Flow

Question 1

What is volume flow rate?

Formula.

Answer 1

volume of the fluid that flows through a cross-section of the tube in time

I_V = ΔV/Δt
or
I_V = (A*Δl)/Δt = A * v

• ΔV = volume of the fluid
• Δt = time
• A = cross-sectional area
• Δl = average displace of the fluid
• v = average speed

Question 2

What is stationary flow?

Answer 2

parameters of the flow (e.g. velocity of the flow, pressure) constant at every place of the flow

Question 3

What does the continuity equation state?

Formula.

Answer 3

in stationary flow of an incompressible fluid the volume flow rate is the same at any point along the tube
⇒ if narrower cross-section → fluid flows faster

I_V = A₁ * v₁ = A₂ * v₂ = const.

• A = cross-sectional area of the tube
• v = average speed at the given cross-section

Question 4

What is viscosity?

It depends on … ?

Unit?

Answer 4

internal friction of fluids and gases
⇒ larger for fluids that flow less easily → more viscous

ƞ in [Pa*s]

depends on:

• material
• temperature

Question 5

When is a fluid referred to as Newtonian?

Give an example.

Answer 5

if the viscosity ƞ does not depend on:

• Δv: velocity change
• Δp: pressure drop

→ water

Question 6

What is laminar flow?

When can it be observed?

Answer 6

at low flow velocities the fluid flows “in layers”

Question 7

How does the speed of the different layers in case of laminar flow vary?

Why?

Answer 7

parabolical decrease in speed of layers due to internal friction (= drag force) as they are approaching the wall

→ v_max in center of the tube (can be turbulent)

Question 8

What is turbulent flow?

When can it be observed?

Formula.

Answer 8

above a critical velocity v_{<strong>crit</strong>} the flow becomes swirling irregularly

v_crit = Re * ƞ/(⍴*r)

• Re = Reynolds number = 1160
• ƞ = viscosity
• ⍴ = density
• r = radius of tube

Question 9

Explain the clinical relevance of turbulent flow.

Answer 9

v_turb < v_lam → volume flow rate decreases → greater demand on heart

• if r_aorta decreased → v_crit increases, v increases even more (due to continuity equation) → turbulent flow
• blood pressure measurement is based on listening to turbulent flow
• pathological decrease of ƞ (incr. hematocrit) may cause turbulent flow

Question 10

What does Hagen-Poiseuille law state?

It is valid for which kind of fluids?

Answer 10

describes stationary laminar flow of Newtonian fluids in rigid tubes

I_V = ΔV/Δt = πr⁴Δp / 8ƞl

⇒ volume flow rate I_V is directly proportional to the pressure drop that maintains the flow and 4th power of r

• ƞ = viscosity
• l = length of tube
• r = radius of tube
• Δp = static pressure drop along the tube

Question 11

The resistance to flow is directly and inversely proportional to … ?

Formula.

Answer 11

R_flow = R_tube = (8*l*ƞ) / (π*r⁴) ~ (l*ƞ)/A²

directly proportional to:

• l = length of the tube
• ƞ = viscosity of the fluid

inversely proportional to:

• r⁴ = A² = fourth power of the radius of the tube = square of the cross-sectional area

_{(REMEMBER: inversely proportional to IV, <em>cf. Hagen-Poiseuille law)</em>}

Question 12

Explain how

• no. of branches
• average diameter
• total cross-sectional area
• average velocity
• blood pressure

of the different types of blood vessels behaves in relation to each other.

Answer 12

• no. of branches: increasing as diameter decreases
• average diameter: decreases
• total cross-sectional area: largest for capillaries since they are the most abundant type of blood vessels
• blood pressure: highest in aorta, then decreasing

Question 13

Can blood be considered as a Newtonian or non-Newtonian fluid?

Answer 13

non-Newtonian bc:

• flow is pulsed, thus not stationary
• walls of blood vessels are not rigid, but elastic