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Unsteady MHD Free Convective Flow of a Visco-Elastic Fluid Past an Infinite Vertical Porous Moving Plate with Variable Temperature and Concentration

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Abstract

The present study considers the effect unsteady MHD free convective flow of a viscoelastic incompressible electrically conducting fluid past a moving vertical plate through a porous medium with time dependent oscillatory permeability and suction in presence of a uniform transverse magnetic field and heat source and chemical reaction along with heat and mass transfer are reported. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The governing equations of the fluid flow, heat and mass transfer are solved by applying multi parameter perturbation technique. Comparison with previously published work have been conducted and the results are found to be in concordance with the previous study. A parametric study is performed on the influence of the visco-elastic fluid parameter, the magnetic field parameter, the permeability parameter, on the fluid velocity. The expressions for transient velocity, temperature, species concentration and non-dimensional skin friction at the plate are illustrated through tables to observe the visco-elastic effect in combination of other flow parameters involved in the solution.

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Acknowledgements

Authors are grateful to the reviewers for their valuable comments and suggestions which helped them to improve the quality of the research paper.

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Authors and Affiliations

  1. Department of Mathematics, Sree Vidyanikethan Engineering College (Autonomous), Tirupati, A. P., India

    K. Ramesh Babu, A. Parandhama & K. Venkateswara Raju

  2. Department of Mathematics, Annamacharya Institute of Technology and Science, Rajampet, A. P., India

    M. C. Raju

  3. Department of Mathematics, VIT University, Vellore, T. N., India

    P. V. Satya Narayana

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  1. K. Ramesh Babu
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  2. A. Parandhama
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  3. K. Venkateswara Raju
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  5. P. V. Satya Narayana
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Correspondence to K. Ramesh Babu.

