Question

Question 24 of Industrial Mathematics: Case Studies in the Diffusion of Heat and Matter

24. The thrust (force) T developed by a ship's propeller in deep water depends on the radius a of the propeller, the number of revolutions per minute n, the velocity V with which the ship advances, the gravitational constant g and the density p of the water. Show that T pa2v2 = 0 (en ca) cene faria o la parte **) for some function of two variables. (Hint: First argue (n) = T.)

Answer 1

The dimension of independent quantities are as follows [a] = 4 [n] = T-1 [V] = LT-1 [8] = LT-2 [p] = ML-3 and, the dimension of dependent quantity is [T] = ' MLT-2 Since I = f(a, n, V, 8, P), the list of relevant parameters are 6, i.e. n = 6.

As a first guess , j = 3, the number of primory dimensions represented in the problem (m, L, and T). If this value of j is conect, the expected number of II's are k = n - - 6 - 3 = 3 Now, choose the three repeating parameters since j=3. The 3 repeating parameters are a, V, and p. Dependent To : 1T, = Tauly Psp t's MOL°T° = (MLT-2)/Ljas (LT-4) P(M1-3)*; = MA*** _ 4+ajt B=3", -2-84 on comparison Y = -1, = -2, 2,= 3 8-B-1 = -3 +2-1 = -2 So, 1 - Ta-? V? pot - I pa²v2 Likewise, first independent I, say IT, Mg = nadzubz plz Mº L°T° = (1-1)(L) 42 (17-1) P2 ( ML-3) "z = MY -3 Y3 +P2 + 22 --P

So, z = 0 , ß, =-1, and Q2 = 372-B, - 1 Tie. TT, = na?vpº - na Likewise, second independent TT, say 12 Iz = gads vP pro MOL° Tº = (LT-2)(L) as (LT-4)B> ( M1-3; ¥3 1 + 03 +33 - 373 1 -2- B3 so, 8z = 0, B3 = -2, 83 = 383 –B3-1 = 1 TT, = ga' v-a pº = ag 12 The relationship between II's is given as TT. = t[ TTg, TT ) on porn flour