Question

Consider a section of a long tube that has an expansion of the tube such as is shown in the Figure below. The area of the tube decreases from A1 at the left end of the tube to an area of A_2 at the right end. Fluid of density p flows though the tube at a steady flowrate Q. The flow can be considered to be inviscid. (i) Find the pressure upstream of the contraction (gage pressure). (ii) How much axial force needs to be supplied to support this contraction and in which direction? Simplify you answer as much as you can.

Answer 1

Let us consider the Bernoulli's equation at upstream and downstream of interaction Flow rate = Q velocity of flow upstream of contraction = Q/p_1 velocity of flow downstream of contraction = Q/p_2 let present at upstream the p p + 1/2 integral(Q/p_1)^2 = p_atm + 1/2 rho (a/P_2) range pressure at upstream = p-p_atm = 1/2 rho[(Q/A_2)^2-(Q/A_1)^2] = 1/2 rho Q^2[1/A^2-1/A_1] Axial force needed to support the contraction = 1/2 rho Q^2 A_1(1/A^2_2-1/A^2_1) = 1/2 rho Q^2(A^2_1-A^2_2/A_1A_2)