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Fluid Dynamics

Let's assume that fluid is incompressible, nonviscous, and that any flow of the liquid is streamlined (irrotational).

Continuity Equation

A_1 V_1 = A_2 V_2

Bernoulli's Equation

p1 and p2 are any two points in a streamlined flow of aliquid of density ρ Then at these two points,

P_1 + \frac{1}{2} \rho v^2_1 + \rho g y_1 = P_2 + \frac{1}{2} \rho v^2_2 + \rho g y_2

Problem 1

The diameter of a horizontal pipe changes along its length. As the water flows from the narrow section into the broader section of the pipe, what can you say about the rate of flow R and the speed of water v in the broader section?

Solution 1

Rate of flow of water R = Av stays the same due to the continuity equation. Hence, as A increases, v must decrease.

Problem 2

A layer of oil with density 800 kg/m3 floats on top of a volume of water with density 1000 kg/m3. A block floats at the oil-water interface with 1/4 of its volume in oil and 3/4 of its volume in water, as shown in the figure above. What is the density of the block, in kg/m3?

Problem 3

An incompressible fluid of density ρ flows through a horizontal pipe of radius r and then passes through a constriction of radius r/2. If the fluid has pressure P0 and velocity v0 before the constriction, the pressure in the constriction is

1. \frac{P_0}{4}
2. P_0 + \frac{3}{2} \rho v_0^2
3. P_0 - \frac{3}{2} \rho v_0^2
4. P_0 + \frac{15}{2} \rho v_0^2
5. P_0 - \frac{15}{2} \rho v_0^2

Problem 4

A balloon will be filled with helium and used to esuspend a mass of 300 kilograms in air. If the mass of the baloon is neglected, what is the volume (in m3) of helium required? The density of air is 1.29 kg/m3 and the density of helium is 0.18 kg/m3.

Buoyant force

F_b = \rho g V

Problem 5

A stream of water of density ρ cross-sectional area A and speed v strikes a wall that is perpendicular to the direction of the stream. The water then flows sideways across the wall. What is the force exerted by the stream on the wall is?