Bulletin of the American Physical Society
84th Annual Meeting of the APS Southeastern Section
Volume 62, Number 13
Thursday–Saturday, November 16–18, 2017; Milledgeville, Georgia
Session D3: Mathematical and Statistical Physics |
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Chair: Laura Whitlock, Georgia College Room: MSU Building Donohoo Lounge |
Thursday, November 16, 2017 4:00PM - 4:12PM |
D3.00001: Analysis of high-speed rotating flow inside gas centrifuge casing Dr. Sahadev Pradhan The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity $\Omega $\textit{\textunderscore i} while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential ($\chi )$, which is derived from the equations of motion in an axisymmetric $(r - z)$ plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15{\%}) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified. [Preview Abstract] |
Thursday, November 16, 2017 4:12PM - 4:24PM |
D3.00002: Construction and Preliminary Investigation of Properties of the Imani Periodic Functions Ronald E. Mickens The Imani periodic functions (IPF) are continuous periodic solutions to the Leah functional equation (LFE) (1) $\frac{y^{2}}{2}+\left( {\frac{3}{4}} \right)x^{\frac{3}{4}}=\frac{3}{4},\quad x(0)=1,\quad y(0)=0,$ where x $=$ x(t) and y$=$y(t), and (2a) $x(t+T)=x(t),\quad y(t+T)=y(t),$ (2b) $x(-t)=x(t),\quad y(-t)=-y(t),$ and, T is a fixed, but arbitrary positive constant. Explicit representations are derived for these functions using nonlinear transformations of the dependent variables. We also obtain several of their mathematical properties. It should be noted that an alternative interpretation of Eq. (1) is that it is the Hamiltonian of a nonlinear oscillator having equation of motion (3) $\begin{array}{l} \frac{d^{2}x}{dt^{2}}+x^{\frac{1}{3}}=0,\quad x(0)=1,\quad \frac{dx(0)}{dt}=0, \\ \\ \end{array}$ where y(t)$=$dx(t)/dt. Finally, while all of the solutions to Eq.(3) are periodic, Eq. (1) may have non-periodic solutions. [Preview Abstract] |
Thursday, November 16, 2017 4:24PM - 4:36PM |
D3.00003: Dynamics of the Hamiltonian H(x,y) $=$ \textbar x\textbar $+$\textbar y\textbar Kale Oyedeji, Ronald E. Mickens We investigate the classical dynamics of the Hamiltonian (1) $H(x,y)=\;\vert x\vert +\vert y\vert $, and normalize the energy value to be H(x,y) $=$ 1. The equations of motion are (2) $\frac{dx}{dt}=\frac{\partial H}{\partial y}=sgn(y),\quad \quad \frac{dy}{dt}=-\frac{\partial H}{\partial x}=-sgn(x).$ In addition to proving all solutions are periodic, we also calculate explicitly the exact analytical solutions to Eq. (2). Further, we show that x(t) and y(t) have many features in common with the standard trigonometric cosine and sine functions. The work is based on the previous results of Mickens [1]. \underline {Reference} [1] R.E. Mickens, ``Some properties of square (periodic) functions''. Proceedings of Dynamic Systems and Applications 7 (2016), 282-286. [Preview Abstract] |
Thursday, November 16, 2017 4:36PM - 4:48PM |
D3.00004: Where Is The Equation Solved? Paul MacNeil Considerations of solution quality for physically-significant equations often focus on observable quantities such as solution geometry, energy levels, and charge density distribution. The degree to which the equation does or does not balance, i.e., the difference between the left and right hand sides (“equation error”) of the equation is a mathematical measure of solution quality. The equation error will be a function of the equation’s independent variables, commonly including space and time. The distribution of the error, and its squared modulus, over these variables, is a quality measure for the solution. Minimization of the squared error modulus (“equation error variance”) integrated/summed over all allowable values off the independent variables can be used to attempt solution of the equation. These considerations are presented with examples from a simple molecular system. The distribution of equation error is visualized. Numerical experiments compare use of equation error variance with the traditional energy minimization via the variational method. The minimization of equation error variance and energy is performed by Particle Swarm Optimization (PSO). Some characteristics of equation error variance minimization become apparent from the results of these experiments. [Preview Abstract] |
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