Abstract
The solutions for velocity and stress are derived by using the methods of Laplace transformation and Modified Bessel’s equation for the rotational flow of Burgers’ fluid flowing through an unbounded round channel. Initially, supposed that the fluid is not moving with t = 0 and afterward fluid flow is because of the circular motion of the around channel with velocity Ω R t p with time positively grater than zero. At the point of complicated expressions of results, the inverse Laplace transform is alternately calculated by “Stehfest’s algorithm” and “MATHCAD” numerically. The numerically obtained solutions in the terms of the Modified Bessel’s equations of first and second kind, are satisfying all the imposed conditions of given mathematical model. The impact of the various physical and fractional parameters are also indeed and so presented by graphical demonstrations.
Highlights
Fluids can be divided into two types i.e., Newtonian and non-Newtonian fluid
Fetecau [13] obtained the analytical solutions for the helical flow of Oldroyd-B fluid
As there are complicated expressions in Equations (16) and (20) with respect to the Laplace transformation, it was not easy to find out final results by using direct inverse Laplace on Equations (16) and
Summary
Fluids can be divided into two types i.e., Newtonian and non-Newtonian fluid. The Newtonian fluids are simple and ideal. The investigation of basic unsteady pipe flow and viscoelastic upper-convected Maxwell fluid in uniform circular cross section, is available in [4]. They used the Fourier Bessel series to obtain the closed form solutions. The exact solutions of different fluids in circular cylinders (may be finite and infinite) can be obtained by applying the Laplace transformation on the system. Fetecau [13] obtained the analytical solutions for the helical flow of Oldroyd-B fluid. Tan et al [24] and Xu et al [25] applied a fractional derivative model to Maxwell and generalized second grade fluids between two parallel plates. The results and discussions of all parameters are given at the end that shows the consistency in obtained results
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