The Rayleigh–Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model

This paper investigates the exact analytic solutions for the Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid. This paper employs the fractional calculus approach to study the flows in an Oldroyd-B fluid. The velocity field corresponding to an incompressible generalized Oldroyd-B fluid with a fractional derivative model within an infinite edge is determined using Fourier sine and Laplace transforms. Two characteristic examples: (i) flow due to an impulsive motion of edge, and (ii) flow due to a uniformly accelerated edge are considered. The solutions that have been obtained reduce to the known solutions of an Oldroyd-B fluid by setting α=β=1. Moreover, the similar solutions for Maxwell and second grade fluids with fractional derivative models and those for the ordinary models appear as the limiting cases of the presented solutions.

[1]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[2]  Constantin Fetecau,et al.  The first problem of Stokes for an Oldroyd-B fluid , 2003 .

[3]  C. Y. Wang,et al.  Onset of oscillatory convection in a porous cylinder saturated with a viscoelastic fluid , 2007 .

[4]  Kumbakonam R. Rajagopal,et al.  A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid , 2007 .

[5]  H. Ersoy,et al.  MHD flow of an Oldroyd-B fluid due to non-coaxial rotations of a porous disk and the fluid at infinity , 2000 .

[6]  Kumbakonam R. Rajagopal,et al.  Exact solutions for some simple flows of an Oldroyd-B fluid , 1995 .

[7]  Constantin Fetecau,et al.  On some helical flows of Oldroyd-B fluids , 2007 .

[8]  Tasawar Hayat,et al.  Exact solutions of flow problems of an Oldroyd-B fluid , 2004, Appl. Math. Comput..

[9]  Masood Khan,et al.  Partial Slip Effects on the Oscillatory Flows of a Fractional Jeffrey Fluid in a Porous Medium , 2007 .

[10]  Tasawar Hayat,et al.  Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space , 2006 .

[11]  Tasawar Hayat,et al.  Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy’s law , 2006 .

[12]  Tan Wen-chang,et al.  A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates , 2003 .

[13]  Wenchang Tan,et al.  Stability analysis of a Maxwell fluid in a porous medium heated from below , 2007 .

[14]  Wenchang Tan,et al.  An anomalous subdiffusion model for calcium spark in cardiac myocytes , 2007 .

[15]  Sharat C. Prasad,et al.  A note on the flow induced by a constantly accelerating edge in an Oldroyd-B fluid , 2007, Int. J. Math. Math. Sci..

[16]  Wenchang Tan,et al.  Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below , 2007 .

[17]  Haitao Qi,et al.  Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel , 2007 .

[18]  Haitao Qi,et al.  Unsteady Rotating Flows of a Viscoelastic Fluid with the Fractional Maxwell Model Between Coaxial Cylinders , 2006 .

[19]  Wenchang Tan,et al.  STOKES FIRST PROBLEM FOR AN OLDROYD-B FLUID IN A POROUS HALF SPACE , 2005 .

[20]  The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid☆ , 2002 .

[21]  Tasawar Hayat,et al.  Erratum to: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate , 2008 .

[22]  Mingyu Xu,et al.  The impulsive motion of flat plate in a generalized second grade fluid , 2002 .

[23]  T. Hayat,et al.  On the MHD flow of fractional generalized Burgers’ fluid with modified Darcy’s law , 2007 .

[24]  B. A. Toms,et al.  Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers , 1948 .

[25]  C. Friedrich.  Relaxation and retardation functions of the Maxwell model with fractional derivatives , 1991 .

[26]  R. Larson.  Instabilities in viscoelastic flows , 1992 .

[27]  Wenchang Tan,et al.  Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below , 2008 .

[28]  I. Podlubny.  Fractional differential equations , 1998 .

[29]  Constantin Fetecau,et al.  Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate , 2009 .

[30]  Cha'o-Kuang Chen,et al.  Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate conditions , 2004 .