It appears that Rosen et al. (1974) first showed in vitro that endothelial cells alter their production of a specific molecule (histamine) in response to altered shear stresses. They accomplished this using cultured cells placed within a parallel-plate device. Their device was 1.3\times 1.3\times 23.5 cm in dimension, with the cells placed in the fully developed region (15 cm from the entrance). Because the flow chamber was not much wider than it was deep, however, the equations to compute the wall shear stress differed from Eq. (9.30). Ensuring that h\ll w allows Eq. (9.30) to be used and facilitates the easy design and interpretation of the experiment. Toward this end, let us consider the work by Levesque and Nerem (1985). Their flow chamber was 0.025\times1.3\times5 cm in dimension; hence, h\ll w and our equations hold. They plotted the morphological measures for the cells versus wall shear stress \tau_{w}, which they computed as \tau_{w}=\frac{\partial \mu Q}{wh^{2}}. (9.30) \tau_{w}=\frac{\partial \mu ^{2}}{\rho h^{2}}Re, where Re is the Reynolds’ number. Show that this relation is correct.
Question 9.1: It appears that Rosen et al. (1974) first showed in vitro th...
As noted earlier, let the Reynolds’ number be given by Re=\rho \overline{\nu }h/\mu . Recall, therefore, that [from Eq. (9.24)]
\overline{\nu }=\frac{Q}{A}=-\frac{h^{2}}{12\mu }\frac{dp}{dz}. (9.24)
\overline{\nu }=-\frac{h^{2}}{12\mu }\frac{dp}{dx}\rightarrow \frac{dp}{dx}=-\frac{12\mu \overline{\nu } }{h^{2}};
hence, from Eq. (9.29),
\tau _{w}=\left|\frac{h}{2}\frac{dp}{dx} \right|, (9.29)
\tau _{w}=\left|\frac{h}{2}\frac{dp}{dx} \right|=\left|\frac{h}{2}\left(-\frac{12\mu \overline{\nu } }{h^{2}} \right) \right|=\left|-\frac{6\mu \overline{\nu } }{h} \right|.
Now, simply multiply by “one,” namely
\tau _{w}=\left|-\frac{6\mu \overline{\nu } }{h}\left(\frac{h}{h}\frac{\rho }{\rho }\frac{\mu }{\mu } \right) \right|=\left|-\frac{6\mu ^{2}}{h^{2}\rho }\left(\frac{\rho \overline{\nu }h }{\mu } \right) \right|=\frac{6\mu ^{2}}{h^{2}\rho }Re,with \rho \gt 0 and Re\gt 0 by definition, thus yielding our desired result.
Some Exact Solutions Note, too, that Levesque and Nerem stated that Re\lt 2,000 ensured a laminar flow, and they subjected the cells to steady shear stresses of 1.0, 3.0, and 8.5 Pa for up to 24 h. Given that the viscosity and density where assumed to equal those of water and that h=250 \mu m, find the exact values of Re for their reported values of \tau _{w} to check if Re\lt 2,000. Finally, note a few of their findings: “After 24 h of exposure at shear stresses of 30 and 85 dyn/cm^{2}, there was a significant reduction in cell surface area, an increase in cell perimeter and length, and a decrease in cell width . . . the more elongated cells have a higher degree of alignment with the flow axis. This effect becomes accentuated with increasing shear stress.” Note that 1 dyn/cm^{2}=0.1 Pa.