Physics 151: Lecture 30 Today’s Agenda Today’s Topic: Fluids in Motion Bernoulli’s Equation and applications

Example / Fluid Statics A beaker of mass mbeaker containing oil of mass moil (density = roil) rests on a scale. A block of iron of mass miron is suspended from a spring scale and completely submerged in the oil as in Figure on the right. Determine the equilibrium readings of both scales.

Fluids in Motion See text: 14.5 Up to now we have described fluids in terms of their static properties: density r pressure p To describe fluid motion, we need something that can describe flow: velocity v There are different kinds of fluid flow of varying complexity non-steady / steady compressible / incompressible rotational / irrotational viscous / ideal

See text: 14.5 Ideal Fluids Fluid dynamics is very complicated in general (turbulence, vortices, etc.) Consider the simplest case first: the Ideal Fluid no “viscosity” - no flow resistance (no internal friction) incompressible - density constant in space and time Simplest situation: consider ideal fluid moving with steady flow - velocity at each point in the flow is constant in time In this case, fluid moves on streamlines streamline

follows from mass conservation if flow is incompressible. See text: 14.6 Ideal Fluids streamlines do not meet or cross velocity vector is tangent to streamline volume of fluid follows a tube of flow bounded by streamlines streamline Flow obeys continuity equation volume flow rate Q = A·v is constant along flow tube. A1v1 = A2v2 follows from mass conservation if flow is incompressible.

Conservation of Energy for Ideal Fluid See text: 14.7 Conservation of Energy for Ideal Fluid Recall the standard work-energy relation Apply the principle to a section of flowing fluid with volume dV and mass dm = r dV (here W is work done on fluid) dV Doesn’t matter whether these particles are distinguishable or not - we are only asking how many are on the left and right. Bernoulli Equation

Lecture 30 Act 1 Continuity A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 1) Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe? a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1

Lecture 30 Act 2 Bernoulli’s Principle A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 2) What is the pressure in the 1/2” pipe relative to the 1” pipe? a) smaller b) same c) larger

DEMO SLIDE The smaller the diameter the lower is the pressure

Example A Pitot tube (see Fig. below) can be used to determine the velocity of air flow by measuring the difference between the total pressure and the static pressure. If the fluid in the tube is mercury, density rHg = 13 600 kg/m3, and h = 5.00 cm, find the speed of air flow. (Assume that the air is stagnant at point A and take rair = 1.25 kg/m3.)

Can we know what is v from what we can measure ? Venturi Meter v = ? Can we know what is v from what we can measure ? h rHg rair

Example A tank containing a liquid of density r has a hole in its side at a distance h below the surface of the liquid. The hole is open to the atmosphere and its diameter is much smaller than the diameter of the tank. What is the speed with of the liquid as it leaves the tank. h r v=?

Example Figure on the right shows a stream of water in steady flow from a kitchen faucet. At the faucet the diameter of the stream is 0.960 cm. The stream fills a 125-cm3 container in 16.3 s. Find the diameter of the stream 13.0 cm below the opening of the faucet.

Example Water is forced out of a fire extinguisher by air pressure, as shown in Figure below. How much gauge air pressure in the tank (above atmospheric) is required for the water jet to have a speed of 30.0 m/s when the water level in the tank is 0.500 m below the nozzle?

Recap of today’s lecture 14.4-7 Streamlines Bernoulli’s Equation and applications