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# Time derivatives following the motion

Let be some property of a fluid, for example the density, . There are two different time derivatives that we must distinguish between. The first is the rate of change of with respect to at a fixed point in space, where the partial derivative is obtained by keeping the other variables constant (in this case the spatial derivatives). The second is which is defined as the rate of change of with respect to time but attached to a particular fluid elemnet. Thus we consider a particular fluid element and as it moves due to a fluid velocity its position also moves. Thus, moving with the fluid, we calculate how changes in time but not at a fixed point in space. This `moving with the fluid'is illustrated in Figure 2.3. Thus, becomes, in the limit of tending to zero, Hence, the final result, as tends to zero is In three dimensional problem with and , the obvious generalisation is simply Finally, this may be written in vector form as (2.6) is also called the total time derivative.

Note, you must be very clear about your notation and know the difference between and .

Example 2. .1Consider the density profile as and the position . If there is a flow so that where the form of has been selected as simplpy . This equation may be solved to give the position as a function of time as where is a constant giving the initial position. Note that is only a function of time and so we do not need to use partial derivatives in calculating . Now we calulate the total time derivative of as Note that we may eliminate from the expression for and get purely as a function of , namely, as before.

Example 2. .2If , use the mass continuity equation to determine satisfying . Now the partial derivatives can be evaluated as Hence, and so Subsections   Next: Incompressible Flow Up: Fluids and Nonmagnetic Plasmas Previous: Continuity Equation
Prof. Alan Hood
2000-11-06