Height-Averaged Navier–Stokes Solver for Hydrodynamic Lubrication

To arrive at a two-dimensional description we take an average of Eq. (1) in z-directionwhere \(\varvec{f}_i\) denotes the i-th column of the flux function matrix \(\mathbb {F}\) (i.e., the flux in direction i), \(h_1(x_1, y_1)\) and \(h_2(x_2, y_2)\) are the height profiles of the lower and upper surface, respectively, and their difference is denoted with h as shown in Fig. 1a. Note that these profiles are expressed in coordinate systems that are attached to the walls, which move relative to the global reference frame with constant velocities \(\mathbf {U}_1=(U_1, V_1, 0)^\top\) and \(\mathbf {U}_2=(U_2, V_2, 0)^\top\).

Applying the Leibniz integral rule for differentiation under the integral sign and the product rule, the l.h.s. of Eq. (4) readsand similarly, for the first term on the r.h.s of Eq. (4), we obtainwhere overbars denote height averages \(\bar{\phi } = \frac{1}{h}\int _{h_1}^{h_2}\phi {\text{d}}z\) (see Appendix A for more details). The second term of Eq. (4) is identical but has x replaced by y. The third term under the integral on the r.h.s. of Eq. (4) can be evaluated directly at the fluid-wall boundariesHence, the averaged scheme is composed of a two-dimensional divergence operator acting on height-averaged flux functions and of terms containing both averaged and unaveraged flux functions, the latter evaluated at the top and bottom walls. The terms outside of the divergence operator can be regarded as geometrical source terms due to the reduction of dimensionality.

The averaged scheme readswithThe total time derivative for the upper (\(i=2\)) and lower (\(i=1\)) rigid surfaces can be written asWithout loss of generality, we assume that the lower wall is flat (\(h_1 = \text{const.}\)) and the upper wall is stationary (\(\mathbf {U}_2 = \mathbf {0}\)). Then, the expression for the source term simplifies significantly toWe use the simplified source term for the numerical tests presented in Sect. 4, but more complicated boundary conditions are generally possible. Hence, for flat channels, where the gradients of the lower wall disappear, the source term only consists of momentum flux contributions in z-direction evaluated at the upper and lower wall, respectively.

The height-averaged scheme that solves the macro problem is entirely formulated in terms of the densities of conserved variables and does not contain any a priori assumption about the constitutive behavior of the lubricant which is encoded in the flux function through the equation of state \(p(\rho )\) and the viscous stress tensor \(\underline{\tau }(\mathbf {q)}\). The functional form of the flux is obtained from the micro problem discussed in Sect. 2.3.

We use a finite-volume discretization of the two-dimensional domain with an explicit time integration scheme to solve Eq. (8). MacCormack’s [10] method is easy to implement and including source terms into the predictor–corrector scheme is straightforward. The discretized version of the system reads where \(\varvec{Q}_{i,j}^*\) and \(\varvec{Q}_{i,j}^{**}\) are intermediate solutions obtained from forward and backward differencing of the discretized flux vectors \(\varvec{F}_{\varvec{x}/\varvec{y};i,j}^{*/n}\) and \(\varvec{S}_{i,j}^{*/n}\) is the discrete source term. Here, subscripts i and j denote the grid cell with side lengths \(\Delta x\) and \(\Delta y\) and the superscript n denotes discrete time points separated by the time step \(\Delta t\). The order of differencing directions is arbitrary and one could also switch between backward and forward differences after every consecutive time step. Although the intermediate steps are first order accurate, the overall scheme is second order accurate in space and in time.

Other Jacobian-free integration schemes such as Richtmyer’s two-step version of the Lax–Wendroff scheme could also be implemented straightforwardly (see Ref. [11] for an overview of modifications to the Lax–Wendroff scheme with source terms). In the linear case (and without source term), MacCormack’s method is equivalent to the Lax–Wendroff scheme [12]. The presented scheme can be modified to be total variation diminishing (TVD) as shown by Davis [13]. This can be of particular use in the presence of large gradients or at discontinuities, since spurious oscillations are damped by adding just the right amount of artificial viscosity. The TVD-correction term to the MacCormack corrector step is shown in appendix B.