Appendix

Appendix

$$\begin{aligned} \hbox {m}_1= & {} \frac{1}{2}\left( {\hbox {S}_{\mathrm{c}} +\sqrt{\hbox {S}_{\mathrm{c}}^{2}+4\hbox {S}_{\mathrm{c}} {\upgamma }}}\right) ,\quad \hbox {m}_2 =\frac{1}{2}\left( {\hbox {S}_{\mathrm{c}} +\sqrt{\hbox {S}_{\mathrm{c}}^{2}+4\hbox {S}_{\mathrm{c}} \left( {{\upgamma }+\hbox {n}}\right) }} \right) , \\ \hbox {m}_3= & {} \frac{1}{2}\left( {\hbox {P}_{\mathrm{r}} +\sqrt{\hbox {P}_{\mathrm{r}}^{2}+4\hbox {P}_{\mathrm{r}} {\upphi }}}\right) , \quad \hbox {m}_4 =\frac{1}{2}\left( {\hbox {P}_{\mathrm{r}} +\sqrt{\hbox {P}_{\mathrm{r}}^{2}+4\hbox {P}_{\mathrm{r}} \left( {{\upphi } +\hbox {n}} \right) }} \right) \\ \hbox {m}_{5}= & {} \frac{1}{2}\left( {1+\sqrt{1+4\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }} \right) , \quad \hbox {m}_6 =\frac{1}{2}\left( {1+\sqrt{1+4\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }} \right) \\ \hbox {m}_7= & {} \frac{1}{2}\left( {1+\sqrt{1+4\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }} \right) ,\quad \hbox {m}_8=\frac{1}{2}\left( {1+\sqrt{1+4\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }} \right) \\ \hbox {A}_1= & {} \frac{\hbox {AS}_{\mathrm{c}} \hbox {m}_1 }{\hbox {m}_1^2 -\hbox {S}_{\mathrm{c}} \hbox {m}_1 -\hbox {S}_{\mathrm{c}} \left( {{\upgamma } +\hbox {n}}\right) },\quad \hbox {A}_2=1-\hbox {A}_1 , \quad \hbox {A}_3 =\frac{-\hbox {Q}_1 \hbox {P}_{\mathrm{r}} }{\hbox {m}_1^2 -\hbox {P}_{\mathrm{r}} \hbox {m}_1-\hbox {P}_{\mathrm{r}} {\upphi } }, \hbox {A}_4 =1-\hbox {A}_3 , \\ \hbox {A}_5= & {} \frac{\hbox {A}\hbox {P}_{\mathrm{r}} \hbox {A}_4 \hbox {m}_3 }{\hbox {m}_3^2 -\hbox {P}_{\mathrm{r}} \hbox {m}_3 -\hbox {P}_{\mathrm{r}} \left( {{\upphi } +\hbox {n}} \right) }, \quad \hbox {A}_6 =\frac{\hbox {A}\hbox {P}_{\mathrm{r}} \hbox {A}_3 \hbox {m}_1 }{\hbox {m}_1^2 -\hbox {P}_{\mathrm{r}} \hbox {m}_1 -\hbox {P}_{\mathrm{r}} \left( {{\upphi }+\hbox {n}} \right) }, \quad \hbox {A}_7 =\frac{-\hbox {Q}_1 \hbox {P}_{\mathrm{r}} \hbox {A}_2 }{\hbox {m}_2^2 -\hbox {P}_{\mathrm{r}} \hbox {m}_2 -\hbox {P}_{\mathrm{r}} \left( {{\upphi } +\hbox {n}} \right) }\\ \hbox {A}_8= & {} \frac{-\hbox {Q}_1\hbox {P}_{\mathrm{r}} \hbox {A}_1 }{\hbox {m}_1^2 -\hbox {P}_{\mathrm{r}} \hbox {m}_1-\hbox {P}_{\mathrm{r}} \left( {{\upphi } +\hbox {n}} \right) }, \quad \hbox {A}_9 =\hbox {A}_6 +\hbox {A}_8 , \hbox {A}_{10} =\hbox {A}_5 +\hbox {A}_7 +\hbox {A}_9 \quad \hbox {A}_{11} =1-\hbox {A}_{10} , \\ \hbox {A}_{12}= & {} \frac{-\hbox {G}_{\mathrm{r}} \hbox {A}_4 }{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }, \quad \hbox {A}_{13} =\frac{-\hbox {G}_{\mathrm{r}}\hbox {A}_3}{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }, \quad \hbox {A}_{14} =\frac{-\hbox {G}_{\mathrm{m}} }{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}}\right) }, \\ \hbox {A}_{15}= & {} \hbox {A}_{13} +\hbox {A}_{14}, \hbox {A}_{16} =\hbox {u}_{\mathrm{p}} -\left( {\hbox {A}_{12}+\hbox {A}_{15} } \right) , \quad \hbox {A}_{17} =\frac{\hbox {A}_{16} \left( {\hbox {m}_5 } \right) ^{3}}{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }, \\ \hbox {A}_{18}= & {} \frac{\hbox {A}_{12} \left( {\hbox {m}_3}\right) ^{3}}{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}} \right) }, \quad \hbox {A}_{19} =\frac{\hbox {A}_{15}\left( {\hbox {m}_1}\right) ^{3}}{\hbox {m}_1^2 -\hbox {m}_1-\left( {\hbox {M}+\frac{1}{\hbox {K}}}\right) } \hbox {A}_{20} =-\left( {\hbox {A}_{17} +\hbox {A}_{18} +\hbox {A}_{19}}\right) ,\\ \hbox {A}_{21}= & {} \frac{-\hbox {Gr}\hbox {A}_{11} }{\hbox {m}_4^2 -\hbox {m}_4-\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}}\right) },\quad \hbox {A}_{22}=\frac{- \hbox {Gr }\hbox {A}_5 }{\hbox {m}_{3}^{2}-\hbox {m}_3-\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}}\right) }, \\ \hbox {A}_{23}= & {} \frac{-\hbox {Gr } \hbox {A}_7}{\hbox {m}_2^2 -\hbox {m}_2-\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\quad \hbox {A}_{24} =\frac{-\hbox {Gr } \hbox {A}_9 }{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \end{aligned}$$
