As noted earlier in this chapter, the primary function of the lungs is to facilitate gas exchange between the atmosphere and the blood. Toward this end, the capillary system in the lungs is very different than that found elsewhere. Conforming to the alveolar geometry (Fig. 10.3), capillary blood

Question

As noted earlier in this chapter, the primary function of the lungs is to facilitate gas exchange between the atmosphere and the blood. Toward this end, the capillary system in the lungs is very different than that found elsewhere. Conforming to the alveolar geometry (Fig. 10.3), capillary blood flow in the lungs is better described as a sheet flow rather than a tube flow; that is, the blood flows within the thin planar walls of the alveoli, which appear as parallel membranes separated by hexagonally positioned posts. Fung and his colleagues sought to quantify the pressure–flow relation in this sheet flow and began with a nondimensionalization. Here, let us perform a similar procedure and compare to that reported by Fung (1993).

Accepted Answer

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

[latex] \Delta p=g( ho ,\mu ,U,\omega ,h,w,\varepsilon ,a, heta ,H,D_{c},E_{c},VSTR).[/latex]

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

[latex]\left[\Delta p ight]=L^{-1}T^{-2}M^{1}, \left[\omega ight]=L^{0} T^{-1}M^{0}, \left[a ight] =L^{1}T^{0}M^{0},[/latex]

[latex]\left[ ho ight]=L^{-3}T^{0}M^{1}, \left[h ight]=L^{1}T^{0}M^{0}, \left[D_{c} ight]=L^{1}T^{0}M^{0},[/latex]

[latex]\left[\mu ight]=L^{-1}T^{-1}M^{1}, \left[w ight]=L^{1}T^{0}M^{0}, \left[E_{c} ight]=L^{-1}T^{-2}M^{1},[/latex]

[latex]\left[U ight]=L^{1}T^{-1}M^{0}, \left[\varepsilon ight]=L^{1}T^{0}M^{0},[/latex]

and, of course, [latex]\left[ heta ight]=\left[H ight]=\left[VSTR ight]=1.[/latex] If we assign length, time, and mass scales (Step 3) as

[latex] L_{s}=h, T_{s}=\frac{h}{U}, M_{s}=( ho hw)h,[/latex]

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

[latex]\pi _{p}=\frac{\Delta p}{ ho U^{2}}\left(\frac{h}{w} ight), \pi _{w}=\frac{w}{h},[/latex]

[latex]\pi _{p}=\frac{h}{w}, \pi _{\varepsilon }=\frac{\varepsilon }{h},[/latex]

[latex]\pi _{\mu }=\frac{\mu }{ ho Uw}=\frac{\mu }{ ho Uh}\left(\frac{h}{w} ight), \pi _{a}=\frac{a}{h}=\frac{a}{\varepsilon }\left(\frac{\varepsilon }{h} ight),[/latex]

[latex]\pi _{U}=1, \pi _{D_{c}}=\frac{D_{c}}{h},[/latex]

[latex]\pi _{\omega }=\frac{\omega }{U}h, \pi _{E_{c}}=\frac{E_{c}}{ ho U^{2}}\left(\frac{h}{\omega } ight), [/latex]

[latex]\pi _{h}=1,[/latex]

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

[latex] \frac{\Delta p}{ ho U^{2}}\left(\frac{h}{w} ight)=\widetilde{g}\left(\frac{h}{w},Re,\frac{w}{h},\frac{\varepsilon }{h},\frac{a}{h},\frac{D_{c}}{h},\frac{\omega }{U}h,\frac{E{_{c}}}{ ho U^{2}}\left(\frac{h}{\omega } ight) , heta , H, VSTR ight),[/latex]

where the Reynolds’ number is [latex]Re= ho Uh/\mu .[/latex] 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)

[latex] \frac{ riangledown ph^{2}}{\mu U}=\frac{(\Delta p/h)h^{2}}{\mu U}=\frac{\Delta ph}{\mu U}\left(\frac{ ho Uw}{ ho Uw} ight)=\frac{\Delta p}{ ho U^{2}}\left(\frac{h}{w} ight)\left(\frac{ ho Uw}{\mu } ight),[/latex]

[latex] \frac{\mu U}{E_{c}h}=\frac{\mu U}{E_{c}h}\left(\frac{ ho U^{2}}{ ho U^{2}} ight)\left(\frac{w}{w} ight)=\frac{\mu }{ ho Uw}\left(\frac{ ho U^{2}}{E_{c}} ight)\left(\frac{w}{h} ight),[/latex]

[latex] \sqrt{\frac{h^{2}\omega ho }{4\mu } }=\frac{1}{2}\sqrt{\left(\frac{\omega }{U}h ight)\left(\frac{ ho Uw}{\mu } ight)\left(\frac{h}{w} ight) },[/latex]

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

[latex] \frac{h}{2}\sqrt{\frac{\omega ho }{\mu } }[/latex]

are both less than unity and thus negligible in this sheet flow. Note, too, that Fung’s parameter [latex]\mu U/E_{c}h[/latex] 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,

[latex] \frac{ riangledown ph^{2}}{\mu U}=-G_{1}\left(\frac{D_{c}}{h},\frac{\mu _{0}U}{E_{c}h},H ight)G_{2}\left(\frac{w}{h} ight)f\left(\frac{h}{\varepsilon },\frac{\varepsilon }{a}, heta, VSTR ight),[/latex]
where
[latex] \mu G_{1}\left(\frac{D_{c}}{h},\frac{\mu _{0}U}{E_{c}h},H ight)\equiv \mu _{a}[/latex]

was taken to be the apparent viscosity, with the form

[latex] \mu _{a}=\mu \left[1+c_{1}\left(\frac{D_{c}}{h} ight)H+c_{2}\left(\frac{D_{c}}{h} ight)H^{2} ight],[/latex]

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

[latex] G_{2}\left(\frac{w}{h} ight)=\frac{12}{1-0.63(h/w)}\approx 12,[/latex]

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

[latex] riangledown p\cong -\frac{12\mu _{a}U}{h^{2}}f\left(\frac{h}{\varepsilon },\frac{\varepsilon }{a}, heta , VSTR ight).[/latex]

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

[latex] \frac{\partial p}{\partial x}=-\frac{12\mu _{a}U}{h^{2}}f_{x}\left(\frac{h}{\varepsilon },\frac{\varepsilon }{a}, heta ,VSTR ight)=-\frac{12\mu _{a}U}{h^{2}}f_{x},[/latex]

[latex] \frac{\partial p}{\partial y}=-\frac{12\mu _{a}V}{h^{2}}f_{y}\left(\frac{h}{\varepsilon },\frac{\varepsilon }{a}, heta ,VSTR ight)=-\frac{12\mu _{a}V}{h^{2}}f_{y},[/latex]

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

[latex] U=-\frac{h^{2}}{12\mu _{a}f_{x}}\left(\frac{\partial p}{\partial x} ight), V=-\frac{h^{2}}{12\mu _{a}f_{y}}\left(\frac{\partial p}{\partial y} ight).[/latex]

Finally, Fung and colleagues suggested that

[latex] h=h_{0}+\alpha \Delta p[/latex]

based on morphometric data, with [latex]h_{0}=4.28 \mu m[/latex] in cat lung and [latex]3.5 \mu m[/latex] in human lung, [latex]α=0.219 μm/cm H_{2}O[/latex] in cat lung for a [latex]\Delta p\sim 10 cm H_{2}O,[/latex] and [latex]α=0.127 μm/cm H_{2}O[/latex] in human lung for a [latex]\Delta p\sim 10 cm H_{2}O.[/latex] 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.

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

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