Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The solutions that have been obtained are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary Maxwell fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the ordinary Maxwell and generalized Maxwell fluids are shown and compared graphically by plotting velocity profiles at different values of time and some important results are remarked.

[1]  Constantin Fetecau,et al.  Flow of a viscoelastic fluid with the fractional Maxwell model between two side walls perpendicular to a plate , 2008, Appl. Math. Comput..

[2]  Tasawar Hayat,et al.  The influence of Hall current on the rotating oscillating flows of an Oldroyd-B fluid in a porous medium , 2007 .

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

[4]  Michael C. Constantinou,et al.  Fractional‐Derivative Maxwell Model for Viscous Dampers , 1991 .

[5]  T. Hayat,et al.  United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS OSCILLATING FLOW OF A BURGERS' FLUID IN A PIPE , 2005 .

[6]  Kumbakonam R. Rajagopal,et al.  Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid , 1983 .

[7]  Marina V. Shitikova,et al.  A new method for solving dynamic problems of fractional derivative viscoelasticity , 2001 .

[8]  R. G. Lerner,et al.  Encyclopedia of Physics , 1990 .

[9]  Shaowei Wang,et al.  Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus , 2009 .

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

[11]  Dengke Tong,et al.  Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe , 2005 .

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

[13]  Constantin Fetecau,et al.  Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder , 2008, Comput. Math. Appl..

[14]  Gary F. Dargush,et al.  Dynamic Analysis of Generalized Viscoelastic Fluids , 1993 .

[15]  J. E. Dunn,et al.  Fluids of differential type: Critical review and thermodynamic analysis , 1995 .

[16]  Yao-hua Zhao,et al.  The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model , 2006 .

[17]  D. Y. Song,et al.  Study on the constitutive equation with fractional derivative for the viscoelastic fluids – Modified Jeffreys model and its application , 1998 .

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

[19]  Mario J. Casarella,et al.  Drag on an Oscillating Rod with Longitudinal and Torsional Motion , 1969 .

[20]  Marina V. Shitikova,et al.  Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders , 2001 .

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

[22]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[23]  W. Shaowei,et al.  Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative , 2006 .

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

[25]  L. Debnath,et al.  Integral Transforms and Their Applications, Second Edition , 2006 .

[26]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[27]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[28]  Tasawar Hayat,et al.  Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model , 2004, Appl. Math. Comput..

[29]  Dengke Tong,et al.  Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe , 2005 .

[30]  T. Hartley,et al.  Generalized functions for the fractional calculus. , 2008, Critical reviews in biomedical engineering.

[31]  C. Fetecau,et al.  Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder , 2006 .

[32]  Kumbakonam R. Rajagopal,et al.  A note on unsteady unidirectional flows of a non-Newtonian fluid , 1982 .

[33]  Tasawar Hayat,et al.  Transient flows of a second grade fluid , 2004 .

[34]  W. Glöckle,et al.  Fractional relaxation and the time-temperature superposition principle , 1994 .

[35]  Tasawar Hayat,et al.  Unsteady motions of a generalized second-grade fluid , 2005, Math. Comput. Model..

[36]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .