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QUESTION 1 In fluid mechanics, the material derivative is used to describe the rate of change of a physical quantity (such as velocity, temperature, or concen- tration) as experienced by a particle moving through a flow field. Given the velocity field of a fluid in two dimensions as

`vec(V)(x,y,t)=u(x,y,t)hat(i)+v(x,y,t)hat(j)`

where

`u(x,y,t)`

and

`v(x,y,t)`

are the velocity components in the

`x`

- and

`y`

-directions, respectively. (a) Define the material derivative

`(D\phi )/(Dt)`

for a scalar quantity

`\phi (x,y,t)`

in terms of the velocity components

`u`

and

`v`

. [2 marks]