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Application of Adomian Method to Model Concentration Near the Surface of a Rotating Disk
Mathematics and Statistics
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The Adomian decomposition method is the recurrent procedure to obtain the approximate solution in the form of the series. It is named after George Adomian, who developed an approximate analytical method for solving nonlinear differential equations, both ordinary and partial. The Adomian decomposition method represents a step toward unified theory for partial differential equations (PDEs). My mentor is Dr. Aleksey Telyakovskiy, and my task is to solve the diffusion type equation using the Adomian decomposition method. The main idea of studying a diffusion equation is to understand how concentration gradients change over time and space. We first derived an expression for the velocity field of the fluid near the disk. It represents a complicated three-dimensional structure with the no-slip condition on the disk surface. Using this expression for the velocity field, we obtained the concentration equation. We then apply Adomian decomposition to construct an approximate solution for the concentration in the vicinity of the rotating disk. We analyze such problem, since in chemical engineering you often need to model processes when there are rotating parts and chemical reaction is happening. We consider infinite rotating disk to make mathematical analysis simpler. It is an approximation of more realistic disk of a finite radius; still we are able to get an idea of the key features of the process. As an example of the specific real-world system, we can use Ethanol as solvent and Sodium Ethoxide (NaC2H5O) and Methyl Iodide (CH3I) as reactants. Moreover, we calculated the Peclet number that represents the ratio of advection to diffusion of a physical quantity for this specific system, and we calculated the diffusion coefficient.