You are building a microchannel of length L with a rectangular cross-section of width, w, and height, h, to study the adhesion of leukocytes to the endothelium. You specify that the width of the channel is 2 cm. You wish to determine the height, h that you would need to generate a wall shear stress, as high as 2 N m^-2, by applying a volumetric flow rate, Q = 2 cm^3 s^-1 with a syringe pump. At x = x0 p = p0, and at x = xL p = pL. The tissue culture medium is Newtonian and has a viscosity of 0.003 g cm^-1 s^-1. You will operate your device under conditions where the flow will be laminar, is at steady-state, and is fully-developed. Let ΔP = p0 - pL a. Draw a diagram of your control volume, choose a coordinate system, label all boundaries, and label all stresses acting on the control volume.
b. Derive the momentum balance for your control volume.
c. Your experimental conditions require that you relate the volumetric flow rate, Q to the stress. You remembered that for your system, the volumetric flow rate is given by: c. Q-I-..[.dady -h/20
Integrate the momentum balance equation and get an expression for v so that you may evaluate the double integral for the volumetric flow rate. State clearly the boundary conditions that you used to obtain the constants of integration and the constitutive equation that relates the shear stress and the shear rate for Newtonian fluids.
d. Sketch the velocity profile in your rectangular channel.
e. Using the result from b), and the numerical values provided in the problem statement, determine the height h that will produce the wall shear stress that you require (4 pts).