The impact of a liquid drop or jet on a rigid obstacle is the fundamental hydrodynamic process that frequently takes place in nature and technology. This process is of practical and scientific interest, as it is related to a wide range of technological applications. Among them we can mention inkjet printing, additive 3D printing technologies, coating, drop erosion of turbine blades, systems of emission filtering, etc. Numerous studies of droplet impact against surfaces of various geometries are well known [1]. Large drops (diameter of 2–4 mm) and low impact velocities (up to 3 m/s) were mainly studied. This can be explained by the relative simplicity of such experiments. The problem becomes more complicated for drops and jets with a diameter of 40–50 µm that move at a velocity of more than 3 m/s. Here, special methods for generating and visualizing the jets are already required. Recently, the relevance of studies on the impact of drops and jets is associated with the spread of COVID19, infection which often occurs as a result of sneezing and coughing by a virus carrier. The infection is transmitted by droplets (microjets) of the oral fluid. Typical process parameters are as follows: the droplet diameter is ~50–100 µm and the droplet velocity is ~10 m/s [2, 3]. On the other hand, an important characteristic of a protective medical mask, whose working element is a mesh of cylindrical fibers, is its ability to delay/slow down drops falling on it.
The aim of the present study is to create a technique and simulate experimentally the collision of microjets with microobstacles for collision parameters close to real situations.
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1 EXPERIMENT
The demo Hewlett Packard ThinkJet printhead operating on the thermal inkjet principle [4, 5] was used to form microjets. Pulsed displacement of the liquid was carried out by a rapidly growing vapor bubble, which was formed as a result of local heating of liquid by a microresistor (Fig. 1). A rectangular pulse with a voltage of 23.3 V and a duration of 4.5 μs was supplied to the microresistor (pulse energy E = 35 × 10–6 J). Outlet nozzle diameter was equal to 60 µm. The working liquid (distilled water) entered the head through a silicone tube from a small (~10 mL) reservoir installed at a height of 10–20 mm relative to the level of the outlet nozzle. The mode of operation of the printhead is determined by its design and did not allow a significant change in the jet ejection velocity, which is v = 9.5–13.3 m/s. The printhead and the flash lamp were controlled by a microprocessor generator. To visualize the ejection and impact of the microjet, the printhead was placed on the stage of a Motic SMZ143 microscope equipped with a Canon 7D camera.
Fig. 1.
Diagram of the experimental facility: print head (1), microjet (2), steam bubble (3), microheater (4), fiber (5), water tank (6), microscope (7), camera (8), control pulse generator with a delay unit (9), flash lamp (10).
A cylindrical glass fiber with a diameter d = 9 and 25 µm served as an obstacle to the microjet. A fiber segment of about 25 mm long was glued to the ends of a wire holder mounted on a 3-axis table for fine adjustment of the fiber position relative to the microjet. The distance from the exit nozzle to the point of impact on the fiber was equal to 0.3–0.5 mm.
The ejection of the microjet and its impact on the fiber were photographed under pulsed illumination. The repeatability of the process and the precisely set delay of the light pulse relative to the start of the ejection made it possible to trace the stages of the jet motion with good temporal and spatial resolution: the step of setting the delay time is equal to 1 μs, the illumination pulse duration is equal to 1 μs, the jet image resolution is equal to 3 pixel/μm.
The experiments were carried out at an ambient temperature of 20–22°C.
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2 RESULTS AND DISCUSSION
2.1 Ejection and Breakup of a Microjet
In Fig. 2 we have reproduced the photos of successive stages of microjet ejection. Initially, the jet has the shape of a mushroom, whose diameter of the cylindrical part is close to the nozzle diameter (on the photographs, only its bevel edge with a diameter of 100 μm is visible, while the nozzle itself has a diameter of 60 μm). Then the jet becomes thinner. This shape of the jet indicates the variability of the fluid outflow velocity. This is evidenced by calculations [6] and direct measurements of the diameter of the displacing vapor bubble [7], as well as by the time dependence of the jet volume V(t) obtained in the present study (Fig. 3b). The jet volume was measured using an image processing program developed on the basis of the MATLAB package. In Fig. 3a we have reproduced an example of jet photo processing. The average jet volume is equal to 2 × 10–4 mm3 and the diameter of a spherical drop of equivalent volume is equal to 73 µm. At the instant of impact on the fiber the maximum jet diameter is D = 40–50 µm.
