1 Introduction

The authors of the paper “Free surface tension in incompressible smoothed particle hydrodynamics (ISPH)” [2] present “a Dirichlet pressure boundary condition for ISPH [...]” and “[...] a new approach to compute the curvature more exactly for three-dimensional cases [...]”. This development is motivated by the claim that the established SPH curvature estimates give wrong results in three dimensions. As we will show below, this claim is based on a straightforward misconception in using the curvature term.

2 Curvature definition

The singular surface-tension force \(\mathbf {F_s}\) at a phase interface considering capillary forces only is given by

$$\begin{aligned} \mathbf {F_s} = \sigma \kappa _f \mathbf {n}, \end{aligned}$$
(1)

where \(\sigma \), \(\kappa _f\) and \(\mathbf {n}\) denote the surface-tension coefficient, the curvature and the surface normal direction, respectively. Assuming constant material properties, the classical Young-Laplace formula for a quiescent spherical drop is simply \(\Delta p = \sigma \kappa _f\).

Fluid mechanical curvature The fluid mechanical curvature is defined as

$$\begin{aligned} \kappa _f = - \nabla \cdot \mathbf {n} = \left( \frac{1}{R_1} + \frac{1}{R_2}\right) =\left( \kappa _1 + \kappa _2\right) ~, \end{aligned}$$
(2)

where \(R_1\) and \(R_1\) are the principal radii of the surface, and \(\kappa _1\) and \(\kappa _2\) its respective principal curvature (see, e.g., [1]). Note, for a sphere in 3D with \(R_1=R_2=R\), the curvature is given by \(\kappa _f=\frac{2}{R}\). In 2D, this curvature is simply \(\kappa _f=\frac{1}{R}\) (considering a cylindrical surface with \(R_1=R\) and \(R_2\rightarrow \infty \)).

Mean curvature The mean curvature or geometrical curvature [5] is mathematically defined as

$$\begin{aligned} \kappa _g = -\frac{1}{2}\nabla \cdot \mathbf {n} = \frac{1}{2}\left( \frac{1}{R_1} + \frac{1}{R_2}\right) =\frac{1}{2}\left( \kappa _1 + \kappa _2\right) ~. \end{aligned}$$
(3)

Here, for a sphere in 3D with \(R_1=R_2=R\), the curvature reduces to \(\kappa _g=\frac{1}{R}\). In 2D, there is only a single principal radius yielding \(\kappa _g=\frac{1}{R}\).

3 Discussion

Obviously, Fürstenau et al have confused the two definitions and compared the numerical approximation for the fluid mechanical curvature (their eq. 34) with the mean curvature. This can be implied from a comparison of the two Figs. 2 and 3 in the article, where the analytical curvature for two bubbles in 2D and 3D is compared with the numerical approximations. Both analytical values are obtained from \(\kappa _g=\frac{1}{R}\) for the given radii. From Fig. 3 showing the numerical curvature \(\kappa _f\) and analytical mean curvature \(\kappa _g = \frac{1}{2} \kappa _f\) the authors conclude:

“The approach was tested by comparing the curvatures of spherical bubbles in 2D and 3D test cases (see Fig. 1). When plotting the curvatures over the width (see Figs. 2, 3) it is obvious that in 2D the difference between ours and the standard approach is small, but in 3D the standard approach overestimates the curvature by a factor of 2 while ours is close to the analytical value.” (p. 493, [2])

This statement or conclusion is wrong and needs to be rectified in order to avoid proliferation. We point out that identical results and erroneous claims also have been published in [3].

The geometrical curvature \(\kappa _g\) coincidences with the fluid mechanical curvature \(\kappa _f\) in 2D. In 3D, however, they differ by definition by a factor of 2. Amongst other references, the following quote from Taylor nicely clarifies this issue:

“1.1 The mean curvature is \(H=\kappa _1+\kappa _2\). The most elementary approach of classical differential geometry [5] is to define principal curvatures \(\kappa _1\) and \(\kappa _2\), and then to define the mean curvature to be \(\left( \kappa _1+\kappa _2\right) /2\). The “mean” in “mean curvature” refers to this idea of the average of the curvatures. But in many ways, as will become clear below, it is much more natural not to divide by that 2, and it has become common to leave it out. Thus we will use \(H=\kappa _1+\kappa _2\)”. [6]

4 Conclusion

We emphasize that the well-known methods to compute the curvature via the divergence of the surface normals (e.g. [1, 4]) give the correct results. Nonetheless, the proposed approach of the authors to extract the mean curvature still is valid and applicable. Yet, this methodology does not improve on the prediction accuracy of existing formulations and does not justify the additional computational effort for the local coordinate transformation with principal curvature extraction.