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

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

The Influence of Magnetohydrodynamic Flow and Slip Condition on Generalized Burgers’ Fluid with Fractional Derivative

         هذا البحث يهدف الى تاثير حقل مغناطيسي هيدروديناميكي للمائع بيركر القابل للانضغاط من خلال ضغط ثابت ولوح متسارع أسي. حيث افتراض عدم الانزلاق بين لوحة التسارع والمائع . حساب  التفاضل والتكامل الكسري استخدم لكتابة  معادلات الحركة لنموذج المائع , باستخدام تحويلات لابلاس وفوريير نحصل على حقل السرعة والاجهاد. اضافة الى ذلك, تم رسم الاشكال الثلاثية الابعاد لعرض تاثير حقل المغناطيسي والمعلمات المختلفة على حقل السرعة.                                                          This paper investigates the effect of magnetohydrodynamic (MHD) of an incompressible generalized burgers’ fluid including a gradient constant pressure and an exponentially accelerate plate where no slip hypothesis between the burgers’ fluid and an exponential plate is no longer valid. The constitutive relationship can establish of the fluid model process by fractional calculus, by using Laplace and Finite Fourier sine transforms. We obtain a solution for shear stress and velocity distribution. Furthermore, 3D figures are drawn to exhibit the effect of magneto hydrodynamic and different parameters for the velocity distribution

Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case

Stokes' first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.Comment: 10 pages, 2 figures; accepted for publication in Mechanics Research Communications; v2 corrects a few typo

An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations

In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is $O(N^{-r}+\tau^2)$, where $N, \tau, r$ are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids

A Numerical Study of Peristaltic Flow Generalized Maxwell Viscoelastic Fluids Through a Porous medium in an Inclined Channel

In this paper presents a study on Peristaltic of generalized Maxwell fluid fluids  through a porous medium  in an inclined channel with slip effect. The governing equation are simplified by assuming long wavelength  and low Reynolds number approximations. The numerical and approximate analytical solutions of the problem are obtained by a semi-numerical technique, namely the homotopy  perturbation method. The influence of the dominating physical parameters such as fractional Maxwell parameter, relaxation time, amplitude ratio, permeability parameter , Froude number, Reynolds number and inclination of channel on the flow characteristics are depicted graphically. Keywords : Peristaltic Transport, fractional generalized Maxwell, Slip effect, Porous Medium, Inclined a      symmetric channel, pimping ,trapping

Viscoelasticity

This book contains a wealth of useful information on current research on viscoelasticity. By covering a broad variety of rheology, non-Newtonian fluid mechanics and viscoelasticity-related topics, this book is addressed to a wide spectrum of academic and applied researchers and scientists but it could also prove useful to industry specialists. The subject areas include, theory, simulations, biological materials and food products among others

Comments on: "Starting solutions for some unsteady unidirectional flows of a second grade fluid," [Int. J. Eng. Sci. 43 (2005) 781]

A significant mathematical error is identified and corrected in a recent highly-cited paper on oscillatory flows of second-grade fluids [Fetecau & Fetecau (2005). Int. J. Eng. Sci., 43, 781--789]. The corrected solutions are shown to agree identically with numerical ones generated by a finite-difference scheme, while the original ones of Fetecau & Fetecau do not. A list of other recent papers in the literature that commit the error corrected in this Comment is compiled. Finally, a summary of related erroneous papers in this journal is presented as an Appendix.Comment: 8 pages, 2 figures (4 images), elsarticle class; accepted for publication in International Journal of Engineering Scienc