# No-slip boundary condition for viscous fluids

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.

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.

• Right. Nature abhors infinities in the same way as she abhors a vacuum. Nov 8, 2014 at 13:23

Usually we think of friction as something like this (not a formal definition, but I think it's close enough to be understood as "friction"):

when two objects move past one another and are in contact, the differential velocity between them leads to a force we call "friction"

At the boundary between a liquid and a solid, if we permit a different velocity (non zero velocity of the liquid at the boundary), we are in essence saying "Although we have differential velocity, there is no force of friction". Because if there were, the liquid would slow down until it reached zero velocity at the boundary.

This is just a different way of stating how we understand viscous flow - if we could have a no-slip condition, there can be no force between the liquid and the wall. And that goes counter to our understanding of friction.

• Isn't it possible that the layer of liquid at the boundary may experience a force greater than the opposing frictional force and thus continue with non-zero velocity ?
– Zam
Aug 17, 2019 at 9:00
• @Zam Not if that implies a discontinuity Aug 17, 2019 at 14:41
• @Foris Could you please elaborate ? [I am actually referring to the line, 'Because if there were, the liquid would slow down until it reached zero velocity at the boundary.' And by force in the previous comment, an example could be pressure difference.]
– Zam
Aug 18, 2019 at 6:04

I found two sources which discuss microscopic explanations for the no-slip condition. The first$$^1$$ is from 1973:

It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale : the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest.

The paper goes on to show how the rough surface mechanism is reasonable from a mathematical angle but does not directly prove it is the only factor responsible for observed no-slip situations.

The second$$^2$$ is from 2002 and offers two:

For many years it has been observed that there is no compelling argument to justify the standard “no-slip” boundary condition of textbook continuum hydrodynamics, which states that fluid at a solid surface has no relative velocity to it . However, this assumption successfully describes much everyday experience... There are two schools of microscopic explanation. The traditional explanation is that since most surfaces are rough, the viscous dissipation as fluid flows past surface irregularities brings it to rest, regardless of how weakly or strongly molecules are attracted to the surface [2–4].

The second microscopic explanation listed by the 2002 paper relates to "whether fluid molecules attract the surface or the fluid more strongly."

The bottom line is that the no-slip condition reflects reality in a multitude of situations and is an incredibly useful tool. As to why, scientists have come up with and analyzed two mechanisms: 1) the microscopic asperities (roughness) on surfaces slow down fluid molecules at the surface and 2) fluid molecules may attract to the surface more than they attract to the fluid. The relative importance of these two factors likely depends highly on the system.

$$^1$$ Richardson, S. (1973). On the no-slip boundary condition. Journal of Fluid Mechanics, 59(04), 707. doi:10.1017/s0022112073001801
$$^2$$ Zhu, Y., & Granick, S. (2002). Limits of the Hydrodynamic No-Slip Boundary Condition. Physical Review Letters, 88(10).