Question

Please write clearly and answer all parts using MATLAB when asked.

The convective heat transfer problem of cold oil (Pr > 10) flowing over a hot surface can be described by the following second-order ordinary differential equations. d^2 T/dx^2 + Pr/2 (0.332/2 x^2) dT/dx = 0 where T is the dimensionless temperature, x is the dimensionless similarity variable, and Pr is called Prandtl number, a dimensionless group that represents the fluid thermos-fluid properties. For oils, Pr = 10 - 1000, depending on the kinds of oils. This problem is a boundary-value problem with the two conditions given on the wall (x = 0) and in the fluid far away from the wall (e.g., x = 5). You are asked to use the shooting method to solve this problem by employing the MATLAB functions such as ode45 and fzero: 4.1 Setup a function file, [dy] = dTdx(x, T, Pr), so you can solve this second-order differential by solve two first-order ordinary differential equations. Note that Pr is a parameter that can be assigned any values. 4.2 Convert the present boundary-value problem to a root-finding problem that is solved with MATLAB fzero. 4.3 Plot the variation of dimensionless temperature T as a function of x for four values of Pr: 10, 100, 200, 500.

Answer 1

syms y(t)
Pr = 100 % assuming as hot body so 100, it can vary from (10 - 1000)

[Vector] = odeToVectorField(diff(y, 2) == -(Pr/2)*((0.332/2)*x*x)*diff(y) % declaring equation
M = matlabFunction(Vector,'vars', {'t','Y'})
answer = ode45(M,[0 500],[2 10]);
fplot(@(x)deval(answer,x,1), [0, 50])

Note: odeToVectorField and matlabFunction are matlab built in libraries.