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Flow due to an infinite flat plate suddenly set into motion in a viscoelastic fluid: first Stokes problem

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Abstract

The problem of flow and heat transfer due to an infinite flat surface suddenly set into motion in an unbounded mass of viscoelastic fluid is investigated under the consideration of time-dependent temperature distribution along the plate surface. A new type of similarity solution is devised that converts the governing partial differential equations into a set of non-linear ordinary differential equations with four physical parameters, viz., viscoelastic parameter k, unsteadiness parameter \(\beta \), Eckert number E and Prandtl number Pr. These equations are then solved numerically by finite-difference method after using the perturbation technique owing to the inherent unavailability of the necessary boundary conditions for solving this type of flow problem. The influences of these parameters on this flow dynamics are graphically analysed. The present analysis discloses that both the velocity and temperature at a given location decrease with the increase of the elasticity in the fluid as well as the unsteadiness of the flow field. The analysis reveals that the elastic property of the fluids causes the back-flow inside the boundary layer after a certain value of the unsteadiness parameter depending upon the presence of elasticity in the fluids. Another important result of this study that comes from the heat transfer analysis is that the elasticity of the fluids reduces the severity of the unwanted effect of the viscous heating.

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Abbreviations

A :

temporal variation of the fluid velocity and wall temperature

C :

dimensionless positive constant

\(c_p\) :

specific heat

\(C_f\) :

skin-friction coefficient

\(e_{ik}\) :

rate-of-strain tensor

E :

Eckert number

\(E_{cr}\) :

critical value of E

f :

similarity function

k :

viscoelastic parameter

\(k_0\) :

viscoelastic coefficient of the fluid

\(Nu_x\) :

local Nusselt number

p :

fluid pressure

Pr :

Prandtl number

\(Re_x\) :

local Reynolds number

t :

time

\(t_{0}\) :

constant reference value of t

T :

fluid temperature

\(T_{0}\) :

constant reference value of T

\(T_w\) :

wall temperature

\(T_{\infty }\) :

ambient fluid temperature

u, v :

velocity components along and normal to the plate surface

U :

plate velocity

\(U_0\) :

velocity scale

x, y :

coordinates along and normal to the plate surface

\(\beta \) :

unsteadiness parameter

\(\lambda \) :

thermal conductivity of the fluid

\(\delta \) :

shear layer thickness (depth of penetration)

\(\eta \) :

dimensionless distance normal to the plate surface

\(\eta _s\) :

corresponding value of \(\eta \) where the wall effect on the fluid has dropped to one percent

\(\mu \) :

dynamic coefficient of viscosity

\(\nu \) :

kinematic coefficient of viscosity

\(\psi \) :

stream function of the fluid

\(\rho \) :

density of the fluid

\(\tau _{ij}\) :

stress tensor

\(\theta \) :

dimensionless temperature of the fluid

0:

reference

cr :

critical

w :

wall

\({\prime }\) :

differentiation with respect to \(\eta \)

\({{\varvec{\dot{}}}}\) :

differentiation with respect to \({{\bar{t}}}\)

\({{\varvec{\bar{}}}}\) :

dimensionless quantities

References

  1. Stokes G G 1856 On the effect of the inertial friction of fluids on the motion of pendulums. Trans. Cambr. Philos. Soc. 9: 8–106

    Google Scholar 

  2. Schlichting H and Gersten K 2000 Boundary-layer theory. Berlin–Heidelberg: Springer-Verlag, pp. 126–128

    Book  Google Scholar 

  3. Currie I G 1993 Fundamental mechanics of fluids. New York: McGraw-Hill, pp. 224–228

    Google Scholar 

  4. Rosenhead L 1963 Laminar boundary layers. Oxford: Clarendon Press, pp. 349–408

    MATH  Google Scholar 

  5. Telionis D P 1981 Unsteady viscous flows. New York: Springer, pp. 79–102

    Book  Google Scholar 

  6. Panton R I 1995 Incompressible flow. New York: Wiley, pp. 166–172

    Google Scholar 

  7. Riley N 1975 Unsteady laminar boundary layers. SIAM Rev. 17: 274–297

    Article  Google Scholar 

  8. Wang C Y 1985 The unsteady oblique stagnation-point flow. Phys. Fluids 28: 2046–2049

    Article  Google Scholar 

  9. Ma K H P and Hui W H 1990 Similarity solutions of the two-dimensional unsteady boundary-layer equations. J. Fluid Mech. 216: 537–559

    Article  MathSciNet  Google Scholar 

  10. Dholey S and Gupta A S 2013 Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate. Phys. Fluids 25: 1–18

    Article  Google Scholar 

  11. Dholey S 2015 The boundary layers of an unsteady separated stagnation-point flow of a viscous incompressible fluid over a moving plate. Fluid Dyn. Res. 47: 1–22

