Answer: π= G[√(u.V/W)]
STEP 1
Given parameters:
Power Input W= FL/T,
Absolute Viscosity u= FT/L²
Basin volume V= V/L³
Velocity gradient G= V/L³
STEP 2
We start by expressing the velocity gradient G as a function of W, u, V
G= G(W,u,V)
To get the pii terms, we use the dimension number formula n=k - r
where n and k are natural numbers representing number of fundamental dimensions and variable present respectively.
n= 4-3=1
STEP 3:
We expressed the pii terms as
π= G.W^a.u^b.V^c
The three fundamental F L T
We can write as
Fⁿ.Lⁿ.Tⁿ= 1/T. (FL/T)^a.(FT/L²)^b.(L³)
Using the exponential rule and by comparing coefficient on both sides;
Fⁿ.Lⁿ.Tⁿ= F^a+b. L^a-2b+3c. T^-a+b-1
Fⁿ= F^a+b = a+b= 0..............I
Lⁿ= L^a-2b+3c=0 = a-2b+3c=0...........ii
Tⁿ=L^-a+b-1=0. -a+b-1=0............iii
From the above equations we have,
a+b =0: b=-a...........iv
putting eq. iv into iii , we have
-a-a-1=0: -2a-1=0: a= -1/2
substituting the above value of a into eq iv, we have
b= 1/2
substituting the value of b above into eq 2, we have,
-1/2-2(1/2)+3c=0
c=1/2.
Lastly, from the pii terms given above we can obtain dimensionless relationship,
π=G(W^-1/2.u^1/2.V^1/2)
We can write this as
π= G[ √1/W.√u. √1/2] = G[(√u.V/√W)] or G[√(u.V/W)].... final answer.