Boundary conditions are implemented using a ghost cell approach, leading to \(N+2\) cells in each direction, due to the three-point stencil of the MacCormack scheme. For Dirichlet-type boundary conditions, the value of the ghost cell is chosen such that the prescribed value of the conserved variable is satisfied at the cell boundary by linear interpolation between the boundary cell and the ghost cell. Inflow and outflow boundary conditions \(\nabla \varvec{q} = \varvec{0}\) are satisfied by setting the ghost cell value to the boundary cell value. Implementation of periodic boundary conditions is also straightforward. One-dimensional examples in Sect. 4 are computed using the full two-dimensional description with periodic boundary conditions in y-direction, leading to a \((N + 2) \times 3\)-grid.

The scheme is implemented in Python using the numpy library [14]. Compared to conventional stationary Reynolds solvers, computation time can be large, since we are resolving the full transient behavior of the flow until steady state is reached, while the time step is limited by the Courant–Friedrichs–Lewy condition. Computational speed-up is achieved by spatial domain decomposition in combination with a parallel implementation using the mpi4py bindings to the message passing interface (MPI) [15].

To benchmark the scaling behavior of our code, we calculated an incompressible flow problem until steady state for \(400 \times 400\) and \(1000\times 1000\) finite-volume grid cells, respectively, on bwForCluster NEMO at the University of Freiburg (2x Broadwell E5-2360v4 at 2.2 GHz per compute node with a 100 Gbit/s OmniPath interconnect). On the smaller grid, the serial implementation reaches only 1.6 million lattice updates per second (MLUPS), which can be improved up to 54.4 MLUPS on 220 MPI processes (11 NEMO compute nodes). On the larger grid, our implementation peaks at 223.7 MLUPS when using 520 MPI processes (26 NEMO compute nodes). The number of time steps until steady state strongly depends on the nature of the problem, ranging from a few hundred steps to more than one million.

The governing equation of the micro problem can be found by applying the splitting approach on Eq. (1)By inserting Eq. (8), the governing equation of the macro problem, we obtainwhere \(\mathbf {s_0}\) is the source term of the macro problem evaluated at the lateral position and time of the macro problem and the Jacobian readsFrom the first row of Eq. (14), we find that \(\partial _z\delta q_4 = \text{const.}\), but since there can be no mass flux through the walls it vanishes everywhere, i.e., \(q_4 = j_z = 0\). Thus, our splitting approach induces laminar flow, which is a reasonable assumption in thin-film flows and a reduced system is given byor simplyThe latter equation governs the local profiles of the remaining conserved variables and illustrates the relation between micro and macro problem. On one hand, local stresses are determined by the macroscopic flow conditions represented through the source term, whereas, on the other hand, these stresses govern the time evolution of the macroscopic variables. In a multiscale simulation, the micro problem could be replaced by a more accurate simulation method, such as molecular dynamics. However, we want to proceed using common assumptions about the viscous stress tensor that lead to the well-known results of continuum lubrication modeling.

For Newtonian fluids, the viscous stress tensor is a linear function of the velocity gradient \(\nabla \vec {u}\) given bywhere \(\eta\) and \(\zeta\) are the coefficients of shear and bulk viscosity, respectively. In order to express the components of the viscous stress tensor \(\underline{\tau }\) in terms of \(\bar{\varvec{q}}\), we simply use our knowledge about \(\underline{\tau }\) in Eq. (17). From the first two rows, we find thatwhere u(z) and v(z) are the streamwise components of the local velocity field \(\vec {u}(x_0, y_0, z) = (u,v,w)^T\), which can be described using second-degree polynomials, The cross-film velocity w(z) vanishes as already shown in Sect. 2.3.

