Similar Solutions For Newtonian Fluids Research Articles

Exact Solution of Unsteady MHD Free Convective Chemically Reacting Visco-Elastic Fluid Flow Past a Moving Vertical Plate Through Porous Medium

The influence of radiation on the unsteady MHD free convection Viscoelastic fluid flow past an exponentially accelerated moving vertical plate through porous medium with chemical reaction is investigated theoretically. The fluid considered here is a gray, absorbing-emitting radiation but non-scattering medium, and the Boussinesq’s approximation is considered for pressure gradient in momentum equation and Taylor’s series expansion to describe the radiative heat flux in the energy equation. The dimensionless governing partial differential equations are solved using Laplace transform technique. They satisfy all imposed initial and boundary conditions and for $$S\rightarrow 0$$ can be reduce to the similar solutions for Newtonian fluids. The corresponding expressions for skin friction, Nusselt number and Sherwood number also evaluated. Numerical results for velocity, temperature and concentration are shown graphically. This study is of fundamental importance and frequently arises in many practical situations such as chemical engineering and polymer extrusion processes. The effect of different parameters like thermal radiation parameter, visco-elastic parameter, chemical reaction parameter, magnetic field parameter, permeability parameter, Schmidt number and Prandtl number are discussed. It is observed that the momentum and thermal boundary layer thickness decreases owing to an increase in the value of the radiation parameter. An increase in the Prandtl number is found to decrease the velocity and temperature but increasing in Nusselt number for air and water.

Interaction of magnetic field with heat and mass transfer in free convection flow of a Walters’-B fluid

The heat and mass transfer phenomenon is studied in the unsteady free convection flow of a Walters’-B fluid in the presence of a transverse magnetic field. The problem is modelled in terms of partial differential equations with some physical conditions. Exact solutions for velocity, temperature and concentration are obtained via Laplace transform method. They satisfy imposed initial and boundary conditions. As a special case for $ \Gamma \rightarrow 0$ , these solutions can be be reduced to the similar solutions for Newtonian fluids. The shear stress at the boundary is evaluated from the velocity solution. Numerical results of velocity and shear stress are portrayed through graphs and discussed for various embedded parameters.

Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium

This article studies the unsteady MHD free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature. The fluid is electrically conducting and passing through a porous medium. This phenomenon is modelled in the form of partial differential equations with initial and boundary conditions. Some suitable non-dimensional variables are introduced. The corresponding non-dimensional equations with conditions are solved using the Laplace transform technique. Exact solutions for velocity and energy are obtained. They are expressed in simple forms in terms of exponential and complementary error functions of Gauss. It is found that they satisfy governing equations and corresponding conditions, and are reduced to similar solutions for Newtonian fluids as a special case. Expressions for skin-friction and Nusselt number are also evaluated. Computations are carried out and the results are analysed for emerging flow parameters.

INFLUENCE OF SIDE WALLS ON THE OSCILLATING MOTION OF A MAXWELL FLUID OVER AN INFINITE PLATE

Starting solutions corresponding to the oscillating motion of a Maxwell fluid between side walls perpendicular to a plate are established using integral transforms. Such solutions are scarce in the literature, the motion of the fluid being due to an oscillating shear on the boundary. The solutions corresponding to the motion over an infinite plate that applies an oscillating shear to the fluid are obtained as limiting cases of general solutions. All solutions are presented as a sum between steady-state and transient solutions and can easily be particularized to give the similar solutions for Newtonian fluids. They describe the motion of the fluid some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of initial conditions. However, they satisfy the boundary conditions and governing equations. Finally, the distance between walls for which the velocity of the fluid in the middle of the channel in unaffected by their presence and the required time to reach the steady-state are graphically determined.DOI: http://dx.doi.org/10.5755/j01.mech.19.3.4665

Starting Solutions for Motion of a Maxwell Fluid Over an Infinite Plate that Applies an Oscillating Shear to the Fluid

Unsteady motion of a Maxwell fluid over an infinite plate that applies an oscillating shear to the fluid is studied by means of integral transforms. The obtained solutions satisfy all initial and boundary conditions. They are presented as a sum of steady-state and transient solutions and can easily be reduced to the similar solutions for Newtonian fluids. They describe the motion of the fluid some time after its initiation. After that time, when the transients disappear, the motion of the fluid is described by the steady-state solutions which are periodic in time and independent of the initial conditions. However, they satisfy the governing equations and boundary conditions. Finally, the required time to reach the steady-state, as well as a comparison between models, is determined by graphical illustrations.

