I am currently dealing with multiphase flows and have to use the non-dimensional form of the Navier-Stokes equations (NSE). In the scientific literature I found various formulations (and almost no explaining o how they get them) but I wanted to get to the point by myself as well. Although the procedure should be simple, the results do not lign up and I cannot wrap around my head on why. NSE can be written for incompressible fluids as: \begin{equation} \frac{\mathrm{D}\mathbb{u}}{\mathrm{D}t}=-\frac{1}{\rho}\mathbb{\nabla}p+\mathbb{g}+\frac{\mu}{\rho}\Delta\mathbb{u} \end{equation} where the ratio $\frac{\mathrm{D}}{\mathrm{D}t}$ represent the material derivative, $\mathbb{u}$ is the fluid velocity, $\rho$ the density, $\mathbb{g}$ the gravitational acceleration and $\mu$ the dynamic viscosity. Also, $\Delta$ is the Laplacian operator.
With surface tension, the equation becomes (CSF formulation of the surface tension) \begin{equation} \frac{\mathrm{D}\mathbb{u}}{\mathrm{D}t}=-\frac{1}{\rho}\mathbb{\nabla}p+\mathbb{g}+\frac{\mu}{\rho}\Delta\mathbb{u}+\frac{\sigma\kappa\hat{\mathbb{n}}\delta_{\varepsilon}}{\rho} \end{equation} where $\sigma$ is the coefficient of surface tension, $\kappa$ is the curvature, $\hat{\mathbb{n}}$ is the normal unit vector to the interface surface and $\delta_{\varepsilon}$ a Dirac delta function (smeared out version).
Introducing the following dimensionless variables (marked with $'$): \begin{equation} x'L=\mathbb{x} \hspace{5mm} u'U=\mathbb{u} \hspace{5mm} g'g=\mathbb{g} \hspace{5mm} \nabla'=\mathbb{\nabla}L \hspace{5mm} \Delta'=\Delta L^2 \hspace{5mm} p=p'\rho U^2 \hspace{5mm} t'=tU/L \end{equation} I get \begin{equation} \frac{U^2}{L} \frac{\mathrm{D}u'}{\mathrm{D}t'}=-\frac{U^2}{\rho}\frac{\nabla'}{L}p'+g'g+\frac{\mu}{\rho}\frac{U}{L^2}\Delta' u'+\frac{\sigma\kappa\hat{\mathbb{n}}\delta_{\varepsilon}}{\rho} \end{equation} and, after multiplying both sides of the equation for $L/U^2$, \begin{equation} \frac{\mathrm{D}u'}{\mathrm{D}t'}=-\frac{1}{\rho}\nabla'p'+\frac{g'gL}{U^2}+\frac{\mu}{\rho}\frac{1}{LU}\Delta' u'+\frac{\sigma\kappa\hat{\mathbb{n}}\delta_{\varepsilon}}{\rho}\frac{L}{U^2} \end{equation} Now, after introducing the well-known Reynolds and Froude numbers \begin{equation} \mathrm{Re}=\frac{\rho LU}{\mu} \hspace{10mm} \mathrm{Fr}=\frac{U}{\sqrt{gL}} \end{equation} it follows that: \begin{equation} \frac{\mathrm{D}u'}{\mathrm{D}t'}=-\frac{1}{\rho}\nabla'p'+\frac{1}{\mathrm{Fr}^2}g'+\frac{1}{\mathrm{Re}}\Delta' u'+\frac{\sigma\kappa\hat{\mathbb{n}}\delta_{\varepsilon}}{\rho}\frac{L}{U^2} \end{equation} Until here no problem.
However, for the manipulation of the surface tension (SF) form, some authors (e.g. "An improved level set method for incompressible two-phase flows". Sussman et al., 1998) use the Weber number: \begin{equation} \mathrm{We}=\frac{\rho LU^2}{\sigma} \end{equation} Plugging this latter in the SF term though, I get: \begin{equation} SF'=\frac{\kappa\hat{\mathbb{n}}\delta_{\varepsilon}L^2}{\mathrm{We}} \end{equation} instead of \begin{equation} SF'=\frac{\kappa\hat{\mathbb{n}}\delta_{\varepsilon}}{\mathrm{We}} \end{equation}
I dare to think that this latter formulation is incorrect due to the incontrovertible principle of dimensional homogeneity.
$\kappa$ is measured (I hereinafter use the S.I. units for simplicity) in [1/m], $\hat{\mathbb{n}}$ is dimensionless [-] (unit vector) and the delta function has as unit of measure the inverse of its argument (which is a distance function) [1/m]. Thus \begin{equation} \mathrm{dimensionless \hspace{3mm}number}[-] \neq \frac{[1/m][-][1/m]}{\mathrm{dimensionless \hspace{3mm}number}[-]}=[1/m^2] \end{equation}
If someone can explain to me if I made some mistake I would greatly appreciate it.
Note: \begin{equation} \hat{\mathbb{n}}=\frac{\mathbb{\nabla}{d}}{|\mathbb{\nabla}{d}|}=[-] \end{equation} \begin{equation} \kappa=\mathbb{\nabla}\cdot\hat{\mathbb{n}}=[1/m] \end{equation}
