INTRODUCTION The laws governing fluid flow around an obstruction can be modeled after a set of two coupled partial differential equations for ψ and ξ: $$\nabla^{2}\psi = \xi$$ $$\nabla^{2}\xi = {\nu}({\partial y}{\partial x} - {\partial x}{\partial y})$$ ψ, or the stream function, represents the path of the fluid when looking at its contour lines. ξ, or the vorticity, represents the curl of the fluid flow, or how much the fluid curves while travelling around this obstruction. In this project, we are interested in computationally modeling the path of a fluid around an obstruction using a two dimensional grid of points. We assume our fluid flowing is symmetric around the obstruction and thus we only consider the top half of our boundary. By modeling our fluid flow on a grid, we are able to use relaxation methods, the general idea of which is to split the matrix into easily invertible and harder to invert points. We can then construct the solution iteratively by using the averages of the four nearest neighbors in order to compute the solution of a grid point in the next iteration. We use a linear combination of updated values using a factor ω which we set to values between 1 and 2. in order to anticipate future corrections. This variant of the Gauss-Sidel method, which allows our solution to converge faster, is called successive overrelaxation. We first model our fluid so that it is in a free flow pattern. We then use a combination of implementation of our boundary conditions and overrelaxation to calculate ψ and ξ values iteratively. This process is repeated until our residuals, which we also compute each time we sweep through the psi and xi matrices, are reasonably small. By plotting the contour lines of ψ, we will be able to see how the fluid flows around the obstruction. We can also plot ξ to see how the fluid swirls around the obstruction. In addition to looking at just how the fluid flows around the obstruction, we are also interested in the parameters that change this fluid flow, particularly its viscosity and initial velocity, encompassed in the Reynolds number. Also, we are interested in how moving the obstruction closer to the edges of the boundaries changes the fluid flow pattern.

INTRODUCTION In this project, we look at a two dimensional Ising model, which explains the behavior of Ferro magnets around a critical temperature, where the magnetization suddenly drops from 1 to 0. Spins inside of a Ferro magnet allow the magnet to have as many states as the square of the number of spins. Each state has a probability determined by the Boltzmann factor. The Boltzmann factor is determined by the temperature and the energy. The energy is very complicated since it involves the interaction between the spins. However, we can simplify it by only considering interactions between nearest neighbors. In order to determine the exact equilibrium for different temperatures, we will need to calculate the energy of all of the states at this particular temperature. For large number of spins, again, the process will be very complicated. Therefore, to approximately calculate the equilibrium, we implement the Metropolis algorithm. This algorithm randomly chooses spins for calculation and avoids calculating states with extremely low probability by using the Boltzmann factor within the random choosing process.