$$\begin{aligned} \hbox {A}_{25}= & {} \frac{-\hbox {Gm }\hbox {A}_2 }{\hbox {m}_2^2 -\hbox {m}_2 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{26} =\frac{-\hbox {Gm } \hbox {A}_1 }{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }\\ \hbox {A}_{27}= & {} \frac{\hbox {A}\hbox {A}_{16} \hbox {m}_5 }{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{28} =\frac{\hbox {A}\hbox {A}_{12} \hbox {m}_3 }{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\\ \hbox {A}_{29}= & {} \frac{\hbox {A}\hbox {A}_{15} \hbox {m}_1 }{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{30} =\hbox {A}_{22} +\hbox {A}_{28} \quad \hbox {A}_{31} =\hbox {A}_{23} +\hbox {A}_{25} \\ \hbox {A}_{32}= & {} \hbox {A}_{24} +\hbox {A}_{26} +\hbox {A}_{29} \quad \hbox {A}_{33} =-\left( {\hbox {A}_{27} +\hbox {A}_{21} +\hbox {A}_{30} +\hbox {A}_{31} +\hbox {A}_{32} } \right) , \\ \hbox {A}_{34}= & {} \frac{\hbox {A}\hbox {A}_{16} \left( {\hbox {m}_5}\right) ^{3}}{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{35} =\frac{\hbox {A}\hbox {A}_{12} \left( {\hbox {m}_3 } \right) ^{3}}{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\\ \hbox {A}_{36}= & {} \frac{\hbox {A}\hbox {A}_{15} \left( {\hbox {m}_1 } \right) ^{3}}{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{37} =\frac{\hbox {A}\hbox {A}_{20} \hbox {m}_6 }{\hbox {m}_6^2 -\hbox {m}_6 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{38}= & {} \frac{\hbox {A}\hbox {A}_{17} \hbox {m}_5 }{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) } ,\quad \hbox {A}_{39} =\frac{\hbox {A}\hbox {A}_{18} \hbox {m}_3 }{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{40}= & {} \frac{\hbox {A}\hbox {A}_{19} \hbox {m}_1 }{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{41}=\frac{\hbox {A}_{33} \left( {\hbox {m}_7 } \right) ^{3}}{\hbox {m}_7^2 -\hbox {m}_7 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{42}= & {} \frac{\hbox {A}_{27} \left( {\hbox {m}_5 } \right) ^{3}}{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\quad \hbox {A}_{43}=\frac{\hbox {A}_{21} \left( {\hbox {m}_4 } \right) ^{3}}{\hbox {m}_4^2 -\hbox {m}_4 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{44}= & {} \frac{\hbox {A}_{30} \left( {\hbox {m}_3 } \right) ^{3}}{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{45} =\frac{\hbox {A}_{31} \left( {\hbox {m}_2 } \right) ^{3}}{\hbox {m}_2^2 -\hbox {m}_2 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{46}= & {} \frac{\hbox {A}_{32} \left( {\hbox {m}_1}\right) ^{3}}{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{47} =\frac{\hbox {n}\hbox {A}_{33}\left( {\hbox {m}_7} \right) ^{2}}{\hbox {m}_7^2 -\hbox {m}_7 \left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\\ \hbox {A}_{48}= & {} \frac{\hbox {n}\hbox {A}_{27} \left( {\hbox {m}_5 } \right) ^{2}}{\hbox {m}_5^2 -\hbox {m}_5 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{49} =\frac{\hbox {n}\hbox {A}_{21} \left( {\hbox {m}_4 } \right) ^{2}}{\hbox {m}_4^2 -\hbox {m}_4 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) },\\ \hbox {A}_{50}= & {} \frac{\hbox {n}\hbox {A}_{30} \left( {\hbox {m}_3 } \right) ^{2}}{\hbox {m}_3^2 -\hbox {m}_3 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{51} =\frac{\hbox {n}\hbox {A}_{31} \left( {\hbox {m}_2} \right) ^{2}}{\hbox {m}_2^2 -\hbox {m}_{2}-\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \\ \hbox {A}_{52}= & {} \frac{\hbox {n}\hbox {A}_{32} \left( {\hbox {m}_1 } \right) ^{2}}{\hbox {m}_1^2 -\hbox {m}_1 -\left( {\hbox {M}+\frac{1}{\hbox {K}}+\hbox {n}} \right) }, \quad \hbox {A}_{53} =\hbox {A}_{41} +\hbox {A}_{47} , \\ \hbox {A}_{54}= & {} \hbox {A}_{34} +\hbox {A}_{38} +\hbox {A}_{42} +\hbox {A}_{48} , \hbox {A}_{55} =\hbox {A}_{43} +\hbox {A}_{49} , \quad \hbox {A}_{56} =\hbox {A}_{35} +\hbox {A}_{39} +\hbox {A}_{44} +\hbox {A}_{50}, \\ \hbox {A}_{57}= & {} \hbox {A}_{45} +\hbox {A}_{51} , \quad \hbox {A}_{58} =\hbox {A}_{36} +\hbox {A}_{40} +\hbox {A}_{46} +\hbox {A}_{52},\\ \hbox {A}_{59}= & {} -\left( {\hbox {A}_{53} +\hbox {A}_{37} +\hbox {A}_{54}+\hbox {A}_{55} +\hbox {A}_{56} +\hbox {A}_{57} +\hbox {A}_{58}}\right) . \end{aligned}$$

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Babu, K.R., Parandhama, A., Raju, K.V. et al. Unsteady MHD Free Convective Flow of a Visco-Elastic Fluid Past an Infinite Vertical Porous Moving Plate with Variable Temperature and Concentration. Int. J. Appl. Comput. Math 3, 3411–3431 (2017). https://doi.org/10.1007/s40819-017-0306-8

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