Exact Solutions for the Axial Couette Flow of a Fractional Maxwell Fluid in an Annulus

Abstract

The velocity field and the adequate shear stress corresponding to the rotational flow of a fractional Maxwell fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. The solutions that have been obtained are presented in terms of generalized G a,b,c ·, t and R a,b ·, t functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the influence of the material parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries

“…Annular flow between concentric cylinders (including purely tangential, purely longitudinal as well as helical deformation) is not new, and has been addressed in the literature for both classical [7,8,9,10,11] and fractional viscoelastic models considering the generalised Maxwell [12,13,14,15,16,17] and the generalised Oldroyd-B [18,19,20,21,22,23,24] models. Apart from the different fractional models considered, these works differ from each other in the boundary conditions used, and the method in which theoretical results are derived.…”

“…• the infinite series solution is not simple, and the proof of its convergence (when it converges) is not an easy task [13];…”

“…• the explicit series solution depends on the roots of a transcendental equation (promoting a decrease in the accuracy of the final solution and making the solution more difficult to obtain) [13,15].…”

“…• the model studied in some analytical works is claimed to come from the simplification of a (unphysical) nonlinear constitutive equation. The model is not frame invariant and this needs to be clarified (see [25,26]), so that both mathematicians and engineers can continue devel-oping new fractional constitutive models capable of describing the response of real materials under large straining deformation [13,15,23].…”

On Unsteady Flow of a Viscoelastic Fluid Through Rotating Cylinders

“…Recently, Fetecau et al [13] and Mahmood et al [14] also studied the flow of fractional Maxwell fluid between coaxial cylinders. There is a vast literature dealing with such fluids, but we shall recall here only a few of the recent papers [15,16,17,18,19,20,21,22,23,24,25,26] in cylindrical domains.…”