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Stability analysis of a mixed convection flow over a moving plate with non-uniform thickness

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Abstract

This paper focuses on the effects of the boundary wall thickness and the velocity power index parameters on a steady 2D mixed convection flow along a vertical semi-infinite moving plate with non-uniform thickness. The effects of diffusion on the velocity, temperature and concentration fields with power-law temperature and concentration distributions at the plate surface are also analyzed. A shooting technique is adopted to obtain dual solutions for the system of nonlinear coupled ordinary differential equations. The significant impacts on the boundary layer development along the boundary surface have been noticed due to the non-flatness of the moving surface. It is noted that the velocity, temperature and concentration profiles admit dual solutions for certain range of unsteadiness parameter. Both, upper and lower branch, solutions are presented to display the effects of the boundary wall thickness and the velocity power index on the flow, thermal and concentration fields. A stability analysis is performed to determine the existence of dual solutions.

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Abbreviations

A :

Physical parameter related with plate surface

\(b,U_{0} \) :

Physical parameter related with moving plate surface

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

Skin friction coefficient

f :

Dimensionless streamfunction expansion

F :

Dimensionless velocity

D :

Mass diffusivity

K :

Thermal conductivity

g :

Acceleration due to gravity

\(\mathrm{Gr}_{x}, \mathrm{Gr}_{x}^{*}\) :

Local Grashof numbers

m :

velocity power index

\(\mathrm{Nu}_{x} \) :

Local Nusselt number

\(\Pr \) :

Prandtl number, \(\Pr ={\nu } /\alpha \)

\(\mathrm{Sc}\) :

Schmidt number, \(\mathrm{Sc}=\nu /D\)

\(\hbox {Re}_{x} \) :

Local Reynolds number based on x, \(\hbox {Re}_{x} ={U_{0} (x)(x+b)^{m+1}} /\nu \)

n :

Plate thickness parameter

\(\mathrm{Sh}_{x} \) :

Local Sherwood number

T :

Fluid temperature in the boundary layer

B :

Characteristic temperature

\(B^{*}\) :

Characteristic concentration

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

Velocity of the moving plate

x, y :

Cartesian coordinates measured along the surface and normal to it, respectively

\(\alpha \) :

Thermal diffusivity

\(\beta \) :

Volumetric coefficient of thermal

\(\beta ^{*}\) :

Volumetric coefficient of expansion for concentration

\(\eta \) :

Similarity variable

\(\theta \) :

Dimensionless temperature

\(\lambda \) :

Buoyancy parameter

\(\mu \) :

Dynamic viscosity

\(\nu \) :

Kinematic viscosity

\(\rho \) :

Fluid density

\(\psi \) :

Dimensional streamfunction

\(\phi \) :

Dimensionless concentration

w:

Condition at the wall

\(\infty \) :

Freestream condition

References

  1. Ali, M.E.: On thermal boundary layer on a power-law stretched surface with suction or injection. Int. J. Heat Fluid Flow. 16, 280–290 (1995)

    Google Scholar 

  2. Atlan, T., Oh, S., Gegel, H.L.: Metal Forming: Fundamentals and Applications. American society for Metals, Metals Park, OH (1983)

    Google Scholar 

  3. Elbashbeshy, E.M.A., Aldawody, D.A.: Heat transfer over an unsteady stretching surface with variable heat flux in the presence of a heat source or sink. Comput. Math. Appl. 60, 2806–2811 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Fang, T., Zhang, J., Zhong, Y.: Boundary layer flow over a stretching sheet with variable thickness. Appl. Math. Comput. 218(5), 7241–7252 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Fisher, E.G.: Extrusion of Plastics. Wiley, New York (1976)

    Google Scholar 

  6. Harris, S.D., Ingham, D.B., Pop, I.: Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman Model with slip. Transp. Por. Med. 77, 267–285 (2009)

    Google Scholar 

  7. Ibrahim, W., Shanker, B.: Unsteady boundary layer flow and heat transfer due to a stretching sheet by Quasilinearization technique. World. J. Mech. 1, 288–293 (2011)

    Google Scholar 

  8. Alam, M.S., Haque, M.M., Uddin, M.J.: Unsteady MHD free convective heat transfer flow along a vertical porous flat plate with internal heat generation. Int. J. Adv. Appl. Math. Mech. 2(2), 52–61 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Ishak, A., Nazar, R., Arifin, N.M., Pop, I.: Dual solutions in mixed convection flow near a stagnation point on a vertical porous plate. Int. J. Ther. Sci. 47, 417–422 (2008)

