THE STEADY MOTION OF A VISCOUS FLUID IN A CURVED TUBE

Abstract

Numerical solutions have been obtained of the equations governing the steady motion of a viscous fluid in a tube of circular cross-section coiled in the form of a circle. Results are presented for the range 96 to 5000 of the parameter D = 4R√(2a/L). Here R = Ga3/4μν, where G is the constant pressure gradient maintaining the motion, a the radius of the cross-section of the tube, and L the radius of the circle in which the tube is coiled. The solutions have been carefully checked for accuracy and the results are compared with previous work on this problem. Substantial discrepancies are found to exist with a recent set of calculations over approximately the same range of D. As D increases it is possible to observe in the solutions a trend towards a structure consisting of a boundary layer near the wall of the tube with an inviscid core in the centre. This type of model has formed the basis of a number of previous asymptotic solutions. The ratio λs/λc of the friction coefficients for a straight tube and a curved tube under the same pressure gradient is expressed as an asymptotic formula from the present results in the form λs/λc ˜ 8.12D-1 – 16.7D-1 and compared with the previous theories. The friction ratio and other properties compare well with experimental measurements.