The Equation of Motion

Newton's law gives

$$\hbox{mass} \times \hbox{accelration} = \hbox{applied force}.$$

For a fluid the acceleration is calculated following a fluid element. Thus, we use the total time derivative. The applied force is due to pressure forces, $-\nabla p$, and any other external forces such as gravity, $\rho {\bf g}$, and magnetic forces, ${\bf j}\times {\bf B}$. Thus, each fluid element with mass $\rho \delta x\delta y\delta z$ satisfies the Equation of Motion

$$\rho {D{\bf v}\over Dt} = -\nabla p +{\bf F},$$ (2.9)

where ${\bf F} = \rho {\bf g} + {\bf j}\times {\bf B}$. In this course we shall not consider viscous forces and so we shall only deal with an inviscid fluid.