Abstract
The unsteady flows of Burgers’ fluid with fractional derivatives model, through a circular cylinder, is studied by means of the Laplace and finite Hankel transforms. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Ut. The solutions that have been obtained, presented in series form in terms of the generalized Ga,b,c(•, t) functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for fractionalized Oldroyd-B, Maxwell and second grade fluids appear as special cases of the present results. Furthermore, the solutions for ordinary Burgers’, Oldroyd-B, Maxwell, second grade and Newtonian performing the same motion, are also obtained as special cases of general solutions by substituting fractional parameters α = β = 1. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison among models, is shown by graphical illustrations.
Highlights
Over the past decade fractional calculus has encountered much success in the description of viscoelastic characteristics
The starting point of the fractional derivative model of non-Newtonian model is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the so-called Riemann-Liouville/Caputo fractional calculus operators
Fractional model of viscoelastic fluids is derived from well known ordinary model by replacing the ordinary time derivatives to fractional order time derivatives and this plays an important role to study the valuable tool of viscoelastic properties
Summary
Over the past decade fractional calculus has encountered much success in the description of viscoelastic characteristics. Some authors[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] have investigated unsteady flows of fractionalized viscoelastic second grade, Maxwell, Oldroyed-B, Burgers’ and generalized Burgers’ model through walls/channel/ annulus and solutions for velocity field and the associated shear stress are obtained by using discrete Laplace transform, Fourier transform, Weber transform and finite Hankel transforms. The ordinary Burgers’, Oldroyd-B, Maxwell, second grade and Newtonian fluids can be obtained form the general solutions using appropriate limits and using fractional parameters α = β = 1. The Maxwell fluid is the swiftest and the second grade fluid is the slowest for fractionalized as well as ordinary fluids
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