An alternative theoretical approach for the derivation of analytic and numerical solutions to thermal Marangoni flows

- A new approach is developed to determine exact and numerical solutions to Marangoni flow.
- The typical kinematic conditions at the free surface are replaced by homogeneous conditions.
- A new class of analytic solutions is determined.
- The approach is also tested against the simulation of 2D thermocapillary and 3D Marangoni-Bénard flows.

The primary objective of this short work is the identification of alternate routes for the determination of exact and numerical solutions of the Navier-Stokes equations in the specific case of surface-tension driven thermal convection. We aim to elaborate a theoretical approach in which the typical kinematic boundary conditions required at the free surface by this kind of flows can be replaced by a homogeneous Neumann condition using a class of 'continuous' distribution functions by which no discontinuities or abrupt variations are introduced in the model. The rationale for such a line of inquiry can be found (1) in the potential to overcome the typical bottlenecks created by the need to account for a shear stress balance at the free surface in the context of analytic models for viscoelastic and other non-Newtonian fluids and/or (2) in the express intention to support existing numerical (commercial or open-source) tools where the possibility to impose non-homogeneous Neumann boundary conditions is not an option. Both analytic solutions and (two-dimensional and three-dimensional) numerical “experiments” (concerned with the application of the proposed strategy to thermocapillary and Marangoni-Bénard flows) are presented. The implications of the proposed approach in terms of the well-known existence and uniqueness problem for the Navier-Stokes equations are also discussed to a certain extent, indicating possible directions of future research and extension.