    Google Scholar 

  10. Ishak, A., Nazar, R., Pop, I.: Dual solutions in mixed convection flow near a stagnation point on a vertical surface in a porous medium. Int. J. Heat Mass Transf. 51, 1150–1155 (2008)

    MATH  Google Scholar 

  11. Ishak, A., Nazar, R., Pop, I.: Dual solutions in mixed convection boundary layer flow of micropolar fluids. Commun. Nonlinear Sci. Numer. Simul. 14, 1324–1333 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Ishak, A.: Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Appl. Math. Comput. 217(2), 837–842 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Khader, M.M., Megahed, A.M.: Boundary layer flow due to a stretching sheet with a variable thickness and slip velocity. J. Appl. Mech. Tech. Phys. 56(2), 241–247 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Khan, W.A., Pop, I.: Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf. 53, 2477–2483 (2010)

    MATH  Google Scholar 

  15. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transf. 48, 2529–2539 (2005)

    MATH  Google Scholar 

  16. Makinde, O.D., Aziz, A.: Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int. J. Ther. Sci. 50, 1326–1332 (2011)

    Google Scholar 

  17. Alam, M.S., Haque, M.M., Uddin, M.J.: Convective flow of nanofluid along a permeable stretching/shrinking wedge with second order slip using Buongiorno’s mathematical model. Int. J. Adv. Appl. Math. Mech. 3(3), 79–91 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Merkin, J.H.: On dual solutions occuring in mixed convection in a porous medium. J. Eng. Math. 20(2), 171–179 (1986)

    MathSciNet  MATH  Google Scholar 

  19. Chand, R., Rana, G.C.: Double diffusive convection in a layer of Maxwell viscoelastic fluid in porous medium in the presence of Soret and Dufour effects. J. fluids. 2014, 479107 (2014)

    Google Scholar 

  20. Alam, M.S., Rahman, M.M.: Dufour and Soret effects on MHD free convective heat and mass transfer flow past a vertical porous flat plate embedded in a porous medium. J. Nav. Arch. Mar. Eng. 2(1), 55–65 (2005)

    Google Scholar 

  21. Ridha, A.: Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations. Z. Angew. Math. Phys (ZAMP) 47(3), 341–352 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Schlichting, H., Gersten, K.: Boundary Layer Theory. Springer, New York (2000)

    MATH  Google Scholar 

  23. Shufrin, I., Eisenberger, M.: Stability of variable thickness shear deformable plates-first order and high order analyses. Thin-Walled Struct. 43(2), 189–207 (2005)

    MATH  Google Scholar 

  24. Soundalgekar, V.M., Ramana Murty, T.V.: Heat transfer in flow past a continuous moving plate with variable temperature. Springer 14, 91–93 (1980)

    Google Scholar 

  25. Subhashini, S.V., Samuel, N., Pop, I.: Double-diffusive convection from a permeable vertical surface under convective boundary condition. Int. Commun. Heat Mass Transf. 38(9), 1183–1188 (2011)

    Google Scholar 

  26. Subhashini, S.V., Nancy, Samuel, Pop, I.: Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface. Int. Commun. Heat Mass Transf. 38(4), 499–503 (2011)

    Google Scholar 

  27. Subhashini, S.V., Sumathi, R., Pop, I.: Dual solutions in a double diffusive convection near stagnation point region over a stretching vertical surface. Int. J. Heat Mass Transf. 55(9–10), 2524–2530 (2012)

    Google Scholar 

  28. Tadmor, Z., Klein, I.: Engineering Principles of Plasticating Extrusion. Polymer science and engineering series. Van Nostrand, Reinhold, New York (1970)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Anna University, Chennai, India

    S. V. Subhashini & S. Lenin Sindhu

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    K. Vajravelu

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  1. S. V. Subhashini
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  3. K. Vajravelu
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Subhashini, S.V., Sindhu, S.L. & Vajravelu, K. Stability analysis of a mixed convection flow over a moving plate with non-uniform thickness. Arch Appl Mech 90, 1497–1507 (2020). https://doi.org/10.1007/s00419-020-01680-9

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