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

Problem
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:
char('Zxj%ymj%uwtizhy%wzqj%tk%inkkjwjsynfynts%y|t%ynrjx%fsi%nsyjlwfyj%ymwjj%ynrjx3'-5)
Background
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

Problem Comments

Solution Comments

Show comments

Problem Recent Solvers2

Suggested Problems

More from this Author261

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!