Fluids - Comprehensive Notes

Introduction to Fluids

• The four common phases of matter are solid, liquid, gas, and plasma. We will not discuss plasma in this course.

• Solids maintain a fixed volume and shape.

• Liquids maintain a fixed volume but not shape.

• Gases can change both volume and shape.

• Liquids and gases are collectively called fluids because they can flow.

• A fluid is a substance that is capable of flowing and takes the shape of its container.

• A fluid is made up of many tiny particles, and we can treat it as a system of objects.

• The macroscopic properties of a fluid are determined by the properties and interactions of the particles that make up the fluid.

• Four attributes of fluids to be investigated: mass, volume, density, and weight.

• Assumptions for liquids:

  • Constant mass (no evaporation).

  • Constant volume (not compressible).

  • Uniform density.

  • Take the shape of their container from the bottom up.

• Assumptions for gases:

  • Constant mass (no leaking out).

  • Variable volume (compressible).

  • Variable density (compressible).

  • Fill their container and therefore also take its shape.

• Two main situations to be investigated:

  • Fluid Statics (hydrostatics): Fluids at rest.

    • Examples: Water in a glass, helium in a balloon, a floating log, different liquids at rest in a container, a coin sinking in a jar, an air bubble rising to the surface.

  • Fluid Dynamics (hydrodynamics): Fluids that are flowing.

    • Calculations will involve pipes with changing diameters or elevation, and fluids undergoing speed or pressure changes.

    • Consideration of energy and mass conservation of flowing fluids.

Density

• Density is an object's mass per unit volume.

• Equation: $\rho = \frac{m}{V}$

  • $\rho$ (rho) is density.

  • $m$ is mass.

  • $V$ is volume.

• SI unit for density is kg/m3, but it's sometimes measured in g/cm3 or kg/L.

• Conversion: To convert from g/cm3 to kg/m3, multiply by 1000.

• Example: Liquid water at standard temperature and pressure has a density of 1 g/cm3. A 1 L water bottle contains 1 kg of water. A cubic meter of liquid water has a mass of 1000 kg (one metric ton).

• To determine the density of an object experimentally, measure mass and volume.

  • Mass can be measured with a balance, spring, or electronic scale.

  • Volume can be measured by measuring the sides or radius with a ruler the sides or radius to calculate volume based on shape.

  • For an object with an unusual shape, volume can be determined by water displacement.

• When plotting data to determine density, linearize the graph to easily find density from the slope.

  • Plot mass (y) versus side length cubed (x) for a cube to get a straight line.

  • The slope m equals the density because $m = \rho V$ takes the same form as $y = mx$, an equation for a straight line.

• When plotting data:

  • Choose good units.

  • Fill the entire graph.

  • Include a line or curve of best fit.

  • Calculate any important slopes.

  • Label the axes.

Pressure

• Pressure is defined as force per unit area.

• Pressure is a scalar quantity.

• Units are in Pascals (Pa).

• $1 Pa = 1 N/m^2$

• Pressure depends on force and area.

• Fluids consist of molecules bouncing off the walls of the container.

• The net force of all molecules is perpendicular to the wall.

• A fluid exerts a perpendicular force on every surface of a submerged object.

• This force, divided by the area of the contact surface, is the pressure exerted by the fluid upon a given part of the object.

• In the presence of gravity, a fluid exerts greater pressure at greater depth.

• Pressure at a given point in a fluid depends only on the density of the fluid, the depth, and the strength of the gravitational field; the shape of the container makes no difference.

• This is true for all incompressible fluids, since an incompressible fluid has a fixed density no matter the depth.

• Equation: $P = \rho gh$

• At sea level on Earth, atmospheric pressure ($P_0$) is about $1.01 \times 10^5$ pascals (Pa), called 1 atmosphere or atm.

• Pressure gauges measure the pressure above or below atmospheric pressure.

• This difference is called gauge pressure ($P_G$).

• In a fluid below the atmosphere, gauge pressure is $P_G = \rho gh$

• Absolute pressure is the sum of atmospheric pressure and gauge pressure: $P = P<em>0 + P</em>G$

• If a U-shaped tube contains two liquids with different densities, the pressure at points on the same horizontal line must be equal.

• The formula for the columns of fluid above points a and b is: $\rho<em>1 g h</em>1 = \rho<em>2 g h</em>2$

• Italian physicist Evangelista Torricelli invented a mercury barometer to measure atmospheric pressure.

• A glass tube is filled with mercury and placed upside down in a reservoir of mercury.

• The mercury level in the glass tube falls, creating a vacuum at the top (no pressure at the top of the tube).

• The barometer works by balancing the pressure due to the mercury in the glass tube against the atmospheric pressure.

• Pascal's Principle

Pascal's Principle

• Pascal's Principle: If an external pressure is applied to a confined and incompressible fluid, the pressure everywhere in the fluid increases by that added amount.

