Matlab代寫 - Math 370 – Mathematical Modeling

時間：2020-12-09

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1. Determine whether the following equation is dimensionally correct? (10 points)
, where
ρ u= ????? ?????wvu ?= ??????? ????
????????
????
????????
????
??????????????? t p ???????? f
Density 3D Velocity Vector gradient time pressure Stress tensor (pressure) force
2. Dimensional Analysis: Suppose V , the speed of a liquid
flowing through a horizontal pipe, depends on the pressure
drop ?P , the length of the pipe l , the diameter D , density
of the liquid ρ , and the viscosity of the liquid μ . Show
that V can be modeled using the Buckingham Π theorem
as
( , ). 2 2 μρ ρ μ P D lD D V ? = ?Φ (10 points)
3. Similitude: Suppose the torque τ (N m) required to rotate a disk of diameter D (cm) submersed in a
fluid with density ρ (kg/m3
), and the bulk viscosity ν (kg/m/s) at the rotation rate ω (Radians/s) can
be modeled, using the Buckingham ∏-theorem as,
( ) 2 5 2 ν ω τ ρω D = D ?Φ [Do not show this!!].
If a scaled model, 50 times smaller, experiences a 5 N-m torque in water with the rotation rate
10 sec
3π rad
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 ????2 ≥ 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)
5) Prescribing Drug Dosage: Doctors may advise some patients to take a low dose of aspirin daily to
prevent blood clots and reduce the risk of stroke and heart attack. Suppose the concentration of a medication
administered in a patient follows the exponential decay model C(t) = C0e ?kt , where C is the concentration ,
C0 is the initial dosage, k is the absorbing rate, and t is time.
a) Derive the concentration residual in the body after administering n consecutive dosages. (10 points)
b) Suppose aspirin is absorbed in the body at the rate k=0.075/hr and each dosage produces 2.5 mg/dl of
aspirin concentration in the body. With such prescription, what is the highest aspirin concentration in the
body? (2pts)
c) How many days does it take to get the aspirin concentration to reach 90% of the highest level? (3pts)
6) Given the differential equation ????????
???????? = (3 ? ????)(1 ? ????)2.
a) Find the equilibria (3 points)
b) Determine the stability at the equilibria (3 points),
c) Draw the solution trajectories in phase space (4 points)
d) Start with u(0)=0.99 and use the Euler method with ????? = 0.1 to get and graph u(t) for the first 5
seconds. (5 points)
7) For each first order system of differential equations, find its equilibrium point(s), draw the phase space near
the equilibrium points, and describe the corresponding stability.
a)
x y dt
dy
y x
dt
dx
(3 2 ) (4 ) = ? = ? (10 points)
b)
x y dt
dy
y x
dt
dx
( 3 2 ) (4 ) = ? + = ? (10 points)

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