Question

3- Through a dimensional analysis, express the following equation for a bubble in a dimensionless form, and show which dimensionless parameters will be finally produced. (Use the characteristic velocity U, and length L for nondimensionalizing procedure.) Pb(t) - po(t) dR 3 R 2 4V dR 2S + P2+- d+2 PL Rdt PLR PL is the density of the surrounding liquid of the bubble R is the radius of the bubble t is the time U is the kinematic viscosity of the surrounding liquid S is the surface tension of the bubble-liquid interface Pb(t) is the pressure within the bubble, and Po(t) is the external pressure infinitely far from the bubble.

Answer 1

solution : - Given thail- characteristic velocily = 40 hength =h chave 'erishe time for bubble => 9 = Ap. 2h - Bolt) - Poolt) ] AR 4 -=[PB (t)- Po(t) Ja Les) @= @'; 4x4o st' are the non- dimentione O alised quantity, Given Equation, Po let) - Pro W)_QD4 + 3 ( ! ) og > Pald-Ralt) – [ Bue)-fulda) en die Pc PL 4. PLR

=) PB (t) - Polt) - R2 LIR! 3.1 ir -> Pg Ct) – Polt) Q PLUOK Here y Re dr? der RI DAT Re" = Luo = Reynolds number and, Polt) – Polt) dimensionales as Pro has ahvor the same unit as pressure 3. The Equation in dimension less forcy becomes | Poste -Heality La final Ans.