# 6.9: Boundary conditions

- Page ID
- 18074

“*Land and water are not really separate things, but they are separate words, and we perceive through words*.” - David Rains Wallace

Boundaries, in the real universe, are figments of the human imagination. We simplify the task of understanding reality by dividing it into parts (e.g., the atmosphere, the ocean) and trying to understand them individually. But our “parts” are not really separate - one blends smoothly into the other. For example, at the ocean surface the properties of the fluid change dramatically over a small distance, but a close look reveals a complex mixture of air, spray, foam, bubbles and water.

The equations of fluid motion remain valid at each point, and therefore any variable that appears differentiated with respect to space must be continuous to avoid infinities. Because the stress tensor is differentiated (as seen explicitly in Equation 6.3.18), it must be continuous everywhere, and in particular at the surface of a body of water. One result of this is that the pressure immediately below the ocean surface must equal atmospheric pressure^{1}. Another is that the transverse stress immediately below the surface must match that exerted by the wind.

We usually model the surface of a lake or ocean as a material surface (i.e., always composed of the same molecules) located where fluid properties change most rapidly. This surface is represented by its height at each horizontal location: \(z=\eta(x,y,t)\), where the average of \(\eta\) is zero. Because the surface is material, we can write

\[\frac{D}{D t}(z-\eta)=0, \text { or }\left.w\right|_{z=\eta}=\frac{D \eta}{D t}=\frac{\partial \eta}{\partial t}+u \frac{\partial \eta}{\partial x}+v \frac{\partial \eta}{\partial y}.\label{eqn:1} \]

A special case of this is a fixed boundary, an example being the lower boundary of the atmosphere at the ground. This boundary can be modeled by Equation \(\ref{eqn:1}\), but with the vertical location \(z = h(x,y)\) representing the Earth’s surface:

\[\frac{D}{D t}(z-h)=0, \text { or } w=\frac{D h}{D t}=u \frac{\partial h}{\partial x}+v \frac{\partial h}{\partial y} \text { at } z=h. \nonumber \]

At a s**olid boundary in a viscous fluid**, we imagine fluid molecules becoming intermingled with boundary molecules (or crystals) and therefore require that the velocity itself be continuous. In the Earth’s reference frame, the fluid velocity must approach zero at the boundary. This is called a “no-slip” boundary condition.

A useful idealization is the **frictionless** boundary, at which the velocity is not necessarily zero but the tangential (shear) stress must vanish. This minimizes the effect of friction on the flow and is realistic in cases where viscosity is unimportant. For example, frictionless flow over a horizontal boundary would obey

\[\frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0. \nonumber \]

This is often called a “free-slip” boundary condition.

^{1}neglecting the effect of surface tension; see exercise 33.