Fluid Dynamics Lecture Notes

Flowing fluid requires:
- A ‘path’: an enclosed region along which the fluid will move.
- A ‘push’: a pressure difference between two locations on the path.
A pressure difference serves as a net force causing acceleration of the fluid.

Definition: Volume flow rate ( extit{F}) is the quantity of fluid that passes through a cross-sectional area of the path per unit time.
This may also be referred to as current.
SI Unit: cubic meters per second (m^3/s).
Mathematical Relation: F = Av
- Where:
- v = velocity of flow
- A = cross-sectional area

The continuity equation states that the quantity of fluid entering a point on the flow path is the same as the quantity of fluid leaving that point.
Continuity Equation:
A_1v_1 = A_2v_2
If the cross-sectional area changes, the velocity will change inversely to maintain the volume flow rate.
Note: In this equation, use small v for velocity and do not confuse it with capital V for volume.

Definition: Poiseuille’s Law states that the volume flow rate ( extit{F}) is directly proportional to the pressure change (drop) ( ext{ΔP}) along the path and inversely proportional to the resistance of the path to flow ( ext{R}).
Mathematical Relation:
F = rac{ ext{ΔP}}{R}
Key Insight: The greater the pressure change, the more work that is done, resulting in more fluid being moved per unit time.

Pressure Formula:
ext{Pressure} = rac{ ext{force}}{ ext{area}}
If we multiply both parts of the fraction by the distance of flow, we obtain:
ext{pressure} = rac{ ext{force} imes ext{distance}}{ ext{area} imes ext{distance}}
Equivalent Relations:
- ext{force} imes ext{distance} = ext{work}
- ext{Area} imes ext{distance} = ext{volume}
Thus, we deduce:
ext{Pressure} = rac{ ext{work}}{ ext{volume}}
ext{Pressure} = rac{ ext{energy imparted}}{ ext{volume}}
This leads to a relation involving kinetic and potential energy of the flowing fluid.

Definition: Flow velocity refers to the distance traveled per unit time by fluid particles.
This is distinct from volume flow rate, which is the quantity of fluid that moves per unit time.

As fluid flows, it possesses mechanical energy originating from the applied pressure.
Energy can be converted between potential and kinetic energy during the flow.
The sum of all energy types in a fluid system remains constant:
ext{pressure energy} + ext{kinetic energy} + ext{potential energy} = ext{constant}
This concept is encapsulated in Bernoulli’s Principle.

Kinetic Energy Formula:
ext{KE} = rac{1}{2} mv^2
Often, kinetic energy density (energy per unit volume) is considered: ext{KE}/V = rac{1}{2} rac{mv^2}{V} = rac{1}{2} ho v^2
- Where:
- m = mass of fluid
- v = velocity of fluid
- ho = density of fluid

Potential Energy Formula:
ext{PE} = mgh
Often, potential energy density (energy per unit volume) is derived: ext{PE}/V = rac{mgh}{V} = ho gh
- Where:
- g = acceleration due to gravity
- h = height above a reference position

The sum of all energy types in a fluid system is constant:
ext{pressure energy} + ext{kinetic energy} + ext{potential energy} = ext{constant}
Bernoulli’s Equation expressed mathematically:
PV + rac{1}{2} mv^2 + mgh = ext{constant}
In energy density form, we write:
P + rac{1}{2}
ho v^2 +
ho gh = ext{constant}
In any system, the sizes of the various energy components will change with conditions, but their sum remains constant.

When the cross-sectional area of the path narrows, the flow velocity must increase to maintain the volume flow rate entering and leaving that location.
As flow velocity increases, kinetic energy increases; thus, pressure must decrease to compensate if the heights of both locations are identical.

The energy per unit volume before is equal to the energy per unit volume after:
P_1 + rac{1}{2} p v_1^2 + p g h_1 = P_2 + rac{1}{2} p v_2^2 + p g h_2
Interpretation: Increased fluid speed results in decreased internal pressure.
In a practical context:
- Given that A_2 < A_1, it follows that v_2 > v_1.

As flow velocity increases, the flow pattern can transition from laminar to turbulent.

Laminar Flow:
- Characterized by regular layers.
- Minimal energy is removed from the system due to low friction.
Turbulent Flow:
- Exhibits chaotic flow patterns.
- Increased energy is removed from the system due to friction, generating heat.
- Turbulence leads to reduced flow rate relative to applied pressure.
- In the cardiovascular system, this means the heart must pump harder to achieve the necessary flow rate.