On the flow of viscoplastic fluids in non-circular tubes

Abstract

Steady flow of the viscoplastic Bingham and Herschel–Bulkley (H–B)​ fluids in tubes of non-circular cross-section is investigated analytically. The solution methodology is general in scope, does not put any restrictions on the Bingham number and thus allows the mapping of the flow field for high Bingham numbers in straight tubes with non-circular axially-symmetric but otherwise arbitrary cross-sectional contours. The circular tube contour is mapped onto an arbitrary non-circular contour on which the no-slip condition is satisfied via a one-to-one and continuous mapping. Governing equations are solved for the full spectrum of axially symmetrical cross-sectional shapes and a specific example is developed for the viscoplastic H–B fluid flow in a tube with an equilateral triangular cross-section. The shape and the extent of the plug zones centered on the tube axis and the stagnant zones in the corners are predicted for both Bingham and H–B fluids. The effect of the shear rate dependent viscosity of the H–B fluids, leading to either shear-thickening or shear-thinning behavior, on the formation of the plug and stagnation zones is examined.