Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid

In recent years, non-Newtonian fluids have received much attention due to their numerous applications, such as plastic manufacture and extrusion of polymer fluids. They are more complex than Newtonian fluids because the relationship between shear stress and shear rate is nonlinear. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluid, which is modelled using terms involving multi-term time fractional diffusion and reaction. In this paper, we consider the application of the finite difference method for this class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete $H^1$ norm and prove that their accuracy is of $O(\tau+h^2)$ and $O(\tau^{\min\{3-\gamma_s,2-\alpha_q,2-\beta\}}+h^2)$, respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.Comment: 19 pages, 8 figures, 3 table

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

The Effect of MHD on a Longitudinal Flow of a Fractional Maxwell Fluid between Two Coaxial Cylinders

     قمنا في هذا البحث بحل معادلة مائع ماكسويل التفاضلية ذات الرتبة الكسرية .كان الحل بصيغة دالة ميتاج- لفلر (Mettag-Leffler) . في حالة  فان حلول مائع ماكسويل غير الكسرية حصلنا عليها كحالة محددة من الحل العام.  اخيراً ، تاثير المعلمات المختلفة في حقل السرعة واجهاد القص تم تحليلها من خلال رسم السرعة واجهاد القص.      In this paper fractional Maxwell fluid equation has been solved. The solution is in the Mettag-Leffler form. For  the corresponding solutions for ordinary Maxwell fluid are obtained as limiting case of general solutions. Finally, the effects of different parameters on the velocity and shear stress profile are analyzed through plotting the velocity and shear stress profile

Transient Stage Comparison of Couette Flow under Step Shear Stress and Step Velocity Boundary Conditions

Couette flow has been widely used in many industrial and research processes, such as viscosity measurement. For the study on thixotropic viscosity, step-loading, which includes (1) step shear stress and (2) step velocity conditions, is widely used. Transient stages of Couette flow under both step wall shear stress and step wall velocity conditions were investigated. The relative coefficient of viscosity was proposed to reflect the transient process. Relative coefficients of viscosity, dimensionless velocities and dimensionless development times were derived and calculated numerically. This article quantifies the relative coefficients of viscosity as functions of dimensionless time and step ratios when the boundary is subjected to step changes. As expected, in the absence of step changes, the expressions reduce to being functions of dimensionless time. When step wall shear stresses are imposed, the relative coefficients of viscosity changes from the values of the step ratios to their steady-state value of 1. but With step-increasing wall velocities, the relative coefficients of viscosity decrease from positive infinity to 1. The relative coefficients of viscosity increase from negative infinity to 1 under the step-decreasing wall velocity condition. During the very initial stage, the relative coefficients of viscosity under step wall velocity conditions is further from 1 than the one under step wall shear stress conditions but the former reaches 1 faster. Dimensionless development times grow with the step ratio under the step-rising conditions and approaches the constant value of 1.785 under the step wall shear stress condition, and 0.537 under the step wall velocity condition respectively. The development times under the imposed step wall shear stress conditions are always larger than the same under the imposed step wall velocity conditions

Effects of MHD on the Unsteady Rotating Flow of a Generalized Maxwell Fluid with Oscillating Gradient Between Coaxial Cylinders

The aim of  this paper is studied the effect of magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two infinite straight circular cylinder .The velocity field and the shear stress are obtained by means of discrete Laplace transform and finite Hankel transform. The exact solution for the velocity field and the shear stress that have been obtained by integral and series form in terms of the generalized G functions and Mitting –leffer function .the graphs are plotted to show the effects of the fractional parameter on the fluid dynamic characteristics with MHD on the velocity and shear stress

Analytical solutions for wall slip effects on magnetohydrodynamic oscillatory rotating plate and channel flows in porous media using a fractional burgers viscoelastic model

A theoretical analysis of magnetohydrodynamic (MHD) incompressible flows of Burger's fluid through a porous medium in a rotating frame of reference is presented. The constitutive model of a Burger's fluid is used based on a fractional calculus formulation. Hydrodynamic slip at the wall (plate) is incorporated and a fractional generalized Darcy model deployed to simulate porous medium drag force effects. Three different cases are considered- namely, flow induced by a general periodic oscillation at a rigid plate, periodic flow in a parallel plate channel and finally Poiseuille flow. In all cases the plate (s) boundary (ies) are electrically-non-conducting and small magnetic Reynolds is assumed, negating magnetic induction effects. The well-posed boundary value problems associated with each case are solved via Fourier transforms. Comparisons are made between the results derived with and without slip conditions. 4 special cases are retrieved from the general fractional Burgers model, viz Newtonian fluid, general Maxwell viscoelastic fluid, generalized Oldroyd-B fluid and the conventional Burger’s viscoelastic model. Extensive interpretation of graphical plots is included. We study explicitly the influence on wall slip on primary and secondary velocity evolution. The model is relevant to MHD rotating energy generators employing rheological working fluids

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