Theoretical Investigation of Creeping Viscoelastic Flow Transition Around a Rotating Curved Pipe

TL;DR

This paper analytically investigates the creeping viscoelastic flow around a rotating torus using a second-order model, focusing on first-order solutions and the influence of geometry on flow behavior.

Contribution

It provides an analytical solution for the first-order flow around a rotating torus using bipolar coordinates and explores the effects of geometry on flow characteristics.

Findings

First-order flow corresponds to Newtonian behavior.

Secondary flow is described by a non-vanishing stream function.

Surface traction distribution depends on toroidal parameters.

Abstract

The study of creeping motion of viscoelastic fluid around a rotating rigid torus is investigated. The analysis of the problem is performed using a second-order viscoelastic model. The study is carried out in terms of the bipolar toroidal system of coordinates where the toroid is rotating about its axis of symmetry (z-axis). The problem is solved within the frame of slow flow approximation. Therefore, all variables in the governing equations are expanded in a power series of angular velocity. A set of successive partial differential equations is obtained. The equations of motion governing the first and second-order are formulated and solved for the first-order only in this paper. However, the solution of the second-order equations will be the subject of a part two of this series of papers. Analytically, Laplace's equation is solved via the usual method of separation of variables. This method shows that, the solution is given in a form of infinite sums over Legendre functions of the first and second kinds. From the obtained solution it is found that, the leading term, W(1), of the velocity represents the Newtonian flow. The second-order term shows that, the only non vanishing term is the stream function, which describes a secondary flow. The distribution of the surface traction at the toroid surface is calculated and discussed. Considering hydrodynamically conditions, the effects of toroidal geometrical parameters on the flow field are investigated in detail.