Chapter 13 Fluids Flashcards

Fluids

Fluids and Density

• A fluid is a substance that flows; liquids and gases are fluids.
• Gases are compressible, meaning their volume can be easily increased or decreased.
• Liquids are nearly incompressible because their molecules are closely packed but can still move around.

Density

• Mass density is the ratio of mass to volume.
• The mass density of an object with mass $m$ and volume $V$ is given by: $ρ = m/V$
• The SI units of mass density are $kg/m^3$.
• Example: Gasoline has a mass density of $680 kg/m^3$, meaning there are 680 kg of gasoline for each 1 cubic meter of the liquid.

Weighing the Air in a Living Room Example

• This section would contain an example problem calculating the mass of air in a living room using its dimensions, but the dimensions and solution were not provided in the transcript.

Pressure

• Liquids exert forces on the walls of their containers.
• Pressure is the ratio of force to the area on which the force is exerted: $P = F/A$
• Fluid pressure acts in all directions, pushing on all parts of the fluid itself.

Pressure in Liquids

• The force of gravity (the weight of the liquid) causes pressure in the liquid.
• Horizontal forces cancel each other out.
• Vertical forces balance each other.
• For a liquid cylinder of cross-sectional area $A$ and height $d$, the mass is $m = ρAd$.
• The pressure at depth $d$ is: $P = ρgd$

The Pressure on a Submarine Example

• A submarine cruises at a depth of 300 m. The problem would calculate the pressure at this depth in both pascals and atmospheres but the solution was not provided.

Pressure in Liquids: Hydrostatic Equilibrium

• A connected liquid in hydrostatic equilibrium rises to the same height in all open regions of the container.
• In hydrostatic equilibrium, the pressure is the same at all points on a horizontal line through a connected liquid of a single kind.

Pressure in a Closed Tube Example

• Water fills a tube. The example problem asks for the pressure at the top of the closed tube, but the solution was not provided in the transcript.

Pascal’s Principle

• If the pressure at one point in an incompressible fluid changes, the pressure at every other point in the fluid changes by the same amount.

Pressure in a Tube with Two Liquids Example

• A U-shaped tube contains water and oil. The problem asks for the gauge pressure at the closed end, but the solution was not provided in the transcript.

Atmospheric Pressure

• Gas is compressible, so air density in the atmosphere decreases with increasing altitude.
• 99% of the air in our atmosphere is below 30 km.
• Atmospheric pressure varies with altitude and weather changes.
• Local winds and weather are largely due to air masses of different pressures.

Measuring Atmospheric Pressure

• A barometer measures atmospheric pressure.
• A glass tube is placed in a beaker of the same liquid; some liquid leaves the tube.
• $P = ρgh$ where $P$ is the pressure due to the weight of the liquid in the tube.
• Equating the two pressures allows atmospheric pressure to be measured by measuring the height of the liquid.

Pressure Units

Buoyancy

• Buoyancy is the upward force of a liquid.
• Pressure in a liquid increases with depth, so the pressure is greater at the bottom of a submerged cylinder than at the top.
• This pressure difference results in a net upward force.

Buoyancy and Static Equilibrium

• If an isolated parcel of fluid is in static equilibrium, the buoyant force (upward) balances the weight force (downward).
• The buoyant force matches the weight of the fluid.
• Replacing the parcel of liquid with an object of the same shape and size does not change the buoyant force.

Archimedes’ Principle

• When an object is immersed in a fluid, it displaces fluid that would otherwise fill that region of space.
• Archimedes’ principle: A fluid exerts an upward buoyant force on an object immersed in or floating on the fluid; the magnitude of the buoyant force equals the weight of the fluid displaced by the object.
• Mathematically: $F<em>B = w</em>{fluid} = m<em>{fluid}g = ρ</em>{fluid}V_{fluid}g$

Is the Crown Gold? Example

• Legend says Archimedes determined if a crown was pure gold by measuring its weight in air and underwater.
• The problem would compare the tension (7.81 N) in a string when a crown weighing 8.30 N is suspended underwater to determine if it's pure gold, but the solution wasn't provided.

Float or Sink?

