# On multiple solutions to the steady flow of incompressible fluids subject to do-nothing or constant traction boundary conditions on artificial boundaries

## TL;DR

This paper investigates the implications of using do-nothing boundary conditions in steady incompressible fluid flow simulations, revealing scenarios with multiple solutions or solution blow-up, highlighting issues with well-posedness.

## Contribution

It provides analytical and numerical examples demonstrating multiple solutions and blow-up phenomena caused by do-nothing boundary conditions in incompressible flow problems.

## Key findings

- Unsteady solutions can blow up or converge depending on initial data.
- Multiple steady solutions can exist under do-nothing boundary conditions.
- Numerical simulations confirm analytical findings in complex geometries.

## Abstract

The boundary conditions prescribing the constant traction or the so-called do-nothing conditions are frequently taken on artificial boundaries in the numerical simulations of steady flow of incompressible fluids, despite the fact that they do not facilitate a well-posed problem, not allowing to establish the standard energy estimates. In a pursuit to understand better the possible consequences of using these conditions, we present a particular set of examples of flow problems, where we find none or two analytical or numerical solutions. Namely, we consider problems where the flow connects two such artificial boundaries. In the simple case of the isotropic radial flows where both steady and unsteady analytical solutions are derived easily, we demonstrate that while for some (large) boundary data all unsteady solutions blow up in finite time, for other data (including small or trivial) the unsteady flows either converge asymptotically to one of two steady solutions, or blow up in finite time, depending on the initial state. We then document the very same behavior of the numerical solutions for planar flow in a diverging channel. Finally, we provide an illustrative example of two steady numerical solutions to the flow in a three-dimensional bifurcating tube, where the inflow velocity is prescribed at the inlet, while the two outlets are treated by the do-nothing boundary condition.