Formula Sheet: Fluids in Motion

Table of Contents
1. Continuity Equation
2. Bernoulli's Equation
3. Special Applications of Bernoulli's Equation
4. Viscosity and Resistance to Flow
5. Poiseuille's Law
6. Fluid Dynamics in Physiology
7. Additional Flow Relationships
8. Important Conceptual Relationships
9. Common Problem-Solving Strategies
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Continuity Equation

Fundamental Principle

• Conservation of Mass: For an incompressible fluid, mass flow rate must remain constant throughout a pipe or tube
• Volume Flow Rate (Q): \[Q = A \times v\] where:
  • \(Q\) = volume flow rate (m³/s)
  • \(A\) = cross-sectional area (m²)
  • \(v\) = fluid velocity (m/s)

Continuity Equation

• Standard Form: \[A_1 v_1 = A_2 v_2\] where subscripts 1 and 2 represent different points along the flow path
• Important Relationship: As cross-sectional area decreases, fluid velocity increases (and vice versa)
• Assumptions: Incompressible fluid, steady flow, single inlet and outlet

Bernoulli's Equation

Full Bernoulli Equation

• Energy Conservation Form: \[P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2\] where:
  • \(P\) = absolute pressure (Pa or N/m²)
  • \(\rho\) = fluid density (kg/m³)
  • \(v\) = fluid velocity (m/s)
  • \(g\) = gravitational acceleration (9.8 m/s²)
  • \(h\) = height above reference point (m)
• Alternative Form: \[P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}\]

Energy Terms in Bernoulli's Equation

• Pressure Energy: \(P\) = static pressure
• Kinetic Energy per Unit Volume: \(\frac{1}{2}\rho v^2\) = dynamic pressure
• Gravitational Potential Energy per Unit Volume: \(\rho g h\)
• Note: All three terms have units of pressure (Pa)

Assumptions and Limitations

• Fluid is incompressible (constant density)
• Flow is steady (non-turbulent, laminar)
• Flow is non-viscous (no internal friction/energy loss)
• Flow occurs along a streamline

Special Applications of Bernoulli's Equation

Horizontal Flow (No Height Change)

• When \(h_1 = h_2\): \[P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2\]
• Implication: As velocity increases, pressure decreases

Torricelli's Theorem (Efflux Speed)

• Speed of fluid exiting a hole: \[v = \sqrt{2gh}\] where:
  • \(v\) = efflux velocity (m/s)
  • \(g\) = gravitational acceleration (9.8 m/s²)
  • \(h\) = depth of hole below fluid surface (m)
• Derivation assumption: Large container (surface velocity ≈ 0), atmospheric pressure at surface and hole

Venturi Effect

• Concept: Fluid pressure decreases in a constricted section of pipe where velocity increases
• Combined with Continuity: Use \(A_1 v_1 = A_2 v_2\) with Bernoulli's equation to solve for pressure or velocity changes

Viscosity and Resistance to Flow

Dynamic Viscosity

• Definition: Measure of a fluid's resistance to flow and internal friction
• Symbol: \(\eta\) (eta)
• Units: Pa·s or N·s/m² or kg/(m·s)
• Temperature dependence: Viscosity of liquids decreases with increasing temperature; viscosity of gases increases with increasing temperature

Laminar vs. Turbulent Flow

• Laminar Flow: Smooth, orderly flow in parallel layers; occurs at low velocities
• Turbulent Flow: Chaotic, irregular flow with eddies; occurs at high velocities
• Reynolds Number (Re): Dimensionless parameter predicting flow type \[Re = \frac{\rho v D}{\eta}\] where:
  • \(\rho\) = fluid density (kg/m³)
  • \(v\) = fluid velocity (m/s)
  • \(D\) = diameter of pipe (m)
  • \(\eta\) = dynamic viscosity (Pa·s)
• Critical values: Re < 2000="" typically="" laminar;="" re=""> 4000 typically turbulent

Poiseuille's Law

Volume Flow Rate for Viscous Flow

• Poiseuille's Equation: \[Q = \frac{\pi r^4 \Delta P}{8 \eta L}\] where:
  • \(Q\) = volume flow rate (m³/s)
  • \(r\) = radius of tube (m)
  • \(\Delta P\) = pressure difference between ends (Pa)
  • \(\eta\) = dynamic viscosity (Pa·s)
  • \(L\) = length of tube (m)
• Key relationship: Flow rate is proportional to \(r^4\) (doubling radius increases flow 16-fold)
• Assumptions: Laminar flow, incompressible Newtonian fluid, rigid cylindrical tube

Resistance to Flow

• Hydraulic Resistance: \[R = \frac{8 \eta L}{\pi r^4}\]
• Flow-Resistance Relationship: \[Q = \frac{\Delta P}{R}\] (analogous to Ohm's Law: \(I = V/R\))
• Note: Resistance is inversely proportional to \(r^4\)

Series and Parallel Resistance

• Resistances in Series: \[R_{total} = R_1 + R_2 + R_3 + ...\]
• Resistances in Parallel: \[\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\]

Fluid Dynamics in Physiology

Blood Flow Applications

• Mean Arterial Pressure (MAP): Average pressure during one cardiac cycle, approximately: \[MAP \approx \frac{P_{systolic} + 2P_{diastolic}}{3}\]
• Cardiac Output (CO): \[CO = HR \times SV\] where:
  • \(CO\) = cardiac output (L/min)
  • \(HR\) = heart rate (beats/min)
  • \(SV\) = stroke volume (L/beat)

Vascular Resistance and Blood Pressure

• Total Peripheral Resistance (TPR): \[TPR = \frac{\Delta P}{CO}\] where \(\Delta P\) is pressure difference across systemic circulation
• Implication of Poiseuille's Law: Small changes in vessel radius (vasoconstriction/vasodilation) dramatically affect resistance and flow

Additional Flow Relationships

Mass Flow Rate

• Definition: \[\dot{m} = \rho Q = \rho A v\] where:
  • \(\dot{m}\) = mass flow rate (kg/s)
  • \(\rho\) = fluid density (kg/m³)
  • \(Q\) = volume flow rate (m³/s)

Power in Fluid Flow

• Power delivered by pressure: \[P_{power} = \Delta P \times Q\] where:
  • \(P_{power}\) = power (W or J/s)
  • \(\Delta P\) = pressure difference (Pa)
  • \(Q\) = volume flow rate (m³/s)

Important Conceptual Relationships

Pressure-Velocity Relationship

• In horizontal flow: high velocity → low pressure (Bernoulli principle)
• In horizontal flow: low velocity → high pressure
• Examples: airplane wing lift, atomizer/sprayer function, Venturi meter

Area-Velocity Relationship

• From continuity: small area → high velocity
• From continuity: large area → low velocity
• Example: blood flows fastest in aorta, slowest in capillaries (due to total cross-sectional area)

Radius-Flow Relationship

• From Poiseuille's Law: Flow ∝ r⁴
• Small decreases in radius cause dramatic decreases in flow
• Clinical relevance: atherosclerosis, vasoconstriction effects

Common Problem-Solving Strategies

Using Continuity and Bernoulli Together

1. Apply continuity equation to relate velocities at two points: \(A_1 v_1 = A_2 v_2\)
2. Substitute into Bernoulli's equation to solve for unknown pressure or velocity
3. Check units and physical reasonableness of answer

Identifying Appropriate Equation

• Ideal fluid (non-viscous), energy conservation: Use Bernoulli's equation
• Real fluid with viscosity, cylindrical tube: Use Poiseuille's Law
• Volume or mass conservation: Use continuity equation
• Hole in container: Use Torricelli's theorem