When dealing with fluid mechanics of viscous fluids, both theoretically and numerically, I've always been told that the boundary condition applied at solid walls has to be a no-slip one. My teachers or textbooks never really explained why, except sometimes "well, the fluid's viscous, so it sticks to solid" which is far from an explanation to me. Therefore I'm reading a little bit to understand the real origin of this condition, and so far there is on thing that I don't understand in what I've found: in Volume II of Modern Developments in Fluid Dynamics by S. Goldstein, it is written that:
"[…]; finally he [Navier] decided on the first [hypothesis on the behaviour of a fluid near a solid body], on the grounds that the existence of slip would imply that the friction between solid and fluid was of a different nature from, and infinitely less than, the friction between two layers of fluid, and also that the agreement with observation of results obtained on the assumption of no slip was highly satisfactory."
I do not understand what is in bold: what does the "different nature" of friction means, and why would it be "infinitely less"?
NB: After this statement, Goldstein refers to a bibliographic entry, but I don't know if it's possible to find such archive on the Internet. Here it is anyway: Trans. Camb. Phil. Soc. 8 (1845), 299, 300; Math and Phys. Papers, 3, 14, 15.
Best Answer
In a viscous fluid the shear stress is proportional to the velocity gradient.
$\sigma=\eta \frac{dv}{dy}$
where $\eta$ is the viscosity, and $v$ is the fluid speed at right angles to the $y$ axis.
Therefore as the small distance $dy$ tends to zero, the change of fluid speed $dv$ also tends to zero, for any non-zero viscosity. Let us now follow Navier and imagine that the wall is also a fluid, but at rest. If we now look at the fluid at distance $dy$ from the wall, we conclude that the speed of the fluid $dv$ tends to zero as $dy$ tends to zero.
This is a long-winded way of saying that an infinite velocity gradient cannot exist - neither in the fluid, nor at the wall.