Channel Flow
In a cylindrical tube, where do we place r=0, and in which direction does z increase?
(r=0) is on the tube’s centerline; (z) increases along the tube from the inlet toward the outlet.at the center of the tube with the origin at the entry face
A small droplet forms on the tip of a needle. The droplet grows until the gravitational force exceeds which force?
Surface tension
Cell type that makes up the majority of cellular content of the blood?
Erythrocytes or RBCs
What distinguishes a Bingham plastic from a Newtonian fluid when a small shear stress is applied?
A Bingham plastic does not flow until its yield stress is exceeded; a Newtonian fluid has no yield threshold.
In a cylindrical tube, the flow looks the same at every angle around the centerline. Which derivatives in the governing equation become zero?
Derivatives with respect to the azimuthal coordinate θ: ∂/∂θ = 0.
In a straight cylindrical tube, what does fully developed flow tell you about how axial velocity changes along the tube?
It no longer changes with axial position: \(\partial v_z/\partial z=0\). The velocity can still vary with radius.
How do you determine the force at which the droplet falls?
Set up a force balance between its weight and the vertical surface-tension force.
What does hematocrit measure? And what's the range for a healthy person?
The fraction of blood volume occupied by red blood cells, 40-47% acceptable range
For an incompressible fluid flowing through a narrowing, what happens to mean velocity if the cross-sectional area decreases while flow rate stays constant?
Mean velocity increases because (Q=AV)
If the flow is fully developed, what can you assume for a cylindrical channel?
Change in velocity is only in r not z
Name three assumptions behind the fully developed, pressure driven flow equation for a Newtonian fluid in a straight cylindrical tube.
steady flow
Netwonian, incompressible fluid, constant density
fully developed flow
laminar flow
velocity independent of azimuthal direction (p not vary in time)
constant delta P/l (p not vary in time)
gravity effects are negligible
If the drop is approximately spherical with radius (R_d), what would the gravitational force equal?
Rho g 4/3 pi Rdrop3
The drop’s gravitational force is simply its weight:
\[ F_g=mg. \]
If we approximate the drop as a sphere of radius \(R_d\), its volume is \(V=\frac{4}{3}\pi R_d^3\). Its mass is density times volume, \(m=\rho V\). Substituting gives
An increase in the deformability of RBCs causes blood viscosity to decrease or increase?
Decrease
What is fractional flow reserve, and when are the pressures compared?
The ratio of pressure distal to a coronary narrowing to pressure proximal to it, measured during maximal flow (hyperemia).
What does 1D flow tell you?
v only changes in one direction
How do we justify the use of laminar flow and full developed flow conditions?
Calculate the entrance length and show that you are beyond Le and the Re is less than 2100
The droplet is attached along a circular contact line of radius r. Assuming surface tension acts vertically there, what is the upward force holding the droplet?
At what droplet radius does its weight balance the surface tension force?
F_surface = gamma × 2πr
Explanation: Surface tension, gamma, is force per unit length. The circular attachment has circumference 2πr, so the total upward force is gamma × 2πr.
Part 2:
rho × g × (4/3)πR_drop³ = gamma × 2πr
R_drop = [3 gamma r / (2 rho g)]^(1/3)
Keep the two radii distinct: R_drop describes the droplet’s volume; r describes where it attaches. If surface tension acts at an angle, only its vertical component contributes, but stating the vertical assumption keeps this appropriate for the review.
For a Newtonian fluid, what would a plot of shear stress versus shear rate look like, and what does its slope represent?
A straight line through the origin; its slope is dynamic viscosity, μ.
Why could microvascular disease make a coronary narrowing appear less severe by FFR?
Impaired downstream dilation limits flow, reducing the pressure drop across the narrowing and making FFR appear higher.
If a fluid is incompressible, what can you assume?
rho is constant
In a cylindrical vessel of radius \(R\), ultrasound measures a velocity (V_{2}). Assume fully developed laminar flow with a parabolic velocity profile. Given density \(\rho\) and viscosity \(\mu\), find the Reynolds number and volumetric flow rate.
Re = rhoV2R2/mu
Q = ½ V2 * pi * R22
A small blood vessel (length=4cm, radius=0.002cm, τy = 0.04 dyn/cm²) can be assumed to have steady, full-developed 1D pressure driven flow. What pressure drop is required to overcome the yield stress of blood within this vessel?
.12mmHg
The pressure drop must produce enough force to overcome the yield stress acting along the vessel wall. Balance the forces:
Pressure force = yield stress force
ΔP × (πR²) = τy × (2πRL)
Therefore, ΔP = 2Lτy/R.
Using L = 4 cm, R = 0.002 cm, and τy = 0.04 dyn/cm²:
ΔP = 160 dyn/cm² ≈ 0.12 mmHg.
This is the minimum pressure drop needed to initiate flow.
For a power law fluid, what value is raised to a power? what type of fluid has the n > 1, and give one example of these fluid.
All three needed for points.
shear rate
shear thickening or dilatant
cornstarch and water or quicksand
A parallel plate flow chamber is being designed. The channel width is 2cm. Determine the channel height that can be used to generate wall shear stresses as high as 20 dyne/cm-2 with flow rates less than 2 cm3/s. The viscosity of the fluid is .0087 g/cms.
h=.051 cm
τw = 6μQ / (wh²)
Solve for channel height:
h = √[6μQ / (wτw)]
Using the maximum allowed flow rate, Q = 2 cm³/s:
h = √[6(0.0087)(2) / (2)(20)]
h = √0.00261
h ≈ 0.051 cm
What assumption breaks down when flow isn't steady?
time derivatives can no longer be negated