on MHD convective flow of an Oldroyd-B liquid toward a stretchy surface. Analysis of thermal relaxation time is made by using Cattaneo–Christov heat flux model. The effects of radiation and convective heating are also taken into account. The ordinary differential equations are retrieved by the help of suitable transformations of governing equations. The analytical solutions are observed by homotopy progress. The velocity, concentration and temperature field are analyzed for various pertinent parameters involved in the study. The graphical results of physical quantities of interest such as skin friction, local Nusselt number and local Sherwood number are presented. A comparative study with existing result indicates excellent agreement.

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Abbreviations

\(A_1\) :

Relaxation time

\(A_2\) :

Retardation time

a :

Stretching rate \(({\mathrm{s}}^{-1})\)

Bi :

Biot number (–)

\(B_0\) :

Constant magnetic field \(({\mathrm{kg}}\,{\mathrm{s}}^{-2}\,{\mathrm{A}}^{-1})\)

c :

Concentration \(({\mathrm{kg}}\,{\mathrm{m}}^{-3}\))

\(c_{\mathrm{p}}\) :

Specific heat \(({\mathrm{J}}\,{\mathrm{kg}}^{-1}\,{\mathrm{K}}^{-1})\)

\(c_\infty\) :

Ambient concentration \(({\mathrm{kg}}\,{\mathrm{m}}^{-3}\))

\(c_{\mathrm{w}}\) :

Fluid wall concentration \(({\mathrm{kg}}\,{\mathrm{m}}^{-3}\))

\(C_{\mathrm{f}}\) :

Skin friction coefficient \((\frac{1+\alpha }{1+\beta }f^{{\prime}{\prime}}(0))\) (–)

Cr :

Chemical reaction parameter (–)

\(D_{\mathrm{m}}\) :

Diffusion coefficient \(({\mathrm{m}}^{2}\,{\mathrm{s}}^{-1})\)

\(D_{\mathrm{f}}\) :

Dufour number (–)

\(f(\eta )\) :

Velocity similarity function (–)

\(h_{\mathrm{f}}\) :

Convective heat transfer coefficient \(({\mathrm{W}}\,{\mathrm{m}}^{-1}\,{\mathrm{K}}^{-1})\)

k :

Thermal conductivity \(({\mathrm{W}}\,{\mathrm{m}}^{-1}\,{\mathrm{K}}^{-1})\)

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

First-order chemical reaction parameter (–)

L :

Auxiliary linear operator (–)

M :

Hartmann number (–)

N :

Nonlinear operator (–)

\(Nu_{\mathrm{x}}\) :

Nusselt number \((-(1+\frac{4}{3}R_{\mathrm{d}})\theta^{\prime}(0) )\) (–)

Pr :

Prandtl number (–)

\(q_{1}\) :

Heat flux \(({\mathrm{W}}\,{\mathrm{m}}^{-2})\)

R d :

Radiation constant (–)

Sc :

Schmidt number (–)

Sr :

Soret number (–)

\(Sh_{\mathrm{x}}\) :

Sherwood number \((-\phi^{\prime}(0))\) (–)

T :

Temperature (K)

\(T_\infty\) :

Ambient temperature (K)

\(T_{\mathrm{w}}\) :

Convective surface temperature (K)

u, v :

Velocity components in (x, y) directions \(({\mathrm{m}}\,{\mathrm{s}}^{-1})\)

\(u_{\mathrm{w}}\) :

Velocity of the sheet \(({\mathrm{m}}\,{\mathrm{s}}^{-1})\)

x, y :

Cartesian coordinates (m)

\(\alpha\) :

Dimensionless relaxation time parameter (–)

\(\beta\) :

Dimensionless retardation time parameter (–)

\(\chi_{\mathrm{m}}\) :

Auxiliary parameter (–)

\(\epsilon_1\) :

Dimensionless first-order slip velocity parameter (–)

\(\epsilon_2\) :

Dimensionless second-order slip velocity parameter (–)

\(\phi (\eta )\) :

Concentration similarity function (–)

\(\gamma\) :

Dimensionless thermal relaxation time (–)

\(\eta\) :

Similarity parameter (–)

\(\lambda_1\) :

First-order slip velocity factor

\(\lambda_2\) :

Second-order slip velocity factor

\(\nu\) :

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

\(\theta (\eta )\) :

Temperature similarity function \((-)\)

\(\rho\) :

Density \(({\mathrm{kg}}\,{\mathrm{m}}^{-3})\)

\(\sigma\) :

Electrical conductivity \(({\mathrm{S}}\,{\mathrm{m}})\)

\(\psi\) :

Stream function \(({\mathrm{m}}^{2}\,{\mathrm{s}}^{-1})\)

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Authors and Affiliations

  1. Department of Mathematics, Erode Arts and Science College, Erode, Tamilnadu, 638009, India

    K. Loganathan & S. Rajan

  2. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    S. Sivasankaran & M. Bhuvaneswari

  3. Department of Mathematics, Kongunadu Polytechnic College, D. Gudalur, Tamilnadu, India

    M. Bhuvaneswari

Authors
  1. K. Loganathan
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  2. S. Sivasankaran
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  3. M. Bhuvaneswari
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  4. S. Rajan
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Correspondence to S. Sivasankaran.

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Loganathan, K., Sivasankaran, S., Bhuvaneswari, M. et al. Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using Cattaneo–Christov heat flux with convective heating. J Therm Anal Calorim 136, 401–409 (2019). https://doi.org/10.1007/s10973-018-7912-5

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