An incompressible fluid is flowing through a vertical pipe with a consteraction.

Question

An incompressible fluid is flowing through a vertical pipe with a consteraction. The wide section is 2cm in diamter and is at the top of the pipe. The pressure of the fluid in the wide section at the top is 200kPa. The velocity of the fluid in the wide section is 4m/s. The narrow section is 1 cm in diamter and is located 2 m below the wide section. ( the density of the fluid is 1,000kg/m^3)
A. what is the pressure of the fluid in the narrow section.
B. What is the diamter of the narrow section? ( if the pressure of the fluid are equal)

Explanation / Answer

Given data:

r1 = 2 cm, V1 = 4 m/s , P1 = 200*10^3 Pa and r2 = 0.5 cm, h1= 2 m, h2 = 0 m

According to equation of continuity,the rate of flow is constant.

i.e A1V1 = A2V2

(2*10^-2)^2*4 m/s = (0.5*10^-2)^2*V2

V2 = 64 m/s

the speed of the flow with which the water exits the nozzle is 64 m/s.

applying the Bernoulli's princeple,

the pressure at nozzle is given by

P2 = 1/2(v1^2 - V2^2) + gh+P1

P2 = 1/2 * 1000*(16-4096) + 1000*9.8*2 + 200*10^3

    = 2220400 Pa.

the pressure of the fluid in the narrow section is 2020400 Pa.

ie the pressure are equal, the diameter of the nozzle is 1 cm.

If the pressure of the wide section is 250kpa then

P2 = 2020400 + 250*10^3

    = 2270400 Pa.

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