Two hoses are connected to the same outlet using a Y-connector, as the drawing s

Question

Two hoses are connected to the same outlet using a Y-connector, as the drawing shows. The hoses A and B have the same length, but hose B has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille's law [Q = ?R4(P2 - P1)/(8?L)] applies to each. In the present situation, Q is the volume flow rate of the water, P2 is the pressure upstream, P1 is the pressure downstream, h is the viscosity of water, and L is the length of a hose. What is the algebraic expression for the ratio vB/vA of the speed of the water in hose B to the speed of the water in hose A? Express your answer in terms of the radius RB of hose B and the radius RA of hose A. (Answer using R_B to be the radius of hose B and R_A to be the radius of hose A). What is the ratio vB/vA of the speed of the water in hose B to the speed of the water in hose A?

Explanation / Answer

P = 8LQ/(pi*r^4)
The pressure difference is equal in both instances, also the length L, the viscosity , radius 1 is 1, radius 2 = 1.21, so

8L*Q1/(pi*1^4) = 8L*Q2/(pi*1.21)^4
Q1 = Q2/1.21^4
Q2 = 2.1434*Q2
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Volume1/s = cross section area 1*length of tube/s with L/s = speed of water in tube
V/s = *L/s:
V1 = 1*L1
v2 = 2*L2
with
V2 = 2.1434*V1 and 2 = pi*1.21^2/pi*2:

L1 = V1/1
L2 = V2/2

L1 = V1/1
L2 = 2.1434*V1/(1*1.21^2)

L1/L2 = 1.21^2/2.1434
L1/L2 = 0.683
or
L2/L1 = speed in B/speed in A = 1/0.683 = 1.464

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