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Three-dimensional unsteady flow of Maxwell fluid with homogeneous–heterogeneous reactions and Cattaneo–Christov heat flux

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Abstract

This article addresses the unsteady three-dimensional flow of Maxwell fluid. Flow is induced by a bidirectional stretching surface. Fluid fills the porous space. Thermal relaxation time is examined using Cattaneo–Christov heat flux. Homogeneous–heterogeneous reactions are also considered. Suitable transformations are used to convert partial differential equations into nonlinear ordinary differential equations. Convergent series solutions are obtained. Effects of appropriate parameters on the velocity, temperature and concentration fields are examined. It is found that increasing value of Deborah number decreases the fluid flow. Larger values of strength of homogeneous reaction parameter decrease the concentration distribution. Also temperature is decreasing function of thermal relaxation time. Present problem is of great interest in biomedical, industrial and engineering applications like food processing, clay coatings, hydrometallurgical industry, fog formation and dispersion.

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Abbreviations

u,  v,  w :

Velocity components along x-, y- and z-axes, respectively (ms\(^{-1}\) )

T :

Temperature (K)

\(T_{w}\) :

Surface temperature (K)

\(T_{\infty }\) :

Ambient fluid temperature (K)

\(k_{c},\) \(k_{s}\) :

Rate constant

A,  B :

Chemical species

k :

Thermal conductivity (WK\(^{-1}\)  m\(^{-1}\) )

\({\hat{k}}\) :

Permeability (m\(^{2}\) )

c,  d :

Stretching constants (s\(^{-1})\)

a, b :

Concentrations of the species A and B

\({\mathbf {q}}\) :

Specific heat flux

\(a_{0}\) :

Positive dimensional constant

\(C_{p}\) :

Specific heat (m\(^{2}\)  s\(^{-2}\) )

\(C_{{\text {f}}x},\) \(C_{{\text {f}}y}\) :

Local skin friction coefficient along x- and y-axes, respectively

\(D_{A}\), \(D_{B}\) :

Diffusion species coefficients

Pr :

Prandtl number

\(u_{w}\) :

Stretching sheet velocity along x-axis (ms\(^{-1})\)

\(v_{w}\) :

Stretching sheet velocity along y-axis (ms\(^{-1})\)

\(Re_{x}, {Re}_{y}\) :

Local Reynolds number

\(A_{1}\) :

Unsteady parameter

Sc :

Schmidt number

K :

Strength of the homogeneous reaction

\(K_{1}\) :

Strength of the heterogeneous reaction

\(\mu\) :

Viscosity (kg m\(^{-2}\,\) s\(^{-1}\) )

\(\upsilon\) :

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

\(\rho\) :

Density (kg m\(^{-3}\) )

\(\lambda _{1}\) :

Heat flux relaxation time

\(\lambda\) :

Retardation time

\(\theta\) :

Dimensionless temperature

\(\xi\) :

Transformed coordinate

\(\alpha\) :

Time constant (s\(^{-1}\) )

\(k_{1}\) :

Porosity parameter

\(\beta _{1}\) :

Deborah number

\(\tau _{{\text {w}}x}\), \(\tau _{{\text {w}}y}\) :

Wall shear stress

\({\mathcal {\gamma }}\) :

Thermal relaxation parameter

\({\mathcal {\beta }}_{2}\) :

Ratio of stretching rates

\(\delta\) :

Ratio of diffusion coefficient

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Authors and Affiliations

  1. Department of Mathematics, University of Wah, Wah Cantt, 47040, Pakistan

    Maria Imtiaz

  2. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad, 44000, Pakistan

    Asmara Kiran & Tasawar Hayat

  3. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Tasawar Hayat & Ahmed Alsaedi

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  1. Maria Imtiaz
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  2. Asmara Kiran
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Correspondence to Maria Imtiaz.

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Technical Editor: Cezar Negrao.

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Imtiaz, M., Kiran, A., Hayat, T. et al. Three-dimensional unsteady flow of Maxwell fluid with homogeneous–heterogeneous reactions and Cattaneo–Christov heat flux. J Braz. Soc. Mech. Sci. Eng. 40, 449 (2018). https://doi.org/10.1007/s40430-018-1360-9

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