Traditional homework often limits meaningful feedback and reflection, undermined by time-consuming feedback processes, penalties for mistakes, and student disengagement once a grade is assigned. Grounded in Zimmerman's cyclical model of self-regulated learning, this study examined the combined effects of staged feedback, structured reflection, and student self-grading across six homework assignments in a college mathematics course. For each assignment, instructors privately recorded a grade before releasing written feedback; students then reviewed that feedback, reflected, and assigned themselves a grade, independent of the instructor's score. Across 1,150 graded cases, student- and instructor-assigned grades agreed 73.6% of the time, with students over-grading by an average of 0.11 points on a 0–4 scale. Survey and reflection data indicated most students grew more confident using feedback, though a subset preferred immediate grades. Findings suggest staged feedback and self-assessment can support self-regulated learning, while highlighting the need for differentiated support to ensure equitable benefits.

In this report, we develop analytical models and implement numerical schemes to represent a coupled mechanical system of deflections and stresses in beams joined together by a continuous spring layer. We begin by considering a one-dimensional and uniform rectangular beam resting on a continuous layer of elastic material coupled with another rectangular beam beneath the elastic material. Following the formulation and nondimensionalization of the governing system, we consider an analysis of steady states, an inverse design problem, and the dynamics of the linearized model. Numerical simulations of the dynamics are carried out for the one- and two-dimensional models. An asymptotic analysis for nonlinearly coupled beams is also considered.

A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches—linear stability analysis, fully nonlinear numerical simulations, and a coarse-grained model—to determine the circumstances that lead to each configuration. The coarse-grained model yields an explicit formula for the transition between deep and shallow convection in the physically relevant limit of small Darcy number. Near the onset of convection, all three of the approaches agree, validating the predictive capability of the explicit formula. The numerical simulations extend these results into the strongly nonlinear regime, revealing novel hybrid configurations in which the flow transitions dynamically from shallow to deep convection. This hybrid shallow-to-deep convection begins with small, random initial data, transitions through a metastable shallow state, and arrives at the preferred steady-state of deep convection. We construct a phase diagram that incorporates information from all three approaches and depicts the regions in parameter space that give rise to each convective state.

The phenomenon of convection is found in a wide variety of settings on different scales—from applications in the cooling technology of laptops to heating water on a stove, and from the movement of ocean currents to describing astrophysical events with the convective zones of stars. Given its importance in these diverse areas, the process of convection has been the focus of many research studies over the past two centuries. However, much less research has been conducted on how the presence of an obstruction in the flow can impact convection. In this work, we find that the presence of an obstruction can greatly affect convection. We note occurrences where the presence of an obstruction yields similar behavior to flow without an obstruction. Additionally, we find cases with markedly different features in comparison to their counterpart without an obstruction—notably, exhibiting long-term periodic behavior instead of achieving a constant steady-state, or the formation of convection cells versus an absence of them.

We perform linear and nonlinear stability analyses for thermal convection in a fluid overlying a saturated porous medium, in addition to conducting novel numerical simulations. We use a coupled system, with the Navier–Stokes equations and Darcy’s equation governing the free-flow and the porous regions, respectively. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid–medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. Within certain regimes, the nonlinear stability thresholds agree closely with the linear ones, and we quantify the differences that exist. We then compare stability thresholds produced by several common variants of the tangential interface conditions, using both numerics and asymptotics in the small Darcy number limit. Furthermore, we investigate the transition between full convection and fluid-dominated convection using both numerics and a heuristic theory. This heuristic theory is based on comparing the ratio of the Rayleigh number in each domain to its corresponding critical value, and it is shown to agree well with the numerics regarding how the transition depends on the depth ratio, the Darcy number, and the thermal-diffusivity ratio. Finally, we detail the numerical methods used to simulate the coupled system. Our analyses and the heuristic theory are then verified with our numerical results.

We perform linear and nonlinear stability analysis for thermal convection in a fluid overlying a saturated porous medium. We use a coupled system in which Navier–Stokes governs the free flow and Darcy's equation governs the porous-medium flow. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid–medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. We find close agreement between the linear and nonlinear stability thresholds, indicating that the more easily obtained linear thresholds indicate global stability. Additionally, we quantify the difference between linear and nonlinear stability curves. We then compare stability thresholds produced by several common variants of the tangential interface conditions, using both numerics and asymptotics in the small Darcy number limit. Finally, we investigate the transition between porous-dominated and fluid-dominated convection. A simple theory based on critical Rayleigh numbers is put forth, and it agrees well with numerics regarding how the transition depends on the depth ratio, the Darcy number, and the thermal-diffusivity ratio.

In this paper, we explore fluid flow caused by the presence of an insoluble surfactant on a thin, incompressible power-law fluid over a horizontal substrate. The gradient in surface tension caused by the surfactant results in fluid flow away from the region where the surfactant was deposited. Work has been conducted with Newtonian fluids and surfactants; however, the extensive effect surfactants have on non-Newtonian fluids has not been studied as thoroughly.

Using the lubrication approximation, we derive a system of coupled nonlinear partial differential equations (PDE) governing the evolution of the height of the fluid and the spreading of the surfactant. We also numerically simulate our system with a finite difference method and vary the power-law index to explore differences in profiles of shear-thickening and shear-thinning fluids. Next, we find significant agreement between our results and previous studies involving Newtonian fluids with power-law relations. Finally, we determine similarity scalings and solutions around the leading edge of the surfactant, which describe the behavior of the fluid and surfactant towards the region of the fluid where the surfactant ends.