Fig. 2.
Ejection and disintegration of a water microjet. The jet velocity is equal to 10 m/s. Here and below, t is the time passed since the beginning of the jet ejection.
Fig. 3.
Profile (a) and volume (b) of the jet obtained by image processing; r is the radius and y is the distance from the nozzle.
From Fig. 3b it follows that the fluid outflow velocity from the nozzle (it is proportional to the fluid flow rate dV/dt) increases in the initial stage of the ejection (t < 2 μs), and then decreases (d2V/dt2 < 0) until its completion at t ≈ 10–15 μs (the completion of the ejection is evidenced by the constancy of the volume of the jet, dV/dt = 0). In the stage of increase in the outflow velocity, the rearguard layers of the fluid overtake the front layers. This leads to accumulation of fluid in the head of the jet, like it occurs in pulsating jets [8] or radial splashes [9]. In the next stage of slowing down and stopping the outflow, the head part of the jet moves more rapidly than the tail part. This leads to jet elongation and thinning. In the end, a drop-shaped jet is formed, in which the main part of the fluid is concentrated in its head (Fig. 2). Note that the law of variation in the outflow velocity is determined by the growth dynamics of the gas bubble that displaces fluid from the printhead.
It is interesting to note that a comparison of the microjet kinetic energy Ek = (ρV)v2/2 = 10–10 J with the electrical energy of the control pulse E = 35 × 10–6 J demonstrates that only a very small part (0.028%) of the energy supplied to the microheater goes to the motion of fluid.
Figure 4 shows the trajectory of the leading edge of the jet. It can be seen that the jet velocity remains approximately constant. Note that a small spread in the fluid outflow velocity inevitably leads to a spread in the position of the jet head, which increases with distance from the nozzle. To improve the repeatability of the impact pattern, the target-fiber should be mounted as close as possible to the exit nozzle.
Fig. 4.
Time dependence of the distance between the leading edge of the jet and the outlet nozzle. Line is an approximating function L = 0.0395+0.00945t. The jet velocity is equal to 9.45 m/s.
The jet is unstable and disintegrates into drops according to the inertial-capillary Rayleigh mechanism [10]. The characteristic breakup time is equal to tb ∼ (ρD3/γ)1/2. In the case of water and D = 50 µm this gives tb ∼ 40 µs. The experimentally observed jet breakup times agree with this estimate in the order of magnitude. It also makes it possible to estimate the distance from the nozzle L0 ~ vtb = v(ρD3/γ)1/2 = 0.4 mm at which an obstacle (target) should be installed so that a still unbroken jet flows onto it.
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2.2 Impact of a Microjet on a Fiber
Figures 5 and 6 show the stages of collision of a microjet with fibers of various diameters. In the case of a frontal transverse impact of a jet on a fiber, in all cases, separation of the jet into two approximately symmetrical parts is observed. In this case, the jet does not flow around the fiber while maintaining its continuity, as it often takes place when freely falling drops of 2–3 mm in diameter hit cylindrical targets [11‐13]. A part of liquid remains on the fiber (the more, the thicker the fiber). Under an off-center (tangential) impact, the jet passes through the fiber almost without changing its shape (Fig. 6b).
Fig. 5.
Collision of a water microjet with glass fiber with a diameter of 9 (a) and 25 µm (b); top view along the fiber axis. The jet velocity is equal to 11.8 (a) and 13.2 m/s (b). The arrow shows the location of the fiber.
Fig. 6.
Observations of the microjet collision from different angles: view perpendicular to the fiber axis (a) and non-central (tangential) impact (b). The fiber diameter is equal to 25 µm. The jet velocity is equal to 10 m/s.
