Just like us, u_2 can do it!

This week we focused on deriving the velocity function u which we had realized was made of two components u=(u1,u2) . As we had learned before the first component u1=0 but we then figured out that the second component which began as a double integral was truly a single integral with an integrand with a natural log. This is what the second component of the velocity field looked like u2=Log{[Cosh(x-L)-Cos(y-t)]/[Cosh(x+L)-Cos(y-t)]}dt  and we attempted to solve the integral by using multiple methods of integration, and with no success, we decided to plot the integrand to gain an idea of the integral value. Fixing x and L and manipulating in y, we found that the Cos(y-t)term in the integrand did not have as large an effect as the Cosh(x-L)term. Simplifying the integrand even further with the expanded form of hyperbolic cosine, we were then able to analyze the behavior of the integrand for certain values of x within the bound [-L, L]. Using the range of these values we were able to create a piecewise function, u2(x)= {2L*2π, x <=-L}, {-2x*2π, -L< x < L}, {-2L*2π, x >=L}. Together, we found the vector plot of the velocity function which tells us that not only is our solution for the strip unique, it is also a steady state solution.

 

Published by isusa

Student?