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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?

Solution 2


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

Solution 3


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.

Solution 4

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?

Solution 5


Problem 6


An open-ended U-tube of uniform cross sectional area contains water (density 1 gram/cm3) standing initially 20cm from the bottom in each arm. An immiscible liquid of density 4.0 grams/cm3 is added to 1 arm until a layer 5 cm high forms. What is the ratio h2/h1 of the heights of the liquid in the 2 arms?

Solution 6