The parameters \(\beta _i\) and \(\gamma _i,\;i=1,2\) are determined from the boundary conditions at the bottom and top wall, namely where \(u_{\text{slip},i}\) and \(v_{\text{slip},i}\) are slip velocities.

From the third row of Eq. (17) we obtain \(\partial _z p = \text{const.}\). If we neglect body forces in z-direction, the pressure at the top and bottom wall must be the same and therefore we find the typical result \(p(z) = \text{const.}\), which is equivalent to a constant mass density across the channel height.

Thus, with the definition of the height-averaged mass fluxwe obtain the remaining parameters \(\alpha _i\), which describe the Poiseuille contribution to the flow. This enables us to describe the viscous stress tensor according to Eq. (18) as a function of the average mass flux and density.

Hence, all entries of the flux matrix are given as a function of averaged conserved variables \(\mathbb {F} = \mathbb {F}(\bar{\varvec{q}})\). Computing the source term also requires unaveraged flux values at the top and bottom wall which are automatically defined by the choice of velocity and density profiles.

Wall slip is typically quantified through the Navier slip length b [16]. It becomes relevant for confined lubricants, when the gap height is in the order of the slip length, such as in EHL contacts or in the boundary lubrication regime. The concept of slip length arises from the assumption that the interfacial shear stress is proportional to the slip velocity with the constant of proportionality quantifying the friction between fluid and solid. Using the Newtonian fluid constitutive equation, the slip length can therefore be described as a combination of a purely bulk property (shear viscosity) with a purely interfacial one (friction) [17]. Geometrically, the slip length can be interpreted as the subsurface distance where the fluid velocity would equal that of the wall when linearly extrapolated, as shown in Fig. 1c. This leads to the following expression for the slip velocities where positive signs correspond to the upper and negative signs to the lower surface respectively. Naturally, many surfaces have heterogeneous surface chemistry, or surface properties can be tailored into particular stick–slip patterns, and therefore the slip length \(b_i\) is a function of the lateral coordinates x, y.

Complex lubricants often deviate from the Newtonian fluid behavior, especially when subjected to high pressures and/or high shear rates. Non-Newtonian effects can be modeled by a generalized Newtonian fluid description, where an effective viscosity \(\eta _\text{eff} = \eta (\dot{\gamma }, p)\) replaces the constant one in the formulation of the Newtonian stress tensor, Eq. (18). For instance, shear-thinning or piezoviscous behavior is then described as a non-linear function of the shear rate \(\dot{\gamma }\) and pressure p, respectively. Effects depending on the deformation history are not captured by the generalized Newtonian fluid model.

Accounting for piezoviscous effects is straightforward, since we have seen that pressure is constant across the gap height as a general result of our splitting approach. A typical model for the pressure dependence of viscosity is Barus equation [18]where \(\alpha\) is the pressure viscosity coefficient, typically ranging between 10 and \(20\,\text{GPa}^{-1}\). Although Eq. (24) is known to overestimate the viscosity for higher pressures and more accurate empirical modifications exist [19, 20], it is still often used.

Conversely, the strain varies across the gap height for general Couette–Poiseuille flow, which leads to velocity profiles that deviate from the Newtonian quadratic ones. Hence, an effective viscosity law depending on a height-averaged strain rate would not give the correct stress, especially at locations where pressure-driven flow dominates.

Here, we employ a power-law fluid model to show that arbitrary constitutive laws can be incorporated into the micro problem. As the name suggests, the shear stress components depend on the shear rate through a power lawwhere n is the flow index, describing shear-thinning (\(n<0\)) or shear-thickening (\(n>0\)) behavior. \(\phi\) is called the flow consistency index and has units of \(\text{Pas}^n\). For \(n=1\), we recover the Newtonian fluid.

