Determine the governing equation for drag-force angles

Determine the equation governing the angles of the competing upper- and lower-region drag forces, \(\bm e^+\) and \(\bm e^-\), in linear granular flows and thereby characterize the associated lift and drag forces.

Background

The proposed model interprets the measured lift force as a vertical component generated by two oppositely directed, tilted drag forces acting on different regions of an intruder. In the simplified linear-flow formulation, the force magnitudes and directions depend on the angle of the regional drag vectors.

Although the model reproduces measured lift and drag trends, the angular dependence is not derived from the underlying granular mechanics. Establishing the governing equation for these angles would provide the missing closure relation needed to predict lift and drag without directly fitting measured angles.

References

However, it is not yet clear what equation governs the drag force angles and this is closely related to the closure problems of section~\ref{sec:mododisperse_flows}.

— A force model for dense granular flows and particle segregation  (2609.09398 - Lantman, 8 Sep 2026) in Section 4.4, subsection “Remarks”

This method could in principle be extended to three dimensional flows in more complex geometries, however it is unclear how the drag forces behave when the layers actually bend and warrants further investigation.

— A force model for dense granular flows and particle segregation  (2609.09398 - Lantman, 8 Sep 2026) in Section 7, “Discussion”

Interestingly, the general structure of the velocity profile is governed by the gravity force, but the specific flow profile follows from a micro-mechanical balance that remains unclear.

— A force model for dense granular flows and particle segregation  (2609.09398 - Lantman, 8 Sep 2026) in Section 8, “Conclusion”