Q. A hypothetical velocity field is given in CartesianCoordinates by: V? = 4z^2t i ? 4x k where i, j and k are the unitbase vectors. t is the time variable and (x, y, z) are theCartesian Coordinates in the ( i, j , k ) directions respectively.The fluid density is constant and given by p = 1000kg/m^3

5(a) Give the (x, y, z) components of V? which are written
(u, v, w)respectively. Check whether the flow satisfies the
appropriate continuity equation and conclude whether this
flow is physically possible. Note that the general continuity
equation is given on the formulae sheet at the end exam
paper.

5(b) Give the general expression of the acceleration fieldfrom
the above velocity field in the Eulerian frame of reference
and calculate the acceleration at (t, x, y, z) = (t, ?1,5,1)and
at (t, x, y, z) = (1, ?1,5,1) .

5(c) Calculate the component of the velocity in the directionof
the vector t = 2 i + a j, written Vt (where a is an unknownconstant) . This can be obtained from:
Vt = V??t/|t |

where the dot indicates the dot product of vectors and the
bold font indicates that we are dealing with vectors.

5(d) Check whether the pressure field p = 2x + y satisfiesthe
appropriate form of the incompressible x momentum
equation which may be derived from the equation given on
the formulae sheet at end of the examination paper. You
can neglect gravity.