AP Physics 2 students spent class exploring fluid dynamics through a series of activities. Using the Venturi meter is one of my favorite as it combines the model from fluid statics with Bernoulli’s equation for fluid dynamics. In addition, it demonstrates the effect of viscosity and one of the limitations of our ideal fluid model. One of the other activities is to blow a dime off the table and into a cup:
In the past, I’ve reinforced a simple (and incorrect) explanation through a misapplication of the Bernoulli Effect along the lines of “the dime lifts off the table because the Bernoulli Effect states that the fast moving air above the dime has lower pressure and the stationary air below the dime has high pressure.” My newly found deeper understanding began when I noticed a single phrase in the definition of the Bernoulli effect in Knight’s College Physics (emphasis mine): “The pressure is higher at a point along a streamline where the fluid is moving slower, lower where the fluid is moving faster.” The Bernoulli effect can only be applied along a streamline and not between streamlines. In addition, the Bernoulli effect focuses on a pressure gradient accelerating a fluid, not just the pressures associated with “fast” and “slow” fluid streams.
So today, I focused on how the streamlines are diverted over the top of the dime. The closer together the streamlines, the faster the speed of the fluid (justified with the equation of continuity – conservation of mass). The velocity of the fluid (air, in this case) increases as it is diverted over the leading edge of the dime and decreases over the trailing edge. The key factor is not that there is fast moving air over the dime but, rather, that the velocity of the air increases as it flows over the leading edge of the dime. Therefore, there must be a region of lower pressure over the dime.
Students were generally pleased with this explanation, but a couple were unsatisfied because we didn’t explain the cause of the region of low pressure over the dime. After reflection, discussion with my colleague, and some research, I think the best conceptual explanation focuses on the path of a particle along a diverted streamline. Since the path is curved, the particle experiences a radial acceleration toward the center of the curve. This radial acceleration requires an unbalanced force and the only force applied to this particle is a pressure difference perpendicular to the particle’s velocity. Therefore, on the concave side of the curve there must be a lower pressure than on the convex side of the curve. The combination of this effect across multiple streamlines results in the low pressure region over the dime.
Tomorrow, I’ll see what my students think about this perspective.