Note that the power law does not correctly describe the fluid behavior in the limiting cases of zero or infinite shear rate. Typically, the effective viscosity converges to finite values (\(\eta _0, \eta _\infty\)) in these limits, which is for instance accounted for in the Carreau [21] or Eyring [22, 23] shear-thinning models. The type of shear-thinning model may also depend on pressure and/or temperature. In a recent molecular dynamics study on model fluids under EHL conditions, Jadhao and Robbins [24] showed that there is a generic transition from power-law behavior at low pressure, and therefore low viscosity, to Eyring behavior at high pressure.

Nevertheless, we use the simple power-law model here, since it can be treated analytically, but emphasize that principally more realistic solutions to the micro problem could be found. Similar to the Newtonian case in Sect. 3.1, we integrate Eq. (17) of the micro problem to obtain the velocity profiles. Davaa et al. [25] showed that for plane Couette–Poiseuille flow of power-law fluids, two cases have to be distinguished: Case 1 requires a piecewise integration from the lower wall \(z=0\) to the position of the maximum velocity \(z=h_1^*\) and from the location of the maximum velocity \(z=h_1^*\) to the upper wall \(z=h\), whereas the velocity profile of case 2 can be written as a single expression that likewise depends on the location of the velocity maximum \(h_2^*\), which not necessarily lies within the gap. The velocity profiles have the formA more detailed description of the integration procedure for the velocity profiles can be found in Appendix C or in Ref. [25]. The parameters \(\beta _1\), \(\gamma _1\), and \(\beta _2\) are obtained in a similar fashion as in the Newtonian case using boundary conditions of the form of Eq. (21a)–(21d). Finally, \(\alpha _1\) and \(\alpha _2\) are obtained from the definition of the height-averaged momentum flux, Eq. (22). However, in both cases, the unknown parameters \(h_1^*\) and \(h_2^*\) are left to be determined. In case 1 we use the continuity condition of the velocity profile at \(z=h_1^*\) as an additional expression for \(\alpha _1\) and in case 2 we use the remaining boundary condition at the top wall to obtain \(\alpha _2\). By equating these two expressions for \(\alpha _i\) in both cases, respectively, we obtain two non-linear equations that determine \(h_1^*\) and \(h_2^*\)where \(\bar{u}\) denotes the height-averaged velocity. Setting \(h^*= 0\) in either of the equations leads to the critical condition \(U/\bar{u} = (1+2n)/(1+n)\) that marks the boundary between the two cases. The resulting velocity profiles, such as the ones shown in Fig. 2, are then used to compute the viscous stress tensor components for the power-law fluid.

Next to the viscous stress tensor another constitutive relation is naturally contained in the flux function. For compressible fluids, the system of continuity equation and momentum balance needs to be closed by an equation of state, relating the pressure in the hyperbolic flux contribution to the mass density. Various types of equations describing the compressibility of the fluid can be used, ranging from the ideal gas to semi-empirical descriptions, such as the isothermal Dowson–Higginson [26, 27] equation of statewith fitting parameters \(C_1\) and \(C_2\), which is commonly used for mineral base oils.

However, such expressions do not reflect the effect of vaporous cavitation, which mainly occurs in diverging contact geometries, where the pressure may fall below the vapor pressure of the liquid. The first model that accounted for mass-conserving cavitation was based on the work of Jakobsson, Floberg, and Olsson (JFO) [28‐31], who formulated boundary conditions for the rupture and reformation of a fluid film under the assumption that the pressure is constant in the cavitated regions. Elrod’s and Adam’s cavitation algorithm [32, 33] builds upon the JFO approach, but added a switch function to the Reynolds equation that suppresses the Poiseuille contribution to the flow in the cavitated regions. The advantage of the Elrod–Adams (EA) approach is that a single equation can be used for the whole domain.

