Exact solutions for unsteady axial Couette flow of a fractional Maxwell fluid due to an accelerated shear

Abstract

The velocity field and the adequate shear stress corresponding to the flow of a fractional Maxwell fluid (FMF) between two infinite coaxial cylinders, are determined by means of the Laplace and finite Hankel transforms. The motion is produced by the inner cylinder that at time t = 0+ applies a shear stress fta (a ≥ 0) to the fluid. The solutions that have been obtained, presented under series form in terms of the generalized G and R functions, satisfy all imposed initial and boundary conditions. Similar solutions for ordinary Maxwell and Newtonian fluids are obtained as special cases of general solutions. The unsteady solutions corresponding to a = 1, 2, 3, ... can be written as simple or multiple integrals of similar solutions for a = 0 and we extend this for any positive real number a expressing in fractional integration. Furthermore, for a = 0, 1 and 2, the solutions corresponding to Maxwell fluid compared graphically with the solutions obtained in [1–3], earlier by a different technique. For a = 0 and 1 the unsteady motion of a Maxwell fluid, as well as that of a Newtonian fluid ultimately becomes steady and the required time to reach the steady-state is graphically established. Finally a comparison between the motions of FMF and Maxwell fluid is underlined by graphical illustrations.

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

“…This Section is dedicated to the discretisation and numerical solution of equations (12) and (13). To test the convergence of the method we compare the numerical results obtained, with the existing analytical solution for the Newtonian case; we develop a numerical code developed for the solution of annular UCM flows, and compare the results obtained from the different numerical methods (considering the fractional model converging to the UCM constitutive equation); finally we study the convergence order of the method by comparing the numerical results with generalised analytical solutions.…”

“…In this subsection we will derive a numerical method for the solution of the system of fractional partial differential equations (12) and 13, with boundary and initial conditions of Dirichlet type:…”

“…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.…”

Thermomechanical Fractional Model of TEMHD Rotational Flow

“…The Generalized second-order fluid with fractional anomalous diffusion studied by Xu et al [24], while the fractional derivative to the constitutive relationship models of Maxwell viscoelastic fluid and second grade fluid had been studied by Wenchang et al [25, 26]. Fractional Maxwell fluid was examined for unsteady Couette flow by Athar et al [27]. The oscillating flows in a generalized second grade fluid was studied by Jamil et al [28].…”

On the existence of solutions to the fractional derivative equations d(alpha)u/dt(alpha)+Au = f, of relevance to diffusion in complex systems

“…[8][9][10][11]), fractional derivative models are of great utility in accurately predicting the rheological behavior of polymer liquids in the glass transition region and beyond (e.g. see [12][13][14][15][16][17][18][19][20][21]; for a review of fractional derivative rheological models see [22]), in interpreting experimental measurements of anomalous diffusion processes in glassy materials [1,23] etc.…”