Consider the axial flow of an incompressible fluid through an annulus as shown in the figure below. In the simplest possible setting, the flow is driven by constant axial pressure gradient " which results in an axisymmetric flow with just one component of velocity active (vz). V. (r ) From the 3D Navier-Stokes equations, one can show that the equations reduce to the simple ODE in cylindrical coordinates 1 d duz "pdr " drJ dp dz' where u is the dynamic viscosity. 1. Solve the above equation analytically with boundary conditions vz(Ro) = Vz(Ri) = 0, and show that it is equal to cr2 Uz (r) = cln(r) R. - R CR- cln (Ri) R. - R. 4ul In (Ri/ Ro) In (Ri/Ro) ) with constant dz ap = c. Hint: Note that " is a constant. You will need to integrate twice, and utilize the given boundary conditions to solve for the resultant integration constants. 2. Derive a 3-point central differencing scheme for the second derivativewhich is second order accurate. 3. Set up the above BVP numerically (using the 3-point central differencing scheme that is second order accurate which you developed in Part 2) on a mesh of N = 12 points in the radial direction (i.e., discretize along Ro and Ri), excluding boundary points. As- sume Ro = 1.0 m, Ri = 0.5 m, pressure gradient - = -2.5 Pa/m, and dynamic viscosity H = 0.00001 (N s) /m2. 4. Use Thomas algorithm to solve the above linear system you set up in Part 3. Compare them against the exact solution. Hint: Solving Part 3 via Thomas algorithm should provide you with approximated solutions for vz at your discretization points j = 1 (boundary), j = 1,..., N-1 (internal/intermediate points), and j = N (boundary). Compare this to your derived exact solution vz(r) from Part 1, evaluating at the appropriate r values based on your discretization