Question 10.8: As noted earlier in this chapter, the primary function of th...

Fung considered the pressure drop $\Delta p$ within a pulmonary capillary to depend on the density and viscosity of the blood $(\rho , \mu),$ mean velocity U, circular frequency of oscillation $\omega,$ sheet thickness h and width w, post diameter $\varepsilon$ and separation distance a, angle between the mean flow and post alignment $\theta ,$ the hematocrit H and red blood cell diameter $D_{c},$ the elastic modulus of the red blood cell $E_{c},$ and a ratio between the vascular space and tissue volume (VSTR). Consistent with Step 1 in our Buckingham Pi approach, we specify

It is easy to see that appropriate fundamental units are L, T, andM, where (Step 2)

and, of course, $\left[\theta \right]=\left[H\right]=\left[VSTR\right]=1.$ If we assign length, time, and mass scales (Step 3) as

Control Volume and Semi-empirical Methods then (Step 4) we can list the computed Pi groups:

and, thus (Step 5), we can express the original equation as

where the Reynolds’ number is $Re=\rho Uh/\mu .$ Hence, Buckingham Pi reduced the number of independent variables from 13 to 10, a slight improvement. Fung (1993) actually chose a few different, equivalent nondimensional parameters; they are related to the present ones via (by multiplying by unity appropriately)

which is to say, our current Pi groups differ from Fung’s only through the Reynolds’ number $\rho Uh/\mu$ and the term h/w. It is interesting that experiments revealed that the Reynolds’ number Re and Womersley’s number

are both less than unity and thus negligible in this sheet flow. Note, too, that Fung’s parameter $\mu U/E_{c}h$ is essentially the ratio of the shear stress in a Couette flow between parallel plates (cf. Example 9.2 of Chap. 9) to the modulus of the red blood cell (RBC), which was interpreted as a RBC membrane shear strain despite the flow not being Couette. Experiments suggested further that, with a minus sign accounting for the pressure gradient being opposite the pressure drop,

$\frac{\triangledown ph^{2}}{\mu U}=-G_{1}\left(\frac{D_{c}}{h},\frac{\mu _{0}U}{E_{c}h},H \right)G_{2}\left(\frac{w}{h} \right)f\left(\frac{h}{\varepsilon },\frac{\varepsilon }{a}, \theta,  VSTR \right),$
where
$\mu G_{1}\left(\frac{D_{c}}{h},\frac{\mu _{0}U}{E_{c}h},H \right)\equiv \mu _{a}$

was taken to be the apparent viscosity, with the form

the effect of  $\mu _{0}U/E_{c}h$ being yet unexplored. The function $G_{2}$ was found to be

whereas the function f was called a geometric friction factor It was found experimentally to vary nearly linearly with $h/\varepsilon$ with values of $f$ from 1.5 to 5 for $h/\varepsilon$ from 1 to 5, with values of $VSTR \sim 91,h\sim 7.4  \mu m,\varepsilon \sim 4  \mu m,$ and $a\sim 12  \mu m,$ and a $\sim 12 \mu m,$ f would equal 1 in the absence of posts. Hence, the semi-empirical relation reduced to

Because shear flow is two-dimensional, in general, Fung and colleagues thus considered Control Volume and Semi-empirical Methods

where U and V are mean velocities in the x and y directions, respectively, and $f_{x}\sim f_{y}\sim 2.5.$  In general, the mean values of 2-D velocities within the capillaries are

Finally, Fung and colleagues suggested that

based on morphometric data, with $h_{0}=4.28  \mu m$ in cat lung and $3.5 \mu m$ in human lung, $α=0.219   μm/cm   H_{2}O$ in cat lung for a $\Delta p\sim 10    cm   H_{2}O,$ and $α=0.127   μm/cm  H_{2}O$ in human lung for a $\Delta p\sim 10  cm  H_{2}O.$ For more details, see Fung (1984, 1993). The take-home message here is simply that Buckingham Pi can often be used advantageously to guide empirical studies, particularly those associated with complex flows as in the pulmonary capillaries.