The modelling approach describes the time evolution of a single droplet on both dry and pre-wetted grinding wheel topographies. It is based on the Navier-Stokes system describing the fluid dynamic of the cooling fluid and its interaction with both the grinding wheel and the surrounding air. Specifically, the framework of the thin-film equation is used: a two-dimensional approximation to the Navier-Stokes system where the free fluid-air interface is characterized by the droplet film height \(h=h\left(t,x\right)\). For details regarding the thin film equations, we refer to the following references [4, 13, 14, 15].

Now, let \(T>0\) be the time horizont of interest and \({\Omega }\) be the surface of the grinding wheel where we observe the drop. The time evolution of this film height can then be described via a balance equation in the form of

$${\partial }_{t}h=-div{ q}_{h}+f\hspace{1em}\text{in }\left(0,T\right)\times \varOmega$$

1

where \({q}_{h}\) denotes the fluid flux density and f a possible volume source density. With this approach, mass conservation is satisfied when \(f=0\). In the unwetted scenario, the source term can therefore be used to model the extraction of fluid into the porous surface structure of the grinding wheel. The flux takes the general form

$${q}_{h}=Q\left(h\right)\left(\nabla \varDelta h\right)$$

2

for a mobility function \(Q\left(h\right)\) leading to a non-linear fourth-order parabolic PDE. The exact form of the mobility function depends on the specific situation considered in the Navier-Stokes system (e.g., boundary conditions, wetted or dry condition). As we are interested in characterizing the slip via the rough, porous substrate of the grinding wheel under dry and pre-wetted conditions, we consider the Navier-slip mobility function given by

$$Q\left(h\right)=\frac{1}{3\mu }\left({h}^{3}+\beta {h}^{2}\right).$$

3

Here, \({\mu }\) denotes the dynamic viscosity of the fluid and \({\beta }\) the slip parameter. Generally speaking, a higher parameter \(\beta\) corresponds to faster wetting behavior.

In all experiments, complete wetting was observed, i.e., there are no stationary contact angles. To avoid the free-boundary part of the problem that comes with explicitly modelling the contact angle evolution, we additionally assume the existence of a small background fluid film \({0<h}_{bg}\ll 1\).

As we are considering a fourth-order parabolic PDE, two sets of boundary conditions are needed in addition to the initial condition for the film height to arrive at a well-posed problem. In our study, we assume both homogeneous Neumann (no in-/outflow)

$$-\nabla h\cdot n =0 \hspace{1em}\text{on }\left(0,T\right)\times \partial \varOmega$$

4

and Dirichlet condition (fixed background film height at the boundary)

$$h={h}_{bg}\hspace{1em}\text{on }\left(0,T\right)\times \partial \varOmega .$$

5

While most parameters in this model are given as part of the setup, the goal is to use the free parameter \(\beta\) to characterize the wetting properties of the different grinding wheels, specifically with regards to the grain size, bonding system, and dry or pre-wetted state. Consequently, experiments investigating these properties are needed. With the thin-film equation, droplet experiments are usually a good starting point as they are easily comparable to the simulation and clearly interpretable.