Blood Flow through Capillaries with a Permeable Wall

Abstract

With a view to obtaining mathematical bases for microcirculation of plasma with filtration or reabsorption the Stokes equations for steady, axisymmetric creeping flow have been applied to a Newtonian fluid flowing through a straight, uniform tube of circular section with permeable walls. As a boundary condition Starling's law is adopted, which states in mathematical version that the velocity component normal to the wall is assumed to be a linear function of the local hydrostatic pressure. It is required that if a dimensionless parameter ε=2ηκ/R (η: viscosity of the fluid, R: radius of the tube, κ: filtration constant) vanishes, the solution will be reduced to the basic Poiseuille flow with a constant pressure gradient. In fact, ε is so small that the radial component of the fluid velocity veries linearly with the axial distance, as is seen from the boundary condition. The solution of the Stokes equations to meet this requirement can be derived in a simple form.The relation between the volume flux through each section of the tube and the pressure gradient is also easily obtained. Although this solution yields a flow pattern with non-vanishing velocity component tangential to the wall, this slip flow is understood to occur in consequence of the penetration.