Incompressible flow in a slightly tapered tube, an incompressible fluid flows through a tube of circular cross section, for which the tube radius changes linearly from R0 at the tube entrance to a slightly smaller value RL at the tube exit. Assume that the Hagen-Poiseuille equation is approximately valid over a differential length, dz, of the tube so that the mass flow rate is this is a differential equation for P as a function of z, but, when the explicit expression for R(z) is inserted, it is not easily solved.

(a) Write down the expression for R as a function of z.

(b) Change the independent variable in the above equation to R, so that the equation becomes.

(c) Integrate the equation, and then show that the solution can be rearranged to give.

Interpret the result. The approximation used here that a flow between nonparallel surfaces can be regarded locally as flow between parallel surfaces is sometimes referred to as the lubrication approximation and is widely used in the theory of lubrication. By making a careful order-of-magnitude analysis, it can be shown that, for this problem, the lubrication approximation is valid as longas4

Transcribed Image Text:

w w 77(Po-P₁)Rp 8μ TIR(z)l'p_d 8μ Part (b) TR'p_dR₁ - Ro e (dB) (R dR [1 W 8μ L dz Part (c) 1+ (R/Ro) + (R₁/R)² - 3(R₁/R)² 1+ (R₂/R) + (R₂/R)² R₁ R(1-(R.)) < <1 Ro