Fluid Dynamics Lecture Notes

Fluid Dynamics Lecture Notes

Session Goals

• By the end of this video, you should be able to:

  • Relate volume flow rate to fluid velocity and cross-sectional area.

  • Explain the concept of continuity of fluid flow in a closed system.

  • Outline Bernoulli’s Principle for the flow of an incompressible fluid.

  • Relate the concept of energy conservation to fluid flow.

  • Apply the concepts of conservation of energy, continuity of flow, and Bernoulli’s Principle in discussing fluid pressure, velocity, and volume flow rate in an enclosed system.

  • Differentiate between laminar and turbulent flow.

  • Perform simple calculations involving the flow of fluids in an enclosed system.

  • Discuss simple clinical and other real-world applications of flowing fluids.

Fluid Dynamics Introduction

• 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.

Volume Flow Rate

• 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

Continuity of Flow

• 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.

Poiseuille’s Law

• 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.

Energy of Fluid Flow

• For fluid to flow, it must possess mechanical energy.

• Implication: Work must have been done on the fluid to induce flow.

• The work originates from the pressure applied along the distance of flow.

Pressure of Flow and Work

• 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.

Flow Velocity

• 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.

Conservation of Flow Energy

• 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 of Flow

• 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

  • $<br>ho$ = density of fluid

Potential Energy of Flow

• 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

Bernoulli’s Equation

• 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:
  $P + rac{1}{2} <br>ho v^2 + <br>ho gh = ext{constant}$

• In any system, the sizes of the various energy components will change with conditions, but their sum remains constant.

Bernoulli Principle Restated

• 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.

Bernoulli Effect

• 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 $v_2 > v_1$.

The Importance of Flow Dynamics

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

Laminar vs Turbulent Flow

• 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.

Summary of Learning Outcomes

• By the conclusion of this session, you should be able to:

  • Relate volume flow rate to fluid velocity and cross-sectional area.

  • Explain the concept of continuity of fluid flow in a closed system.

  • Outline Bernoulli’s Principle for flow of an incompressible fluid.

  • Relate the concept of energy conservation to fluid flow.

  • Perform simple calculations involving the flow of fluids in an enclosed system.

  • Differentiate between laminar and turbulent flow.

  • Discuss simple clinical applications of flowing fluids.