Doubly excited pulse waves on thin liquid films flowing down an inclined plane: An experimental and numerical study

The interaction patterns between doubly excited pulse waves on thin liquid films flowing down an inclined plane are studied both experimentally and numerically. The effect of varying the film flow rate, interpulse interval, and substrate inclination angle on the pulse interaction patterns is examined. Our results show that different interaction patterns exist for these binary pulses, which include solitary wave behavior, partial or complete pulse coalescence, and pulse noncoalescence. A regime map of these patterns is plotted for each inclination angle examined, parametrized by the film Reynolds number and interpulse interval. Finally, the individual effect of the system parameters mentioned above on the coalescence distance of binary pulses in the “complete pulse coalescence” mode is studied; the results are compared to numerical simulations of the two-dimensional Navier-Stokes equations yielding good agreement.

Figure 1

Schematic of the experimental setup for doubly excited pulse waves on flowing liquid films, comprising the falling film rig, and a pulse-excitation and high-speed imaging system. The items are 1: glass substrate; 2: high-speed camera; 3: computer; 4: reservoir tank; 5: pump; 6: solenoid valve; 7: ultrasonic flow meter; 8: halogen lamp; 9: light diffuser.

Figure 2

Generated pulse-wave excitation signal for valve control: $A$ is the pulse duration, $B$ is the interpulse interval, $T$ is the period of the pulse, and $t_0$ is the interval prior to wave excitation.

Figure 3

Shadowgraphic images showing the evolution and propagation of the pulse waves in the solitary wave behavior mode. The snapshots in (a–c) have been captured at 0.19, 0.28, and 0.47 s, respectively. The parameter values are $\mathrm{Re}=111,\mathrm{We}=4.27,\beta =5^{\circ },A=0.04\mathrm{s},B=0.04\mathrm{s}$, and $T=0.08\mathrm{s}$. See the Supplemental Material [46], Movie 1.

Figure 4

Space-time plot obtained from high-speed images for the parameters in Fig. 3. The propagation line of the wave hump has been highlighted here for emphasis.

Figure 5

Numerical results showing computed spatiotemporal propagation of doubly excited pulse waves in the solitary wave behavior mode at 0.06 s interval. The parameters here correspond to those used to generate Fig. 3. See the Supplemental Material [46], Movie 2.

Figure 6

Shadowgraphic images showing the evolution and propagation of the pulse waves in the complete pulse coalescence mode. The snapshots in (a–c) have been captured at 0.42, 0.50, and 0.71 s, respectively. The parameter values are $\mathrm{Re}=83.3,\mathrm{We}=5.47,\beta ={10}^{\circ },A=0.04\mathrm{s},B=0.08\mathrm{s}$, and $T=0.12\mathrm{s}$. See the Supplemental Material [46], Movie 3.

Figure 7

Space-time plot obtained from high-speed images. The propagation lines of the wave humps have been thickened here for emphasis.

Figure 8

Numerical results showing computed spatiotemporal propagation of doubly excited pulse waves in the coalescence mode with 0.06 s interval. The parameters here correspond to those used to generate Fig. 6. The points labeled A and B show the onset and completion of coalescence, respectively. See the Supplemental Material [46], Movie 4.

Figure 9

Variation of the coalescence point with the pulse interval in the coalescence mode, expressed in dimensionless variables. The red, blue, and green points trace the coalescence points at substrate inclination angles of $5^{\circ },{10}^{\circ }$, and ${15}^{\circ }$, respectively. The pulse duration, $A$ has been maintained at 0.04 s. The inset is a dimensional plot of coalescence point against dimensional pulse interval.

Figure 10

Variation of the coalescence point with the Reynolds number for different substrate inclination angles: $5^{\circ },{10}^{\circ }$, and ${15}^{\circ }$ at a constant interpulse interval.

Figure 11

Shadowgraphic images showing the evolution and propagation of the pulse waves in the partial coalescence mode. The snapshots in (a–c) have been captured at 0.38, 0.68, and 0.80 s, respectively. The parameter values are $\mathrm{Re}=83.3,\mathrm{We}=5.47,\beta ={10}^{\circ },A=0.04\mathrm{s},B=0.12\mathrm{s}$, and $T=0.16\mathrm{s}$. See the Supplemental Material [46], Movie 5.

Figure 12

Space-time plot obtained from high-speed images. The propagation lines of the wave humps have been thickened here for emphasis.

Figure 13

Numerical results showing computed temporal propagation of doubly excited pulse waves in the partial coalescence mode at 0.06 s interval. The parameters here correspond to those used to generate Fig. 11.

Figure 14

Shadowgraphic images showing the evolution and propagation of the pulse waves in the total noncoalescence mode. The snapshots in (a–c) have been captured at 0.15, 0.30, and 0.38 s, respectively. The parameter values are $\mathrm{Re}=83.3,\mathrm{We}=4.79,\beta ={15}^{\circ },A=0.04\mathrm{s},B=0.12\mathrm{s}$, and $T=0.16\mathrm{s}$. See the Supplemental Material [46], Movie 7.

Figure 15

Space-time plot obtained from high-speed images. The propagation line of the wave humps have been thickened here for emphasis.

Figure 16

Numerical results showing computed temporal propagation of doubly excited pulse waves in the pulse noncoalescence mode at 0.06 s interval. The parameters used correspond to those used to generate Fig. 14. See the Supplemental Material [46], Movie 8.

Figure 17

Flow regime map of the pulse-wave interactions based on the film Re and dimensionless interpulse interval at $5^{\circ },{10}^{\circ }$, and ${15}^{\circ }$ corresponding to (a–c), respectively. The black triangle, green square, red circle, and blue star represent the different interaction modes which are solitary mode, coalescence, partial coalescence, and total noncoalescence, respectively.