Chapter 7 - Some Mathematics:
The Equations of Motion

7.8 Solutions to the Equations of Motion

Equations (7.12) and (7.19) are four equations, the three components of the momentum equation plus the continuity equation, with four unknowns: u, v, w, p. In principle, we ought to be able to solve the equations with appropriate boundary conditions.

Note, however, that these are non-linear partial differential equations. Conservation of momentum, when applied to a fluid, converted a simple, first-order, ordinary, differential equation for velocity (Newton's Second Law), which is usually easy to solve, into a non-linear partial differential equation, which is almost impossible to solve.

Boundary Conditions:
In fluid mechanics, we generally assume:

1. No velocity normal to a boundary, which means there is no flow through the boundary; and
2. No flow parallel to a solid boundary, which means no slip at the solid boundary.

Solutions
We expect that four equations in four unknowns plus boundary conditions give a system of equations that can be solved in principle. In practice, solutions are difficult to find even for the simplest flows. First, as far as I know, there are no exact solutions for the equations with friction. There are very few exact solutions for the equations without friction. Those who are interested in ocean waves might note that one such exact solution is Gerstner's solution for water waves (Lamb, 1945: 251). Because the equations are almost impossible to solve, we will look for ways to greatly simplify the equations. Later, we will find that even numerical calculations are difficult.

Analytical solutions can be obtained for much simplified forms of the equations of motion. Such solutions are used to study processes in the ocean, including waves. Solutions for oceanic flows with realistic coasts and bathymetric features must be obtained from numerical solutions. In the next few chapters we seek solutions to simplified forms of the equations. In Chapter 15 we will consider numerical solutions.

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