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.