4. For a given constant v > 0, the Navier-Stokes equations, which describe the flow of a viscous, incompressible fluid i
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4. For a given constant v > 0, the Navier-Stokes equations, which describe the flow of a viscous, incompressible fluid in a pipe 12, are given by au +(u: Vàu + Vp = v Au, at where the pipe 1 R3 is given in set notation by ER3 : yż + y} <R ] with R >O is the radius of the pipe. (a) Sketch the set 12 which models the pipe. (2 marks) (b) Assuming that the pressure p is constant, verify that the time-independent vector field u: 1 → R3 defined by u(x):= (°*") a solution of the Navier-Stokes equations (1) in for some fixed constant DER the pipe 12. (5 marks) (c) By solving the following initial-value problem dF (t;xo) = u(F(t; xo), t), dt F(0; xo) = Xo for the flow operator F, find the location at time t = 1 of a tracer particle P in the pipe whose initial position is at Xo = (0,0,0). (13 marks)