The time derivatives

The most important difference between those two methods that describes the fluid motion comes in the time rate of change. So in Lagrange notation the total derivative Df/Dt of some property f of the fluid is the time rate of change of f in a portion of the fluid, as it moves though the point x at time t. In contrast, in the Euler notation the partial derivative ∂f/∂t is the change of f at the fixed point x, as the fluid streams past. Thus ∂f/∂t includes not only the time rate of change of f in the moving fluid, but also the change in f at x because the fluid is moving past, if f differs from point to point in the fluid.

Both methods have some advantages. The main advantage of Lagrange description lies in the time derives. So total derivative dρ/dt measures fluid expansion or compression and du/dt measures true acceleration of a portion of the fluid, and thus is proportional to the net force acting on the fluid at x,t. Euler description of the fluid motion has many other advantages so it will be useful to find the measure of the difference between the two time derivatives.

To find the total derivative of some property of the fluid, in terms of the partial derivatives at t,x, we compare the value of f at point x at time t with its value at time t+dt at the point x+udt to which the fluid has moved in time dt. Accoridng to the definitions of derivatives, this is:

$$\begin{align} & \frac{df}{dt}dt=f(x+udt,t+dt)-f(x,t) \\ & \frac{df}{dt}dt=f(x,t)+\frac{\partial f}{\partial x}udt+\frac{\partial f}{\partial t}dt-f(x,t) \\ & \frac{df}{dt}=\frac{\partial f}{\partial t}+u\frac{\partial f}{\partial x} \\ \end{align}$$

 

The actual change in f, as the fluid moves past x with velocity u, is the change in f with time at the fixed point x, plus the spatial change in f times its velocity u of flow past x. Thus the true acceleration of the fluid at x at time t is given, in terms of the Eulerian velocity function u(x,t), by

$$\frac{du}{dt}=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}$$

We see that the correction term u(∂u/∂x) is second order in the fluid velocity, and therefore, if u is small (as it often is for sound waves), du/dt = ∂u/∂t to the first order of approximation. The distinction cannot be forgotten, however, because the sound energy and intensity are second-order expressions, and the correction term may have to be included.