Unsteady hydromagnetic rotating flow of a conducting second grade fluid

The purpose of this work is to investigate the hydromagnetic oscillatory flow of a fluid bounded by a porous plate, when the entire system rotates about an axis normal to the plate. The fluid is assumed to be non-Newtonian (second grade), incompressible and electrically conducting. The magnetic field is applied transversely to the direction of the flow. Such a flow model has great significance not only of its theoretical interest, but also for applications to geophysics and engineering. The resulting initial value problem has been solved analytically for steady and unsteady cases. The analysis of the obtained results showed that the flow field is appreciably influenced by the material parameter of the second grade fluid, the applied magnetic field, the imposed frequency, rotation and suction and blowing parameters. It is observed in a second grade fluid that a steady asymptotic hydromagnetic solution exists for blowing and resonance which is different from the hydrodynamic situation.     

Oscillatory flows of second grade fluid in a porous space

The exact analytical solutions are developed for the magnetohydrodynamic (MHD) flows of the second grade fluid in a porous medium. The analysis is performed using modified Darcy’s law and takes into account the effect of the Hall current. Closed form solutions are given for three problems using the Fourier sine transform. Comparison has been made with the existing results and are found to be in excellent agreement. The graphs are plotted for various emerging parameters and discussed.     

Unsteady flow and heat transfer of a second grade fluid over a stretching sheet

The problem of unsteady boundary layer flow of a second grade over a stretching sheet is investigated in this paper. The governing equations of motion are reduced into a partial differential equation with two independent variables by using similarity transformations. The heat transfer analysis has been also carried out for two heating processes namely the prescribed surface temperature (PST case) and prescribed surface heat flux (PHF case). The series solutions of the problem are developed by employing homotopy analysis method (HAM). Convergence of the obtained series solutions are analyzed. It is noted that the present solutions of a second grade are valid for all dimensionless times. Finally, the results are obtained and discussed through graphs for various parameters of interest.     

Couette flow of a third-grade fluid with variable magnetic field

This investigation deals with the analytic solution for the time-dependent flow of an incompressible third-grade fluid which is under the influence of a magnetic field of variable strength. The fluid is in an annular region between two coaxial cylinders. The motion is induced due to an inner cylinder with arbitrary velocity. Group theoretic methods are employed to analyse the nonlinear problem and a solution for the velocity field is obtained analytically.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model

In the present work, the exact analytic solutions for some oscillating flows of a generalized second grade fluid are investigated using Fourier sine and Laplace transforms. A more appropriate model is presented for fluid material between viscous and elastic to introduce the fractional calculus approach into the constitutive relationship. This paper employs the fractional calculus approach to study second grade fluid flows. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method has been used. Similar solutions for second grade fluid appear as the limiting cases of our solutions. The influence of pertinent parameters on the flows is delineated and appropriate conclusions are drawn.     

Unsteady Couette flow of viscous fluid under a non-uniform magnetic field

The aim of this letter is to construct the analytic solution for unsteady Couette flow in the presence of an arbitrary non-uniform applied magnetic field. The flow is induced by a generalized velocity given to the lower plate. The perturbed eigenfunction expansion method is employed to develop a series solution for small magnetic field.     

Couette flow of a third grade fluid with rotating frame and slip condition

An incompressible third grade fluid occupies the porous space between two rigid infinite plates. The steady rotating flow of this fluid due to a suddenly moved lower plate with partial slip of the fluid on the plate is analysed. The fluid filling the porous space between the two plates is electrically conducting. The flow modeling is developed by employing a modified Darcy’s law. A numerical solution of the governing problem consisting of a non-linear ordinary differential equation and non-linear boundary conditions is obtained and discussed. Several limiting cases of the arising problem can be obtained by choosing suitable parameters.     

Some analytical solutions for second grade fluid flows for cylindrical geometries

This paper deals with some unsteady flow problems of a second grade fluid. The flows are generated by the sudden application of a constant pressure gradient or by the impulsive motion of a boundary. The velocities of the flows are described by the partial differential equations. Exact analytic solutions of these differential equations are obtained. The well known solutions for a Navier–Stokes fluid in the hydrodynamic case appear as the limiting cases of our solutions.     

Helical flows of second grade fluid due to constantly accelerated shear stresses

The helical flows of second grade fluid between two infinite coaxial circular cylinders is considered. The motion is produced by the inner cylinder that at the initial moment applies torsional and longitudinal constantly accelerated shear stresses to the fluid. The exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions. In the limiting case when α → 0, the solutions for Newtonian fluid are obtained for the same motion. The large-time solutions and transient solutions for second grade fluid are also obtained, and effect of material parameter α and kinematic viscosity ν is discussed. In the last, the effects of various parameters of interest on fluid motion as well as the comparison between second grade and Newtonian fluids are analyzed by graphical illustrations.     

Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative

This paper deals with the unsteady helical flows of a generalized Oldroyd-B fluid between two infinite coaxial cylinders and within an infinite cylinder. The fractional calculus approach is used in the constitutive relationship of fluid model. Exact analytical solutions are obtained with the help of integral transforms (Laplace transform, Weber transform and finite Hankel transform). The corresponding solutions for generalized second grade and Maxwell fluids as well as those for the Newtonian and ordinary Oldroyd-B fluids are also given in limiting cases. Finally, the influence of model parameters on the velocity field is also analyzed by graphical illustrations.     

Couple stress fluid flow with variable properties: A second law analysis

The present work examines the combined influence of variable thermal conductivity and viscosity on the irreversibility rate in couple stress fluid flow in between asymmetrically heated parallel plates. The dimensionless fluid equations are solved by using homotopy analysis method (HAM) and validated with Runge‐Kutta shooting method (RKSM). The convergent series solution is then used for the irreversibility analysis in the flow domain. The effects of thermal conductivity and viscosity variation parameters, couple stress parameter, Reynolds number, Grashof number, Hartmann number on the velocity profile, temperature distribution, entropy production, and heat irreversibility ratio are presented through graphs, and salient features of the solutions are discussed. The computations show that the entropy production rate decreases with increased magnetic field and thermal conductivity parameters, whereas it rises with increasing values of couple stress parameter, Brinkman number, viscosity variation parameter, and Grashof number. The study is relevant to lubrication theory.     

Unsteady MHD Couette flow in a rotating system

The unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system has been considered. An exact solution of the governing equations has been obtained by using a Laplace transform. Solutions for the velocity distributions as well as shear stresses have been obtained for small times as well as for large times. It is found that for large times the primary velocity decreases with increase in the rotation parameter K² while it increases with increase in the magnetic parameter M². It is also found that with increase in K², the secondary velocity v₁ decreases near the stationary plate while it increases near the moving plate. On the other hand, the secondary velocity decreases with increase in the magnetic parameter.     

Creeping flow of a second grade fluid in a corner

A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields.     

Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field

In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,

(i)

Couette flow

(ii)

Poiseuille flow and

(iii)

generalized Couette flow.

Homotopy analysis method (HAM) is used to solve the non-linear differential equation analytically. It is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants (non-Newtonian parameters) which is different from the model of Oldroyd 3-constant fluid. Numerical solutions are also given and compared with the solutions by HAM.     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

Heat transfer in unsteady axisymmetric second grade fluid

This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers.     

混合微通道中不稳定的广义Couette流

在一个平行板通道中,部分充满了均匀的多孔介质,部分为纯流体的流动区,对其微通道中完全发展的不稳定层流进行了数值分析,流动由其中一块板的运动和压力梯度所引起.多孔介质区域的流动,采用扩展的Brinkman模型,即Darcy模型,纯净流动区域的流动,采用Stokes方程.还对稳定的完全发展流进行了理论分析,给出了分界面速度、边界板处的速度和表面摩擦的闭式解.通过数值计算发现,稳定完全发展流的闭式解,和不稳定流动的数值解,在所有时间点上得到很好地吻合.     

Radiative magnetogasdynamic Couette flow with variable parameters

Couette flow of an electrically conducting compressible fluid with uniform magnetic field applied transversely between flat walls of arbitrary electrical conductivity, radiative emissivity and temperature is analysed. The equations of steady flow are formulated with the introduction of temperature dependent coefficients of viscosity and thermal and electrical conductivity, together with absorption coefficient in addition dependent upon the density. Profiles of velocity, induced magnetic field, radiative flux and temperature are derived numerically under a one-dimensional modelling. The parameter dependence, expressed as a power law, is found to have a marked effect upon the velocity and magnetic field but to be of little import for the thermal profiles. Similar influence is found for the effect of wall electrical conductivity and emissivity. On the other hand, changes in the strength of the fluid thermal conductivity significantly affect the temperature profile in the presence of electromagnetic interaction.

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Résumé Pour un fluide conducteur, rayonnant et compressible sous l'influence d'un champ magnétique constant et transverse l'écoulement de Couette entre deux surfaces planes conductrices et rayonnantes avec des températures arbitraires est analysé. On formule les équations d'un écoulement permanent quand la viscosité, la conductibilité thermique et la conductibilité électrique dépendent de la température et le coffficient d'absorption de la densité. On calcule la vitesse, le champ magnétique induit, le rayonnement et la température pour des écoulements qui ne sont fonction que de la cordonnée transverse.La vitesse et le champ magnétique dépendent de façon significative de ce paramètre, mais pas la température ni le rayonnement. On obtient des résultats similaires pour la conductibilité électrique et l'emissivité de rayonnement des surfaces planes. D'autre part quand la conductiblité électrique du fluide n'est pas nulle, une variation de la conductibilité thermique modifie la température d'une manière significative.