Here, we incorporate mass-conserving cavitation directly through the equation of state, by either fixing the pressure to a constant value \(p_{\text{cav}}\) at densities lower than the saturation density, which is conceptually similar to the EA algorithm or using a unique equation of state describing the behavior of vapor, liquid, and vapor–liquid mixture as in the model of Bayada and Chupin [34].

Their model assumes constant compressibility both in the liquid and the vapor phases, defined by the velocities of sound \(c_\text{l}\) and \(c_\text{v}\), respectively. The assumption that both phases of the vapor–liquid mixture have the same velocity allows to homogenize the dynamics of the flow in the cavitated region such that the same set of equations as in the full-film region can be used. The pressure–density relation for the mixture of vapor bubbles and liquid is defined in terms of the vapor fraction \(\alpha =(\rho -\rho _\text{l})/(\rho _\text{v}-\rho _\text{l})\), in which \(\rho _\text{v}\) and \(\rho _\text{l}\) are the density of the vapor and the liquid at the wet point and the bubble point, respectively. Bayada and Chupin used a relation proposed by Van Wijngaarden [35]relating the fluid’s velocity of sound \(c_\text{f}\) with the vapor fraction. Integrating equation \({\text{d}}p/{\text{d}}\rho =c_\text{f}^2\) in each of the regions and requiring continuity of the pressure at the transition from vapor to mixture (\(\alpha =1\)) as well as from mixture to liquid (\(\alpha =0\)) and that \(p(0) = 0\), one findswith pressure at the transition points from vapor to mixtureand from mixture to liquid (i.e., the cavitation pressure)Note that, for convenience, in Eq. (30) the definition of \(\alpha\) is extended to values larger than one or smaller than zero. Obviously, the interpretation as the vapor fraction is then no longer valid in these regimes. Fig. 3 shows an example of Eq. (30) for material parameters that lead to a cavitation pressure of \(0.061\,{\text{MPa}}\). For comparison, an EOS following the Dowson–Higginson relation in the liquid phase and having constant pressure everywhere else is given, which represents a realization of the EA algorithm through the EOS.

The pressure–density relation of Bayada and Chupin [34] is accompanied by a viscosity model that interpolates between the vapor and liquid viscosity, the simplest one being a linear interpolation in terms of the vapor fraction \(\alpha\)

As a first test for the transient numerical scheme, we use an inclined slider geometry with an ideal gas EOS and compare the steady-state solution with the results of Ref. [8] obtained from a compressible Reynolds equation. The height profile is given by \(h(x) = h_{\text{max}} - s x\) for \(x\in [0,L]\), where s is the slope of the pad. Here, we test the scheme at different sliding speeds U for a bearing with \(L = 0.1\,\text{m}\), \(h_{\text{max}}=66\,\mu{\text{m}}\), and \(s=5.6\cdot 10^{-4}\). The equation of state is given aswith ambient pressure \(p_0 = 101325\,{\text{Pa}}\) and ambient density \(\rho _0=1.1853\,{\text{kg/m}}^3\). The shear viscosity is assumed to be constant at value of \(\eta = 18.46\cdot 10^{-6}\,{\text{Pas}}\) and the bulk viscosity \(\zeta\) is set to zero. The obtained pressure profiles for three different velocities of the gas-lubricated slider bearing are shown in Fig. 4a. A substantial increase of load-bearing capacity with sliding speed can be observed and the results match perfectly with the ones obtained in Ref. [8].

The simulation results for the inclined slider geometry, as well as all other tests in the remainder of this section, are obtained under the assumption that convective (non-linear) inertial terms in the momentum equations can be neglected, which is a reasonable assumption in most cases. However, as shown in Ref. [8] for slider bearings and the ideal gas EOS, these necessary additional assumptions can lead to significant errors at high sliding speeds. Since the convective acceleration term is generally included into the definition of the flux function in Eq. (2), we have tested its influence for the inclined slider geometry for three different velocities. In Fig. 4b, the relative difference between the height-averaged Stokes solution (without non-linear term, \(p_{\text{s}}\)) and the height-averaged Navier–Stokes solution (\(p_{\text{NS}}\)) is shown. The maximum pressure error due to neglection of inertial terms for a sliding speed of \(U=125\,{{\text{m/s}}}\) goes up to 8% and agrees with the error reported in Ref.[8]. We want to highlight that both linear and non-linear solutions are obtained using the same numerical framework.

