Analytical Solution of Laminar Velocity Distribution in Annular Sector Duct

Velocity distribution, Vz of laminar flow in a noncircular duct which has a cross-sectional shape of an annular sector (radius of inner cylinder : R1, and of outer one : R2, and opening angle : 2α) has been obtained by an analytical method. Result is shown as Vz=-R22/μ·dP/dZ·2/Inη ∞∑m=1 1-(-1)mη2/km(km2+4) sin (kmInr) · (1-coshkmθ/coshkmα) where r=R/R2, η=R1/21, km=mπ/1nη. Then an equivalent diameter, Deq, which is defined as -dP/dZ=32μVz/Deq2 by a similar method as for a circular tube, has been given by Deq=R2 [-128/1-(η2)Inη ∞∑m=1 (1-(-1)mη2/km2+4)2 (1-tanhkmα/kmα)]1/2 Analytical result of Deq is confirmed by the experimental result on the pressure gradient for some cross-sectional shapes of ducts.