Chapter 8 - Equations of Motion With Viscosity

Chapter 8 Contents

Throughout most of the interior of the ocean and atmosphere friction is relatively small, and we can safely assume that the flow is frictionless. At the boundaries, friction, in the form of viscosity, becomes important. This thin, viscous layer is called a boundary layer. Within the layer, the velocity slows from values typical of the interior to zero at a solid boundary. If the boundary is not solid, then the boundary layer is a thin layer of rapidly changing velocity whereby velocity on one side of the boundary changes to match the velocity on the other side of the boundary. For example, there is a boundary layer at the bottom of the atmosphere, the planetary boundary layer we described in Chapter 3. In the planetary boundary layer, velocity goes from many meters per second in the free atmosphere to tens of centimeters per second at the sea surface. Below the sea surface, another boundary layer, the Ekman layer described in Chapter 9, matches the flow at the sea surface to the deeper flow.

In this chapter we consider the role of friction in fluid flows, and the stability of the flows to small changes in velocity or density.

8.1 The Influence of Viscosity

In the last chapter we wrote the x-component of the momentum equation for a fluid in the form (7:12a):

(8.1)

where Fx 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 8.1). 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 8.1 Molecules colliding with the wall and with each other transfer momentum from the fluid to the wall, slowing the fluid velocity.

Molecular viscosity is the ratio of the stress Tx tangential to the boundary of a fluid and the shear of the fluid at the boundary. So the stress has the form:

(8.2)

for flow in the ( x, z ) plane within a few millimeters of the surface, where ν is the kinematic molecular viscosity. Typically ν = 10-6m2/s for water at 20°C.

Generalizing (8.2) 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 nd the details in Lamb (1945: 328) or Kundu (1990: p. 93). For an incompressible fluid, the frictional force per unit mass in (8.1) takes the from:

(8.3)

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