2 - Conservation of mass and the Reynolds transport theorem

Summary

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 n_EDGF (production rate per cell area) that depends on the local wall shear stress according to n_EDGF = kτ_wall, 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 τ_wall = 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.)