Definition of a fluid
A material that is unable to sustain an applied shear load at rest
Re and flow characteristics
0<Re<1 - extremely viscous laminar flow dominated by viscosity
1<Re<2000 - Laminar flow, both viscous and inertial terms important
Re>2000 - Turbulent flow, viscosity relatively unimportant
Critical Re for onset of turbulence depends on geometry and surface features
Blood flow characteristics
- most blood flow is laminar
- might be turbulent in aorta at peak systole, at stenosis
- turbulent when heart valves leak and when using a sphygmomamnometer
Turbulent shear stress
- density x product of mutually perpendicular components of velocity
Sphygmomanometry
- the measuring of blood pressure
Method: - place cuff around arm
- rest arm on surface just below heart level
- place stethoscope on hollow of elbow over brachial artery
- inflate cuff to pressure where blood does not flow
- release until K sounds are heard - systolic pressure
- release until no sound is heard - diastolic pressure
Viscosity
- the constant of proportionality linking shear stress to shear strain rate
- fluids that exhibit this behaviour are known as Newtonian fluids
- viscosity can vary within fluids with change in temperature due to interactions at smaller length scales
Variation in fluid behaviour for different fluid types
Plastic fluids
- high viscosity at low shear rates but ‘softens’ at higher shear rates’
Newtonian fluids
- constant viscosity (shear stress proportional to shear rate)
Dilatant fluids
- low viscosity at low shear rates but ‘stiffens’ at higher shear rate
Frictionless
- no resistance to applied shear / no viscosity
Hematocrit definition
Percentage of blood volume occupied by red blood cells (typically 50%)
Blood fluid properties
- inhomogeneous
- plasma (no hematocrit) is a Newtonian fluid with viscosity close to water
- as hematocrit increases viscosity increases
- at low shear rates RBCs can clump together (Rouleux formation) - this increases viscosity at low shear rates
- For 50% hematocrit viscosity is relatively constant in range 1 - 100 s^-1 (4 x thicker than water)
Fahreus lindquist effect
- as blood of constant haematocrit flows into smaller tubes a larger proportion of the volume is close to walls
- this results in a cell free layer close to walls, reducing overall haematocrit in tube
- viscosity of blood will therefore also decrease as diameter decreases
Affects of shear stress on RBCs and Platelets
10 - 100 dyne/cm2: damages adhered RBC, platelets activated
> 500 dyne/cm2 turbulent shear stress: loss of bioconcavity
>3000 dyne/cm2: haemolysis (RBC rupture)
Viscous vs inviscid flow
Inviscid = viscous effects neglected
Inviscid flow assumed when fluid flow is far from a boundary (usually external to a body)
Incompressible vs compressible flow
- when variation of density is small we assume flow is incompressible
- this is quantified by a Mach number < 0.3
Ma = u/a where u is velocity of flow relative to medium and a is speed of sound in medium - speed of sound in water is 1484m/s and resulting flows Ma«0.3
Ratio of distal and proximal diameter
Ratio of distal and proximal area
Ratio of distal and proximal velocity
Ratio of distal and proximal Reynolds number
- D(d) = 0.8D(p)
- A(d) = 1.28A(p)
- v(d) = 0.781v(p)
- Re(d) = 0.625Re(p)
Flow development:
- temporal
- spatial
Temporal: refers to time it takes for a flow to become fully developed
Spatial: when flow is disturbed a fully developed flow is not observed immediately, space is required for the viscous effects at the wall to act on the fluid to produce a parabolic profile
- entrance length for flow to become fully developed is given by:
- L = 0.03 D Re
Why (experimentally) we might not see a parabolic velocity profile in a tube
Flow could be:
- temporally undeveloped (not enough time passed under constant pressure gradient)
- spatially undeveloped (not far enough away from flow disturbance for flow to be parabolic)
- flow is turbulent
What is the equation for Reynolds number
See equation sheet
What is the equation for poiseulle flow
See equation sheet
Derive poiseuille flow
You heard me
what is the womersley and strouhal number
- characterise periodicity of flow
- womersley number applies to straight tubes
- see equation sheet for equations
