Unsteady Flows of a Generalized Fractional Burgers’ Fluid between Two Side Walls Perpendicular to a Plate

Abstract

The unsteady flows of a generalized fractional Burgers’ fluid between two side walls perpendicular to a plate are studied for the case of Rayleigh-Stokes’ first and second problems. Exact solutions of the velocity fields are derived in terms of the generalized Mittag-Leffler function by using the double Fourier transform and discrete Laplace transform of sequential fractional derivatives. The solution for Rayleigh-Stokes’ first problem is represented as the sum of the Newtonian solutions and the non-Newtonian contributions, based on which the solution for Rayleigh-Stokes’ second problem is constructed by the Duhamel’s principle. The solutions for generalized second-grade fluid, generalized Maxwell fluid, and generalized Oldroyd-B fluid performing the same motions appear as limiting cases of the present solutions. Furthermore, the influences of fractional parameters and material parameters on the unsteady flows are discussed by graphical illustrations.

Unsteady Helical Flows of a Size-Dependent Couple-Stress Fluid

“…In order to find the solution of problem (15)-(25), we use the Laplace transform with respect to the variable time and finite Hankel transform with respect the radial coordinate [24,25].…”

“…Applying the Laplace and finite Hankel transforms to (16) and 17, using the initial and boundary conditions (22)- (25) and Lemmas 1 and 2, we obtain the following transformed equations:…”

Numerical modeling and experimental validation of anomalous time and space subdiffusion for gas transport in porous coal matrix

“…The operators @ 1Àk =@t 1Àk and @ b =@r b are the Riemann-Liouville fractional derivatives which are defined as [29,30] @ 1Àk Cðr; tÞ…”

Fractional Burgers wave equation on a finite domain

“…In the case of models belonging to the first class, the highest differentiation order of strain µ + η ∈ [1,2] , with η ∈ {α, β} , is greater than the highest differentiation order of stress, that is either γ ∈ [0, 1] in the case of Model I, in addition to 0 ≤ α ≤ β ≤ γ ≤ µ ≤ 1 and η ∈ {α, β, γ} , or γ ∈ [1,2] in the case of Models II -V, in addition to 0 ≤ α ≤ β ≤ µ ≤ 1 and (η, γ) ∈ {(α, 2α) , (α, α + β) , (β, α + β) , (β, 2β)} , while for models belonging to the second class differentiation orders of stress β ∈ [0, 1] and β + η ∈ [1,2] coincide with the highest differentiation orders of strain in addition to 0 ≤ α ≤ β ≤ 1, so that η = α, in the case of Model VI; η = β in the case of Model VII; and α = η = β, ā1 = a 1 + a 2 , and ā2 = a 3 in the case of Model VIII. Similar forms of the fractional Burgers models are checked for the thermodynamical consistency in [3,10], while the classical and different variants of fractional Burgers models, describing the flow of viscoelastic fluids in various geometries, are considered in [27,28,29,30,31,32,33,34].…”