We further test the scheme for fluids with varying compressibility using the Dowson–Higginson EOS (Eq. (28)). Here, we choose the parameters reported in Ref. [36] (\(C_1 = 2.22\,\text{GPa}\), \(C_2 = 1.66\)) and a slider geometry with inlet gap height \(h_{\text{max}} = 1.5\,\mu{\text{m}}\), sliding speed \(U = 0.1\,{\text{m/s}}\), and three different slopes. The obtained pressure profiles are shown in Fig. 5 and are compared with results obtained from the compressible Reynolds solver reported in Codrignani et al. [37]. As in the previous case, the obtained pressure profiles match the Reynolds reference solution.

We have introduced the power-law fluid in Sect. 3.3 as an example for a generalized Newtonian fluid. The availability of analytical expressions for the velocity profiles allows us to implement the relevant stress components as a result of the micro problem for power-law fluids. This highlights the advantage of our splitting approach, since the numerical solution of the macro problem remains untouched.

We test our implementation of power-law stresses for an infinitely long journal bearing geometry which can be described by a sinusoidal height profilewith radial clearance \(c=R_b - R_j\) and eccentricity e. \(R_b\) and \(R_j\) are the radii of the bearing and journal, respectively. The eccentricity ratio is defined as \(\varepsilon =e/c\).

The resulting pressure profiles for four different flow indices are shown in Fig. 6. All simulations were performed for a journal bearing with eccentricity ratio \(\varepsilon =0.6\), bearing radius \(R_b = 1/2\pi\, {\text{mm}}\), clearance \(c=0.01R_b\), and at a sliding speed of \(U=5\,{{\text{m/s}}}\). The flow consistency index \(\phi =0.0794\,{\text{Pas}}^n\) is the same in all four cases and can be interpreted as the Newtonian shear viscosity for \(n=1\). For a better comparison we show the dimensionless pressure profiles \(\tilde{p} =(p-p_0)(c/U)^n c/(2\pi R_b\phi )\). The effect of non-Newtonian lubricants with shear-thinning (\(n<1\)) or shear-thickening (\(n>1\)) effect is directly visible from the normalized pressure profiles. However, this example cannot be used to describe an increase or decrease of load-bearing capacity due to the non-Newtonian behavior of the lubricant, since cavitation is not considered in the diverging part of the geometry. The implementation of cavitation models is validated in the following examples.

To study the effect of cavitation in hydrodynamic lubrication, we use a converging–diverging height profile, where we expect cavitation bubble formation in the diverging part of the bearing. Therefore, we use a parabolic slider geometry, which has been widely used to benchmark cavitation models. The height profile of the one-dimensional bearing is given bywith maximum gap height \(h_{\text{max}} = 50.8\,\mu{\text{m}}\), minimum gap height \(h_{\text{min}} = 25.4\,\mu{\text{m}}\), and length \(L_x = 76.2\,\text{mm}\). The parameters describing the lubricant through Eq. (30) are given in Tab. 1. The resulting pressure profiles for two different discretizations (\(N=100\) and \(N=200\)) are shown in Fig. 7a and the relative density or saturation \(\rho /\rho _l\) is shown in Fig. 7b. Both agree very well with the profiles presented in Ref. [38].

The cavitation pressure \(p_{\text{cav}}\), or, in the notation of Bayada and Chupin, \(p_\text{ml}\), for the presented results is approximately \(0.061\,{\text{MPa}}\). As shown in their paper, the obtained results can be directly compared to solutions using the JFO/EA formalism, when using the same cavitation pressure and a constant compressibility EOS in the full-film region.

