M. A. Rana

Application of He's Homotopy Perturbation Method to Sumudu Transform

Abstract In this work, an application of He’s homotopy perturbation method is proposed to compute Sumudu transform. The method, in contrast of usual methods which need integration, requires simple differentiation. The results reveal that the method is very effective and simple. Keywords: Perturbation methods; Homotopy perturbation method; Sumudu transform 1. Introduction Various perturbation methods [1,2] have been widely applied by scientists and engineers to solve nonlinear problems. The traditional perturbation techniques are based on the existence of small parameter. These techniques are so powerful that sometimes small parameter is introduced artificially into a problem having no parameter and then finally set equal to unity to recover the solution of the original problem. He [3-6] proposed a new method called homotopy perturbation method (HPM) in 1998. The HPM, in fact, is a coupling of the traditional perturbation method and homotopy in topology. The HPM method, without demanding a small parameter in equations, deforms continuously to a simple problem which is easily solved. This method yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. This new method was further developed and improved by He and applied to non-linear oscillators with discontinuity [7], asymptotology [8], non-linear wave equations [9], bifurcation of nonlinear problems [10], limit cycle [11], delay-differential equations [[12], and boundary values problems [13]. He’s method is a universal one which can solve various types of non-linear problems. For example, it was applied to the non-Newtonian flow by Siddiqui et al. [14-15], to Volterra’s integro-differential equation by El-Shahed [16], to Helmholtz equation and fifth-order KdV equation by Rafei et al. [17], to nonlinear oscillator by Cai et al. [18], and to compute the Laplace transform by Abbasbandy [19]. A complete review on HPM’s applications is given in [20-21]. Sumudu transform was probably first time introduced by Watugala in his work [22]. Its simple formulation and direct applications to ordinary differential equations immediately sparked interest in this new tool. This new transform was further developed and applied to many problems by various workers. Asiru [23,24] applied to integro-differential equations, Watugala [25,26] extended the transform to two variables with emphasis on solution to partial differential equations and applications to engineering control problem, and its fundamentals properties were established by Belgacem et al. [27-29]. The Sumudu transform has very special and useful properties and can

Application of Homotopy Perturbation and Numerical Methods to the Unsteady Squeezing MHD Flow of a Second Grade Fluid between Circular Plates

The unsteady two-dimensional flow of an electrically conducting, incompressible second grade fluid between two circular plates approaching or receding from each other symmetrically is studied. A similarity transformation is used to reduce the system of partial differential equations to a single fifth-order non-linear differential equation. The resulting non-linear boundary value problem is solved using homotopy perturbation method and numerical method, and the obtained results are compared graphically for different values of relevant parameters. The total resistance to the upper plate is also calculated.

Calendering analysis of a third-order fluid

In this paper, the study of a non-Newtonian material when it is dragged through the narrow region between two co-rotating rolls is carried out. The conservation of mass and momentum equations based on lubrication theory are nondimensionalized and solved for the velocity and pressure fields using the perturbation technique. By considering the influence of the material parameter, the dimensionless leave-off distance in the calendering process is determined. The leave-off distance is expressed in terms of eigen value problem. Quantities of engineering interest like maximum pressure, the roll-separating force, and the power transmitted to the fluid by rolls are calculated. It is observed that the material parameter has great influence on detachment point, velocity, and pressure distribution, which are useful for the calendering process.

Effect of magnetohydrodynamics on Newtonian calendering

In this paper, an analysis has been presented for the calendering process of incompressible magnetohydrodynamics Newtonian fluid. The lubrication approximation is used to simplify the equations of motion. Exact solutions for velocity profile, pressure gradient, flow rate per unit width, rate of strain, shear stress, maximum shear rate and shear stress at the roll surface are obtained. The value of λ, the distance from the nip to the point where the sheet leaves the rolls, is calculated using Newton–Cotes formula along with the regula-falsi method. Numerical results are presented for pressure distribution, power transmitted to the fluid by both rolls, force separating the two rolls using Newton–Cotes formula with regula-falsi method and Simpson’s rule for different values of magnetic parameter, M , and the corresponding values of λ. Some results are shown graphically. It is found that the magnetic field provides the controlling parameter to increase or decrease power transmission, separation force and distance between attachment and detachment points, which are useful for the calendaring process.

