urgent! please help

Mammalian cells, which lack the rigid cell wall of bacterial cells, will rupture if subjected to excessive shear. Consequently, fermentors for culturing mammalian cells must be designed to minimize the shear stress applied to the cells. A typical lab-scale mammalian cell fermentor is shown below. The vessel is a cylinder of radius R. Mixing and aeration is achieved by introducing air bubbles through a sparger centered at the bottom of the vessel. As the UROP student at CBE in charge of this fermentor, you are asked to determine the cell death rate due to shear forces by estimating the maximum shear stress induced by the bubbles. To simplify the calculation, you adopt the following approximation (refer to Fig. B). The bubbles entrain the fluid near the center of the vessel and the entrained fluid travels upward as a plug of radius Rp at velocity U. The core region contains 50% bubbles and 50% liquid broth by volume. In the annular region, there are no bubbles. Develop a formula for the maximum shear stress experienced by the fluid in the annular region in terms of R,Rp,U and fluid properties. The annular flow profile away from the top and the bottom of the vessel is fully developed and the flow re-arrangement in the top and bottom should be ignored in the calculation. You are not required to solve all equations to get the final expression for maximum shear stress. Instead, you are asked to answer the following questions: (i) List all plausible flow assumptions and state the assumed function of the velocity profile in the annular region and check for consistency with the continuity equation. Hint: similar flow properties and assumptions were used in many in-class examples. (15pts) (ii) Clearly state and simplify all relevant equations of motion including the expression for the velocity profile and maximum shear stress (show non-zero terms only). Hint: you will find that three boundary conditions are needed to solve the three unknown constants (15 pts) (iii) List all required boundary conditions based on the number of variables needed to be solved from (ii). Two BCs should be straightforward and the 3rd one is more subtle. Hint: volumetric flow rate in an annular region could be expressed as: Q= 2RpRvzrdr.vz is the flow velocity in the z direction, heading upward (10 pts) (iv) Sketch the flow pattern in the annular region as realistically as possible, indicating the place or places where you think the maximum shear stress is located (10 pts) Airlift fermentor system (Fig. A) Flow pattern in the fermentor (Fig. B)