Computing Change with Partial Differential Equations

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.

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.

Towards the beginning of the week, our team decided to take a closer look at some of the literature provided to us in order to understand some context to the problem we will be working on. We also looked at new functions and their hyperbolic behavior, Sinh[x] and Cosh[x]. These two functions were pivotal to our understanding of the Stream Function (Ψ(x,y) = Γ∗θ(x,y) =∫1/2log(cosh(x−ξ1)−cos(y−ξ2))θ(ξ1,ξ2)dξ1dξ2) and its relation to velocity and vorticity.

For the remainder of the week, our team took a further look at convolution and its relevance to the Stream Function. This process involved us taking several partial derivatives of the Stream Function with respect to different variables. Once we took a look at this, we decided to take the Laplacian of the Stream Function and it surprisingly yielded something that had some relation to the velocity u(x,y). Moreover, some of the literature that was provided demonstrated some u (velocity) that we are trying to prove is a steady state solution to the partial differential equation that we are working with. As of now we are continuing to prove that the function u(x,y) (velocity) is a steady state solution to the differential and we are doing so by using a Dirac function.

Part of the process this week was to gain a better understanding of what the three functions (Stream Function, velocity, and vorticity) mean in relation to each other and how to move forward in our project.