Fluids - Hydrodynamics Physics 6B Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

With the following assumptions, we can find a few simple formulas to describe flowing fluids: Incompressible – the fluid does not change density due to the pressure exerted on it. No Viscosity - this means there is no internal friction in the fluid. Laminar Flow – the fluid flows smoothly, with no turbulence. With these assumptions, we get the following equations: Continuity – this is conservation of mass for a flowing fluid. Bernoulli’s Equation - this is conservation of energy per unit volume for a flowing fluid. Here A=area of the cross-section of the fluid’s container, and the small v is the speed of the fluid. Notice that there is a potential energy term and a kinetic energy term on each side. Some examples will help clarify how to use these equations: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Water travels through a 9.6cm diameter fire hose with a speed of 1.3m/s. At the end of the hose, the water flows out through a nozzle whose diameter is 2.5 cm. what is the speed of the water coming out of the nozzle? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We use continuity for this one. We have most of the information, but don’t forget we need the cross-sectional areas, so we need to compute them from the given diameters. 1 2 slower here faster here Note: we don’t really need to change the units of the areas, as long as both of them are the same, the units should cancel out. Plugging in the numbers, we get:Using the shortcut, we get:

At one point in a pipeline, the water’s speed is 3 m/s and the gauge pressure is 40 kPa. Find the gauge pressure at a second point on the line that is 11 m lower than the first if the pipe diameter at the second point is twice that of the first. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1 2 y 1 =11 m We need Bernoulli’s Equation for this one (really it’s just conservation of energy for fluids). Notice we set up the y-axis so point 2 is at y=0. y 2 =0 Here’s Bernoulli’s equation – we need to find the speed at point 2 using continuity, then plug in the numbers. Continuity Equation: This is the ratio of the AREAS – it is the square of the ratio of the diameters Plugging in the numbers to the Bernoulli Equation:

A medical technician is trying to determine what percentage of a patient’s artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 12 kPa, while in the region of blockage it is 11.5 kPa. Furthermore, she knows that the blood flowing through the normal artery just before the blockage is traveling at 30 cm/s, and the density of the patient’s blood is 1060 kg/m 3. What percentage of the cross-sectional area of the patient’s artery is blocked by plaque? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB It seems like there is a lot going on in this problem, but it is really just like the last one. In fact, it’s even easier if we assume the artery is horizontal (they don’t mention any vertical distances so this is probably ok). We’ll use Bernoulli’s Equation to find the speed just after the blockage, then continuity will tell us the ratio of the areas. these will be 0 solve for this speed Now use continuity: So the artery is 73% blocked (the blood is flowing through a cross-section that is only 27% of the unblocked area)

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB The Bernoulli ‘Effect’ Fast Flow=Low Pressure ↔ Slow Flow=High Pressure Airplane Wing Atomizer Hurricane Damage Curveballs, Backspin, Topspin Motorcycle Jacket Attack of the Shower Curtain

Example: Hurricane a)If the speed of the wind is 50 m/s (that’s about 100 miles per hour), and the density of the air is 1.29 kg/m 3, Find the reduction in air pressure due to the wind. b)If the area of the roof measures 10m x 20m, what is the net upward force on the roof?