• Pascal's Barrel: A 10-meter-long tube was inserted into a barrel filled with water. When water was poured into the tube, the increase in pressure caused the barrel to burst.

• Pascal's Principle implies that the weights of layers of fluids above you (including the atmosphere!) will add to the pressure you feel.

• Small force (Fin) applied to a small area (Ain) results in a large force (Fout) applied to a large area (Aout).

• The work done is the same at each end, so Fin is applied over a greater distance than Fout.

• Hydraulic lift: Pressure applied via a piston at one surface of a fluid is transmitted through the fluid to another piston, where it can do work, like a lever.

• The equation for Pascal's Principle is: $P = \frac{F<em>{in}}{A</em>{in}} = \frac{F<em>{out}}{A</em>{out}}$

Buoyancy and Archimedes' Principle

• The pressure due to a fluid increases with depth, so there is greater pressure at the bottom of a submerged object than at the top.

• This results in a net force directed upwards on the object, called the buoyant force ($F_B$).

• Archimedes' Principle: The upward buoyant force on an object immersed in a fluid, partially or completely, is equal to the weight of the displaced fluid.

• Assume the beach ball is stationary (not rising or sinking).

• The buoyant force upwards is cancelled out by two downward forces: the bear pushing down, and the earth pulling down on the ball and the air inside it.

• The buoyant force on the ball is equal to the weight of the displaced fluid.

• The volume of water displaced by the ball is equal to the volume of the ball, since the entire ball is underwater.

• $F<em>B = \rho</em>F V g$

  • $\rho_F$ is the density of the fluid.

  • $V$ is the volume of the displaced fluid.

• The buoyant force is the weight of the displaced fluid!

• The net force on an immersed object is the difference between the buoyant force and the gravitational force.

• Remember: for a motionless object, $F_{net} = 0$

• The buoyant force is not dependent on the object's depth in the fluid, since only the difference in height of the top and bottom matter.

• Any floating object displaces its own weight of fluid: $F<em>B = m</em>{fluid}g$

• Consider an object of volume $V<em>0$ and density $\rho</em>0$ placed in a fluid of density $\rho_F$

• If the density of the object is less than the density of the fluid, there will be a net upward force until the buoyant force balances the objects weight, and only part of the object's volume remains submerged

• The fraction of volume submerged is given by the ratio of the object's density to that of the fluid

• Specific Gravity

  • The ratio of the density of a substance to that of a standard substance is known as specific gravity.

  • The standard substance almost always used for liquids is water at its densest (4°C), with a density of 1000 kg/m³.

  • A substance with a specific gravity less than one means that it is less dense than water and will float on water, and a substance with a specific gravity greater than one means that it is more dense than water and will sink in water.

• You can compare the densities of different objects by seeing how much of the object is submerged while floating.

Fluids in Motion & Bernoulli's Principle

• Laminar flow: A fluid that flows smoothly, with no friction.

• Mass flow rate: The mass that passes a given area per unit time ($\Delta t$).

• The mass flow rate must be the same as it crosses any area as long as no fluid is added or removed.

• The equation for Mass Flow Rate is: $\frac{\Delta m<em>1}{\Delta t} = \frac{\Delta m</em>2}{\Delta t}$

• $\Delta m = \rho V$

• where upper-case V is the volume of one fluid element, and lower-case v is the speed of the fluid.

• Since the mass that flows past any point is given by $\rho<em>1 A</em>1 v<em>1 = \rho</em>2 A<em>2 v</em>2$

  • If the fluid is incompressible, $\rho<em>1 = \rho</em>2$, and the continuity equation becomes: $A<em>1v</em>1 = A<em>2v</em>2$

• Density does not typically change in liquids, and this means that where a pipe is wider, the flow is slower.

• Bernoulli's Equation

• The total mechanical energy of the moving fluid remains constant unless work is done by a net force, or pressure difference.

• Bernoulli's Equation: $P<em>1 + \frac{1}{2} \rho v</em>1^2 + \rho g y<em>1 = P</em>2 + \frac{1}{2} \rho v<em>2^2 + \rho g y</em>2$

• This equation comes in handy in any situation where you have a pipe that is changing shape and/or elevation

• If horizontal fluid flow occurs: As fluid speed increases, the pressure decreases.

• note: h isn't changing

• $P + \frac{1}{2} \rho v^2 = constant$

Torricelli's Theorem

• We can use Bernoulli's Principle to find the speed of a fluid coming out the spigot of an open tank.

• The result is called Torricelli's Theorem.

• Since both points 1 and 2 are exposed to air, they have the same pressure, $P<em>0$ (atmospheric pressure): $P</em>1 = P<em>2 = P</em>0$

• If $v_1 = 0$

• The velocity with which the liquid leaves the container horizontally at point 2 is exactly the same velocity that an object would have if dropped from a height of $y= y<em>1-y</em>2$

• Torricelli's Theorem equation: $v_2 = \sqrt{2g\Delta y}$