Events
Department of Mathematics and Statistics
Texas Tech University
We generalize Einstein’s probabilistic method for the Brownian motion to study compressible fluids in porous media. The multi-dimensional case is considered with general probability distribution functions. By relating the expected displacement per unit time with the velocity of the fluid, we derive an anisotropic diffusion equation in non-divergence form that contains a transport term. Under the Darcy law assumption, a corresponding nonlinear partial differential equations for the density function is obtained. The classical solutions of this equation are studied, and the maximum and strong maximum principles are established. We also obtain exponential decay estimates for the solutions for all time, and particularly, their exponential convergence as time tends to infinity. Our analysis uses some transformations of the Bernstein-Cole–Hopf type which are explicitly constructed even for very general equations of state. Moreover, the Lemma of Growth in time is proved and utilized in order to achieve the above decaying estimates.
This is joint work with Akif Ibragimov (Texas Tech University, and Oil and Gas Institute of the Russian Academy of Science).
 | Wednesday
Feb. 18
|
| Algebra and Number Theory
No Seminar
|
Abstract: This talk explores the aerodynamics of complete aircraft geometries at the initial conceptual design stage and the importance of understanding the propulsion-related aerodynamics and performance of ducted fans and open rotors for their integration. A ducted rotor system was used to produce turbulent jets with a Reynolds number up to 5.97x105 and Mach number of 0.222 based on mean streamwise velocity. Three rotors with a diameter of 11.8 cm were manufactured and tested inside a duct with a 1 mm tip clearance at a speed up to 30, 000 revolutions per minute (rpm). three different blade planform shapes were used including a rectangular shape with constant chord, trapezoidal shape with a taper ratio of 0.5, and elliptical shape where the trailing edge of the blade is expressed with an elliptical function. The rotor thrust and electric power were measured, and the thrust coefficient and figure of merit was computed. The flow-field produced by the ducted rotors was measured in the near-field using laser Doppler velocimetry techniques. Time-averaged contours of cross-stream vorticity reveal intense hub and blade tip vortex structures, which are impacted by the shape of the blade, particularly in the blade tip region. Tip vorticity as well as streamwise turbulence intensity and turbulent kinetic energy in this region were mitigated for the rotors with trapezoidal and elliptical blades. In a related study, the blades of a 12 cm diameter ducted rotor system were coated with a sharkskin- inspired surface with diverging tip micropillars. Surfaces containing 40 μm and 70 μm tall micropillars were applied on the rotor blades in order to study their role on fan aerodynamics and downstream jet flow. The effect of the micropillar coatings on the rotor blades marginally increases the mean streamwise velocity and rotor figure of merit due to mitigating boundary layer separation at higher rotor speeds. Moreover, this occurs due to the micropillar’s ability to increase wall-normal turbulence intensity in the boundary layer when the pillar height is scaled appropriately to the boundary layer thickness. The rotor hub and blade tip vortex structures become diffused and undergo breakup into smaller structures accompanied with an acceleration in the decay of absolute mean cross-stream vorticity.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this link
Direct Link that embeds meeting and ID and passcode.
- Choice #2: join meeting using this link
Join Meeting, then you will have to input the ID and Passcode by hand:
* Meeting ID: 949 9288 2213
* Passcode: Applied
Abstract pdf
The Biomath seminar may be attended virtually Friday at 11:00 AM CST (UTC-6) via this Zoom link.
Meeting ID: 938 8653 3169
Passcode: 883472
abstract noon CST (UTC-6)
Zoom link available from Dr. Brent Lindquist upon request.