An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by Collocation Method

Abstract

An analytical investigation is applied for unsteady motion of a rigid spherical particle in a quiescent shear-thinning power-law fluid. The results were compared with those obtained from Collocation Method (CM) and the established Numerical Method (Fourth order Runge–Kutta) scheme. It was shown that CM gave accurate results. Collocation Method (CM) and Numerical Method are used to solve the present problem. We obtained that the CM which was used to solve such nonlinear differential equation with fractional power is simpler and more accurate than series method such as HPM which was used in some previous works by others but the new method named Akbari-Ganji’s Method (AGM) is an accurate and simple method which is slower than CM for solving such problems. The terminal settling velocity—that is the velocity at which the net forces on a falling particle eliminate—for three different spherical particles (made of plastic, glass and steel) and three flow behavior index n, in three sets of power-law non-Newtonian fluids was investigated, based on polynomial solution (CM). Analytical results obtained indicated that the time of reaching the terminal velocity in a falling procedure is significantly increased with growing of the particle size that validated with Numerical Method. Further, with approaching flow behavior to Newtonian behavior from shear-thinning properties of flow (n → 1), the transient time to achieving the terminal settling velocity is decreased.

Keywords

• Analytical solution
• Non-Newtonian fluid
• Spherical particle
• Collocation Method (CM)