Analytical Solutions for a General Mixed Boundary Value Problem Associated with Motions of Fluids with Linear Dependence of Viscosity on the Pressure

D. Vieru 1 , C. Fetecau 2 ,  and C. Bridges 3
  • 1 Technical University of Iasi, , Romania
  • 2 Academy of Romanian Scientists, , 050094, Bucharest, Romania
  • 3 Department of Engineering, Texas Christian University, Fort Worth, TX-76129

Abstract

An unsteady flow of incompressible Newtonian fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates is analytically studied. The fluid motion is induced by the upper plate that applies an arbitrary time-dependent shear stress to the fluid. General expressions for the dimensionless velocity and shear stress fields are established using a suitable change of independent variable and the finite Hankel transform. These expressions, that satisfy all imposed initial and boundary conditions, can generate exact solutions for any motion of this type of the respective fluids. For illustration, three special cases with technical relevance are considered and some important observations and graphical representations are provided. An interesting relationship is found between the solutions corresponding to motions induced by constant or ramptype shear stresses on the boundary. Furthermore, for validation of the results, the steady-state solutions corresponding to oscillatory motions are presented in different forms whose equivalence is graphically proved.

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  • [1] Stokes G.G. (1845): On the theories of the internal friction of fluids in motion, and motion of elastic solids. – Trans. Camb. Phil. Soc., vol.8, pp.287-305.

  • [2] Bridgman P.W. (1931): The Physics of High Pressure. – Macmillan.

  • [3] Hron J., Malek J. and Rajagopal K.R. (2001): Simple flows of fluids with pressure dependent viscosities. – Proc. Roy. Soc., London, Ser. A, Mathematical Physical and Engineering Sciences, vol.457, pp.1603-1622.

  • [4] Rajagopal K.R. and Szeri A.Z. (2003): On an inconsistency in the derivation of the equations of elastodynamic lubrication. – Proc. Roy. Soc., London, Ser. A, vol.459, pp.2771-2786.

  • [5] Vasudevaiah M. and Rajagopal K.R. (2005): On fully developed flows of fluids with a pressure dependent viscosity in a pipe. – Application of Mathematics, vol.50, pp.341-353.

  • [6] Rajagopal K.R. (2004): Couette flows of fluids with pressure dependent viscosity. – Int. J. of Applied Mechanics and Engineering, vol.9, No.3, pp.573-585.

  • [7] Kannan K. and Rajagopal K.R. (2005): Flows of fluids with pressure dependent viscosities between rotating parallel plates. – In: P. Fergola et al. (Eds.), New Trends in Mathematical Physics, World Scientific, Singapore, pp.172-183.

  • [8] Rajagopal K.R. (2008): A semi-inverse problem of flows of fluids with pressure-dependent viscosities. – Inverse Problems in Science and Engineering, vol.16, No.3, pp.269-280.

  • [9] Massoudi M. and Phuoc T.X. (2006): Unsteady shear flow of fluids with pressure-dependent viscosity. – Int. J. Eng. Sci., vol.44, pp.915-926.

  • [10] Srinivasan S. and Rajagopal K.R. (2009): Study of a variant of Stokes’ first and second problems for fluids with pressure dependent viscosities. – Int. J. Eng. Sci., vol.47, pp.1357-1366.

  • [11] Rajagopal K.R. and Saccomandi G. (2006): Unsteady exact solution for flows of fluids with pressure-dependent viscosities. – Mathematical Proceedings of the Royal Irish Academy, vol.106A, No.2, pp.115-130.

  • [12] Prusa V. (2010): Revisiting Stokes first and second problems for fluids with pressure-dependent viscosities. – Int. J. Eng. Sci., vol.48, pp.2054-2065.

  • [13] Rajagopal K.R., Saccomandi G. and Vergori L. (2013): Unsteady flows of fluids with pressure dependent viscosity. – J. Math. Anal. Appl., vol.404, pp.362-372.

  • [14] Kalogirou A., Poyiadji S. and Georgiou G.C. (2011): Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity. – J. Non-Newtonian Fluid Mech., vol.166, pp.413-419.

  • [15] Housiadas K.D., Georgiou G.C. and Tanner R.I. (2015): A note on the unbounded creeping flow past a sphere for Newtonian fluids with pressure-dependent viscosity. – Int. J. Eng. Sci., vol.86, pp.1-9.

  • [16] Renardy M. (1988): Inflow boundary condition for steady flow of viscoelastic fluids with differential constitutive laws. – Rocky Mountain Journal of Mathematics, vol.18, pp.445-453.

  • [17] Renardy M. (1990): An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions. – J. Non-Newtonian Fluid Mech., vol.36, pp.419-425.

  • [18] Waters N.D. and King M.J. (1970): Unsteady flow of an elastico-viscous liquid. – Rheol. Acta, vol.9, No.3, pp.345-355.

  • [19] Debnath L. and Bhatta D. (2007): Integral Transforms and Their Applications, Second Ed. – Chapman and Hall/CRC Press, Boca Raton.

  • [20] Toki C.J. and Tokis J.N. (2007): Exact solutions for unsteady free convection flows on a porous plate with timedependent heating – Z. Angew. Math. Mech., vol.87, No.1, pp.4-13.

  • [21] Fetecau C. and Agop M.: Exact solutions for unsteady motion between parallel plates of some fluids with powerlow dependence of viscosity on the pressure. – Sent for publication to Ann. Acad. Rom. Sci. Ser. Math. Appl.

  • [22] Hristov J. (2019): Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels. – Eur. Phys. J. Plus., vol.134, pp.293-323.

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