I'm close to the modeling, simulation, and experimental work in fluid dynamics, but I've been wondering about mathematicians that do analysis and PDEs: for example, what do physical quantities such as the viscosity mean to the mathematicians? In modeling, simulations and experiments, there's a great deal of focus on physical intuition -- one would describe the quantities physically, e.g. describe physically the drag or lift force, the viscosity, the torque, etc. For someone in analysis and PDEs, is it necessary for them to have the same level of physical intuition, or is something such as the viscosity simply viewed as an input, a number, to the mathematician?
I guess generally I am wondering whether mathematicians that work in analysis and pdes have to know any physics at all - is a yes / no answer possible here?
(In a math department, I've often seen many grad-level topics courses in PDEs and fluid dynamics that specify "no physics background is assumed". The requirements were mathematical, e.g. real analysis, functional analysis, and harmonic analysis.)
Thanks
I guess I can chime in on my current experience. I am a current undergraduate at University of California, Los Angeles that is doing research on fluid dynamics in the Math Department (with a higher emphasis in the Applied Portion); in particular, on inertial flow in microfluidic devices.
I have also taken a graduate course in the math department on Fluid Dynamics in the Math Department. In my line of research, we mainly focus on the incompressible Navier Stokes e.g. we assume $\nabla \cdot \textbf{u} = 0$, where $\textbf{u} $ is the fluid velocity. Then we have the following equation \begin{align} \mu \nabla^2 \textbf{u} - \nabla p = \rho(\frac{\partial \textbf{u}}{\partial t} + \textbf{u} \cdot \nabla \textbf{u}) \end{align} \begin{align} \nabla \cdot \textbf{u} = 0\end{align} where we assumed our forces are negligible, and that the viscosity $\mu$ and density $\rho$ are constants. With these assumptions, we can non-dimensionalize the Navier Stokes Equation to obtain \begin{align}\nabla^2 \textbf{u} - \nabla p = Re(\frac{\partial \textbf{u}}{\partial t} + \textbf{u} \cdot \nabla \textbf{u}) \end{align} with $Re$ being a dimensionless constant known as the Reynolds Number.
This is very powerful as this allows us to reduce the physical intuition needed to understand the equations. Indeed, in the dimensionless formulation, we can view the Reynolds Number as the balancing term between the viscous and pressure terms compared to the inertial terms.
We use mainly analytical tools such as asymptotic expansion, dominant balance, numerical solvers, and the weak formulation of the incompressible dimensionless Navier Stokes to approximate the flow. E.g., we don't use many physical properties about the fluid. Even in my graduate course, the only physics we really used were deriving the Navier Stokes equation.
However, I'm sure there are many more sophisticated tools from analysis that one can use to study these PDEs in much more detail, but I have only taken up to Measure Theory and a basic introduction to PDE course. But, I hope this answers sheds some light on how mathematicians view Fluid Mechanics.
For reference on the physics question, I have not taken any physics since high school physics so far. Sometimes it's a struggle to understand the underlying physics, but it is at an accessible level who puts in the effort.