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Unsteady Solutions in a Third-Grade Fluid Filling the Porous Space
T. Hayat,H. Mambili-Mamboundou,F. M. Mahomed
Mathematical Problems in Engineering , 2008, DOI: 10.1155/2008/139560
Abstract: An analysis is made of the unsteady flow of a third-grade fluid in a porous medium. A modified Darcy's law is considered in the flow modelling. Reduction and solutions are obtained by employing similarity and numerical methods. The effects of pertinent parameters on the flow velocity are studied through graphs.
Unsteady Magnetohydrodynamic Free Convection Flow of a Second Grade Fluid in a Porous Medium with Ramped Wall Temperature  [PDF]
Samiulhaq, Sohail Ahmad, Dumitru Vieru, Ilyas Khan, Sharidan Shafie
PLOS ONE , 2014, DOI: 10.1371/journal.pone.0088766
Abstract: Magnetic field influence on unsteady free convection flow of a second grade fluid near an infinite vertical flat plate with ramped wall temperature embedded in a porous medium is studied. It has been observed that magnitude of velocity as well as skin friction in case of ramped temperature is quite less than the isothermal temperature. Some special cases namely: (i) second grade fluid in the absence of magnetic field and porous medium and (ii) Newtonian fluid in the presence of magnetic field and porous medium, performing the same motion are obtained. Finally, the influence of various parameters is graphically shown.
Stokes First Problem for an Unsteady MHD Third-Grade Fluid in a Non-Porous Half Space with Hall Currents  [PDF]
Haider Zaman, Arif Sohail, Ubaidullah  
Open Journal of Applied Sciences (OJAppS) , 2014, DOI: 10.4236/ojapps.2014.43010
Abstract: The well-known problem of unidirectional plane flow of a fluid in a non-porous half-space due to the impulsive motion of the rigid plane wall it rests upon is discussed in the context of an unsteady MHD third-grade fluid in presence of Hall currents. The governing non-linear partial differential equations describing the problem are converted to a system of non-linear ordinary differential equations by using the similarity transformations. The complex analytical solution is found by using the homotopy analysis method (HAM). The existing literature on the topic shows that it is the first study regarding the effects of Hall current on flow of an unsteady MHD third-grade fluid over an impulsively moving plane wall. The convergence of the obtained complex series solutions is carefully analyzed. The effects of dimensionless parameters on the velocity are illustrated through plots and the effects of the pertinent parameters on the local skin friction coefficient at the surface of the wall are presented numerically in tabular form.
A Modified Homotopy Perturbation Transform Method for Transient Flow of a Third Grade Fluid in a Channel with Oscillating Motion on the Upper Wall  [PDF]
Mohammed Abdulhameed,Rozaini Roslan,Mahathir Bin Mohamad
Journal of Computational Engineering , 2014, DOI: 10.1155/2014/102197
Abstract: A new analytical algorithm based on modified homotopy perturbation transform method is employed for solving the transient flow of third grade fluid in a porous channel generated by an oscillating upper wall. This method incorporates the He’s polynomial into the HPM, combined with Laplace transform. Comparison with HPM and OHAM analytical solutions reveals that the proposed algorithm is highly accurate. This proves the validity and great potential of the proposed algorithm as a new kind of powerful analytical tool for transient nonlinear problems. Graphs representing the solutions are discussed, and appropriate conclusions are drawn. 1. Introduction The equations describing the motion of non-Newtonian fluids are strongly of nonlinear higher order than the Navier-Stokes equation for Newtonian fluids. These nonlinear equations form a very complex structure, with a small number of exact solutions. Mostly, numerical methods have largely been used to handle these equations. The class of problems with known exact solution is related to the problem for infinite flat plate. The related studies in the recent years are as follows: Fakhar et al. [1] examine the exact unsteady flow of an incompressible third grade fluid along an infinite plane porous plate. They obtained results by applying a translational type of symmetries combined with finite difference method. Danish and Kumar [2] analysed a steady flow of a third grade between two parallel plates using similarity transformation. Abdulhameed et al. [3] consider an unsteady viscoelastic fluid of second grade for an infinite plate. They applied Laplace transform together with the regular perturbation techniques to obtain the exact solution. Ayub et al. [4] analysed the problem of steady flow of a third grade fluid for an infinite plate porous plate using homotopy analysis method (HAM). Homotopy perturbation method developed by He [5] for solving linear and nonlinear initial-boundary value problem merges two techniques, the perturbation and standard homotopy. Recently, the homotopy perturbation method has been modified by some scientists to obtain more accurate results and rapid convergence and also to reduce the amount of computation. Ghorbani [6] introduced He’s polynomials based on homotopy perturbation method for nonlinear differential equations. The homotopy perturbation transform method (HPTM) introduced by Khan and Wu [7] is a combination of the homotopy perturbation method and Laplace transform method that is used to solve various types of linear and nonlinear systems of partial differential equations. The
A Note on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls  [PDF]
H. Zaman,Z. Ahmad,M. Ayub
ISRN Mathematical Physics , 2013, DOI: 10.1155/2013/705296
