The first step in inflammation is for immune cells $($leukocytes$)$ to roll to a stop on the inside of the blood vessel wall near the site to be inflamed. This is accomplished by molecules of E$-$selectin, which are produced by and incorporated into the endothelial cells lining the blood vessel wall, binding to molecules of sialyl$-$Lewis$($x$),$ which are produced by and incorporated into the leukocytes. Dr$.$ Dan Hammer $($of the Univ. of Pennsylvania$)$ and his students have produced glass beads to which they have covalently bound sialyl$-$Lewis$($x$).$ They flow these between a microscope slide and cover glass. E$-$selectin has been covalently conjugated to the microscope slide. They then flow the glass beads over the microscope slide at various flow rates and measure the relative numbers of free$-$flowing, rolling, and stopped beads. Imagine it is your first day in his lab as a student researcher, and Dr$.$ Hammer asks that, instead of reporting your data in terms of flow rate, report your data in terms of shear rate at the surface of the microscope slide.

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Note: The fluid flows between the two plates. That is$,$ there is an upper plate and a lower plate, the fluid flows between the two plates.

Step $1$: Using the control volume method to derive the appropriate momentum balance $($shell balance for momentum$)$ with all important terms. Write all applicable boundary conditions.

Step $2$: Solve for the velocity profile of the fluid $($simplify as much as you can but don't substitute in numbers yet$).$

Step $3$: Let the flow rate $(Q)$ be $)=(1\mathrm{.}67{10}^{-7}\frac{m^3}{s},$ and solve for the pressure gradient necessary to achieve that flow rate. Use information from the diagram and the viscosity of water at room temperature to fill in numerical values for all of the other parameters. $[$Hint: $Qv$ A because $v=f(z)]$

Step $4$: What is the shear stress where the fluid touches the bottom plate?