An alternative to the no-slip boundary condition at a stationary solid, proposed by Navier in the 1820s and revived in recent years for modeling liquid flows in extremely small channels, is to suppose that the tangential fluid velocity is proportional to the shear rate. For fully developed flow in a tube of radius R this may be written as

where the proportionality constant L_s is called the slip length.

Derive an expression for the volume flow rate Q in a tube with radius R, slip length L_s, and a given pressure drop per unit length. How is the dependence of Q on R affected by the ratio L_s/R?

Transcribed Image Text:

v₂(R) = Ls dv₂ (R) dr (P7.5-1)