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.* Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB *We will see later how to deal with cases where these assumptions are not valid.

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. Here A=area of the cross-section of the fluid’s container, and the small v is the speed of the fluid. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB *We will see later how to deal with cases where these assumptions are not valid.

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. Here A=area of the cross-section of the fluid’s container, and the small v is the speed of the fluid. Bernoulli’s Equation - this is conservation of energy per unit volume for a flowing 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 *We will see later how to deal with cases where these assumptions are not valid.

Example 1: 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 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 1: 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?

Note: We didn’t really need to change the units of the areas - as long as both of them are the same, the units will cancel out. 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. slower here faster here 1 2 Example 1: 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?

Note: We didn’t really need to change the units of the areas - as long as both of them are the same, the units will cancel out. 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. slower here faster here 1 2 Plugging in the numbers, we get: Example 1: 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?

Note: We didn’t really need to change the units of the areas - as long as both of them are the same, the units will cancel out. 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. slower here faster here 1 2 Plugging in the numbers, we get:Using the shortcut, we get: Example 1: 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?

Example 2: 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 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 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 2: 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 that is 11 m lower than the first if the pipe diameter at the second point is twice that of the first.

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. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 2: 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 that is 11 m lower than the first if the pipe diameter at the second point is twice that of the first.

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 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 2: 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 that is 11 m lower than the first if the pipe diameter at the second point is twice that of the first.

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: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 2: 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 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 Example 3: 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?

There is a lot going on in this problem, but it is really just like the last one. In fact, it’s easier if we assume the artery is horizontal. We’ll use Bernoulli’s Equation to find the speed just after the blockage, then continuity will tell us the ratio of the areas. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 3: 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?

There is a lot going on in this problem, but it is really just like the last one. In fact, it’s easier if we assume the artery is horizontal. 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 V 1 = 30 cm/s V 2 = ? cm/s Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 3: 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?

There is a lot going on in this problem, but it is really just like the last one. In fact, it’s easier if we assume the artery is horizontal. 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 V 1 = 30 cm/s V 2 = 102 cm/s Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 3: 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?

Example 3: 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? There is a lot going on in this problem, but it is really just like the last one. In fact, it’s easier if we assume the artery is horizontal. 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 70% blocked (the blood is flowing through a cross-section that is only 30% of the unblocked area) V 1 = 30 cm/s V 2 = 102 cm/s 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 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: 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? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: Hurricane For part a) we need to use Bernoulli’s equation. 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? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: Hurricane For part a) we need to use Bernoulli’s equation. We can assume (as in the last example) that y 1 =y 2 =0. We can also assume that the wind is not blowing inside. Take point 1 to be inside the house, and point 2 to be outside. these will be 0 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? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: Hurricane For part a) we need to use Bernoulli’s equation. We can assume (as in the last example) that y 1 =y 2 =0. We can also assume that the wind is not blowing inside. Take point 1 to be inside the house, and point 2 to be outside. these will be 0 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? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: Hurricane For part a) we need to use Bernoulli’s equation. We can assume (as in the last example) that y 1 =y 2 =0. We can also assume that the wind is not blowing inside. Take point 1 to be inside the house, and point 2 to be outside. these will be 0 Part b) is just a straightforward application of the definition of pressure. 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? Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Example 4: 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? For part a) we need to use Bernoulli’s equation. We can assume (as in the last example) that y 1 =y 2 =0. We can also assume that the wind is not blowing inside. Take point 1 to be inside the house, and point 2 to be outside. these will be 0 Part b) is just a straightforward application of the definition of pressure. Assuming the roof is flat, we just multiply: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Viscosity and Turbulence Earlier we made assumptions that our fluid has low viscosity (i.e. friction) and laminar (not turbulent) flow. But many fluids do not behave so nicely. To measure the importance of viscosity on a flowing fluid we can calculate its Reynolds number. If the Reynolds number is low, then viscosity plays a significant role, and the fluid will exhibit laminar flow, but Bernoulli’s equation will not be satisfied. In this case we can use a different formula to relate the flow rate to the pressure difference. This is Poiseuille’s equation. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Viscosity of the fluid In cases where the Reynolds number is high (above 2000 or so) the flow will become turbulent and complicated.

Water flows at mL/s through a horizontal tube that is 30.0cm long and has an inside diameter of 1.50mm. Determine the pressure difference required to drive this flow if the viscosity of water is 1.00mPa·s. Is it reasonable to assume laminar flow in this case? Hint: calculate the Reynolds number. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 5: Viscous water flow (from textbook)

Water flows at mL/s through a horizontal tube that is 30.0cm long and has an inside diameter of 1.50mm. Determine the pressure difference required to drive this flow if the viscosity of water is 1.00mPa·s. Is it reasonable to assume laminar flow in this case? Hint: calculate the Reynolds number. For now, let’s assume laminar flow so we can use Poiseuille’s equation. Be careful with units. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Example 5: Viscous water flow (from textbook)

Water flows at mL/s through a horizontal tube that is 30.0cm long and has an inside diameter of 1.50mm. Determine the pressure difference required to drive this flow if the viscosity of water is 1.00mPa·s. Is it reasonable to assume laminar flow in this case? Hint: calculate the Reynolds number. For now, let’s assume laminar flow so we can use Poiseuille’s equation. Be careful with units. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To find the Reynolds number we need the flow speed. We can get that from our original formulation of flow rate: Q=Av. Example 5: Viscous water flow (from textbook)

Water flows at mL/s through a horizontal tube that is 30.0cm long and has an inside diameter of 1.50mm. Determine the pressure difference required to drive this flow if the viscosity of water is 1.00mPa·s. Is it reasonable to assume laminar flow in this case? Hint: calculate the Reynolds number. For now, let’s assume laminar flow so we can use Poiseuille’s equation. Be careful with units. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To find the Reynolds number we need the flow speed. We can get that from our original formulation of flow rate: Q=Av. Example 5: Viscous water flow (from textbook) Now we can use the formula for Reynolds number.

Water flows at mL/s through a horizontal tube that is 30.0cm long and has an inside diameter of 1.50mm. Determine the pressure difference required to drive this flow if the viscosity of water is 1.00mPa·s. Is it reasonable to assume laminar flow in this case? Hint: calculate the Reynolds number. For now, let’s assume laminar flow so we can use Poiseuille’s equation. Be careful with units. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To find the Reynolds number we need the flow speed. We can get that from our original formulation of flow rate: Q=Av. Example 5: Viscous water flow (from textbook) Now we can use the formula for Reynolds number. This is well below 2000, so laminar flow is reasonable.