Question

Q3. Drag measurements were taken for a 5 cm diameter sphere in water at 20°C to predict the drag force of a 1 m diameter balloon rising in air with standard temperature and pressure. Given kinematic viscosity of water (v) = 10 X 106 m²/s and kinematic viscosity of air (v) = 1.46 X 10-5 m2/s. 1. Perform the Buckingham Pi theorem to generate a relationship for Fo as a function of the independent variables. Assume the drag Fo is a function of diameter (d), velocity (V), fluid density () and kinematic viscosity (V) 2. Determine the sphere velocity if the balloon was rising at 3 m/s (50 Marks)

Answer 1

solution Cinen Fo is the function of independent variables o, vit, & where fo: Drag force, Va velouty -le density v kinematic viscosity d = diameter Mathematically Fo=fld, V, fov) be written as Lobis filio, d, V, P, v) = 0 ou it con 7:5 Total number of voulable Alumber of Bundamental olimensions on=3 fm is obtained by couiting dimension of e cich varable as Fo = MLT-2 VELI-2 V = 421-1 DEL P: ML-3 Thus all fundamental dimensjovy are M, Lond I Hence m= - 3] NO. of dimensionlegg 7 teims - n-m = 5-3 Thus a Ta terme , say T, and I are formed Hence f, cald) = 0 mt 1 vanables each T term contains Repeating Variables 1/4

choosing_d_kend pas repeating axloble We have C, ED TE d" vbre Ta = d az vb. p. (2. v. .-5-2 Cit1 First teim Substituting dimensione on both side we get MOLOTO : [ d[LT 1]bEML3JC MLT-2 MOLOTO = 9, + bi-39, +4 Equating powers C+1=0 91761 - 36+1=0 Si=-1 a = -2 by=-2 2=0 substituting a, b, c, in Ay - of? 12.6-750 지 1 Second a term Subcituting cimenting on both side MOLOTO [lr-4]b2 [MC3][13) MOLOTO M²2 192+52-36, +2 Equating powers. -4 C2=0 C₂ = 0 sor a2+ b2-362+2=0 92=-1 -b2-2=0 b2=-2 2/4

A2 = d 1-1 you T2 = dov from Equation and @ 7, = fo & T2 = 0 d²u2p div Sube Htuting I, and fi in equation A fo ful 1 -o dov or PER Ans مدي 2 which is the creation of drag force fo with Velocity, diameter, viscosity and density. 3/4

Tao NO.: Date: γουν (6) Given, diameter of_sphere d=5cm = 0.05m diameter of ballon D=1m. kinematic viscosity of water z Vo = 1.0x10mts Kinematic viacority of air V = 1.464 165 m/ sphere velocity Viets? bet Velocity of ballon V= 3 m/s dlo.05 K 2 D=4m Vm= 2 V=3mis Sphere modes in water ballon model in air Dimensional Onalysis D PV202 bf va V Similarity requirement VO Ind Vw. Vm= May (8)xx (1.0x10-6 3 xóms)x 1.46 x 10-5 E V =4 tsats 4-1096 mle Velocity of sphere jp #samo 4011 mls 4/4