A Note on New Exact Solutions for Some Unsteady Flows of Brinkman- Type Fluids over a Plane Wall

Flows of a Brinkman fluid due to a plane boundary moving in its plane are studied using Laplace transform. The solutions that have been obtained for the velocity are presented in simple forms in terms of the complementary error function erfc(.). They satisfy all imposed initial and boundary conditions and can easily be reduced to the similar solutions for Newtonian fluids

EXACT SOLUTION FOR RAYLEIGH−STOKES PROBLEMS OF AN OLDROYD-B FLUID IN A POROUS MEDIUM AND ROTATING FRAME

The aim of this paper is to determine the new exact solution of a magnetohydrodynamic (MHD) and rotating flow of the Maxwell fluid induced by a suddenly moved plate in its own plane. This is accomplished by using the Fourier sine and Laplace transforms. Based on the modified Darcy’s law, the expression for the velocity field is obtained. It is found that similar solutions for Newtonian fluid appear as limiting cases of our solutions. Finally some graphical results of the velocity profiles are presented for different values of the material constants.

General Solutions for the Unsteady Flow of Second-Grade Fluids over an Infinite Plate that Applies Arbitrary Shear to the Fluid

General solutions corresponding to the unsteady motion of second-grade fluids induced by an infinite plate that applies a shear stress ƒ (t) to the fluid are established. These solutions can immediately be reduced to the similar solutions for Newtonian fluids. They can be used to obtain known solutions from the literature or any other solution of this type by specifying the function ƒ (.). Furthermore, in view of a simple remark, general solutions for the flow due to a moving plate can be developed.

New exact solution for Rayleigh–Stokes problem of Maxwell fluid in a porous medium and rotating frame

The aim of this paper is to determine the new exact solution of a magnetohydrodynamic (MHD) and rotating flow of the Maxwell fluid induced by a suddenly moved plate in its own plane. This is accomplished by using the Fourier sine and Laplace transforms. Based on the modified Darcy’s law, the expression for the velocity field is obtained. It is found that similar solutions for Newtonian fluid appear as limiting cases of our solutions. Finally some graphical results of the velocity profiles are presented for different values of the material constants.

Starting solutions of a viscoelastic fluid with Maxwell model between two oscillating cylinders

In this paper, we have determined the exact starting solutions of velocity field and associated shear and normal stresses corresponding to the unsteady flow of a Maxwell fluid, due to torsional oscillations of two infinite coaxial circular cylinders by means of Laplace and Hankel transforms. The fluid is situated at rest in an annular region between the cylinders. Suddenly both cylinders start torsional sine oscillations around their common axis, with different angular frequencies of their oscillations. The starting solutions that have been obtained satisfy the governing equation of motion and all imposed initial and boundary conditions. Furthermore, these solutions, presented as a sum of steady-state and transient solutions, reduce to the similar solutions for Newtonian fluid as the limiting case. They describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the unsteady solutions tend to the steady-state solutions which are periodic in time and independent of initial conditions. Finally, numerical results and concluding remarks are given.

Unsteady flow of a second grade fluid between two side walls perpendicular to a plate

Exact solutions for the unsteady flow of a second grade fluid induced by the time-dependent motion of a plane wall between two side walls perpendicular to the plane are established by means of the Fourier sine transforms. The similar solutions for Newtonian fluids, performing the same motions, are obtained as limiting cases for α 1 → 0 . The steady solutions, the same for Newtonian and non-Newtonian fluids, are also obtained as limiting cases for t → ∞ . In the absence of the side walls, all solutions that have been obtained reduce to those corresponding to the motion over an infinite plate. Graphical illustrations show that the diagrams corresponding to the velocity field in the middle of channel and the shear stress at the bottom wall for a second grade fluid are going to be those for a Newtonian fluid if the normal stress module α 1 → 0 .