Question 13 A fluid is flowing in-between two cylinders. The inner cylinder radius is ri and the outer cylinder radius is r2. The outer cylinder is stagnant while the inner cylinder is moving at a steady velocity of up. a) Develop the expression for the velocity using the relevant Navier-Stokes equation assuming that the pressure drop per unit length is negligible. b) If the inner cylinder is moving steadily at 2.5 m/s, what force is applied to cylinder (inner) in order to attain this steady velocity? Assume the viscosity of the fluid is 0.1 Pa.s. The diameter of the inner cylinder is 4 cm while that of the outer cylinder is 9 cm, and the length of the cylinder is 2 m. Useful Equations: F = tw X Aw Tw=-X dr re Navier-Stokes Equations: In rectangular co-ordinates , X-component au cu +u, Al Ou: +4 +U at ax Oy OP (au, au, au +44 du, ax ax @ya +pg dz cu ou Share y-component + ou, ay + + Ou Oz =-+ Oy ( ou ou ou au, Oz? + + Pg at ax ar? Oy Ou , ou au au . ar 2-component au + + , +u ax + + role + P8 ay az Oz 2 + +u In cylindrical co-ordinates Ou, Ou, u, ou, u, ou, u? Ou, ap + + Il ar ar r de r Oz or r-component a la 1 ou 2 ou ou 41 (ru) + + or rar rara Oz? r + Hlet (.) + + Pg Question 13 A fluid is flowing in-between two cylinders. The inner cylinder radius is fi and the outer cylinder radius is r2. The outer cylinder is stagnant while the inner cylinder is moving at a steady velocity of up. a) Develop the expression for the velocity using the relevant Navier-Stokes equation assuming that the pressure drop per unit length is negligible. b) If the inner cylinder is moving steadily at 2.5 m/s, what force is applied to cylinder (inner) in order to attain this steady velocity? Assume the viscosity of the fluid is 0.1 Pa.s. The diameter of the inner cylinder is 4 cm while that of the outer cylinder is 9 cm, and the length of the cylinder is 2 m. Useful Equations: F = tw X Aw du Tw = -ux dr/rea Navier-Stokes Equations: In rectangular co-ordinates , X-component Ou + at . ax +u , + ay = OP +u ax (ou ou ou 1+ Pg Ox Oy Oz? + + dz au . ou au Y-component + + audu au +4 Oy ar? Oz? + Pg at ax @y az Oy? here hotele ++) au ap Z-component + u au ax +u au +1 @y , Oz au a'u au ar? Pg at In cylindrical co-ordinates . , + ot u, u, ou, ug ap +11 au, az + rao =- ar r ar r-component (2017 a 1 ou 2 ou du + ru + + Pg