This week started off with brainstorming questions to see where we wanted to go with the solution that we reviewed that was published by Denisov and Beichman. Two of the questions that we began to ask were how does time become a factor in the stability of the solution and how much do we want to alter the boundaries of the patch. For the remainder of the week our team decided to alter the boundaries of the patch a few different ways. We started off with making the boundary of the solution really wiggle and made sure to have the left-most boundary -L+(1-Cos(ωy)) and the right-most boundary L+(1-Cos(ωy)). Omega is the angular frequency of the function and the very first alteration made sure to make ω=1. In this case, when analyzing the integral to attempt in finding parts of the velocity field u=(u1,u2), we were happy to come to the conclusion that u1 was 0. This great realization was great until we had realized that when looking at the integrand and plotting it, there was dependency on both x and y, since then it has been a challenge in finding patterns for which we can use to find u2.
After figuring that the previous attempt was a challenge we moved on to altering the boundaries of the path differently. We took a look at altering the boundary where instead of it oscillating too much, it would just be one bump. Therefore, we decided to make the boundaries -L+A(y^2+2πy) and L+A(y^2+2πy), where the amplitude, A, we looked at was ¼. Finally, we ran into challenges with this one as well and one next step we took was to look at a profile that is an even extension of a linear function of y. While we were juggling between profiles, Jenn let us know that we had to account for a variable very important in convolution. Therefore, next week we will take a look at the profiles to make sure all the variables needed are accounted for.