Here, we proceed with another numerical test using an EOS with varying compressibility in combination with the EA algorithm, as for instance presented in Ref. [36]. The so-called twin-parabolic slider is used, consisting of two neighboring “bumps” with a parabolic shape and the same minimum and maximum gap heights as in the previous example. The length and sliding speed of the bearing are also the same as before. The pressure boundary conditions at the inlet and outlet are \(p_{\text{in}} = 3.36\,p_0\) and \(p_{\text{out}} = p_{\text{cav}}\), respectively, with ambient pressure \(p_0 = 10^5\,{\text{Pa}}\) and cavitation pressure \(p_{\text{cav}} = 0\,{\text{Pa}}\). The pressure profile is shown in Fig. 8 which agrees well with the result of Ref. [36].

Surface roughness plays an important role in lubrication, since small-scale height variations can lead to non-zero load-bearing capacities even when the contacting surfaces are macroscopically flat, which was first observed experimentally by Hamilton et al. [39]. Hence, we benchmark the presented numerical scheme on geometries that mimic surface roughness. Again, we use an example presented in Ref. [36], with a height profile given byWe study two realizations of this profile, with amplitude parameter \(a=h_0 / 2\) and two different frequencies. The number of periods n affects the gap height at the inlet, where \(n=5\) leads to a diverging inlet and \(n=4.75\) to a converging inlet, as shown in Fig. 9a. We perform simulations for gap height \(h_0 = 10\,\mu{\text{m}}\), bearing length \(L = 0.1\,{\text{m}}\), sliding speed \(U = 0.25\,{{\text{m/s}}}\), and the fluid’s viscosity is \(\eta = 0.04\,{\text{Pas}}\). The resulting pressure profiles do not only depend on the gap height and geometry of the inlet, but also on the pressure boundary condition.

In this example, we use both a pressurized inlet (\(p_{\text{in}}=2\,p_{\text{cav}}\)) and a starved inlet (\(p_{\text{in}}=p_{\text{cav}}\)) with \(p_{\text{cav}}=10^5\,{\text{Pa}}\). The former case leads to a pressure profile with load-bearing capacity for both type of geometries as can be seen in Fig 9b. Also, if the inlet geometry is converging and the ambient pressure is the cavitation pressure, a load-bearing capacity can be achieved, since incoming fluid is immediately compressed and further surface corrugations do not lead to cavitation. The obtained pressure profiles are similar for all three cases and agree very well with the results of Ref. [36].

The last case, a starved inlet with diverging geometry, is different from the previous ones as it does not generate a load-bearing pressure profile. However, the results of the transient simulation reveal that initially a substantial pressure build-up is produced, which is similar to the other three situations. With increasing time, a small cavitation zone grows from the inlet into the domain and the pressure peak decreases. Furthermore, as shown in the time evolution of the pressure profile in Fig. 9c, the velocity with which the cavitation zone expands decreases until it almost comes to a standstill at \(t = 8\,{\text{s}}\), with a remaining non-zero load-bearing capacity. This example highlights an advantage of the explicit solution algorithm in comparison to the mostly implicit solutions of the stationary Reynolds equation, since transient phenomena in lubrication are automatically resolved.

Surface slip reduces friction and the possibility to control the surface chemistry [40] opens a wide range of applications in lubrication, e.g., by tailoring the confining walls of a flat channel into slipping and sticking domains [41, 42]. Here, we study a flat channel of length \(L=2\lambda\) and height h with heterogeneous wetting properties at the top surface as reported in Ref. [43]. The lower surface sticks and is sheared at constant velocity \(U_1\). For better comparison, we here present the results for the pressure in dimensionless form, i.e., \(\tilde{p} = p/p_0\), with \(p_0 = \eta U_1 \lambda / h^2\). The sticking and slipping domains have equal length \(\lambda\), and periodic boundary conditions are applied in the x- and y-direction (infinitely long bearing).

