Abdelhalim Ebaid

Peristaltic transport in an asymmetric channel through a porous medium

The problem of peristaltic transport of an incompressible viscous fluid in an asymmetric channel through a porous medium is analyzed. The flow is investigated in a wave frame of reference moving with velocity of the wave under the assumptions of long-wavelength and low-Reynolds number. An explicit form of the stream function is obtained by using Adomian decomposition method. The analysis showed that transport phenomena are strongly dependent on the phase shift between the two walls of the channel. It is indicated that the axial velocity component U in fixed frame increases with increasing the permeability parameter. In the case of high permeability parameter (as K->~), our results are in agreement with Mishra and Ramachandra Rao [M. Mishra, A. Ramachandra Rao, Peristaltic transport of a Newtonian fluid in an asymmetric channel, ZAMP 53 (2003) 532] and Eytan and Elad [O. Eytan, D. Elad, Analysis of intra-uterine fluid motion induced by uterine contractions, Bull. Math. Biol. 61 (1999) 221]. The results given in this paper may throw some light on the fluid dynamic aspects of the intra-uterine fluid flow through a porous medium.

A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method

Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced in this research to investigate linear and non-linear singular two-point BVPs. The effectiveness of the proposed approach is verified by several linear and non-linear examples.

Exact Analytical Solution of the Peristaltic Nanofluids Flow in an Asymmetric Channel with Flexible Walls and Slip Condition: Application to the Cancer Treatment

In the cancer treatment, magnetic nanoparticles are injected into the blood vessel nearest to the cancers tissues. The dynamic of these nanoparticles occurs under the action of the peristaltic waves generated on the flexible walls of the blood vessel. Studying such nanofluid flow under this action is therefore useful in treating tissues of the cancer. In this paper, the mathematical model describing the slip peristaltic flow of nanofluid was analytically investigated. Exact expressions were deduced for the temperature distribution and nano-particle concentration. In addition, the effects of the slip, thermophoresis, and Brownian motion parameters on the temperature and nano-particle concentration profiles were discussed and further compared with other approximate results in the literatures. In particular, these results have been obtained at the same values of the physical examined parameters that was considered in Akbar et al., “Peristaltic flow of a nanofluid with slip effects,” 2012. The results reveal that remarkable differences are detected between the exact current results and those approximately obtained in the literatures for behaviour of the temperature profile and nano-particles concentration. Accordingly, the current analysis and results are considered as optimal and therefore may be taken as a base for any future comparisons.

Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel

In applied science, the exact solution (when available) for any physical model is of great importance. Such exact solution not only leads to the correct physical interpretation, but also very useful in validating the approximate analytical or numerical methods. However, the exact solution is not always available for the reason that many authors resort to the approximate solutions by using any of the analytical or the numerical methods. To ensure the accuracy of these approximate solutions, the convergence issue should be addressed, otherwise, such approximate solutions inevitably lead to incorrect interpretations for the considered model. Recently, several peristaltic flow problems have been solved via the homotopy perturbation method, which is an approximate analytical method. One of these problems is selected in this paper to show that the solutions obtained by the homotopy perturbation method were inaccurate, especially, when compared with the exact solutions provided currently and also when compared with a well known accurate numerical method. The comparisons reveal that great remarkable differences have been detected between the exact current results and those approximately obtained in the literatures for the temperature distribution and the nano-particle concentration. Hence, many similar problems that have been approximately solved by using the homotopy perturbation method should be re-investigated by taking the convergence issue into consideration, otherwise, the published results were really incorrect.

Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions

A new straightforward approach for solving ordinary and partial second-order boundary value problems with Neumann boundary conditions is introduced in this research. This approach depends mainly on the Adomian decomposition method with a new definition of the differential operator and its inverse, which has been modified for Neumann boundary conditions. The effectiveness of the proposed approach is verified by several linear and nonlinear examples.

New types of exact solutions for nonlinear Schrödinger equation with cubic nonlinearity

Although, many exact solutions were obtained for the cubic Schrodinger equation by many researchers, we obtained in this research not only more exact solutions but also new types of exact solutions in terms of Jacobi-elliptic functions and Weierstrass-elliptic function.

Exact Analytical Solution for Suction and Injection Flow with Thermal Enhancement of Five Nanofluids over an Isothermal Stretching Sheet with Effect of the Slip Model: A Comparative Study

We introduced a direct and effective approach to obtain the exact analytical solution for the nanoparticles-water flow over an isothermal stretching sheet with the effect of the slip model. In particular, we examined and compared the effect of the existence of five metallic and nonmetallic nanoparticles, namely, Silver, Copper, Alumina, Titania, and Silicon Dioxide, in a base of water. The most interesting physical parameters were then discussed in the presence of no-slip model, first order slip, and second order slip parameters. It is found that, with no-slip effect, the present exact solutions are in a very good agreement with the previous published results. On the other hand, with the effect of the slip model, increase in the nanoparticle volume friction decreases the velocity for the high density of nanoparticles, increases it for the low density of them, and increases the temperature for all investigated nanoparticles. Further, increase in the wall mass decreases the velocity and temperature; however, it increases the local skin friction. Furthermore, increase in the slips slows down the velocity, increases the temperature with an impressive effect in the injection case, and decreases the local skin friction and the reduced Nusselt number. It was also demonstrated that, as the nanoparticle becomes heavier, this results in increase and decrease in reduced skin friction coefficient and reduced Nusselt number, respectively, with significant effect in the presence of the second slip. Finally, Silver is the suitable nanoparticle if slowing down the velocity and increasing the temperature are needed; Silicon Dioxide is the appropriate nanoparticle if different behavior is to be considered.

Generalization of He's Exp-Function Method and New Exact Solutions for Burgers Equation

A generalization of He’s Exp-function method for nonlinear equations is introduced in this research. New exact solutions are obtained for Burgers equation

An advanced study on the solution of nanofluid flow problems via Adomian’s method

Abstract Nanofluid flow is one of the most important areas of research at the present time due to its wide and significant applications in industry and several scientific fields. The boundary layer flow of nanofluids is usually described by a system of nonlinear differential equations with boundary conditions at infinity. These boundary conditions at infinity cause difficulties for any of the series method, such as Adomian’s method, the variational iteration method and others. The objective of the present work is to introduce a reliable method to overcome such difficulties that arise due to an infinite domain. The proposed scheme, that we will introduce, is based on Adomian’s decomposition method, where we will solve a system of nonlinear differential equations describing the boundary layer flow of a nanofluid past a stretching sheet.

Effect of Velocity Slip Boundary Condition on the Flow and Heat Transfer of Cu-Water and TiO2-Water Nanofluids in the Presence of a Magnetic Field

In nanofluid mechanics, it has been proven recently that the no slip condition at the boundary is no longer valid which is the reason that we consider the effect of such slip condition on the flow and heat transfer of two types of nanofluids. The present paper considers the effect of the velocity slip condition on the flow and heat transfer of the Cu-water and the TiO2-water nanofluids over stretching/shrinking sheets in the presence of a magnetic field. The exact expression for the fluid velocity is obtained in terms of the exponential function, while an effective analytical procedure is suggested and successfully applied to obtain the exact temperature in terms of the generalized incomplete gamma function. It is found in this paper that the Cu-water nanofluid is slower than the TiO2-water nanofluid for both cases of the stretching/shrinking sheets. However, the temperature of the Cu-water nanofluid is always higher than the temperature of the TiO2-water nanofluid. In the case of shrinking sheet the dual solutions have been obtained at particular values of the physical parameters. In addition, the effect of various physical parameters on such dual solutions is discussed through the graphs.