No-Slip Boundary Condition

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the no-slip boundary condition

The velocity of the fluid immediately adjacent to a solid is the same as the velocity of the solid itself.

Life in Moving Fluids,
by Steven Vogel.

This seemingly simple condition is not obvious. To learn more, let’s consult Steven Vogel’s masterpiece Life in Moving Fluids: The Physical Biology of Flow.

The No-Slip Condition

The properly skeptical reader may have detected a peculiar assumption in our demonstration of viscosity: the fluid had to stick to the walls… in order to shear rather than simply slide along the walls. Now fluid certainly does stick to itself. If one tiny portion of a fluid moves, it tends to carry other bits of fluid with it—the magnitude of that tendency is precisely what viscosity is about. Less obviously, fluids stick to solids quite as well as they stick to themselves. As nearly as we can tell from the very best measurements, the velocity of a fluid at the interface with a solid is always just the same as that of the solid. This last statement expresses something called the “no-slip condition”—fluids do not slip with respect to adjacent solids. It is the first of quite a few counterintuitive concepts we’ll encounter in this world of fluid mechanics; indeed, the dubious may be comforted to know that the reality and universality of the no-slip condition was heatedly debated through most of the nineteenth century. Goldstein (1938) devotes a special section at the end of his book to the controversy. The only significant exception to the condition seems to occur in very rarefied gases, where molecules encounter one another too rarely for viscosity to mean much.

The reference to a book by Sydney Goldstein

Goldstein, S. (1938) Modern Developments in Fluid Dynamics. Reprint. New York: Dover Publications, 1965.

The no-slip boundary condition is important not only a low Reynolds number but also (and more surprisingly) at high Reynolds number. When discussing a solid sphere moving through a fluid, Russ and I say

At very high Reynolds number, viscosity is small but still plays a role because of the no-slip boundary condition at the sphere surface. A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient and therefore significant viscous drag.

Vogel also addresses this point

Most often the region near a solid surface in which the velocity gradient is appreciable is a fairly thin one, measured in micrometers or, at most, millimeters. Still, its existence requires the convention that when we speak of velocity we mean velocity far enough from a surface so the combined effect of the no-slip condition and viscosity, this velocity gradient, doesn’t confuse matters. Where ambiguity is possible, we’ll use the term “free stream velocity” to be properly explicit.

Many fluid problems in IPMB occur at low Reynolds number, where thin boundary layers are not relevant. However, at high Reynolds number the no-slip condition causes a host of interesting behavior. Russ and I write

A thin layer of fluid, called the boundary layer, sticks to the solid surface, causing a large velocity gradient… At extremely high Reynolds number, the flow undergoes separation, where eddies and turbulent flow occur downstream from the sphere.

Turbulence! That’s another story.


See y’all next week for more coronavirus bonus posts.