BIOE 360 Case 1
Team #: Names
Introduction
Case information:
Module Lessons
Questions:
What are the patient circumstances for the diagnosis?
What is the pressure difference in healthy ears versus ears with tubes?
Does the size of tube make a difference? Would this system display capillary flow?
What role does surface tension play?
Would the type of fluid make a difference? Water, bath water, chlorinated water, etc.
Does the shape of the ear canal change the fluid dynamics?
Is there a correlation between fluid and infection?
Is there a difference in fluid dynamics, other outcomes, due to material of the tube?
Anatomical different between kids and adults?
Would the earplug actually make it worse?
Patient behaviors post-op? Swimming, diving, bathing, etc. What post-op instructions were
given?
Learning outcomes:
At the end of this case, students will be able to…
Draw and label a diagram of the system.
Make assumptions related to the biological and physical parameters of the case.
Calculate and explain different flow situations related to pressure driven flow.
Describe the role of surface tension in fluid flow.
Present findings and defend in a mock trial.
Lessons:
Basic fluid flow (Navier Stokes and Bernoulli’s equations)
Flow constriction (stenosis and orifice flow)
Scaling, dimensional analysis
Surface tension and viscosity relations
Problems:
1. Set up a symbolic solution to the problem of one-dimensional, laminar, incompressible flow in
a circular tube with radius, R using the continuity equation and Navier-Stokes equation, both in
cylindrical coordinates. Define boundary conditions and express in terms of velocity profile as a
function or r, 𝑣𝑧 (𝑟).
2. A stenosis is a narrowing of a blood vessel or valve. Stenosis of a blood vessel arises during
atherosclerosis could occlude an artery, depriving the tissue downstream of oxygen. Further,
the fluid shear stresses acting on the endothelial cells lining blood vessels may affect the
expression of genes that regulate endothelial cell function. Such gene expression can influence
whether the stenosis grows or not. Consider a symmetric stenosis as shown in Figure below.
Assume that the velocity profile within the stenosis of radius Ri(z) has the same shape as the
profile outside the stenosis and is represented as
𝑟2
𝑣𝑧 (𝑟) = 𝑣𝑚𝑎𝑥 (1 − 2 )
𝑅
Outside the stenosis, the radius equals Ro and the maximum velocity is constant. Within the
stenosis, the radius of the fluid channel R(z) equals
𝑧 2 1/2
𝑅(𝑧) = 𝑅0 {1 − 0.5 [1 − 4 ( ) ] }
𝐿
The origin of the z axis is the midpoint of the stenosis
(a) Develop an expression for Vmax in a stenosis in terms of the volumetric flow rate Q,
cylindrical tube of radius R0, and distance along the stenosis zlL.
(b) Compute the shear stress acting on the surface of the stenosis (r =Ri) at z = 0 relative to the
value outside the stenosis.
3. Consider a catheter of radius Rc placed in a small artery of radius R as shown in the figure
below. The catheter moves at a constant speed V. In addition, blood flows through the annular
region between Rc and R under a pressure gradient Δp/L that only varies in the z direction. We
want to determine the effect of the catheter upon the shear stress at r = R. Assume steady, fully
developed flow of a Newtonian fluid.
(a) State the momentum balance and boundary conditions.
(b) Sketch the velocity profile and provide a justification for its shape.
(c) Solve the momentum balance; substitute Newton’s Law of Viscosity and solve. Determine
the velocity.
(d) Calculate the shear stress acting on the blood vessel surface, r=R, using the following
𝑔
𝑑𝑦𝑛𝑒⁄
values: R=0.17cm; Rc=0.15 cm; V=10 𝑐𝑚⁄𝑠; µ=0.03 ⁄𝑐𝑚 𝑠; Δp/L=100
𝑐𝑚3
1
4. Using Poiseuille's law, 〈𝑣〉 = 2 𝑣𝑚𝑎𝑥 and 𝑣𝑚𝑎𝑥 =
∆𝑝𝑅2
4𝜇𝐿
,show that the Fanning friction factor, a
friction factor used characterize pressure drop regardless of dimension expressed as 𝑓 =
∆𝑝 𝐷
16
, for laminar flow in a cylindrical tube is simply 𝑅𝑒.
2𝜌〈𝑣〉2 𝐿
5. A drug is being injected at a constant infusion rate of Q from a syringe of diameter D into a
needle of diameter d (Figure below). The fluid viscosity is µ and density ρ Determine the force
on the syringe. Assume that the flow through the catheter is laminar and that the blood pressure
is p. State all other assumptions made.
6. Pulmonary banding is performed in infants with congenital heart defects that make them
susceptible to pulmonary hypertension. A band is placed around the pulmonary artery to induce
a stenosis and thereby reduce flow and pressure in the pulmonary artery. For the following
conditions, determine the area reduction and radius of the stenosis needed to produce a
pressure drop of 15 mmHg from pt 1 to pt 2. (Hint: Be sure to calculate the Reynolds number in
each region.)
Flow Rate
Upstream Pulmonary artery diameter
Density of Blood
Conversion Factor
Viscosity of Blood
3
2000 𝑐𝑚 ⁄𝑚𝑖𝑛
1.3 cm
𝑘𝑔
1070 ⁄ 3
𝑚
133.32 𝑃𝑎⁄𝑚𝑚𝐻𝑔
𝑔
0.035 ⁄𝑐𝑚 𝑠
Assessments
Problem solutions: Attach neatly handwritten, and/or LaTex, or Matlab solutions to each
problem and explain how it helped you reach the learning outcomes for the unit. No Wolfram
Alpha.
Case Solution: Attach neatly handwritten, and/or LaTex, or Matlab solutions to the case and
conclusions drawn relating to the case questions.
Team assessment
Preparedness: How much time and effort did you put into the unit?
Thoroughness: Did your entire group give an adequate summary of the case and solution using
concepts from the learning activities?
Effectiveness: How useful was your discussion in helping understand the material?
Connections: How significant and helpful was your connection between the case and the
learning activities?
Creativity: How innovative were you in making the material “come alive”?