The apparatus consists of two concentric horizontal tubes. One streams goes through the inner tube $($flow in a pipe$)$ and the other stream goes through the annular region. The inner tube inside radius is ro$,$ the inner tube outside radius is r$1$and the outer tube inside radius in r$2.$ The length of the tubes is L$.$ The flow of each stream is driven by its own pressure difference $(-$P$).$ Each stream has its own properties: the stream in the pipe $(1,\backslash$rho $1)$ and the stream in the annulus$(2,\backslash$rho $2).$

Summarize the key aspects of the flow model for the annular stream and provide the main results. The development of this model should be included in Appendix A $($as explained below$).$In this section you should include just a summary, including:

a$.$ Description of this part of the system.

b$.$ List of assumptions.

c$.$ Simplified GDE and BC$$s$.$

d$.$ Expressions for shear stress and velocity profiles.

e$.$ Expressions for average velocity and volume flow rate.