T. Haroon

Exactly satisfying initial conditions neural network models for numerical treatment of first Painlevé equation

A new stochastic intelligence method is developed to solve first Painleve equation.Design of three unsupervised ANN models that satisfying exactly initial conditions.Optimization capability of SQP is exploited for training of design parameter of ANNs.Accuracy and convergence are validated in term of various performance criterions.Impact on effectiveness of the models is investigated by varying neurons in ANNs. In this paper, novel computing approach using three different models of feed-forward artificial neural networks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painleve equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models.

Unsteady MHD flow due to non-coaxial rotations of a porous disk and a fluid at infinity

SummaryAn exact solution of the unsteady three-dimensional Navier-Stokes equations is derived for the case of flow due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field. An analytical solution of the problem is established by the method of Laplace transform, and the velocity field is presented in terms of the tabulated functions. It is found that the boundary layer thickness in the cases of suction/blowing decreases with the increase in the magnetic parameter.

Unsteady MHD flow of a non-newtonian fluid due to eccentric rotations of a porous disk and a fluid at infinity

SummaryAn exact solution of the unsteady flow of a second-order fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is investigated. It is once again shown that for uniform suction or uniform injection at the disk an asymptotic profile exists for the velocity distribution. The effects of the magnetic field, the material parameters of the second-order fluid, suction and injection on the velocity distribution are studied. Further, from the solution of a rigid disk, it is found that for parameter β>.01, a non-Newtonian effect is present in the velocity field. However, for β<.01 the velocity field becomes a Newtonian one.

Torsional flow of third grade fluid using modified homotopy perturbation method

This paper considers the steady flow of an incompressible third grade fluid between two vertical concentric rotating cylinders of infinite lengths. Modified Homotopy perturbation method is used to reduce the volume of tedious calculations involved to solve second order nonlinear differential equation. Special cases with one cylinder, inner or outer at rest (bounded domain case), and fluid flow at a rotating cylinder (unbounded domain case) are obtained. The effect of @b, the non-dimensional number related to the material constants, @w, rotation of the cylinder and R, the ratio between radii of outer and inner cylinders, on the velocity profile are discussed and are shown graphically.

MHD flow of a third-grade fluid due to eccentric rotations of a porous disk and a fluid at infinity

Abstract The problem of magnetohydrodynamics (MHD) flow of a conducting, incompressible third-grade fluid due to non-coaxial rotations of a porous disk and a fluid at infinity in the presence of a uniform transverse magnetic field is considered. An exact analysis is carried out to model the governing non-linear partial differential equation. A numerical solution of the third-order non-linear partial differential equation has been obtained. Several graphs and tables have been drawn to show the influence of porosity e, magnetic parameter N, material parameters α and β on the velocity distribution.

Wire Coating Extrusion in a Pressure-type Die in Flow of a Third Grade Fluid via Homotopy Perturbation Method

A mathematical model for third grade fluid is developed for the wire coating, using a conical unit, where the flow is dependent on the wire velocity and the geometry of the unit. The nonlinear constitutive equations are formulated in cylindrical coordinates and theoretical results are obtained by using the homotopy perturbation method. Explicit analytical expressions for velocity field, radius of the coated wire, volumetric flow rate, shear stress on the wire surface and force on the total wire surface have been derived.

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.

A comparison of the adomian and homotopy perturbation methods in solving the problem of squeezing flow between two circular plates

Abstract The objective of this paper is to compare two methods employed for solving nonlinear problems, namely the Adomian Decomposition Method (ADM) and the Homotopy Perturbation Method (HPM). To this effect we solve the Navier‐Stokes equations for the unsteady flow between two circular plates approaching each other symmetrically. The comparison between HPM and ADM is bench‐marked against a numerical solution. The results show that the ADM is more reliable and efficient than HPM from a computational viewpoint. The ADM requires slightly more computational effort than the HPM, but it yields more accurate results than the HPM.