Non-Newtonian flow in pipes of non-circular cross section

Rectilinear flow through pipes of non-circular cross section in the case of elastico- viscous liquids is not possible in general and some secondary flow in the cross section of the pipe is to be expected. The present paper contains a detailed theoretical and experimental study of the problem for pipes of square and rectangular cross section. Our main concern is the form that the secondary flows take and their influence on the flow rate through the pipe.

The governing equations, which are non-linear partial differential equations, are solved numerically using finite difference methods used in conjuction with S.L.O.R. It is shown, that, for both the square and rectangular cross sections, there are eight vortices present. These have the same strength for the square cross section, but four of the eight become progressively weaker as the ratio of the sides of the rectangle is increased.

The effect of a variable (shear-dependent) viscosity on the flow rate is found to be sub- stantial but the effect of the secondary flows on the flow rate is predicted to be small unless the fluid has some rather unusual properties

The experimental results are in general agreement with the theoretical predictions.