The mathematical model for a physical process is usually a partial differential or an ordinary differential equation prescribed over a physical domain, which has boundaries. The behaviour of the function representing the physical process needs to be defined on the boundaries, and these are the boundary conditions. For example, the flow around a body translating at constant speed $\vec{U}$ in an ideal incompressible fluid. The velocity field of the flow is given as the gradient of a potential function $\phi(x,y,z)$, which satisfies the Laplace's equation


\begin{displaymath}\nabla^{2}\phi = 0 \notag
\end{displaymath}  

and has to satisfy the kinematic boundary condition


\begin{displaymath}(\nabla\phi-\vec{U}).\vec{n} = 0 \mbox{\ on the body surface}.\notag
\end{displaymath}  

If the fluid domain is unbounded, we need to specify the behavior of $\phi$ as $r = \sqrt{x^{2}+y^{2}+z^{2}} \rightarrow \infty$, which is


\begin{displaymath}\nabla\phi \rightarrow 0 \mbox{\ as \ } r \rightarrow \infty \notag
\end{displaymath}  



Timothy S Choe
2003-03-15