Orthogonal Cubic Spline Collocation Method for the Nonlinear Parabolic Equation Arising in Non-Newtonian Fluid Flow

Abstract

Using the orthogonal cubic spline collocation method, solution for the nonlinear parabolic equation arising in magneto-hydrodynamic unsteady Poiseuille flow of the generalized Newtonian fluid (Carreau rheological model) is obtained. Also, using the Lyapunov functional, a bound for the maximum norm of the semi-discrete solution is derived. Moreover, optimal error estimates are established for the semi-discrete solution. Numerical results thus obtained are presented graphically and the salient features of the solution are discussed, for various values of the parameters. The results obtained reveal many interesting behaviors that warrant further study on the parabolic equations related to non-Newtonian fluid phenomena. Furthermore the analysis can be used to study the mathematical models that involve the flow of viscous fluids with shear rate-dependent properties: For example, models dealing with polymer processing, tribology and lubrication, and food processing. © 2006 Elsevier Inc. All rights reserved.