The obtained pressure profiles for three different sliding velocities are compared to a Reynolds model including wall slip for an incompressible fluid in Fig. 10a. Since our method is generally designed for compressible fluids, we arbitrarily choose an extremely stiff EOS to mimic incompressibility. Here, we use again the Dowson–Higginson EOS and choose parameters ensuring that deviations from the reference density are negligibly small. The Reynolds model predicts a pressure gradientwith \(\lambda _1\) and \(\lambda _2\) being the size of the sticking and slipping domain, respectively. Thus, for \(\lambda _1 = \lambda _2 = L/2\), the non-dimensional pressure gradient readswith \(\tilde{x}=x/\lambda\) and \(\kappa =5b/(2h + 5b)\). The dimensionless pressure profiles normalized by the wall slip quantifying parameter \(\kappa\) agree perfectly with the prediction of Eq. (37). However, both the analytical model and the numerical results predict unphysical negative pressures, and a non-zero load-bearing capacity in symmetric systems with periodic boundary conditions can only be achieved by considering cavitation.

Therefore, we proceed with parameters that represent the n-decane system studied with molecular dynamics (MD) simulations in Ref. [43]. The channel has length \(L=143.9\,{\text{nm}}\), height \(h=5.5\,{\text{nm}}\), and sliding velocity \(U_1=10\,{\text{m/s}}\). We use the van der Waals equation as EOSwith \(a=5.273\,{\text{Pa}}\,{\text{m}}^6/{\text{mol}}^2\), \(b=3.04\cdot 10^{-4}\,{\text{m}}^3/{\text{mol}}\), \(M=142.29\,{\text{g/mol}}\), \(T=303\,{\text{K}}\), and the shear viscosity is \(\eta =0.3\,{\text{mPas}}\). To avoid negative pressure and the unphysical increase of pressure with volume in the subcritical van der Waals loop, we use the aforementioned EA approach with a cutoff at \(P_{\text{cav}}=1\,{\text{MPa}}\), which agrees with the external pressure applied in the MD simulations. Note, that in case of the van der Waals equation, the EA approach is similar to the well-known Maxwell construction [44], except that our choice of saturation pressure is not based on the equal area rule.

The dimensionless pressure profiles for the n-decane system are shown in Fig. 10b for two different slip lengths. For both systems a cavitation zone forms directly at the transition from stick to slip behavior at the top wall and the size of the cavitation zone decreases with increasing slip length. We extract the size of the full-film regions \(\lambda _1\) and \(\lambda _2\) from our simulation to compute the Reynolds solution according to Eq. (38). The analytical and numerical pressure profiles agree reasonably well. The slight curvature of the numerical pressure profiles can be accounted to the compressibility of the van der Waals equation in contrast to the incompressible analytical result.

The inset of Fig. 10b shows the non-dimensional load-bearing capacity per unit widthfor various slip lengths. The simulation results suggest that with increasing stick–slip contrast the system’s ability to support an external load increases linearly for small slip lengths \(b<<h\) until it reaches a stable value as b is in the order of the gap height or larger, which agrees with the analytical Reynolds model.

The MD simulations of Savio et al. [43] predict a substantially larger cavitation zone and, consequently, lower pressure excursion in the remaining full-film region. The pressure falls below the external pressure in front of the stick–slip boundary, where it reaches its minimum before it goes to zero in the cavitated zone. Such complex behavior governed by the nucleation, lifetime, and collapse of vapor bubbles cannot be addressed with simple macroscopic cavitation models.

This example foreshadows the potential application of the presented solver in multiscale frameworks, such as the combined use of MD simulations and continuum methods. Molecular interactions govern effects, such as the nucleation of cavitation bubbles or local variations of surface slip, that are important to accurately describe lubrication phenomena on experimental length and time scales.