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An analysis of Maxwell fluid through a shrinking sheet with thermal slip effect: a Lie group approach

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Abstract

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid passing through the shrinking sheet is considered. With the impact of thermal slip, thermal radiation and heat source-sink conditions, the UCM fluid model is integrated. The method of the Lie scaling group is used to transform the strongly nonlinear governing partial differential equations (PDEs) into the ordinary differential equations (ODEs). The transformed ODEs are numerically solved using NDSolve command of MATHEMATICA and graphically presented with their results. The Deborah number’s influence on the velocity profile \(f^{\prime } (\eta )\) is studied for different values and different behavior observed. The Hartmann number M and the mass transfer parameter S have decreased the boundary layer thickness. The Prandtl number has increased the temperature profile \(\theta (\eta )\). In contrast, the thermal boundary layer thickness was decreased by the heat source-sink parameter Q , the radiation parameter R and the thermal slip parameter L. Table 1 shows the verification of the results.

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Abbreviations

\(B_{0}\) :

Uniform magnetic field strength (T)

\(D_{1}\) :

Shrinking sheet constant

k :

Thermal conductivity [W/(m K)]

\(L_{1}\) :

Slip constant

\(Q_{\mathrm{o}}\) :

Heat source-sink parameter

\(R^{^{\prime }}\) :

Rate of chemical reaction

T :

Temperature of the fluid (K)

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

Temperature at the wall (K)

\(T_{\infty }\) :

Free stream temperature (K)

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

Shrinking sheet velocity (m/s)

\({\overline{u}}\) :

x velocity component (m/s)

\(V_\mathrm{r}\) :

Free stream velocity of the fluid (m/s)

\(V_{0}\) :

Free stream velocity constant (m/s)

\({\overline{v}}\) :

y velocity component (m/s)

\({\overline{x}}\) :

Direction along the sheet

\({\overline{y}}\) :

Direction perpendicular to the sheet

\(\lambda \) :

Relaxation time (s)

\(\nu \) :

Kinematic viscosity (m\(^{2}\)/s)

\(\sigma \) :

Electric conductivity

\(\sigma _{1}\) :

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

\(\rho \) :

Density (kg/m\(^{3})\)

\(\eta \) :

Similarity parameter

k :

Mean absorption coefficient (1/m)

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

Specific heat capacity

\(\gamma \) :

Thermal radiation parameter

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Acknowledgements

I am very thankful to reviewers and editor for keeping this manuscript in new shape with their useful suggestions and recommendations.

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Authors and Affiliations

  1. Department of Mathematics, University of Management and Technology Lahore, Sialkot Campus, Sialkot, 51310, Pakistan

    M. N. Tufail & M. Saleem

  2. Department of Mathematics, University of Engineering and Technology, Lahore, 54890, Pakistan

    M. Saleem & Q. A. Chaudhry

  3. Department of Mathematics, College of Science, University of Hail, Hail, Saudi Arabia

    Q. A. Chaudhry

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  1. M. N. Tufail
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  2. M. Saleem
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  3. Q. A. Chaudhry
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Correspondence to M. Saleem.

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Tufail, M.N., Saleem, M. & Chaudhry, Q.A. An analysis of Maxwell fluid through a shrinking sheet with thermal slip effect: a Lie group approach. Indian J Phys 95, 725–731 (2021). https://doi.org/10.1007/s12648-020-01745-z

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