Hybrid numerical approach of finite difference and asymptotic interpolation methods for non-newtonian fluids flow

Abstract

Previous research in the mathematical and physics fields has used computational or empirical approaches to analyse fluid flow problems. Therefore, in this thesis a hybrid numerical approach for non-Newtonian third- and fourth-grade fluid flow problems using the finite difference method and the asymptotic interpolation method are presented. The hybrid method is important for finding accurate results as the size of the problem domain increases to infinity. The finite difference method is used to discretize the nonlinear partial differential equation into a linear system. An asymptotic interpolation method is used to estimate nodal value as the size of the domain tends to infinity. The algorithm is coded using the MATLAB program. A polynomial function that fits the hybrid solution is used to calculate the error of the equation. Theoretical error analysis using truncation error in the finite difference method, right-hand side perturbation linear system, and right perturbation theorem is conducted to determine the norm and range of errors. An implicit numerical scheme of modified fluid problems with an exact solution has been achieved by adding an extra term to the partial differential equation. The norm of error between the hybrid method and exact solution is less than the norm of error between the finite difference method and exact solution. The theory of stability for third-grade fluid is carried out, and the numerical scheme is stable provided that the condition of modulus of the amplifier holds. The hybrid method is used to solve the constant acceleration of an unsteady magnetohydrodynamic third-grade fluid in a rotating frame. The analyses show that the increment of the magnetic and rotating parameters decreases the speed of motion and thus the velocity. The velocity increases with an increase in time. The unsteady magnetohydrodynamic fourth-grade fluid problem in the rotating frame is investigated. Increasing the elastic parameters increases the velocity of the fluid. The problem of heat transfer for third-grade non-Newtonian fluid flow with magnetic effect is addressed. The temperature drops by increasing the Prandtl number. It is noted that increasing the Grashof number increases the temperature and velocity. The obtained results have shown that the hybrid method is consistent, stable, and converges to the solution.