Download A thin layer of an incompressible fluid flows steadily over a

Name: ------------------
Quiz: No. 9
Time: 15 minutes
ME:5160, Fall 2015
Fluids-ID:---------------------------------------------------------------------------------------------------------------------------------------------------------The exam is closed book and closed notes.
A thin layer of an incompressible fluid flows steadily over a horizontal smooth plate as shown in
the Figure. The fluid surface is open to the atmosphere, and an obstruction having a square cross
section is placed on the plate as shown. The dimensional variables for this problem are V, g, d, ρ,
μ, and surface tension σ. Taking V, d, and ρ as repeating variables, one can find that the
dimensionless parameters are Reynolds, Froude, and Weber numbers. (a) Write down the
Reynolds and Froude numbers (or find if don’t have them memorized) and use Pi theory to find
out the equation for Weber number. (b) A model with a length scale of ¼ and a fluid density scale
of 2.0 is to be designed to study the mean velocity V at upstream. Find the scale ratios for velocity,
viscosity, and surface tension.
Name: ------------------
Quiz: No. 9
Time: 15 minutes
(2)
ME:5160, Fall 2015
Fluids-ID:----------------------------------------------------------------------------------------------------------------------------------------------------------
Solution:
KNOWN: dimensional parameters,
(1)
FIND: Weber number,
𝑉𝑉𝑚𝑚 𝜇𝜇𝑚𝑚 𝜎𝜎𝑚𝑚
𝑉𝑉𝑝𝑝
,
𝜇𝜇𝑝𝑝
,
𝑑𝑑𝑝𝑝
𝑑𝑑𝑚𝑚
= 4,
𝜌𝜌𝑝𝑝
𝜌𝜌𝑚𝑚
=
1
2
𝜎𝜎𝑝𝑝
ASSUMPTIONS: the problem is only a function of the above dimensional variables
ANALYSIS:
(a)
𝜎𝜎 = 𝑓𝑓𝑓𝑓𝑓𝑓(𝑑𝑑, 𝐿𝐿, 𝑉𝑉, 𝜌𝜌, 𝜇𝜇)
𝑛𝑛 = 6
𝑗𝑗 = 3
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 = 𝑉𝑉, 𝑑𝑑, 𝜌𝜌
𝑘𝑘 = 𝑛𝑛 − 𝑗𝑗 = 3
𝜌𝜌𝜌𝜌𝜌𝜌
𝜋𝜋1 = 𝑅𝑅𝑅𝑅 =
𝜇𝜇
𝜋𝜋2 = 𝐹𝐹𝐹𝐹 =
(1)
𝑉𝑉
(1)
�𝑑𝑑𝑑𝑑
𝜋𝜋3 = 𝜎𝜎𝑉𝑉 𝑎𝑎 𝑑𝑑 𝑏𝑏 𝜌𝜌𝑐𝑐 = {(𝑀𝑀𝑇𝑇 −2 )(𝐿𝐿𝑇𝑇 −1 )𝑎𝑎 (𝐿𝐿)𝑏𝑏 (𝑀𝑀𝐿𝐿−3 )𝑐𝑐 } = {𝑀𝑀0 𝐿𝐿0 𝑇𝑇 0 }
𝑎𝑎 = −2; 𝑏𝑏 = −1; 𝑐𝑐 = −1
𝜋𝜋3 =
𝜎𝜎 (0.5)
𝜌𝜌𝑉𝑉 2 𝑑𝑑
(0.5)
Name: ------------------
Quiz: No. 9
Time: 15 minutes
ME:5160, Fall 2015
Fluids-ID:----------------------------------------------------------------------------------------------------------------------------------------------------------
(b)
𝐹𝐹𝐹𝐹𝑚𝑚 = 𝐹𝐹𝐹𝐹𝑝𝑝 →
𝑉𝑉𝑚𝑚
�𝑑𝑑𝑚𝑚 𝑔𝑔𝑚𝑚
𝑔𝑔𝑚𝑚 = 𝑔𝑔𝑝𝑝 ;
=
𝑉𝑉𝑝𝑝
(0.5)
�𝑑𝑑𝑝𝑝 𝑔𝑔𝑝𝑝
𝑑𝑑𝑚𝑚 1
=
𝑑𝑑𝑝𝑝 4
𝑉𝑉𝑚𝑚
𝑑𝑑𝑚𝑚 1
=�
=
𝑉𝑉𝑝𝑝
𝑑𝑑𝑝𝑝 2
(0.5)
(0.5)
𝜌𝜌𝑚𝑚 𝑉𝑉𝑚𝑚 𝑑𝑑𝑚𝑚 𝜌𝜌𝑝𝑝 𝑉𝑉𝑝𝑝 𝑑𝑑𝑝𝑝
𝑅𝑅𝑅𝑅𝑚𝑚 = 𝑅𝑅𝑅𝑅𝑝𝑝 →
=
𝜇𝜇𝑚𝑚
𝜇𝜇𝑝𝑝
(0.5)
𝜌𝜌𝑚𝑚
(0.5)
=2
𝜌𝜌𝑝𝑝
𝜇𝜇𝑚𝑚 𝑉𝑉𝑚𝑚 𝑑𝑑𝑚𝑚 𝜌𝜌𝑚𝑚 1 1
1
=
×
×
= × ×2=
𝜇𝜇𝑝𝑝
𝑉𝑉𝑝𝑝 𝑑𝑑𝑝𝑝 𝜌𝜌𝑝𝑝 2 4
4
𝜌𝜌𝑚𝑚 𝑉𝑉𝑚𝑚 2 𝑑𝑑𝑚𝑚 𝜌𝜌𝑝𝑝 𝑉𝑉𝑝𝑝 2 𝑑𝑑𝑝𝑝
=
𝑊𝑊𝑊𝑊𝑚𝑚 = 𝑊𝑊𝑊𝑊𝑝𝑝 →
𝜎𝜎𝑚𝑚
𝜎𝜎𝑝𝑝
2
(0.5)
(0.5)
𝜎𝜎𝑚𝑚
𝑉𝑉𝑚𝑚
𝑑𝑑𝑚𝑚 𝜌𝜌𝑚𝑚 1 1
1
=� � ×
×
= × ×2=
𝜎𝜎𝑝𝑝
𝑉𝑉𝑝𝑝
𝑑𝑑𝑝𝑝 𝜌𝜌𝑝𝑝 4 4
8
(0.5)