Abstract
In the present work, the exact solutions of Stokes second problem for a Burgers’ fluid are investigated. The expressions for the velocity field and the corresponding tangential stress are obtained when the relaxation times satisfy the condition γ < λ2/4. The solutions have been determined by means of Laplace transform. Only one initial condition is necessary for velocity and these solutions presented in the forms of simple or multiple integrals in terms of Bessel functions. The corresponding solutions for a Newtonian fluid as well as Oldroyd-B fluid appear as the limiting cases of the presented results. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. Moreover, a comparison for velocity is made with Oldroyd-B and Newtonian fluids.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fosdick R.L., Rajagopal K.R.: Anomalous features in the model of second order fluids. Arch. Rat. Mech. Anal. 70, 145–152 (1978)
Bandelli R., Rajagopal K.R.: Start-up flows of second grade fluid in domain with one finite dimension. Int. J. Non-Linear Mech. 30, 817–839 (1995)
Rajagopal K.R.: A note on unsteady unidirectional flows of a non-Newtonian fluid. Int. J. Non-Linear Mech. 17, 369–373 (1982)
Hayat T., Khan M., Ayub M.: On the explicit analytic solutions of an Oldroyd 6-constant fluid. Int. J. Eng. Sci. 42, 123–135 (2004)
Hayat T., Khan M., Asghar S.: Homotopy analysis of MHD flow of an Oldroyd 8-constant fluid. Acta Mech. 168, 213–232 (2004)
Hayat T., Khan M.: Homotopy solutions for a generalized second grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005)
Hayat T., Abbas Z., Sajid M., Asghar S.: The influence of thermal radiation on MHD flow of a second grade fluid. Int. J. Heat Mass Transf. 50, 931–941 (2007)
Hayat T., Ellahi R., Asghar S.: Unsteady magnetohydrodynamic non-Newtonian flow due to non-coaxial rotations of disk and a fluid at infinity. Chem. Eng. Commun. 189, 37–49 (2007)
Fetecau C., Corina Fetecau, Vieru D.: On some helical flows of Oldroyd-B fluids. Acta Mech. 189, 53–63 (2007)
Erdogan M.E., Imrak C.E.: On some unsteady flows of a non-Newtonian fluids. Appl. Math. Model. 31, 170–180 (2007)
Burgers J.M.: Mechanical considerations-model systems-phenomenological theories of relaxation of viscosity. In: Burgers, J.M. (eds) First Report on Viscosity and Plasticity, Nordemann Publishing company, New York (1935)
Krishnan J.M., Rajagopal K.R.: A thermodynamic framework for the constitutive modeling of asphalt concrete: theory and application. J. Mater. Civil Eng. 16, 155–166 (2004)
Lee A.R., Markwick A.H.D.: The mechanical properties of bituminous surfacing materials under constant stress. J. Soc. Chem. Ind. 56, 146–156 (1937)
Tan B.H., Jackson I., Gerald J.D.F.: High-temperature viscoelasticity of sine-grained polycrystalline olivine. Phys. Chem. Miner. 28, 641–664 (2001)
Yuen D.A., Pelter W.R.: Normal modes of the viscoelastic earth. Geophys. J. R. Astronom. Soc. 87, 1113–1141 (1986)
Ravindran P., Krishnan J.M., Rajagopal K.R.: A note on the flow of a Burgers’ fluid in an orthogonal rheometer. Int. J. Eng. Sci. 42, 1973–1985 (2004)
Fetecau C., Vieru D., Hayat T., Fetecau C.: On the first problem of Stokes for Burgers’ fluid, I. Case γ < λ2/4. Nonlinear Anal. Real World Appl. 10, 2183–2194 (2009)
Khan M., Ali S.H., Fetecau C.: Exact solutions of accelerated flows for a Burgers’ fluid. I. The case γ < λ2/4. Appl. Math. Comput. 203, 881–894 (2008)
Hayat T., Maqbool M., Khan M.: Hall and heat transfer effects on the steady flow of a generalized Burgers’ fluid induced by sudden pull of eccentric rotating disks. Nonlinear Dyn. 51, 267–276 (2008)
Hayat T., Hussain M., Khan M.: Effects of Hall current on the flows of a Burger fluid through a porous medium. Transp. Porous Med. 68, 249–263 (2007)
Rajagopal K.R., Bhatnagar R.K.: Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech. 113, 233–239 (1995)
Chen C.I., Chen C.K., Yang Y.T.: Unsteady unidirectional flow of a second grade fluid between the parallel plates with different given volume flow rate conditions. Appl. Math. Comput. 137, 437–450 (2003)
Tan W.C., Masuoka T.: Stokes’ first problem for an Oldroyd-B fluid in a porous half space. Phys. Fluids 17, 023101 (2005)
Khan M., Ali S.H., Fetecau C., Hayat T.: MHD flow of a second grade fluid between two side walls perpendicular to a plate through a porous medium. Int. J. Non-Linear Mech. 43, 302–319 (2008)
Khan M., Naheed E., Fetecau C., Hayat T.: Exact solutions of arbitrary flows for second grade fluid in a porous medium. Int. J. Non-Linear Mech. 43, 868–879 (2008)
Khan M., Ali S.H., Qi H.: Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders. Nonlinear Anal. Real World Appl. 10, 1775–1783 (2009)
Qi H.T., Xu M.Y.: Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model. Acta Mech. Sin. 23, 463–469 (2007)
Qi H.T., Xu M.Y.: Unsteady flow of viscoelastic fluid with fractional Maxwell model. Mech. Res. Commun. 34, 210–212 (2007)
Roberts G.E., Kaufman H.: Table of Laplace Transform. W.B. Saunders Company, Philadelphia (1968)
Fetecau C., Vieru D., Fetecau C.: A note on the second problem of Stokes for Newtonian fluids. Int. J. Non-Linear Mech. 43, 451–457 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khan, M., Anjum, A. & Fetecau, C. On exact solutions of Stokes second problem for a Burgers’ fluid, I. The case γ < λ2/4. Z. Angew. Math. Phys. 61, 697–720 (2010). https://doi.org/10.1007/s00033-009-0025-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-009-0025-z