Math 370 – Mathematical Modeling

1.确定以下等式在尺寸上是否正确？ （10分）
2.尺寸分析：假设V，液体的速度

V可以用白金汉Π定理建模

（）。 2
2
μ
ρ
ρ

d
d
V ∆ =⋅Φ（10分）
3.相似性：假设旋转直径为D（cm）的圆盘浸入水中时所需的扭矩τ（N m）。

），并且在转速ω（弧度/ s）下的堆积粘度ν（kg / m / s）可以

If a scaled model, 50 times smaller, experiences a 5 N-m torque in water with the rotation rate
10 sec
ω = and
the real-sized model is tested in oil. Assuming the two models are similar and water kinematic viscosity = 1.2
oil kinematic viscosity and water density = 3 oil density,
a) What is the rotation rate for the real-size model? (5 points)
b) At that rotation rate, what is the real-sized torque? (5 points)
4) Modeling with data: Given the data set populations versus the mean velocities over a 50-foot course for
15 locations on Table 4.5 of page138, find two non-polynomial models that can describe the relation between
P and V with 𝑅𝑅2 ≥ 92%. For each model, provided the mathematical equation with the parameters specified
along with the graph. You must include the R-square values. (10 pts for each model)

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