On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities

We demonstrate a close analogy between a viscoelastic medium and an electrically conducting fluid containing a magnetic field. Specifically, the dynamics of the Oldroyd-B fluid in the limit of large Deborah number corresponds to that of a magnetohydrodynamic (MHD) fluid in the limit of large magnetic Reynolds number. As a definite example of this analogy, we compare the stability properties of differentially rotating viscoelastic and MHD flows. We show that there is an instability of the Oldroyd-B fluid that is physically distinct from both the inertial and elastic instabilities described previously in the literature, but is directly equivalent to the magnetorotational instability in MHD. It occurs even when the specific angular momentum increases outwards, provided that the angular velocity decreases outwards; it derives from the kinetic energy of the shear flow and does not depend on the curvature of the streamlines. However, we argue that the elastic instability of viscoelastic Couette flow has no direct equivalent in MHD.Comment: 21 pages, 3 figures, to be published in J. Fluid Mec

Numerical study of chemical reaction effects in magnetohydrodynamic Oldroyd B oblique stagnation flow with a non-Fourier heat flux model

Reactive magnetohydrodynamic (MHD) flows arise in many areas of nuclear reactor transport. Working fluids in such systems may be either Newtonian or non-Newtonian. Motivated by these applications, in the current study, a mathematical model is developed for electrically-conducting viscoelastic oblique flow impinging on stretching wall under transverse magnetic field. A non-Fourier Cattaneo-Christov model is employed to simulate thermal relaxation effects which cannot be simulated with the classical Fourier heat conduction approach. The Oldroyd-B non-Newtonian model is employed which allows relaxation and retardation effects to be included. A convective boundary condition is imposed at the wall invoking Biot number effects. The fluid is assumed to be chemically reactive and both homogeneous-heterogeneous reactions are studied. The conservation equations for mass, momentum, energy and species (concentration) are altered with applicable similarity variables and the emerging strongly coupled, nonlinear non-dimensional boundary value problem is solved with robust well-tested Runge-Kutta-Fehlberg numerical quadrature and a shooting technique with tolerance level of 10−4. Validation with the Adomian decomposition method (ADM) is included. The influence of selected thermal (Biot number, Prandtl number), viscoelastic hydrodynamic (Deborah relaxation number), Schmidt number, magnetic parameter and chemical reaction parameters, on velocity, temperature and concentration distributions are plotted for fixed values of geometric (stretching rate, obliqueness) and thermal relaxation parameter. Wall heat transfer rate (local heat flux) and wall species transfer rate (local mass flux) are also computed and it is observed that local mass flux increases with strength of heterogeneous reactions whereas it decreases with strength of homogeneous reactions. The results provide interesting insights into certain nuclear reactor transport phenomena and furthermore a benchmark for more general CFD simulations

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

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

Analog of Astrophysical Magnetorotational Instability in a Couette-Taylor Flow of Polymer Fluids

We report experimental observation of an instability in a Couette-Taylor flow of a polymer fluid in a thin gap between two coaxially rotating cylinders in a regime where their angular velocity decreases with the radius while the specific angular momentum increases with the radius. In the considered regime, neither the inertial Rayleigh instability nor the purely elastic instability are possible. We propose that the observed "elasto-rotational" instability is an analog of the magnetorotational instability which plays a fundamental role in astrophysical Keplerian accretion disks.Comment: 4 pages, 1 figur

Magnetohydrodynamic free convection boundary layer Flow of non-Newtonian tangent hyperbolic fluid from a vertical permeable cone with variable temperature

The nonlinear, non-isothermal steady-state boundary layer flow and heat transfer of an incompressible tangent hyperbolic non-Newtonian (viscoelastic) fluid from a vertical permeable cone with magnetic field are studied. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using the second-order accurate implicit finite difference Keller-box technique. The numerical code is validated with previous studies. The influence of a number of emerging non-dimensional parameters, namely a Weissenberg number (We), rheological power law index (m), surface temperature exponent (n), Prandtl number (Pr), magnetic parameter (M) suction/injection parameter (fw) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime, is examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is observed that velocity, surface heat transfer rate and local skin friction are reduced with increasing Weissenberg number, but temperature is increased. Increasing m enhances velocity and surface heat transfer rate but reduces temperature and local skin friction. An increase in non-isothermal power law index (n) is observed to decrease the velocity and temperature. Increasing magnetic parameter (M) is found to decrease the velocity and increase the temperature. Overall, the primary influence on free convection is sustained through the magnetic body force parameter, M, and also the surface mass flux (injection/suction) parameter, fw. The rheological effects, while still prominent, are not as dramatic. Boundary layers (both hydrodynamic and thermal) are, therefore, most strongly modified by the applied magnetic field and wall mass flux effect. The study is pertinent to smart coatings, e.g., durable paints, aerosol deposition processing and water-based solvent thermal treatment in chemical engineering

Free Convective Heat Transfer in Radiative MHD Casson Fluid Flow over a Stretched Surface of Variable Thickness

This study deals with the free convective heat transfer in a 2D magnetohydrodynamic flow of Casson fluid over a non-uniform thickness stretching sheet in the presence of thermal radiation and non-uniform heat source/sink effects. The governing equations of the flow and heat transfer are transformed as the asset of nonlinear ODEs and solved numerically using bvp4c Matlab package. The effect of pertinent parameters, namely, magnetic field parameter, Casson parameter, thermal radiation parameter, non-uniform heat source/sink parameters on the flow and heat transfer is investigated with the assistance of graphs. Numerical results are computed for the friction factor and reduced Nusselt number. It is observed that the thermal radiation has tendency to enhance the temperature field of Casson fluid. Keywords: MHD, Casson fluid, Thermal radiation, Slendering sheet, non-Uniform heat source/sink

On Hydromagnetic Channel Flow of an Oldroyd-B Fluid induced by Pulses of Longitudinal Impulses

A flow problem concerning motion of an incompressible electrically conducting Oldroyd-B fluid in a channel bounded by two infinite rigid non-conducting parallel plates in presence of an external magnetic field acting in a direction normal to the plates has been solved in this paper. The unsteady motion is supposed to generate impulsively from rest in the fluid due to pulses of longitudinal impulses applied periodically on the upper plate in its own plane with the lower plate held fixed. There is no external electric field acting on the system and the magnetic Reynolds number is very small. The operational method is used to obtain the exact solutions for the velocity field and the skin-friction on the walls. The influence of the magnetic field and the fluid elasticity on the flow as well as on the skin-frictions are examined quantitatively. Solutons for the hydromagnetic and hydrodynamic situations are derived as special cases of the present analysis

Effect of Thermal Radiation on MHD Casson Fluid Flow over a Stretched Surface of Variable Thickness

This study deals with the free convective heat transfer in a 2D magneto hydrodynamic flow of Casson fluid over a non-uniform thickness stretching sheet in the presence of thermal radiation and non-uniform heat source/sink effects. The governing equations of the flow and heat transfer are transformed as the asset of nonlinear ODEs and solved numerically using bvp4c Mat lab package. The effect of pertinent parameters, namely, magnetic field parameter, Casson parameter, thermal radiation parameter, non-uniform heat source/sink parameters on the flow and heat transfer is investigated with the assistance of graphs. Numerical results are computed for the friction factor and reduced Nusselt number. It is observed that the thermal radiation has tendency to enhance the temperature field of Casson fluid. Keywords: MHD, Casson fluid, Thermal radiation, Slendering sheet, non-Uniform heat source/sink