Starting solutions for some simple oscillating motions of second-grade fluids

The exact starting solutions corresponding to the motions of a second-grade fluid, due to the cosine and sine oscillations of an infinite edge and of an infinite duct of rectangular cross-section as well as those induced by an oscillating pressure gradient in such a duct, are determined by means of the double Fourier sine transforms. These solutions, presented as sum of the steady-state and transient solutions, satisfy both the governing equations and all associate initial and boundary conditions. In the special case when α1→0, they reduce to those for a Navier-Stokes fluid

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

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

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 f t a (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 i

General solutions for the unsteady flow of second-grade fluids over an infinite plate that applies arbitrary shear to the fluid,” Zeitschrift fur Naturforschung—Section A:

General solutions corresponding to the unsteady motion of second-grade fluids induced by an infinite plate that applies a shear stress f (t) to the fluid are established. These solutions can immediately be reduced to the similar solutions for Newtonian fluids. They can be used to obtain known solutions from the literature or any other solution of this type by specifying the function f (·). Furthermore, in view of a simple remark, general solutions for the flow due to a moving plate can be developed

Time-fractional free convection flow near a vertical plate with Newtonian heating and mass diffusion

The time-fractional free convection flow of an incompressible viscous fluid near a vertical plate with Newtonian heating and mass diffusion is investigated in presence of first order chemical reaction. The dimensionless temperature, concentration, and velocity fields, as well as the skin friction and the rates of heat and mass transfer from the plate to the fluid, are determined using the Laplace transform technique. Closed form expressions are established in terms of Robotnov-Hartley and Wright functions. The similar solutions for ordinary fluids are also determined. Finally, the influence of fractional parameter on the temperature, concentration and velocity fields is graphically underlined and discussed