Analytical Solution of Laminar Velocity Distribution in Annular Sector Duct

Abstract

Velocity distribution, V_z 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
V_z=-R₂²/μ·dP/dZ·2/Inη ∞∑m=1 1-(-1)^mη²/k_m(k_m²+4) sin (k_mInr) · (1-coshk_mθ/coshk_mα)
where r=R/R₂, η=R₁/2₁, k_m=mπ/1nη. Then an equivalent diameter, D_eq, which is defined as
-dP/dZ=32μV_z/D_eq²
by a similar method as for a circular tube, has been given by
D_eq=R₂ [-128/1-(η²)Inη ∞∑m=1 (1-(-1)^mη²/k_m²+4)² (1-tanhk_mα/k_mα)]^1/2
Analytical result of D_eq is confirmed by the experimental result on the pressure gradient for some cross-sectional shapes of ducts.