Pressure gradient in an incompressible fluid as a reaction force and the preservation of the principle of ‘cause and effect’

Abstract

When considering the motion of an incompressible fluid, it is common practice to take the curl on both sides of the Navier-Stokes (or Euler) equations and cancel the pressure force. The governing equations are sufficient to derive the velocity field of the fluid \textit{without} any knowledge of the pressure. In fact, the pressure is only calculated \textit{after} obtaining the velocity field. This raises a number of conceptual problems. For instance, why is the pressure unnecessary for obtaining the velocity field? Traditionally, forces have been considered as the ``causes`` of motion, and the resulting acceleration as the ``effect``. However, the acceleration (the effect) and the resulting velocity field can be obtained without any recourse to the pressure (the cause), seemingly violating the principle of ``cause`` and ``effect``. We address these questions by deriving the pressure force of an incompressible fluid, starting from d'Alembert's principle of virtual work, as a ``reaction force`` that maintains the incompressibility condition. Next, we show that taking the curl on both sides of the Navier-Stokes (or Euler) equations is \textit{equivalent} to using d'Alembert's principle of virtual work, which cancels out the virtual work of the pressure gradient. This shows that abstract procedures, such as taking the curl on both sides of an equation, can actually be tacit applications of rich physical principles, without one realizing it. This can be quite instructive in a classroom of undergraduate students.