Differential Equations and Analyzing Fluid Flow Assignment

Assignment Task

Submit a single pdf file consisting of a scan of your neatly handwritten solutions (solutions may also be typed or written electronically using a tablet). Include full working for all solutions; you must demonstrate that you can find your solutions without the aid of a calculator, computer algebra system or similar (e.g. Wolfram Alpha).

For MATLAB, you should include both your script and the plots created. It is recommended you do so by publishing your script as a pdf and combining this pdf with your working

Question 1

(a) Find the maximum velocity by doing the following:

i) find all the critical points of the function v(x, y)

ii) identify the single critical point that is within the cross section area A depicted above (including, possibly, on the boundary).

iii) show that this critical point is a local maximum using the Hessian determinant test,

iv) evaluate the value of v at this maximum

(b) The flow rate of water Q (metres3/second) through the channel is defined as the integral

where A is the cross-section depicted in the figure above. For h = 1, calculate the flow rate by doing the following:

• describe the fluid region A mathematically, with x as the outer variable and y as the inner variable,
•  Set up and evaluate the double integral. If you want, for fun (but no bonus marks) you can try to find the flux for general depth h

(c) Use MATLAB to create a contour plot of the velocity v(x, y) for h = 1. Make sure to include a few (three or more) contours (level curves) between zero and the maximum value you found in Q1a.

Question 2

(a) Assume the resistance is constant: R = 100. Check by direct substitution that the function VC (t) = 9(1 – e^−10t ) satisfies the ODE and the initial condition.

(b) Now assume that the resistor slowly degrades, so that its resistance increases with time:

R = R(t) = 100 + 2t. Solve the ordinary differential equation for VC (t) with this resistance, using either separation of variables, or the integrating factor method. Make sure to also include the initial condition.

(c) use MATLAB to create a plot of both solutions from Q2a and Q2b (in the same figure). You can either use the explicit solutions that come from solving the ODEs, or use ode45 to generate numerical solutions to the ODEs. Make sure you have chosen an interval to plot over so that the behaviour of the solutions can be clearly seen.

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