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Consider the horizontal, laminar flow of a Newtonian fluid of viscosity =f()== 5102Pas that is subjected to a constant pressure gradient of 300Pa/m in a pipe of circular cross-section (radius R=5cm ). Performing a momentum balance on an annular fluid element (length dz and thickness dr ) leads to the differential equation (DE) drd(rrz)=rdzdp where r is the radial coordinate. The no-slip boundary condition applies at the pipe wall (i.e., r=R), whereas the strain rate vanishes at r=0. 1. Convert the given DE to a second-order ODE in which velocity is the dependent variable. (Naturally, you need to use the appropriate shear stress-shear strain relationship.) [5 marks ] 2. With the ODE from Part 1 above, use the shooting method with simple Euler integration from r=R to r=0, where r=0 corresponds to the pipe centreline, to compute the velocity profile of the fluid, using r=0.5cm. Report the velocities in tabulated form at each node in SI units. Compare your calculated values with those obtained from the analytical expression for the velocity profile by calculating the root-mean-square error (RMSE) between the two profiles, defined as RMSE=Ni=1N(viv^i)2 where N is the total number of nodes excluding the node at x=R and v^ is the analytical velocity at node i. [10marks] 3. Using the ODE given in the question i.e., Equation (2) above, solve it simultaneously with the appropriate shear stress-shear strain relationship, again from r=R to r=0, where r=0 corresponds to the pipe centreline, to compute the velocity profile of the fluid, again using r=0.5cm. Of course, use simple Euler integration. Report the velocities in tabulated form at each node in SI units