The material derivative is a useful concept, which expresses the rate of change measured by an…

The material derivative is a useful concept, which expresses the rate of change measured by an observer moving with the local fluid velocity u(x, t). The material derivative (to be discussed in lecture), usually denoted D/Dt, is defined as D Dt = ? ?t + u · ? . (27) (a) In one dimension, with u = u(x, t)ex and D Dt = ? ?t + u ? ?x , show b Db Dt = 1 2 Db2 Dt . Note that this result is the same as for the ordinary derivative. Solution: Simply differentiate to obtain ?b2 ?t = 2b ?b ?t and ?b2 ?x = 2b ?b ?x . Hence, we see that 1 2 Db2 Dt = b ?b ?t + u ?b ?x = b Db Dt