Abstract: This work is concerned with the influence of uniform suction or injection on flow and heat transfer analysis of unsteady incompressible magnetohydrodynamic (MHD) fluid with slip conditions. The resulting unsteady problem for velocity and heat transfer is solved by means of Laplace transform. The characteristics of the transient velocity, overall transient velocity, steady state velocity and heat transfer at the walls are analyzed and discussed. Graphical results reveal that the magnetic field, slip parameter, and suction (injection) have significant influences on the velocity, and temperature distributions, which also changes the heat transfer behaviors at the two plates. The results of Fang (2004) are also recovered by keeping magnetic field and slip parameter absent. 1. Introduction Navier-Stokes equations are the basic equations of fluid mechanics. Exact solutions of Navier-Stokes equations are rare due to their inherent nonlinearity. Exact solutions are important because they serve as accuracy checks for numerical solutions. Complete integration of these equations is done by computer techniques, but the accuracy of the results can be established only by comparison with exact solutions. In the literature, there are a large number of Newtonian fluid flows for which exact solutions are possible [1–6]. The effects of transverse magnetic field on the flow of an electrically conducting viscous fluid have been studied extensively in view of numerous applications to astrophysical, geophysical, and engineering problems [7–15]. If the working fluid contains concentrated suspensions, then the wall slip can occur [16]. Khaled and Vafai [3] studied the effect of the slip on Stokes and Couette flows due to an oscillating wall. However, the literature lacks studies that take into account the possibility of fluid slippage at the walls. Applications of these problems appear in microchannels or nanochannels and in applications where a thin film of light oil is attached to the moving plates or when the surface is coated with special coating such as a thick monolayer of hydrophobic octadecyltrichlorosilane [17]. Yu and Ameel [18] imposed nonlinear slip boundary conditions on flow in rectangular microchannels. Erdogan [6] studied deeply the solution to the Stokes problem under nonslip conditions at the wall. Ayub and Zaman [19] studied the effects of suction and blowing for orthogonal flow impinging on a wall. Khan et al. [20] discussed the flow of Sisko fluid through a porous medium. Ariel et al. [21] considered the flow of elasticoviscous fluid with partial slip.
Hall Effects on Unsteady Magnetohydrodynamic Flow of a Third Grade Fluid
K Fakhar,XU Zhen-Li,CHENG Yi,

中国物理快报 , 2007,
Abstract: The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible third grade fluid bounded by an infinite porous plate is studied with the Hall effect. An external uniform magnetic field is applied perpendicular to the plate and the fluid motion is subjected to a uniform suction and injection. Similarity transformations are employed to reduce the non-linear equations governing the flow under discussion to two ordinary differentialequations (with and without dispersion terms). Using the finite difference scheme, numerical solutions represented by graphs with reference to the various involved parameters of interest are discussed and appropriate conclusions are drawn.
Closed Form Solutions for Unsteady Free Convection Flow of a Second Grade Fluid over an Oscillating Vertical Plate  [PDF]
Farhad Ali, Ilyas Khan, Sharidan Shafie
PLOS ONE , 2014, DOI: 10.1371/journal.pone.0085099
Abstract: Closed form solutions for unsteady free convection flows of a second grade fluid near an isothermal vertical plate oscillating in its plane using the Laplace transform technique are established. Expressions for velocity and temperature are obtained and displayed graphically for different values of Prandtl number Pr, thermal Grashof number Gr, viscoelastic parameter α, phase angle ωτ and time τ. Numerical values of skin friction τ0 and Nusselt number Nu are shown in tables. Some well-known solutions in literature are reduced as the limiting cases of the present solutions.
Unsteady Couette Flow through a Porous Medium in a Rotating System  [PDF]
Maitree Jana, Sanatan Das, Rabindra Nath Jana
Open Journal of Fluid Dynamics (OJFD) , 2012, DOI: 10.4236/ojfd.2012.24016
Abstract: An investigation has been made on an unsteady Couette flow of a viscous incompressible fluid through a porous me- dium in a rotating system. The solution of the governing equations has been obtained by the use of Laplace transform technique. It is found that the primary velocity decreases and the magnitude of the secondary velocity increases with an increase in rotation parameter. The fluid velocity components are decelerated by an increase of Reynolds number. An increase in porosity parameter leads to increase the primary velocity and the magnitude of the secondary velocity. It is also found that the solution for small time converges more rapidly than the general solution. The asymptotic behavior of the solution is analyzed for small as well as large values of rotation parameter and Reynolds number. It is observed that a thin boundary layer is formed near the moving plate of the channel and the thicknesses of the boundary layer increases with an increase in porosity parameter.
Laplace transform in spaces of ultradistributions  [PDF]
Bojan Prangoski
Mathematics , 2013,
Abstract: The Laplace transform in Komatsu ultradistributions is considered. Also, conditions are given under which an analytic function is a Laplace transformation of an ultradistribution.
Analytical Solution of Thermal Radiation and Chemical Reaction Effects on Unsteady MHD Convection through Porous Media with Heat Source/Sink
Abdel-Nasser A. Osman,S. M. Abo-Dahab,R. A. Mohamed
Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/205181
Abstract: This paper analytically studies the thermal radiation and chemical reaction effect on unsteady MHD convection through a porous medium bounded by an infinite vertical plate. The fluid considered here is a gray, absorbing-emitting but nonscattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The dimensionless governing equations are solved using Laplace transform technique. The resulting velocity, temperature and concentration profiles as well as the skin-friction, rate of heat, and mass transfer are shown graphically for different values of physical parameters involved.
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