Question

(25pts) In studying sand transport by ocean waves, it was postulated that the threshold wave-induced bottom shear stress t required to move particles depends on gravity g, particle size d, particle density Pp , water viscosity u and water density p. Find all the dimensionless, i.e., pi groups for this problem. Consider p, g, d as repeating variables. Write all the quantities in terms of the primary dimensions, i.e. M, L, T. For example, the dimension of viscosity u is MLT t=f(g, d, Pp, u, p)

Answer 1

dimension for :

= [ML^-1T^-2]

7

g = [LT^-2]

d = [L]

$\rho$ = $\rho$ _P = [ML^-3]

$\mu$ = [ML^-1T^-1]

T-f(a, g, d, Sp 18) Total no. of variables -n- 6 No. of fundamental dimensions used, m-3 CM,L,t) No. of dimensionless T-terms are nam 6-3 -3 Let these terms are to Ta and Tz. No. of repeating variables m given that to use particle size, gravity and water- denility as repeating verlattes. Put the dimensions of each variable- [MPL* +9 - [4 Cht-27Cm() @Cmeta] [M] C*' [LJA+R-36-'[+] 28-2 compare the powers of fundamental dimension of RoHS & L.H.S.

-28-20 Cti C--1 A+B-3C-I- A-1+3-1O B = -( A-1 so, て fdg Sesond T-term- Ty da go zelu but the dimensions- [Mex") - [LC] [May °CML-**-] -(mft' WA+B-34(+3-20-1 .-28-10 A+B-30-40 B-1/2 A-4+342 (A-2 cho 4 So, Id Idg As Third I-term- T3 = [MOL°°) - [L]^ [lt-2] [Mc-a). [ML-2] [Mg C*' [Ly9+B-3C-3 [+]-28 SO) Pe ctino -26 20 773 BO A+B-36-30 A+ 0+3-3 -6 Aso