• Whether an object floats or sinks depends on whether the upward buoyant force is larger or smaller than the downward weight force.
• Average density is defined as: $ρ_{avg} = m/V$
• The weight of a compound object can be written as: $w = mg = ρVg$

Float or Sink? (cont.)

• An object will float if the density of the fluid is larger than the average density of the object, and sink if it's smaller.
• If the densities are equal, the object is in static equilibrium, known as neutral buoyancy.
• Steel sinks because it is denser than water; oil floats because it is less dense than water.

Measuring the Density of an Unknown Liquid Example

• A wooden block floats with 4.6 cm submerged in an unknown liquid and 5.8 cm submerged in water. The problem asks to calculate the density of the unknown liquid but does not give the solution in the transcript.

Boats and Balloons

• The hull of a boat is a hollow shell, displacing a large volume of water.
• The boat sinks until the weight of the displaced water matches the boat’s weight, achieving static equilibrium.
• Balloons cannot be filled with regular air because it has the same weight as the displaced air.
• For a balloon to float, it must be filled with a gas that has a lower density than air.

Example Problem: Hot Air Balloon

• A hot air balloon with a volume is flying in Fort Collins, Colorado, where the air density is known.
  • A. Calculate the mass of air displaced by the balloon.
  • B. If heated, the air inside the balloon has a density of 80% of the surrounding air; find the mass of air in the balloon.
  • C. Determine how much mass the balloon can lift.

Fluids in Motion

• For fluid dynamics, we use a simplified model of an ideal fluid.
  • The fluid is incompressible (good assumption for liquids and reasonably good for moving gases).
  • The flow is steady, meaning the fluid velocity at each point is constant (laminar flow), distinguished from turbulent flow.

Laminar vs. Turbulent Flow

• Laminar flow is smooth, while turbulent flow is chaotic.
• A laminar-to-turbulent transition is not uncommon in fluid flow.
• Our model of fluids can only be applied to laminar flow.

The Role of Viscosity

• Viscosity is resistance to flow.
• Reynolds number (ratio of inertial forces to viscous forces) can be applied to the motion of fluids.
• If the Reynolds number is large, viscous forces can be ignored; if small, viscous forces dominate.
• Reynolds number also determines transition between laminar and turbulent flow.

Viscosity Examples

• For water flowing at 1 m/s in a river 2 m deep, viscosity can be ignored.
• Blood moving in an arteriole is dominated by viscosity.

The Equation of Continuity

• When an incompressible fluid enters a tube, an equal volume of fluid must leave the tube.
• The velocity of the molecules will change with different cross-section areas of the tube.

Equation of Continuity Formula

• The equation of continuity relates the speed $v$ of an incompressible fluid to the cross-section area $A$ of the tube in which it flows: $A<em>1v</em>1 = A<em>2v</em>2$
• The volume of an incompressible fluid entering one part of a tube must be matched by an equal volume leaving downstream.
• A consequence of the equation of continuity is that flow is faster in narrower parts of a tube and slower in wider parts.

Volume Flow Rate

• The rate at which fluid flows through a tube (volume per second) is called the volume flow rate $Q$.
• The SI units of $Q$ are $m^3/s$.
• We can also say that the volume flow rate is constant at all points in the tube: $Q = Av = constant$

Speed of Water through a Hose Example

• A garden hose with an inside diameter of 16 mm fills a 10 L bucket in 20 s.
  • A. Find the speed of the water out of the end of the hose.
  • B. Determine the diameter of a nozzle that would cause the water to exit with a speed 4 times greater than the speed inside the hose.

Streamlines and Fluid Elements

• A streamline is the path or trajectory followed by a “particle of fluid”.
• A fluid element is a small volume of a fluid, a volume containing many particles of fluid.
• A fluid element has a shape and volume; the shape can change, but the volume is constant.

Fluid Dynamics

• A fluid element changes velocity as it moves from the wider part of a tube to the narrower part.
• This acceleration is caused by a force.
• The fluid element is pushed from both ends by the surrounding fluid, that is, by pressure forces.
• To accelerate the fluid element, the pressure must be higher in the wider part of the tube.
• A pressure gradient is a region where there is a change in pressure from one point in the fluid to another.
• An ideal fluid accelerates whenever there is a pressure gradient.

