Problem 51087. Solve an ODE: equation for a 2D laminar jet

In the solution for a two-dimensional laminar jet, the following nonlinear ordinary differential equation arises
f’’’ + 2(f’2+ff”) = 0
where primes denote differentiation with respect to the independent variable η. The problem has the conditions f(0) = f''(0) = 0 and f'(oo) = 0 and the constraint
integral of f'^2 from -infinity to infinity = a
where a is a constant.
Write a function to solve this problem—that is, return values of f at specified values of η. The test suite allows MATLAB’s functions for numerical solution of ODEs, but the equation has a relatively simple analytical solution. If you would like a hint for the analytical solution, execute this command:
The physical problem involves a jet in the x, y plane emanating from a source at (x,y) = (0,0). It can be solved by expressing the velocities u and v in terms of a streamfunction and employing a similarity solution that combines the spatial coordinates into a single variable η. In the problem above, f(eta) is proportional to the streamfunction. The conditions at eta = 0 define the line y = 0 as a line of symmetry; the transverse velocity v and the shear stress are zero there. The condition f'(oo) = 0 states that the streamwise velocity u is zero far from the source. The integral constraint states that the momentum flux of the jet is constant. The value a = 4/3 corresponds to the physical problem of the jet. Other values are included for variety.

Solution Stats

66.67% Correct | 33.33% Incorrect
Last Solution submitted on Apr 28, 2022

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