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.

Basic understanding of the flows for non-Newtonian fluids are of great importance in a number of practical engineering applications, such as the extrusion of polymer fluids, exotic lubricant, animal bloods, heavy oils, and colloidal and suspension solutions [

More recently, the fractional calculus has achieved much success in the description of complex dynamic system and is widely applied to many fields [

The availability of exact solutions for non-Newtonian fluids is of significance because such solutions not only can explain the physics of some fundamental flows, but also can be used as a benchmark for complicated numerical codes that have been developed for much more complex flows. However, exact solutions for the unsteady flows of viscoelastic fluids are very rare and difficult to obtain due to the nonlinearity of their constitutive equations. When the fractional calculus approach is introduced in the constitutive equations, the solvability becomes more difficult even though the problems are one-dimensional in case of simple geometries such as single plate or disk. The literature survey indicates that the Rayleigh-Stokes’ first and second problems for flows between two side walls perpendicular to a plate are two of few problems that can be analytically solved. Fetecau et al. [

In this work, we study the unsteady flows of a Burgers’ fluid between two side walls perpendicular to a plate with a fractional derivative model. The following two cases are studied: (i) the flow induced by the impulsive motion of the bottom plate (Rayleigh-Stokes’ first problem) and (ii) the flow induced by the periodic oscillation of the bottom plate (Rayleigh-Stokes’ second problem). The exact solutions for the two problems are obtained in terms of generalized Mittag-Leffler function by using integral transform technique.

The momentum and continuity equations for an incompressible fluid are given by

For an ordinary Burgers’ fluid, the extra stress tensor

We consider an incompressible Burgers’ fluid occupying the space above an infinite flat plate and between two side walls perpendicular to this plate, as shown in Figure

The schematic diagram of system considered here.

According to (

The governing equations corresponding to a generalized fractional Burgers’ fluid performing the same motion can be obtained from (

Then, from (

Initially, the fluid is at rest and then the plate is suddenly brought to a steady velocity

Introducing the dimensionless parameters

It should be noted that in order to solve a well-posed problem for (

For the sake of brevity and convenience, we omit the asterisks “

To solve the partial differential equation (

Taking the transform (

To obtain an exact solution of (

Let

For a well presentation of the final results, (

Taking the inverse Laplace transform to (

Finally, inverting (

In view of the following formulae [

It is easy to find that the velocity field

Making

In some limiting cases, the present solution for a generalized Burgers’ fluid can be reduced to those corresponding to a generalized second-grade fluid, Maxwell fluid, and Oldroyd-B fluid.

If one takes

Applying the inverse Laplace transform term by term on (

If one takes

Applying the inverse Laplace transform term by term on (

If one takes

Consider the flow is caused by the plate whose velocity is of the form

Based on the result obtained in Section

First,

Then, we attempt to find the solution by means of the temporal Fourier transform. The transform and its inverse transform are defined by

Having in mind (

In course of obtaining (

Using the transform pair (

Finally, inverting (

Equation (

After substituting the corresponding Fourier coefficients of (

By letting

Furthermore, the solutions for a generalized second-grade fluid, Maxwell fluid, and Oldroyd-B fluid performing the same motion can be retrieved by taking corresponding limiting cases of the parameters

In this section, we plot the velocity fields according to the exact solutions obtained in the last section. For clarity, the symbols

The influences of the fractional parameters

The influence of the fractional-order parameter

The influence of the fractional-order parameter

The effect of the material parameter

The influence of the material parameter

The Reynolds number

The influence of the Reynolds number Re on the velocity profile

Figure

The variation of velocity profile

The objective of this paper is to provide exact solutions of unsteady flows for a generalized fractional Burgers’ fluid between two side walls perpendicular to a plate. The unsteady flows are induced by the impulsive motion or general periodic oscillations of the plate, which are, respectively, termed as the Rayleigh-Stokes’ first and second problems. The analytic solutions of the two problems are obtained by using Fourier sine and Laplace transform methods in terms of Mittag-Leffler function. Moreover, the effects of various parameters are analyzed by plotting the velocity profiles according to the exaction solutions. The fractional constitutive model is more flexible and useful than the convectional model for characterizing the property of viscoelastic fluids, so it is expected that the present results will be of significance to fundamental research and practical applications in this field.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (Grant nos. 2014XT02 and 2014ZDPY03), the National Natural Science Foundation of China (Grant no. 11402293), the China Postdoctoral Science Foundation (Grant no. 2014M560458), the Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT13098), A Program Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, The Team Project Funded by 2014 Jiangsu Innovation and Entrepreneurship Program, and the Qing Lan Project of Jiangsu Province.