Modeling of non-isothermal flow of a magnetohydrodynamic, viscoplastic fluid during calendering

We have modeled an incompressible, non-isothermal flow of a magnetohydrodynamic (MHD) viscoplastic fluid when it passes through the tapered area between two co-rotating rolls. The basic conservation equations like mass, momentum, and energy based on the lubrication approximation theory (LAT) are developed, non-dimensionalized, and solved exactly for the velocity-, pressure gradient-, and temperature-distribution. The value of λ, where the sheet leaves the rolls, is calculated using Simpson’s 1/3 formula for numerical integration along with the modify regula-falsi method. Moreover, quantities of engineering interest such as extreme pressure, the roll separation force and the power transferred to the fluid by the rolls are also computed. The results show that the inclusion of the viscoplastic parameter affects the velocity field, pressure gradient-, and temperature-distribution substantially. It is found that the magnetic field provides a mechanism to control power transmission, separation force, and distance between attachment and detachment points, which are very useful for the calendering process.

A Variant of the Classical Von Kármán Flow for a Couple Stress Fluid

We present an attempt to study the influence of couple stresses on the flow induced by an infinite disk rotating with a constant angular velocity. The governing equations of motion in three dimensions are treated analytically yielding the derivation of exact solutions which differ from those corresponding to the classical Von Karmans Bow. The analysis reveals that a boundary layer structure develops near the surface of the disk, whose far-field behaviour is distinct from the near-wall solution. The velocity and vorticity components for various values of the dimensionless parameters, associated with the flow, are presented graphically.

Roll coating analysis of a second-grade material

In this study, the lubrication approximation theory is used to provide numerical results in roll coating over a moving flat porous web. The rate of injection at the roll surface is assumed equal to the rate of suction on the web. The second-grade fluid is used, which reduces with appropriate modifications to the Newtonian fluid model. Results are obtained in such quantities as coating thickness, split location, pressure distribution, stresses, forces, the power input to the roll, and adiabatic temperature rise between the coating roll and the coated web. Some of these results are shown graphically. It is found that material parameters β and Reynolds number are the parameters to control flow rate, coating thickness separation points, separation force, power input, and pressure.

Roll coating analysis of a third grade fluid

This paper studies the roll-coating process of an incompressible viscoelastic fluid, where the roll and the web have equal velocities. The lubrication approximation theory is used to simplify the equations of motion. Solutions for velocity profile, pressure gradient, flow rate per unit width, and shear stress at the roll surface are obtained by using a regular perturbation method. Integrated quantities of engineering interest are also calculated. These include the maximum pressure, separation point, roll/sheet separating force, power transmitted to the fluid by the roll, and coating thickness. It is found that these quantities increase substantially and monotonically as the fluid’s material constant increases.

Unsteady unidirectional MHD flow of an Erying Powell model between two parallel infinite porous plates with variable suction/injection

The unsteady unidirectional MHD flow of an incompressible Erying Powell model between two parallel infinite porous plates with variable suction/injection velocity under the action of uniform magnetic field and couple stresses is analyzed analytically and numerically. The change appearing in model due to various non-dimensional parameters is discussed and presented graphically. It is predicted that in electrically conducting polar fluids, couple stress effects may be large.

Numerical solution of thin film steady MHD flow of a Jhonson-Segalman fluid on a vertical surfaces

The steady MHD thin film flow of a Johnson-Segalman fluid on a vertical surface is modeled as well as analyzed numerically. Two problems regarding lifting and drainage are investigated. The effect of different parameters such as Stokes number St, Weissenberg number We, the ratio of viscosities φ, slip parameter α, and applied magnetic fiel M2 emerging in the model on velocity fields for the both case are studied graphically.