8.1: The Influence of Viscosity

Viscosity is the tendency of a fluid to resist shear. In the last chapter I wrote the \(x\)-component of the momentum equation for a fluid in the form \((7.6.5)\): \[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + 2 \Omega \ v \sin \varphi + F_{x} \nonumber \]where \(F_{x}\) was a frictional force per unit mass. Now we can consider the form of this term if it is due to viscosity.

Molecules in a fluid close to a solid boundary sometime strike the boundary and transfer momentum to it (figure \(\PageIndex{1}\)). Molecules further from the boundary collide with molecules that have struck the boundary, further transferring the change in momentum into the interior of the fluid. This transfer of momentum is molecular viscosity. Molecules, however, travel only micrometers between collisions, and the process is very inefficient for transferring momentum even a few centimeters. Molecular viscosity is important only within a few millimeters of a boundary.

Figure \(\PageIndex{1}\): Molecules colliding with the wall and with each other transfer momentum from the fluid to the wall, slowing the fluid velocity.

Molecular viscosity \(\rho \nu\) is the ratio of the stress \(T\) tangential to the boundary of a fluid and the velocity shear at the boundary. So the stress has the form: \[T_{xz} = \rho \nu \frac{\partial u}{\partial z} \nonumber \]

for flow in the \((x, z)\) plane within a few millimeters of the surface, where \(\nu\) is the kinematic molecular viscosity. Typically \(\nu = 10^{-6} \ \text{m}^{2}/\text{s}\) for water at 20\(^{\circ}\)C.

Generalizing \((\PageIndex{2})\) to three dimensions leads to a stress tensor giving the nine components of stress at a point in the fluid, including pressure, which is a normal stress, and shear stresses. A derivation of the stress tensor is beyond the scope of this book, but you can find the details in Lamb (1945: §328) or Kundu (1990: p. 93). For an incompressible fluid, the frictional force per unit mass in \((\PageIndex{1})\) takes the form: \[F_{x} = \frac{\partial}{\partial x} \left[\nu \frac{\partial u}{\partial x}\right] + \frac{\partial}{\partial y} \left[\nu \frac{\partial u}{\partial y}\right] + \frac{\partial}{\partial z} \left[\nu \frac{\partial u}{\partial z}\right] = \frac{1}{\rho} \left[\frac{\partial T_{xx}}{\partial x} + \frac{\partial T_{xy}}{\partial y} + \frac{\partial T_{xz}}{\partial z}\right] \nonumber \]