Ternary phase-field simplified multiphase lattice Boltzmann method and its application to compound droplet dynamics on solid surface in shear flow

A ternary phase-field simplified multiphase lattice Boltzmann method (TPF-SMLBM) is developed and employed in numerical investigation of a compound droplet placed on solid substrate in shear flow at moderate Reynolds numbers. The TPF-SMLBM utilizes the simplified multiphase lattice Boltzmann method as the flow solver and the ternary phase-field model as the interface tracking algorithm. Compared with conventional lattice Boltzmann method, this method preserves advantages in memory cost and boundary treatment. The numerical investigations recover three major modes of motion, namely the quasisteady sliding, tumbling-sliding, and tumbling-detachment. The quasisteady sliding dynamics are analyzed from the perspective of energy conversion, which explains the exponential shrinking rate of the wetting length in early evolution stage. By using the force-balance analyses, the onset of tumbling motion is firstly elaborated as the breakup of the balance in quasisteady sliding. A dimensionless parameter named as the tumbling number $\mathrm{Tu}=\mathrm{Ca}/{\mathrm{Re}}^{-0.5}$, where Ca and Re are, respectively, capillary and Reynolds numbers, is proposed to identify the mode transition towards tumbling. The physical rational of this parameter can be established from both the force analyses and the balance of timescales, and is further validated by abundant simulation results which show that its criticality is roughly on the order of $O$(1) in the low-density–viscosity ratio scenarios. For the tumbling-sliding mode, the re-adhesion of tumbling droplet portion generates extra diffusive flow behavior and shrinks the wetted length of the sliding droplet portion. The dynamics of detachment is also investigated, showing that the critical capillary number of detachments can be scaled by ${\mathrm{Ca}}_c,{}_{\mathrm{detach}}\sim {\mathrm{Re}}^{-0.3}$.

Figure 1

Schematic diagram of a compound droplet placed on a substrate in a shear flow.

Figure 2

Grid-independence tests at equal density and viscosity ratios with Re=50 and Ca=0.10. The black and the red curves represent the results obtained on mesh sizes of 801 × 201 and 1601 × 401, respectively. The solid and the dashed lines are interface locations of fluid 1 and fluid 2, respectively. Note that the left and the right boundaries of the computational domain are periodic.

Figure 3

Illustration of three kinematic modes for a compound droplet moving in the shear flow. Presented in each subfigure are interface locations at dimensionless time of $T$=5 (black lines), 15 (red lines), 30 (blue lines), and 50 (green lines). The solid lines represent the interface location of fluid 1, and the dashed lines depict the interface of fluid 2. Simulation results are from case series 1. Note that the channel is periodic on its left and right boundaries. The reference kinematic modes from Ref. [31] are also included as the small window figures for comparison.

Figure 4

Pressure contours and interface locations at dimensionless time $T$=1 for cases with different contact angles of ${\theta }_{13}={60}^{\circ }$ (top), ${90}^{\circ }$ (middle), and ${150}^{\circ }$ (bottom). Density and viscosity ratios are both 1:1:1. Re=50, Ca=0.10.

Figure 5

Time histories of the wetted length on the substrate for the droplet portion of fluid 1 at Re = 50. Data are collected in case series 1. Inserted interface locations are for the case of Re = 50 and Ca = 0.05 at two dimensionless time of $T$=10 and 20.

Figure 6

Time histories of the wetted length on the substrate for the droplet portion of fluid 1 at Ca=0.04. Data are collected in case series 1.

Figure 7

Evolution of the upstream contact-line velocity of the droplet portion of fluid 1 in quasisteady sliding mode for some typical examples in case series 1.

Figure 8

Dimensionless contact-line speed ${\mathrm{Ca}}_{cl}$ as a function of the capillary number Ca for cases at Re=15. Data are collected in test series 1.

Figure 9

Vorticity contours and streamlines at Re=50 and with different capillary numbers of Ca=0.03 (top), 0.04 (middle), and 0.05 (bottom). Results are from case series 1.

Figure 10

Vorticity contours and streamlines at Ca=0.03 and with different Reynolds numbers of Re=10 (top), 40 (middle), and 100 (bottom). Results are from case series 1.

Figure 11

Critical capillary number for the onset of tumbling mode as a function of Reynolds number.

Figure 12

Phase diagram of kinematic modes for case series 1. The coordinates are capillary and Reynolds numbers in log scale.

Figure 13

Phase diagram of kinematic modes for case series 2. The coordinates are capillary and Reynolds numbers in log scale.

Figure 14

Phase diagram of kinematic modes for case series 3. The coordinates are capillary and Reynolds numbers in log scale.

Figure 15

Critical tumbling numbers at each Reynolds number for all case series. The values of averaged critical tumbling numbers are attached at the end of fitted lines. The upper limit of the error bar is the lowest tumbling number for the tumbling mode, while the lower limit of the error bar represents the highest tumbling number for the quasisliding mode.

Figure 16

Snapshots of interface locations and zoom-in views of vorticity contours and streamlines at Re=50 and Ca=0.06 in case series 1. The top subfigure snapshots the detachment of droplet portion of fluid 1 from the substrate; the middle subfigure shows the re-adhesion of the droplet portion of fluid 1 to the substrate; and the bottom subfigure illustrates the equilibrium state of quasisteady sliding. Note that the volume-averaged sliding velocity of the compound droplet has been deducted from the initial velocity field, based on which the vorticity and stream functions are calculated. Note that the left and the right boundaries of the computational domain are periodic.

Figure 17

Evolutions of the relative locations of the contact lines and the wetted length at Re=50 and Ca=0.06. Note that the contact-line location refers to the relative location to its initial position. Data are collected from case series 1.

Figure 18

Snapshots of detachment criticality for two typical cases. Zoom-in views are the local vorticity fields and streamlines near the detachment area. The volume-averaged horizontal velocity of the compound droplet has been deducted from the velocity field. The solid lines are interface of the droplet portion of fluid 1, and the dashed lines depict the interface location of the droplet portion of fluid 2. Results are from case series 1. Note that the left and the right boundaries of the computational domain are periodic.

Figure 19

Vorticity contours of the snapshots before detachment for two typical cases. The solid lines are interface of the droplet portion of fluid 1, and the dashed lines depict the interface location of the droplet portion of fluid 2. Results are from case series 1. Note that the left and the right boundaries of the computational domain are periodic.

Figure 20

Evolutions of the relative locations of the contact lines and the wetted length of the droplet portion of fluid 2 for cases of: (a) Re=10, Ca=0.12; (b) Re=100, Ca=0.12. Note that the contact-line location refers to the relative location to its initial position. Data are collected from case series 1.

Figure 21

Critical capillary numbers of detachment as the function of Reynolds number in log scale. The fitted lines are with the slope of $k$ = −0.3.

Figure 22

Schematic diagram of the spreading liquid lens. The solid lines and the dashed lines represent the interfaces of fluid 1 and fluid 2, respectively.

Figure 23

Density contours and interface locations for the spreading of a liquid lens. The solid lines and the dashed lines represent the interfaces of fluid 1 and fluid 2, respectively.

Figure 24

Schematic diagrams of the simulation of a compound droplet on substrate.

Figure 25

Computed equilibrium shapes of compound droplet deposited on substrate.