The acceleration of a liquid particle

The velocity of a small element of liquid, or liquid particle, has been discussed in section 4.3. This is not the velocity of the individual molecules, but the drift velocity of a small region of liquid and is approximately that of a small, light solid particle carried along with the liquid. For a complete description of the motion of the liquid the velocity u must be known as a function of the space variables and also the time since, at any fixed point in the liquid, the velocity is not, generally, independent of time. 

Set up fixed Cartesian axes within a liquid undergoing flow and consider the behaviour of a liquid particle. If the Uquid particle is at a point A, say, at a time t and at a point B at a time r + 6f, the average acceleration of the liquid particle over the time interval ht is  

This expression may be put into a more useful and instructive form by considering the behaviour of a liquid particle in the neighbourhood of the point (x,y, z). 

Let the fluid particle under consideration be that contained in an element with edges dx, dy, 8z, respectively, and with one corner at the point (x, y, z) in space, as in Fig. 5.2. Further, let 

This differentiation is often called differentiation following the motion of the liquid and, to bring out both the time and space dependence, it is often written D//Df D/Dr is sometimes called the mobile operator or the convective derivative . If, for example, / denotes the density p of the liquid, t)p/Dr gives the rate of change of density of a particular liquid particle, but dp/df gives the rate of change of density at a particular point in space, i.e. part of the total change in p results from the lapse of time and part from the movement of the particle to a different place. 

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