Physics of Flow and Viscosity in the Circulatory System

Physics of Flow and Viscosity

Circulation

The circulatory system consists of two main circuits:

• Pulmonary Circuit: Involves gas exchange in the capillary beds of the lungs.

  • Includes pulmonary arteries and pulmonary veins.

• Systemic Circuit: Involves gas exchange in the capillary beds of all body tissues.

  • Includes the aorta, branches, and venae cavae.

• Blood flows from the right atrium to the right ventricle, then to the pulmonary circuit, back to the left atrium, then to the left ventricle, and finally to the systemic circuit.

• The velocity of blood flow depends on the total cross-sectional area. Velocity is highest in the aorta and venae cavae and lowest in the capillaries.

Learning Outcomes

After this lecture, you should be able to:

1. Describe and explain the changes in pressure that are seen in various parts of the circulatory system.

2. Use the continuity equation for incompressible fluid flow.

3. Describe viscosity and sketch its effect on flow near a surface.

4. Apply Poiseuille’s Law to determine the effect of changes of viscosity, tube length, and radius on flow for a given pressure difference.

5. Recognize Bernoulli’s equation and apply it to explain pressure changes for moving fluids.

Flow Rate

• Mass flow rate: The mass $\Delta m$ of fluid that passes per unit time $\Delta t$. Units: kg/s.

  $\frac{\Delta m}{\Delta t} = \rho A v$

  Where:

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

  • $A$ is the cross-sectional area.

  • $v$ is the velocity of the fluid.

• Volume flow rate: The volume $\Delta V$ of fluid that passes per unit time $\Delta t$. Units: m³/s, liters/s.

  $Q = \frac{\Delta V}{\Delta t} = A v$

• Consider a cylinder of fluid with density $\rho$, cross-sectional area $A$, and length $\Delta l$, moving with velocity $v$.

• $\Delta m = \rho \Delta V$

Equation of Continuity

• Conservation of mass or volume dictates that the flow rate through two different areas ($A<em>1$ and $A</em>2$) must be equal.

• For mass flow:

  $\rho<em>1 A</em>1 v<em>1 = \rho</em>2 A<em>2 v</em>2$

• For volume flow:

  $A<em>1 v</em>1 = A<em>2 v</em>2$

  This is the equation of continuity.

Viscosity

• Real fluids exhibit internal friction, known as viscosity.

• Friction: The resistance encountered when one surface or object moves over another.

• Viscosity: A frictional force between adjacent layers of fluid as they move past one another (laminar flow).

• Viscosity is expressed by the coefficient of viscosity $\eta$ (eta).

• Model to derive viscous force F involved smooth(laminar) flow. F =ηA $\frac{v}{L_l}$

• Blood $\eta \approx 0.004$ Pa·s

• Water $\eta \approx 0.0018$ Pa·s

• Honey $\eta \approx 2$ Pa·s

Laminar and Turbulent Flow

• Laminar Flow: Smooth flow where fluid moves in layers.

• Turbulent Flow: Irregular flow with eddies (swirling motions).

• Reynolds Number ($Re$): A dimensionless number that predicts whether flow will be laminar or turbulent.

  $Re = \frac{\rho v L}{\eta}$

  Where:

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

  • $v$ is the velocity of the fluid.

  • $L$ is a characteristic length (e.g., diameter of the tube).

  • $\eta$ is the dynamic viscosity of the fluid.

• If Re is large, the flow is turbulent; if Re is small, the flow is laminar.

• Rule of thumb: Re > 2000 suggests turbulent flow.

Bernoulli’s Equation

• As a fluid flows, its height $y$, speed $v$, and pressure $P$ can change. These variables are related by Bernoulli’s equation for flow along a streamline (laminar flow):

  $P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$

  or,

  $P<em>1 + \frac{1}{2} \rho v</em>1^2 + \rho g y<em>1 = P</em>2 + \frac{1}{2} \rho v<em>2^2 + \rho g y</em>2$

  Where:

  • $P$ is the pressure.

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

  • $v$ is the speed of the fluid.

  • $g$ is the acceleration due to gravity.

  • $y$ is the height.

• As speed increases, pressure decreases (for constant height).

Bernoulli’s Principle

• The principle behind the wing, which allows airplanes to fly. Pressure difference overcomes the force of gravity

• A person with a constricted subclavian artery may suffer a temporary lack of blood flow to the brain – a TIA.

• This is known as subclavian steal syndrome.

Flow in Tubes and Viscosity

• Without viscosity, a fluid could flow through a level tube or pipe (or human body) without a force being applied.

• Viscosity acts like friction (“resistance to flow”).

• A pressure difference is needed for any steady flow.

• Thus, the rate of flow depends on pressure and viscosity.

• Calculating the rate of flow can be complicated

Poiseuille’s Equation

• The rate of flow of a fluid in a cylindrical tube depends on the viscosity $\eta$ of the fluid, the pressure difference $\Delta P$, and the dimensions of the tube (assuming the tube has smooth internal surfaces).

  $Q = \frac{\Delta P \pi r^4}{8 \eta L}$

  Where:

  • $Q$ is the volume flow rate.

  • $\Delta P$ is the pressure difference between the ends of the tube ($\Delta P = P<em>1 - P</em>2$).

  • $r$ is the radius of the tube.

  • $L$ is the length of the tube.

• Assumptions: Laminar flow and incompressible flow.

• L >> r

Effects of Parameter Variation on Volume Flow Rate

Using the Poiseuille's Equation: $Q = \frac{\Delta P \pi r^4}{8 \eta L}$

• If $\Delta P \rightarrow 2 \Delta P$, then $Q \rightarrow 2Q$

• If $L \rightarrow 2L$, then $Q \rightarrow Q/2$

• If $\eta \rightarrow 2 \eta$, then $Q \rightarrow Q/2$

• If $r \rightarrow 2r$, then $Q \rightarrow 2^4 \cdot Q = 16Q$

Blood Flow and Poiseuille’s Equation

• Restrictions in arteries have a major effect on blood flow.

• If the radius $R$ is halved, the pressure $\Delta P$ has to increase by a factor of $(2)^4 = 16$ to maintain the same flow rate $Q$.

• As the body’s organs & muscles require a given blood flow, hence blood pressure increases in response to the heart.