1. The effect of temperature on viscosity can be modelled using the so-called Andrade equation
μ = DeB/T (1)
where μ is the dynamic viscosity (measured in Ns/m2), T if the absolute temperature (measured in K) and D and B are parameters. In order to estimate the parameters D and B, the measurements of the following table will be used.
|
T [K]
|
0.1
|
5
|
10
|
20
|
30
|
40
|
|
μ [Ns/m2]
|
1.787
|
1.519
|
1.307
|
1.002
|
0.7975
|
0.6529
|
a) Write the necessary Matlab functions and/or scripts to estimate the parameters D and B of the Andrade equation.
b) In the report, answer to the following questions:
i. Describe in detail the strategy followed to solve the problem. Do not describe the code or show any code in the report, just the ideas that you followed to solve the problem in a clear and concise manner.
ii. Which interpolation or approximation method do you recommend to use? Why?
iii. Produce an approximated value of the dynamic viscosity for a fluid with temperature T = 80 K using your Matlab program/s.
2. Bernoulli's equation for fluid flow in an open channel with a small bump is
Q2/2gb2h02 + h0 = Q2/2gb2h2 + h + H
where Q is the volume rate of flow, g is the gravitational acceleration, b is the width of the channel, h0 is the upstream water level, H is the height of the bump and h is the water level about the bump.
a) Write the necessary Matlab functions and/or scripts to obtain the water level about the bump h for a volume rate flow of Q = 1.2m3/s assuming g = 9.81m/s2, b= 1.8m, h0 = 0.6m and H = 0.0075m.
b) In the report, answer to the following questions:
i. Describe in detail the strategy followed to solve the problem. Do not describe the code or show any code in the report, just the ideas that you followed to solve the problem in a clear and concise manner.
ii. Which numerical method do you recommend to use? Why?
3. The equations describing the natural convection along a heated vertical plate in contact with a cooler fluid is given by the following coupled system of high-order differential equations
d3f/dz3 + 3f(d2f/dz2) - 2(df/dz)2 + T = 0, z ∈ [0, 8] (2)
d2T/dz2 + 3f Pr (dT/dz) = 0, z ∈ [0, 8] (3)
where f is the stream function, T is the temperature and Pr = 0.7 is the Prandtl number. The initial conditions are
f (0) = 0, df/dz(0) = 0, d2f/dz2(0) = 0.68 (4)
T(0) = 1, dT/dz = - 0.50 (5)
a) Write the necessary Matlab functions and/or scripts to solve the system of the stream function and temperature in the interval z ∈ [0, 8].
b) In the report, answer to the following questions:
i. Write in detail all steps required to transform the system of two high-order differential equations to a system of five first-order differential equations.
ii. Display a single Matlab figure with the stream function and the temperature.
iii. Explain in one paragraph the results obtained. What evidence do you have to believe that the results are correct?