Abstract
The unsteady pulsatile flow of blood through porous medium in an artery has been studied under the influence of periodic body acceleration and slip condition by considering blood as incompressible Newtonian electrically conducting fluid in the presence of magnetic field. In this paper, a new technique of differential quadrature method is introduced to find numerical solution of non-linear partial differential equations such as the equation of motion of this problem “Navier-Stokes equation”. The presence of the nonlinearity in the problem leads to severe difficulties in the solution approximation. In construction of the numerical scheme “a new algorithm” a generalized differential quadrature method (GDQM) is to use for derivatives with respect to space variables of differential equations and for the time derivative applying fourth order RungeKutta Method (RKM). The GDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by 4th order RKM. This combination of DQM and 4th order RKM gives a very good numerical technique for solving time dependent problems. The algorithm is coded in Matlab 7.14.0.739 and the simulations are run on a Pentium 4 CPU 900 MHz with 1 GB memory capacity. The effects of slip condition, magnetic field, porous medium, and body acceleration have been discussed. The numerical results show that the proposed method is more accurate and convergent than other numerical methods in literature. The method is illustrated and compared with the exact and analytical solutions and it is found that the proposed method gives a better accuracy and is quite easy to implement.
Highlights
Numerical approximation methods for solving partial differential equations have been widely used in many fields of science, in physics, engineering, chemistry and finance, and are fundamental for the mathematical formulation of continuum models
The present approach of generalized differential quadrature method (GDQM) used the method of directly Substitutes the Boundary Conditions into the Governing Equations is referred to as (SBCGE); this method overcomes the drawbacks of the δ-technique in implementing the derivative conditions
The problem is reduced by using GDQM to a system of ordinary differential equations (ODEs) that is solved by the 4th order Runge-Kutta Method (RKM)
Summary
Numerical approximation methods for solving partial differential equations have been widely used in many fields of science, in physics, engineering, chemistry and finance, and are fundamental for the mathematical formulation of continuum models. Many mathematical models have already been investigated by several research workers to study the unsteady pulsatile blood flow through porous medium under the influence of periodic body acceleration in the presence of magnetic field with slip boundary condition numerically and analytically. Sanyal and Debnath [26] presented a mathematical model for studying the effect of magnetic field on pulsatile blood flow through an inclined circular tube with periodic body acceleration. Eldesoky [32] studied the slip effects on the unsteady MHD pulsatile blood flow through porous medium in an artery under the influence of body acceleration. GDQM used for studying the effect of slip condition on unsteady pulsatile blood flow through a porous medium under the influence of periodic body acceleration in the presence of magnetic field
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