## Fluid Dynamics

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

## Continuity Equation

## Bernoulli's Equation

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

### 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/m^{3} floats on top of a volume of water with density 1000 kg/m^{3}. 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/m^{3}?

### Solution 2

Answer

### 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 P_{0} and velocity v_{0} before the constriction, the pressure in the constriction is

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

### Solution 3

Answer

### 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 m^{3}) of helium required? The density of air is 1.29 kg/m^{3} and the density of helium is 0.18 kg/m^{3}.

### Solution 4

Buoyant force

### 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?

### Solution 5

Answer

### Problem 6

An open-ended U-tube of uniform cross sectional area contains water (density 1 gram/cm^{3}) standing initially 20cm from the bottom in each arm. An immiscible liquid of density 4.0 grams/cm^{3} is added to 1 arm until a layer 5 cm high forms. What is the ratio h_{2}/h_{1} of the heights of the liquid in the 2 arms?