    Article  MathSciNet  Google Scholar 

  12. Dholey S 2018 Unsteady separated stagnation-point flow and heat transfer of a viscous fluid over a moving flat surface. Phys. Fluids 30: 1–11

    Article  Google Scholar 

  13. Dholey S 2019 An unsteady separated stagnation-point flow towards a rigid flat plate. ASME J. Fluids Eng. 141: 1–11

    Google Scholar 

  14. Dholey S 2019 Unsteady separated stagnation-point flow over a permeable surface. Z. Angew. Math. Phys. 70:10 : 1–19.

    MathSciNet  MATH  Google Scholar 

  15. Rossow V J 1960 On Rayleigh’s problem in magnetohydrodynamics. Phys. Fluids 3: 395–398

    Article  MathSciNet  Google Scholar 

  16. Soundalgekar V M 1965 Hydromagnetic flow near an accelerated plate in the presence of a magnetic field. Appl. Sci. Res. 12: 151–156

    Google Scholar 

  17. Pop I 1967 On the hydromagnetic flow due to an accelerated plate. Rev. Roum. Phys. 12: 865–867

    Google Scholar 

  18. Beard D W and Walters K 1964 Elastico-viscous boundary-layer flows. Proc. Camb. Philos. Soc. 60: 667–674

    Article  Google Scholar 

  19. Tanner R I 1962 Note on the Rayleigh problem for a visco-elastic fluid. Z. Angew. Math. Phys. 13: 573–580

    Article  MathSciNet  Google Scholar 

  20. Teipel I 1981 The impulsive motion of a flat plate in a visco-elastic fluid. Acta Mech. 39: 277–279

    Article  Google Scholar 

  21. Rajagopal K R and Na T Y 1983 On Stokes problem for a non-Newtonian fluid. Acta Mech. 48: 233–239

    Article  Google Scholar 

  22. Phan-Thien N and Chew Y T 1988 On the Rayleigh problem for a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 28: 117–127

    Article  Google Scholar 

  23. Fetecau C and Fetecau C 2003 A new exact solution for the flow of a Maxwell fluid past an infinite plate. Int. J. Non-Lin. Mech. 38: 423–427

    Article  MathSciNet  Google Scholar 

  24. Fetecau C and Zierep J 2003 The Rayleigh–Stokes-problem for a Maxwell fluid. Z. Angew. Math. Phys. 54: 1086–1093

    Article  MathSciNet  Google Scholar 

  25. Christov C I and Jordan P M 2009 Comment on “ Stokes’ first problem for an Oldroyd-B fluid in a porous half space” [Phys. Fluids 2005 17: 023101]. Phys. Fluids 21: 1–5

  26. Khan M 2013 Stokes’ first problem for an MHD Burgers fluid. Commun. Theor. Phys. 59: 99–104

    Article  Google Scholar 

  27. Choubey K R 1985 The impulsive motion of a flat plate in a viscoelastic fluid in the presence of a transverse of a magnetic fluid. Indian J. Pure Appl. Math. 19: 931–937

    MATH  Google Scholar 

  28. Mahapatra T R, Dholey S and Gupta A S 2007 Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid towards a stretching surface. Meccanica 42: 263–272

    Article  Google Scholar 

  29. Mahapatra T R, Dholey S and Gupta A S 2007 Oblique stagnation-point flow of an incompressible visco-elastic fluid towards a stretching surface. Int. J. Non-Lin. Mech. 42: 484–499

    Article  Google Scholar 

  30. Dholey S 2019 Unsteady hydromagnetic flow and heat transfer of a viscous fluid near a suddenly accelerated flat surface. Sadhana 44(3): 68. https://doi.org/10.1007/s12046-018-1044-2

    Article  Google Scholar 

  31. Fletcher C A J 1998 Computational techniques for fluid dynamics, vol. 1. Berlin–Heidelberg–New York: Springer

    Book  Google Scholar 

  32. Dholey S 2017 Instabilities of a thin viscoelastic liquid film flowing down an inclined plane in the presence of a uniform electromagnetic field. Rheol. Acta 56: 325–340

    Article  Google Scholar 

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Acknowledgements

The author is very thankful to the editors and referees for their constructive suggestions that essentially improved the presentation of the paper. The author would also like to thank Shreya Dholey and Anita Dholey for their kind cooperation during the work. This work has been supported by SERB (No. EMR/2016/005533) of India.

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  1. Department of Mathematics, Maharajadhiraj Uday Chand Women’s College, Burdwan, 713 104, India

    S Dholey

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Dholey, S. Flow due to an infinite flat plate suddenly set into motion in a viscoelastic fluid: first Stokes problem. Sādhanā 44, 118 (2019). https://doi.org/10.1007/s12046-019-1098-9

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