fluid dynamics Please do part 5 part 4 is given below here t

fluid dynamics

Please do part 5. part 4 is given below.

here the answer to part 1

Problem 3 The cone-and-plate bioreactor is a device used in biomedical research to expose cells to a well- controlled shear stress environment (see figure below). The system consists of an inverted cone (radius R, angle a) rotating at angular speed do on top of a flat stationary plate. The gap between the cone and the plate is filled with fluid (density p, viscosity Au). The distance between the cone apex and the plate is h p, Al a h(r) We wish to determine the shear stress produced by this device on the surface of the stationary plate. 1- Using the Navier Stokes and continuity equations worksheet in cylindrical coordinates, simplify the equations using the following assumptions steady flow (1), purely circumferential flow (2), and rotationally symmetric (axisymmetric) flow (3). Assume further no body forces (4) and no imposed pressure gradient in the tangential direction (5). 2- In this question, we wish to simplify the reduced form of the Navier Stoles equation along 6 by comparing the order of magnitude of its different terms. Given the geometry of the problem, the following scales can be constructed: r. ve* Roo where all star quantities are dimensionless and of order 1. Express the derivatives ajar, ajat and aa/a-2 in terms of the scaled variables r and z* Rewrite the reduced form of the Navier Stokes equation along 6 in terms of the scaled Show that if the liquid layer is very thin (ho R), the reduced and v quantities r form of the equation becomes 0. 3- Integrate the equation obtained in 2) and express the tangential velocity component in terms of two constants of integration and C

Solution

The solution given for part 1 is erroneous: it assumes viscosity is zero which is not so here. There are viscosity terms also.

5) The answer is given already as d²V_theta/dz² =0

(in reduced coordinates), integrate twice Vtheta =

AZ +B

apply BC at Z=0, no slip ==> Vtheta =0, so B=0,

At z=h Vtheta = Romega A

Soln Vtheta/Romega = AZ/ho