# An example of framing influencing resources

With finals wrapping up I was thinking back to my stat mech final in grad school. I realized that my behavior on one of the questions was a clear example of how a student’s framing of a problem can influence which resources the student draws on to solve the problem. The problem was something like the following:

Consider a long cylindrical pipe of diameter

dm filled with flowing water. If the pressure driving the flow is suddenly turned off, how long will it take for the water to come to rest? The viscosity of the water is ν kg/(s*m), the density of water is ρ kg/m^{3}, and the coefficient of thermal expansion is α m^{3}/K.

I first read this question and thought about it as a relaxation to equilibrium problem. Relaxation to equilibrium had been the last topic of the semester and unfortunately it was a topic I found quite confusing. I thought about the problem for a few minutes and with no clear path forward I decided to skip it and finish the rest of the exam.

When I came back to the problem, it suddenly occurred to me that this is a fluid dynamics problem. I had taken fluid dynamics the previous semester and this seemed like a relatively simple problem in the context of a fluid dynamics class. I started by constructing an approximate solution based on dimensional analysis. The relaxation time should decrease with increasing viscosity (more friction with the walls) and should increase with increasing diameter and density (more moving mass for a given amount of friction). The relaxation time should not depend on temperature (at least to a very good approximation). The only way to combine d, ν, and ρ and get units of time is ρ*d^{2}/ν. This must be the relaxation time (aside from a dimensionless multiplicative constant).

Framing the problem as a fluid dynamics problem suddenly made the problem much more physical to me, allowing me to reason qualitatively about how different variables should affect the solution. In addition, fluid dynamics makes frequent use of dimensional analysis arguments which likely cued me to think about constructing a solution based on dimensional analysis. (Further, my expectations about what types skills are valued in physics led me to believe that a solution based on dimensional analysis would be worth at least half credit even though I didn’t use “stat mech” in my solution.)

After constructing the approximate solution I went on to write down the Navier-Stokes equations and solve the problem exactly (doing so exactly as I would have in my fluid dynamics class).

I clearly possessed the knowledge necessary to solve this problem, the challenge was accessing that knowledge. Initially, I was only drawing on my stat mech knowledge and was trying to remember the assumptions and equations we used in stat mech. My initial framing of the problem as a stat mech question prevented me from reasoning physically about the scenario (a skill that was emphasized much more often in my fluids class than in stat mech) and prevented me from considering other viable ways of obtaining a solution. Once I shifted my framing, I was able to draw on a different collection of knowledge elements as well as different procedures for constructing a solution. I don’t know what prompted my shift in framing but I do remember that it was a very sudden realization and not a gradual shift in viewpoint.