Vascular endothelial cells are cultured on the inside of a 10-cm long hollow tube that has an internal diameter of 3 mm. Culture medium flows through the tube at Q = 1 ml/s. The cells produce a cytokine, EDGF, at a rate nEDGF (production rate per cell area) that depends on the local wall shear stress according to nEDGF = kτwall, where k is an unknown constant with units of ng/dyne per s. The flow in the tube is not fully developed, such that the shear stress is known to vary with axial position according to τwall = τ0(1 – βx), where β = 0.02 cm−1, τ0 = 19 dyne/cm2, and x is the distance from the tube entrance. Under steady conditions a sample of medium is taken from the outlet of the tube, and the concentration of EDGF is measured to be 35 ng/ml in this sample. What is k?

Flow occurs through a layer of epithelial cells that line the airways of the lung due to a variety of factors, including a pressure difference across the epithelial layer (ΔP = P0) and, in the case of transient compression, to a change in the separation between the two cell membranes, w2, as a function of time. We consider these cases sequentially below. Note that the depth of the intercellular space into the paper is L, and the transition in cell separation from w1 to w2 occurs over a length δ much smaller than H1 and H2. (See the figure overleaf.)

Consider a membrane of thickness 10 μm that has a number of tiny cylindrical pores (of radius 10 nm) passing through it. The density of pores in the membrane is such that the porosity (fractions of water-filled space) of the membrane is 0.1%.

The graph shown in the figure overleaf is adapted from a 1927 paper [16] in which Landis proved the existence of Starling’s phenomenon by occluding capillaries. The ordinate is the volume of fluid leaking out of (or re-entering) the capillary per unit capillary wall area, j. Although it is not precisely true, for the purposes of this question you may assume that the reflection coefficient of this capillary wall to plasma proteins is unity.

Fluid often passes through pores in cell membranes or cell layers. The dimensions are small and the velocities low, so viscous forces dominate (low Reynolds number). Use dimensional analysis, or an approximate method of analysis based on the viscous flow equations, to determine the scaling law that expresses the dependence of the pressure drop across a pore (ΔP) on the flow rate through it (Q). The other parameters that are given include the pore radius, R, and the viscosity of the fluid, μ. The membrane itself should be considered infinitesimally thin so that its thickness does not influence the pressure drop.

Using a stroboscope, it has been observed that freely falling water drops vibrate. The characteristic time for this vibration does not depend on the viscosity of the water (except for very, very small drops). Determine what parameters you expect this vibration time to depend on, and find a relationship between the vibration time and these parameters. Estimate the characteristic vibration time for a water droplet of diameter 2 mm at a temperature of 25° C. (Hint: this time-scale is the same for a droplet inside of a rocket in space as it is for a droplet falling on the Earth.)

When blood is taken out of the body for processing into an extracorporeal device, a major concern is that the level of shear stress to which the blood is exposed should be less than a critical level (roughly 1000 dyne/cm2). For exposure to shear-stress levels higher than this, lysis of the red blood cells can occur, together with platelet activation and initiation of the clotting process.

Consider the flow of blood through a device that has a set of parallel tubes each with a diameter of 1 mm and a length of 10 cm. What is the maximum pressure drop that should be used for such a device if the highest shear levels in the device occur in these tubes? (Blood has a viscosity about five times that of water.) You may neglect entry effects and treat the flow as fully developed.

A parallel-plate flow chamber is to be designed to study the effects of shear stress on adhesion of leukocytes to endothelia. However, endothelial cells can be damaged by shear stress greater than 400 dyne/cm2. The width of the flow channel is to be 1 cm and its length 5 cm. The flow is to be driven by gravity, and a fluid column 1 m in height is available. The system must work both for saline and for blood. What should be the maximum separation (s ≪ 1 cm) between the two plates such that the endothelial cells are not damaged? The schematic diagram below is not to scale. You may neglect entry effects and treat the flow as fully developed.

A fluorescently labeled molecule (with a diffusion coefficient of 1 × 10−6 cm2/s) is released into the upstream end of a small blood vessel of diameter 100 μm and length 2 mm. The flow rate of blood passing through this vessel is 0.3 μl/min. Roughly estimate how long it should take before the molecule can be detected at the downstream end of the blood vessel.

Transport of nutrients, growth factors, and other molecules to tissues frequently takes place through the capillary wall. In many capillaries, there are tiny gaps between endothelial cells that allow both diffusion and convection of solutes across the vessel wall. Consider a particular endothelium in which the gaps between the cells are characterized by the following dimensions: L = 1 μm long, h = 200 nm high, and W = 10 nm in width (the last dimension is the distance between the two cells; see the figure below). The fluid in this gap is at 37 °C, and has the same properties as physiologic saline.

Nanoparticles can be used to probe the intracellular environment. By tracking their motion one can draw conclusions regarding transport inside a cell.

An investigator has placed a nanoparticle of diameter 100 nm inside of a Xenopus oocyte. The cytoplasm of this cell behaves like a viscous fluid with a viscosity 20 times that of water. Over a period of 20 s (at a temperature of 18 °C), the particle travels (on a somewhat erratic path) over a distance of approximately 3 μm from the periphery of the cell toward the nucleus in the center of the cell.

The investigator concludes that there is a preferential motion or “flow” from the periphery of the cell toward the nucleus. He would now like to plan a full study to examine what causes this “flow.” Does this seem like a reasonable next step? If so, justify why. If not, explain what next step you would suggest.

Fibrinogen has a diffusion coefficient in saline of approximately 2 × 10–7 cm2/s at 25 ˚C. It is a rod-shaped molecule whose length is roughly 10 times its radius. Estimate the length of this molecule.

We want to determine how fast the alveolar CO2 concentration can change in response to changes in blood CO2 concentration. Assume a spherically shaped alveolus (of radius 0.015 cm) with a spatially uniform internal gaseous composition at time t = 0. Calculate the time necessary to achieve 95% equilibration when the CO2 concentration in the alveolar wall is suddenly changed at t = 0. You may neglect any mass-transfer effects of the wall tissue and the liquid film in the alveolus. The diffusion coefficient of CO2 in air is 0.14 cm2/s.

Considering your answer, how important is the diffusion resistance to mass transport through the gas contained within the alveolus?

Fluorescein is a small fluorescent tracer molecule that is used in a wide variety of physiologic studies. In the eye, it is used in a technique known as fluorophotometry to characterize the transport characteristics of aqueous humor, the fluid that fills the anterior chamber behind the cornea.

A drop of fluorescein (50 μl of 0.005%, by mass, fluorescein in saline) is placed onto the cornea and spreads evenly over the corneal surface to create a thin film. It then diffuses through the cornea (D = 1 × 10–6 cm2/s) and enters the aqueous humor on the back side of the cornea. The thickness of the cornea is about 0.05 cm and its radius is 0.5 cm.

This book arose out of a need that frequently faced us, namely coming up with problems to use as homework in our classes and to use for quizzes. We have found that many otherwise excellent textbooks in transport phenomena are deficient in providing challenging but basic problems that teach the students to apply transport principles and learn the crucial engineering skill of problem solving. A related challenge is to find such problems that are relevant to biomedical engineering students.

The problems included here arise from roughly the last 20–30 years of our collective teaching experiences. Several of our problems have an ancestry in a basic set of fluid mechanics problems first written by Ascher Shapiro at MIT and later extended by Ain Sonin, also at MIT. Roger Kamm at MIT also generously donated some of his problems that are particularly relevant to biomedical transport phenomena. Thanks are due to Zdravka Cankova and Nirajan Rajkarnikar, who helped with proof-reading of the text and provided solutions for many of the problems.