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Summary lecture 1-3 transport in biological systems for biomedical engineering students (LS&T old cohord) at the RUG
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Transport in biological systemsLecture 1
10-5-2021
1.2 fluid characteristics and viscosity
Fluid = substance, liquid or gas, that deforms continuously under applied shear stress. It doesn’t
matter if the stress is small. A fluid does not have a preferred shape.
Solid does not deform continuously when a shear stress is applied and it relaxes back to a
preferred shape when stress is removed.
Fluid properties:
- Density = mass per unit volume of a substance (SI unit; kg/m 3).
- Specific weight = weight per unit volume of a substance (SI unit; N/m 3).
- Viscosity
Viscosity is the additional property that determines the difference in behavior of the fluid.
When looking at the figure below plate A is fixed, while plate b can move.
Case of fluid in between the plates: A constant force F is applied to plate B, causing it to
move along at a constant velocity V B with respect to the fixed plate. The plate moves
continuously. After a time dt, the line of the fluid at t=0 will move to a new position, the
velocity profile. The angle between the line of the fluid at t=0 and t=t+dt = dt is defined as
shearing strain, dγ.
V B ⅆt
o tan ( ⅆγ )= = opposite / adjacent
h
VB
o Rate of shearing strain, dγ/dt: ⅆγ ∕ ⅆt =
h
VB
o =velocity gradient=dV / dy , here y is the distance from plate A to some
h
arbitrary point above the plate.
The fluid that touches plate B moves with the same velocity at that of plate B. These molecules
adhere to the plate and do not slide along its surface (= no slip condition). The velocity of the
fluid at any point between the plates varies linearly between V = 0 (at plate A) and V = V B (at
plate B). The velocity gradient is defined as the change in fluid velocity with respect to y. For a
linearly varying velocity profile, like shown below, the velocity gradient is equal to V B/h as seen in
the third bullet point.
Case of solid in between the plates: force F creates a displacement d, a shear stress τ and a
shear strain γ. The motion of the upper plate would stop after a small displacement.
, The ‘cube’ seen on the left is a part of the fluid at some
arbitrary point between the plates. When the shear
stress on one side is bigger, it will rotate. Since the fluid
element will be moving at a constant velocity (so no
rotation), the shear stress on the element τ’ must be the
same as the shear stress τ.
dτ / dy = 0 and τA = τB = τwall.
The shearing stress at the wall may also be represent by
τA = τB = force / plate area.
Rate of shearing strain = γ̇ . The relating between the
shear strain rate (deformation) γ̇ and shear stress
(loading) τ is determined by the viscosity μ. If you plot
shear stress vs shear strain rate, the slope is the
viscosity. For common fluids like oil, water and
air, the viscosity does not vary with shearing
rate. For Newtonian fluids, shear stress and rate of
shearing strain may be related by the following
equation:
τ =μ ⋅ γ̇
Newtonian fluid = “a fluid in which the viscous
stresses that arise from its flow, at every point, are
linearly correlated to the locate strain rate (=the
rate of change of its deformation over time).”
For non-Newtonian fluids, shear stress and shear rate are
not linearly related. In that case, viscosity can be defined
as the instantaneous slope of the shear stress/shear rate
curve. The slope of the shear stress/shear rate curve is
not constant. There can be shear thickening or shear
thinning.
Shear thinning fluids = non-Newtonian fluids whose
apparent viscosity decreases as shear rate increases. An example is latex paint. The viscosity is low
when someone is painting, but it becomes higher when there is no shearing force present. Shear
thickening fluids = non-Newtonian fluids whose apparent viscosity increases when the shear rate
increases. Quicksand/cornstarch in water are good examples. As you try to move slowly in it, the
viscosity is low, but if you try to move quickly, the viscosity increases and the movement becomes
difficult.
The lines starting at the origin are fluids with no yield stress. The ones starting higher have a yield
stress, like toothpaste, mayonnaise or blood. A Bingham plastic I neither a fluid nor a solid. It can
withstand a finite shear load and flow like a fluid when that shear stress is exceeded.
Summarized
Name Symbol Meaning
Shear(ing) strain dy The ratio of change in deformation to its original length
perpendicular to the axes of the
member due to shear stress (like dy
in the 2nd image or the angle between
D and D’ in the first image).
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