On a Generalization of the Poiseuille Law

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Abstract
From Newton''s relation for the viscous drag of a moving fluid, the differential equation for the flow of fluids with variable viscosity through cylindrical tubes having non-rigid walls is developed. It is yd(pr2)/dx = d([eta]ydv/dy)/dy, in which p is the fluid pressure, r is the tube radius, v is the velocity and [eta] is the viscosity of the fluid at a point x along the tube. The dimen-sionless parameter y varies between 0 and 1. On solving for the rate of flow Q of the fluid through a cross-section of the tube, the generalized form of the Poiseuille law is obtained, namely, [image] dv, in which v'' is the velocity of the fluid along the axis of the tube at the section of radius r. If the viscosity is constant,[image]. This result is applied to tubes (1) with rigid walls, (2) with thin walls for which Hooke''s law applies, and (3) with thin elastomeric walls. The last application is of special significance in studying blood flow in the larger vessels of the body. In this case, [image]. Here p'' is the pressure in the fluid where the radius is a, p is the pressure where the radius is r at a distance x downstream from the section of radius a, and po is the pressure on the outside surface of the tube.

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