![]() ![]() Since we know that both Velocity Potential and Electric Potential similarly obey Laplace’s Equation, and that there is an analogous relationship between Fluid Velocity and Electric Field, I thought it would be interesting to use this relationship to model Fluid Flow through the application of the Gauss-Seidel method, a method we also covered in Chapter 5 of our textbook. ![]() ![]() ![]() The negative derivative value for the Electric and Velocity Potential for each Cartesian coordinate is equivalent to each respective component of the Electric Field and Velocity. In class, we studied Laplace’s Equation in Chapter 5 of the textbook as it applied to Electrostatic conditions where we were solving for Electric Potential and the Electric Field instead.īy solving for the Electric Potential in a similar fashion to how we solved for the Velocity Potential, we can see how these potentials are algebraically the same.Īdditionally, we can find an analogous relationship between the negative derivative of each potential. It tells us that the value of potential at any point is the average of its neighboring points. This equation can be simplified by assuming that delta_ x=delta_ y=delta_ z. The second-derivative can be expressed as follows:īy plugging the second-derivative for each Cartesian coordinate into Laplace’s equation, we can find the equation to solve for the Velocity Potential. It can be written in a variety of ways as seen below: The first derivative is essentially the change in potential with respect to the appropriate Cartesian coordinate, which in this case is x-coordinate. In order to create the equation to solve for the Velocity Potential, we must first determine the first-order derivative of each function. Using Laplace’s Equation, we can move toward solving for the Velocity Potential. Laplace’s equation states that the sum of the second-order partial derivatives of a function, with respect to the Cartesian coordinates, equals zero: In completing research about Fluid Dynamics, I gained a better understanding about the physics behind Fluid Flow and was able to study the relationship Fluid Velocity had to Laplace’s Equation and how Velocity Potential obeys this equation under ideal conditions. I found this topic to be particularly fascinating since fluid dynamics is a type of mechanical physics that we do not have a chance to explore in our curriculum and for the simple fact that modeling invisible interactions is always a cool topic to explore. One example of this showed the application of these techniques onto devices that aid in the study of ocean surface currents and allowed for more accurate modeling of fluid dynamics. Kristy Schlueter-Kuck, a Mechanical Engineer whose research focuses on the applications of coherent pattern recognition techniques to needed fields to aid in solving a variety of problems. My interest in investigating Fluid Dynamics stemmed from a lecture given on campus in early February by Dr. I used the Gauss-Seidel Method to model velocity/electric field changes using vectors that correlate to changes in velocity/electric potential which depend on the points proximity to metal conductors/walls of pipes. My Computational Physics final project models fluid flow by relying on the analogous relationship between Electric Potential and Velocity Potential as solved through Laplace’s Equation. ![]()
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