Asymptotic analysis of Stokes flow in a tortuous vessel

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 $\epsilon$, 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 $O\left ( {{\epsilon ^2}} \right )$ 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 $O\left ( {{\epsilon ^4}} \right )$ 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.