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## Homework Statement

A tank of cross-sectional area A is initially filled with fluid of density ρ and viscosity μ to height hi. The pressure above the fluid in the tank is atmospheric, Patm. At the base of the tank there are two pipes, which are both open to teh atmosphere, such that fluid flows out of them. The pipe on the left has diameter D1, length L1, and flow rate Q1. The pipe on the right has diameter D2, length L2, and flow rate Q2.

A. Write an expression for the initial pressure that drives the flows through pipes 1 and 2.

B. What is the ratio of the initial flow rates Q1/Q2?

C. What is the ratio of the initial Reynold's numbers for the flow in the two pipes, Re1/Re2?

D. Where is the shear force lowest and highest in the pipes? (a. At the wall, b. At the center line, c. Where the pipe meets the tank, d. Where the fluid flows out of the pipe to the atmosphere)

## Homework Equations

0 = ΔP + ρgΔh + (1/2)ρ(Δu

_{avg})

^{2}

Q =(ΔPπR

^{4})/(8μL) = Au

Re = ρuD/μ

## The Attempt at a Solution

A. P

_{init}= ρg(hi) + (Patm)

B. Q

_{1}=(ΔPπ(D

_{1}/2)

^{4})/(8μL

_{1})

Q

_{2}=(ΔPπ(D

_{2}/2)

^{4})/(8μL

_{2})

Q

_{1}/Q

_{2}= (D

_{1}/D

_{2})

^{4}(L

_{2}/L

_{1})

C. When I got to this one, I realized that I don't know how to find the velocity of the fluid, since I had assumed it to initially be zero, so I probably did A and B wrong.

Maybe I'm supposed to get it from Bernoulli's equation:

0 = ΔP + ρgΔh + (1/2)ρ(Δu

_{avg})

^{2}

u = √((-2/ρ)(ΔP + ρgΔh))

u = √((-2/ρ)(Patm + ρg(hi)))

But I don't know why this would be the case, and it also isn't taking into account he difference in area or length between the pipes. Could you please send me in the right direction? Am I missing an equation?

D. I think the shear stress would be highest where the pipes meet the tank and lowest at the center line, but I don't know why.