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Impact of Nonlinear Rosseland Approximation on Flow of Newtonian Fluid with Unequal Diffusivities of Chemically Reactive Species

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Abstract

This investigation studies the flow of a Newtonian fluid close to stagnation point past a stretching/shrinking plate embedded in a porous media. Heat transfer analysis with nonlinear thermal radiation (TR) is also carried out. Homogeneous-heterogeneous reactions with unequal diffusivities of reactants take place in the fluid. The controlling partial differential equations (DEs) of the existing model are reduced to dimensionless ordinary DEs through similarity transformations. These subsequent equations are numerically tackled with the guide of the shooting procedure. The velocity, temperature and concentration’s behaviour has been inspected for specific estimations of the including parameters. The viscous fluid (VF) temperature increases with nonlinear TR. The change in the concentration profile due to the diffusion coefficient rate \( \delta \) is more prominent in the concentration of specie D compared to specie C. Heterogeneous reaction parameter intensity is very helpful in reducing the concentration of bulk fluid \( \phi_{1}(\eta) \) as well as increasing the surface catalyst concentration \( \phi_{2}(\eta) \)

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Abbreviations

\( x,y \) :

Spatial coordinates [m]

\( q_{r} \) :

Thermal radiation

\( u,v \) :

Velocity components in \( x \) and \( y \) directions, [ms−1]

\( p \) :

Pressure \( [{\text{kgm}}^{ - 1} {\text{s}}^{ - 2} ] \)

\( \rho \) :

Density \( [{\text{kgm}}^{ - 3} ] \)

\( c,d \) :

Concentrations of species \( C \)and \( D \)

\( \theta \) :

Temperature \( [K] \)

\( k_{i} \left( {i = e,s} \right) \) :

Rate constants

\( \theta_{w} ,\theta_{\infty} \) :

Wall and free stream temperatures \( [K] \)

\( q_{r} \) :

Thermal radiation

\( \varsigma \) :

Permeability of porous zone

\( k \) :

Thermal conductivity \( [{\text{Wm}}^{ - 1} {\text{K}}^{ - 1} ] \)

\( \upsilon \) :

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

\( C_{p} \) :

Specific heat capacity \( [{\text{Jkg}}^{ - 1} {\text{K}}^{ - 1} ] \)

\( \sigma^{ * } \) :

Stephen Boltzmann constant

\( D_{C}^{*} ,D_{D}^{*} \) :

Diffusion coefficients

\( \psi \) :

Stream function

\( U_{w} ,U_{\infty } \) :

Surface and free stream velocities \( [{\text{ms}}^{ - 1} ] \)

\( \eta \) :

dimensionless variable \( [ - ] \)

\( k^{*} \) :

Wall mean absorption

\( \varTheta_{w} \) :

Temperature ratio \( [ - ] \)

\( K_{s} \) :

Heterogeneous reaction strength

\( \beta \) :

Porosity parameter \( [ - ] \)

\( Sc_{C} ,\,Sc_{D} \) :

Schmidt numbers \( [ - ] \)

\( \lambda \) :

Stretching/shrinking parameter \( [ - ] \)

\( K \) :

Homogenous reaction strength

\( \delta \) :

Ratio of diffusion coefficients \( [ - ] \)

\( Rd \) :

Radiation parameter \( [ - ] \)

\( \varTheta \) :

Dimensionless fluid temperature \( [ - ] \)

\( \Pr \) :

Prandtl number \( [ - ] \)

\( \phi_{1} ,\phi_{2} \) :

Dimensionless concentrations \( [ - ] \)

\( f,f^{\prime} \) :

Dimensionless velocities \( [ - ] \)

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Acknowledgements

We are thankful to the reviewers for their encouraging comments and constructive suggestions to improve the quality of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Sialkot, Sialkot, 51310, Pakistan

    M. Sheikh

  2. Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, 63100, Pakistan

    Z. Abbas

  3. Department of Computer Sciences, Bahria University, Islamabad, 44000, Pakistan

    J. Hasnain

  4. Department of Mathematical Sciences, Federal Urdu University of Art, Science and Technology, Islamabad, 44000, Pakistan

    A. Nadeem

  5. Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad, 44000, Pakistan

    M. Sajid

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  1. M. Sheikh
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Correspondence to J. Hasnain.

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Sheikh, M., Abbas, Z., Hasnain, J. et al. Impact of Nonlinear Rosseland Approximation on Flow of Newtonian Fluid with Unequal Diffusivities of Chemically Reactive Species. Arab J Sci Eng 46, 2711–2719 (2021). https://doi.org/10.1007/s13369-020-05216-9

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