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.)