Asymptotic analysis of Stokes flow in a tortuous vessel
Richard S. Chadwick · Quarterly of Applied Mathematics · 1985
Steady slow viscous flow is considered inside a vessel with circular cross section. The centerline curvature is specified as a function of arc length. The Stokes equations are written in orthogonal curvilinear coordinates. The primary small parameter is the slenderness ratio ϵ \epsilon , which is the ratio of vessel radius to vessel length or wavelength. The product of centerline curvature and vessel length is assumed to be of order unity. A transverse drift appears at O ( ϵ 2 ) O\left ( {{\epsilon ^2}} \right ) that is proportional to the rate of change of curvature. Contours of axial velocity show a primary peak shifted toward the inside wall and a secondary peak grows toward the outside wall as curvature is increased. The flux ratio or relative hydrodynamic conductance is calculated to O ( ϵ 4 ) O\left ( {{\epsilon ^4}} \right ) and includes the effect of variable curvature. The present calculations tend to indicate that the sinusoidal mode of buckled micro-vessel could offer substantially more resistance to flow than the helical buckled mode.