Apply unsteady Bernoulli/Euler by determining the time derivative of the velocity potential during wind instrument playing

Determine the time derivative of the velocity potential required to apply the unsteady Bernoulli equation to airflow during wind instrument playing, or equivalently solve the Euler equations for the unsteady flow in this context, so that an unsteady-flow Bernoulli formulation can be used for modeling brass performance.

Background

In critiquing the use of steady Bernoulli’s principle at the lip aperture, the paper argues that its assumptions (laminar, steady flow) are violated during brass performance. The authors note that an unsteady form of Bernoulli’s principle exists but has not been applied in this setting because it requires knowledge of the time derivative of the velocity potential.

Establishing or computing this quantity—potentially via solving the Euler equations—would enable the unsteady Bernoulli framework to be brought to bear on modeling the airflow that drives the lip vibrations, addressing a key gap identified by the authors.