Question

Problem 3. Consider a pipe containing a steadily flowing inviscid fluid. It has one inlet and branches into two arms so that there are two outlets (see Fig. 1). Flow can be considered uniform and parallel to the walls when entering and exiting the pipe Inlet Pi Outlet ρ2 A2 p, Outlet Figure 1: Flow of fluid through a "T" -junction in a pipe, shown from above (not to scale) Part A (a) The Continuity equation, as given on the formula sheet, can be used to describe the flow in such a configuration. Explain the physical basis of the Continuity equation including mention of the meaning of each of the ters in the equation. (b) Show how Gauss's Theorem and the Continuity equation can be used to relate the mass flow (pvA) entering the junction to the mass flow out. You should use symbols shown in the diagram in developing your equation. (c) Suppose the flow is incompressible and that AiA2-As. Show that the speed of the fluid flowing out each of the exit ports must have half of the speed of the in-low Part B (Advanced) Pascal's law describes the variation of pressure with depth in a stationary body of fluid. It is commonly quoted as where P) is the pressure at the surface of the fluid, ρ s the density of the fluid, g is the acceleration due to gravity, and h is the distance below the surface Can Pascal's law be derived from the Navier-Stokes equation? If so, then carry out the derivation. If not, then explain why not

Answer 1

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