Name :_________________ Quiz: No. 6 Time: 15 minutes Student ID# : ____________ Course: ME 5160, Fall 2021
--- The exam is closed book and closed notes.
When a fluid flows slowly past a vertical plate of height 𝒉 and width 𝒃, pressure develops on the face of the plate. Assume that the pressure,
𝒑, at the midpoint of the plate is a function of plateheight and width, the approach velocity
𝑽, and the fluid viscosity 𝝁 and fluid density 𝝆. Makeuse of dimensional analysis to determine how the pressure,
𝒑, will change when the fluidvelocity, 𝑽, is doubled.
Hint: find the relation of 𝜋0= 𝑓(𝜋1, 𝜋2)
If 𝜋1 𝑎𝑛𝑑 𝜋2 are remain as same values, 𝜋0 also must be same regardless of any condition changing.
Name :_________________ Quiz: No. 6 Time: 15 minutes Student ID# : ____________ Course: ME 5160, Fall 2021
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Solution:
KNOWN: dimensional parameters FIND: Pi groups
ASSUMPTIONS: the problem is only a function of the given dimensional variables ANALYSIS:
𝑝 = 𝑓(ℎ, 𝑏, 𝑉, 𝜇, ρ) 𝑛 = 6
𝑝 = {𝑀𝐿
−1𝑇
−2}; ℎ = {𝐿}; 𝑏 = {𝐿};
𝑉 = {𝐿𝑇
−1}; 𝜇 = {𝑀𝐿
−1𝑇
−1}; 𝜌 = {𝑀𝐿
−3};
𝑗 = 3 → 𝑘 = 𝑛 − 𝑗 = 6 − 3 = 3
The repeating variables are 𝑏, 𝑉, 𝜌 adding each remaining variable in turn, we find the Pi groups:
Π
0= 𝑏
𝑎𝑉
𝑏𝜌
𝑐𝑝 = {(𝐿)
𝑎(𝐿𝑇
−1)
𝑏(𝑀𝐿
−3)
𝑐(𝑀𝐿
−1𝑇
−2)} = {𝑀
0𝐿
0𝑇
0} 𝑎 = 0, 𝑏 = −2, 𝑐 = −1
Π
0= 𝑝 𝑉
2𝜌
Π
1= 𝑏
𝑎𝑉
𝑏𝜌
𝑐ℎ = {(𝐿)
𝑎(𝐿𝑇
−1)
𝑏(𝑀𝐿
−3)
𝑐(𝐿)} = {𝑀
0𝐿
0𝑇
0} 𝑎 = −1, 𝑏 = 0, 𝑐 = 0
Π
1= ℎ 𝑏
Π
2= 𝑏
𝑎𝑉
𝑏𝜌
𝑐𝜇 = {(𝐿)
𝑎(𝐿𝑇
−1)
𝑏(𝑀𝐿
−3)
𝑐(𝑀𝐿
−1𝑇
−1)} = {𝑀
0𝐿
0𝑇
0} 𝑎 = −1, 𝑏 = −1, 𝑐 = −1
Π
3= 𝜇
𝜌𝑏𝑉 → 𝜌𝑉𝑏
𝜇 = 𝑅𝑒
(1)
(0.5) (1)
(1)
(1) (0.5)
(0.5)
(0.5) (1)
Name :_________________ Quiz: No. 6 Time: 15 minutes Student ID# : ____________ Course: ME 5160, Fall 2021
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Thus the arrangement of the dimensionless variables is:
Π
0= 𝑓( Π
1, Π
2)
𝑝
𝑉
2𝜌 = 𝑓 ( ℎ
𝑏 , 𝜌𝑉𝑏 𝜇 )
𝑝 = 𝜌𝑉
2× 𝑓 ( ℎ
𝑏 , 𝜌𝑉𝑏 𝜇 )
So, if the fluid velocity 𝑉 is doubled, the pressure incresed by 4 times
(2)
(1)
