Physics: Fluids

Things to note:

1. Fluids are described by density 'ρ'. The mass of a fluid is m = ρV
2. When an object is floating, masses are equal - the mass of the object equals the mass of fluid displaced. When an object is submerged, volumes are equal - the volume of the object equals the volume of fluid displaced. 
3. It's good to think of still fluids as forces in equilibrium, and moving fluids in terms of energy.
4. A fluid is a liquid or a gas.

Equations: 

• ρ = m/v
• PV = nRT, where R = 0.082 atm*L/mole*K
• P = F/A (where P = pressure, and A = Area)
• P = ρgh (fluid at rest)
• Fb = ρVg (b = buoyant force) 
• V = AΔh
• Q = Av
• I = ρQ = ρAv

The Specific Gravity (S.G.) of a substance is the density of that substance compared to the density of water. 

S.G. = (ρsubstance/ρwater)

Obviously, things with an S.G. >1 will sink and things with an S.G. < 1 will float. 

ρwater = 1000 kg/m^3

ρwater = 1g/cm^3

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The fluid pressure is defined as P = F/A, and the S.I. unit of pressure is the Pascal, which is scalar. 

For a fluid at rest with uniform density, P is given by: 

P = ρgh

Knowing these equations, one can rearrange it to give: 

ρgh = mg/a , where a = area 

Ptotal = pgh1 + pgh2 + pgh3 etc.

In any fluid open to the atmosphere, the pressure can be found from P = ρgh + Patm

• Patm = 101,000 Pa (Pascals)

Pascal's Principle states that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container.

     -It is illustrated mathematically as P1 = P2, or F1/A1 = F2/A2

Absolute pressure is pressure measured relative to a vacuum. 

Pabs = Pgauge + Patm 

A standing fluid exerts a buoyant force on any object that is floating, submerged, or sunk in the fluid. 

The buoyant force is characterized by: 

Fb = ρVg (Fb = ρfluid Vfluid g)

So this means: 

That if, Fb = ρVg,

and V = AΔh, 

then it follows that Fb = ρgAΔh

Since, Fb/A = ρgΔh

then, ΔP = ρgΔh

Also, since Fb = ρVg, 

and Fb = (m/V)⋅V⋅g,

then Fb = mg (where m = mass of fluid)

Fb = mg shows that the upward buoyant force is equal in magnitude to the weight of the displaced fluid. This is known as Archimedes Principle.

Remember that the upward buoyant force is equal in magnitude to the weight of the displaced fluid. This is as given by Fb = m⋅g (where m = the mass of the fluid)

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Floating Objects:

For a floating object: 

           -This equation simplifies to mfluid = mobject

And so: 

In sum: 

Submerged Object that is also Floating:

For a submerged object in general, we can infer: 

For a fully submerged object that is ALSO floating: Vfluid = Vobject, and ρfluid = ρobject

Submerged Object that is Sunken:

A sunk object displaces a volume of fluid equal to its own volume and experiences an upward buoyant force of lesser magnitude than the downward gravitational force. 

This is because Fb < Fg, and mfluid < mobject. 

     -And since Vfluid = Vobject, then it follows that an object sinks when its density is greater than the density of the surrounding fluid. 

For a sunk object:

Apparent weight of object = Fn

Actual weight of the object = mg

Therefore, apparent weight Fn = Fg - Fb 

The apparent weight is always less than the actual weight in this case. 

Illustration of Fn + Fb = Fg 

Summary of all Three Cases: 

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Center of buoyancy: 

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Characteristics of Ideal Fluids: 

1. Ideal fluids have no viscosity. (Syrup has a greater viscosity than water) 
2. Ideal fluids are incompressible with uniform density. This can be assumed for any liquid, but not for gases. 
3. Ideal fluids lack turbulence. In other words, they experience steady laminar flow. 
4. Ideal fluids experience irrotational flow. Any object moving with an ideal fluid will continue to point in one direction regardless of the direction of flow. 

The Continuity Equation

• Q = Av (where A = cross-sectional area, and Q = volume flow rate) 
• Remember that v = d/t

The mass flow rate is simply the continuity equation multiplied by density: 

• I = ρQ = ρAv 

This entails that smaller diameters (lower A) give rise to higher velocities (higher v) 

Bernoulli's equation restates conservation of energy in terms of fluids: 

An example using Bernoulli's equation: 

Since fluid flowing into regions of constriction is associated with higher velocity, and since higher velocities are associated with decreases in pressure, then by definition, the venturi effect is the effect that describes the decrease in pressure that occurs when a fluid flows into a constricted region of a pipe. 

***Keep in mind that even though moving to more constricted regions are associated with higher velocities, and higher velocities are associated with decreases in pressure, the reason why vasoconstriction is associated with an increase in blood pressure in physiology is because vasoconstriction is concerned with non-ideal fluids, or real fluids, while the venturi effect is describing ideal liquids via the bernoulli's equation. It is important to note this, because ideal fluids experience no turbulence. 

Real Fluids: (Non-ideal fluids) 

In a horizontal pipe of constant cross-sectional area, fluid will flow from high pressure to low pressure according to the following equation: 

ΔP = QR

Where: 

Q = flow rate 

R = resistance to flow

Poiseuille's Law predicts the flow rate of real fluids: 

Where: 

L = pipe length

η = viscosity 

Surface Tension and Capillary Action:

Surface tension (intermolecular forces) = Cohesive force

Forces b/w the molecules of the tube and the fluid molecules = Adhesive force

If the cohesive forces are stronger, a convex surface is formed as the liquid is pulled downward by the cortical component of the surface tension. 

If the adhesive forces are stronger, a concave surface is formed as the fluid is pulled upward by the vertical component of the surface tension.