Bernoulli Effect

• The pressure is higher at a point along a streamline where the fluid is moving slower, and lower where the fluid is moving faster.
• This property was discovered by Daniel Bernoulli and is called the Bernoulli effect.
• The speed of a fluid can be measured by a Venturi tube.

Applications of the Bernoulli Effect

• As air moves over a hill, streamlines bunch together, increasing air speed.
• This creates a zone of low pressure at the crest of the hill.

Lift

• Lift is the upward force on the wing of an airplane that makes flight possible.
• Wing design increases air speed above the wing (low pressure) and decreases air speed below the wing (high pressure).
• The high pressure below the wing pushes more strongly than the low pressure above, causing lift.

Bernoulli’s Equation

• Bernoulli’s equation is derived from the principle of conservation of energy.
• As a fluid moves through a tube of varying widths and heights, energy is conserved within the fluid segment.

Derivation of Bernoulli's Equation

• As a fluid moves through a constriction, the system moves out of cylindrical volume $V<em>1$ and into $V</em>2$.
• The kinetic energies are: $K<em>1 = \frac{1}{2} m</em>1 v<em>1^2$ and $K</em>2 = \frac{1}{2} m<em>2 v</em>2^2$
• The net change in kinetic energy is: $\Delta K = \frac{1}{2} m v<em>2^2 - \frac{1}{2} m v</em>1^2$

Gravitational Potential Energy and Work

• The net change in gravitational potential energy is: $\Delta U = m g y<em>2 - m g y</em>1$
• The positive and negative work done are: $W<em>1 = P</em>1 V<em>1$ and $W</em>2 = -P<em>2 V</em>2$
• The net work is: $W = P<em>1 V - P</em>2 V$

Bernoulli’s Equation

• Combining the equations for kinetic energy, potential energy, and work done, we obtain Bernoulli’s equation:
• $P<em>1 + \frac{1}{2} \rho v</em>1^2 + \rho g y<em>1 = P</em>2 + \frac{1}{2} \rho v<em>2^2 + \rho g y</em>2$

Bernoulli's Equation Usage

• Bernoulli’s equation relates ideal-fluid quantities at two points along a streamline.

Pressure in an Irrigation System Example

• Water flows through pipes. The water’s speed through the lower pipe is 5.0 m/s and a pressure gauge reads 75 kPa. The goal is to find the pressure gauge reading on the upper pipe, but a solution to the example was not provided.

Viscosity and Poiseuille’s Equation

• Viscosity measures a fluid’s resistance to flow.
• In the absence of viscosity, a fluid would “coast” at constant speed with no change in pressure.
• Real (viscous) fluids require a pressure difference to flow at a constant speed.

Viscosity and Pressure Difference

• The pressure difference needed to keep a fluid moving is proportional to viscosity ($\eta$) and tube length $L$, and inversely proportional to cross-section area $A$.
• $\eta$ is the coefficient of viscosity with units of $Pa \cdot s$

Fluid Speed in Viscous Fluids

• In an ideal fluid, all particles move at the same speed. For a viscous fluid, the speed is highest in the center of the tube and decreases towards the walls, where it is 0.

Poiseuille’s Equation

• The average speed of a viscous fluid is: $v_{avg} = \frac{\Delta P}{8 \eta L} r^2$
• The volume flow rate for a viscous fluid is: $Q = \frac{\Delta P \pi r^4}{8 \eta L}$
• This viscous flow rate equation is called Poiseuille’s Equation.

What Causes the Pressure Drop? Example

• This example discusses a scenario of taking a shower while a roommate flushes a toilet, affecting water pressure.
• The problem describes water traveling 40 m through a pipe of radius 0.0072 m with viscosity $1.1 \times 10^{-3} Pa \cdot s$ at $20^{\circ} C$.
• The gauge pressure at the showerhead is initially 200 kPa with a flow rate of $2 \times 10^{-4} m^3/s$.
• When the toilet refills, the total flow becomes $7 \times 10^{-4} m^3/s$.
• The problem asks for the new pressure at the showerhead, assuming no appreciable change in height, but the solution was not provided.