We can propose the following qualitative impact model. The fiber slows down the primary jet flowing onto it and splits it into two parts (secondary jets). The fiber pushes these parts apart, initiating their transverse expansion. After the secondary jets are behind the fiber, they continue to move in the transverse direction at some velocity vx. At the same time, under the action of surface tension, parts (halves) of the jet take on a drop-like shape.
We will evaluate the factors that influence the impact process, which is a competing interaction of inertial, capillary and viscous forces. First of all, we can note that the small value of the Weber number Wea = ρ\({{{v}}^{2}}\)D/γ ~ 0.08, based on the air density ρa = 1.2 kg/m3, indicates the insignificance of aerodynamic effects. The low value of the Ohnesorge number Oh = µ/(ρDγ)1/2 ~ 0.017 indicates the predominance of capillary forces over the viscous ones. For the studied experimental parameters the Weber and Reynolds numbers are as follows: We = ρ\({{{v}}^{2}}\)D/γ = 60–126 and Re = ρ\({v}\)D/μ = 420–680, where ρ = 103 kg/m3, μ = 10–3 Pa s, γ = 0.072 N/m are the water density, viscosity, and surface tension, respectively. The high values of the Weber number show that at the instant of collision of the jet with the fiber, the inertial effects prevail over the capillary ones.
In Fig. 7 we have reproduced some examples of the time dependences of the distance between the primary (L) and two secondary jets (L1, L2) on the nozzle, as well as the dependence of the transverse distance between the secondary jets (l). The approximations of the experimental dependences make it possible to determine the longitudinal velocities of the primary and secondary jets, as well as the transverse expansion velocity of the secondary jets. The different speed of the secondary jets is a consequence of the imperfect centering of the jet impact on the fiber (which, obviously, is not achievable in principle). The data obtained indicate strong deceleration of the jet by the fiber. For a fiber with a diameter of d = 9 µm, decrease in the longitudinal velocity is equal to \({{{v}}_{y}}\)/\({v}\) = 0.54 ± 0.03 (mean value and standard deviation), and for a fiber with a diameter of d = 25 µm, we have \({{{v}}_{y}}\)/\({v}\) = 0.53 ± 0.1. Thus, the expected growth of jet deceleration with increase in the fiber diameter was not detected. This can be explained by the fact that mainly only the central part of the jet is decelerated, while its peripheral sections pass through the fiber with slight deceleration.
Fig. 7.
Time dependences of the distance of the leading edges of the primary jet L (1) and two secondary jets L1 (2) and L2 (3) from the nozzle, as well as the transverse distance between the secondary jets l (4) for a fiber with a diameter of 9 (a) and 25 μm (b). Lines correspond to the approximating functions L = 0.0192 + 0.01148t, L1 = 0.178 + 0.00683t, L2 = 0.184 + 0.00616t, l = 0.00218 + 0.00174t (a); L = 0.328 + 0.01326t, L1 = 0.144 + 0.00845t, L2 = 0.161 + 0.00755t , l = –0.062 + 0.00758t – 8.159 × 10–5t2 (b). Horizontal dotted line shows the location of the fiber.
The relative transverse expansion velocity of the secondary jets for a fiber with the diameter d = 9 µm is \({{{v}}_{x}}\)/\({v}\) = 0.07 ± 0.01, and for a fiber with the diameter d = 25 µm we have \({{{v}}_{x}}\)/\({v}\) = 0.15 ± 0.01. In this case, the expansion of the secondary jets can be slowed down and even be replaced by their convergence (Fig. 7b). This indicates the presence of a sufficiently strong (with respect to inertial forces) interaction between the secondary jets. In some frames of impact, the formation of a short-lived film connecting the secondary jets was observed. Apparently, this is the reason for the slowdown in their expansion. The mechanism of formation of this film requires individual consideration and is the subject of further studies.
We will now estimate the decrease in the longitudinal velocity of the jet using the equation for change in the momentumwhere M and \(v\) are the mass and velocity of the primary jet, respectively, \({{v}_{y}}\) is the velocity of the secondary jet after impact, and F is the force exerted on the jet from the fiber during the time Δt. Neglecting the effects of viscous friction, we have
$$M(v - {{v}_{y}}) = F\Delta t,$$
(2.1)
$$F\sim \left( {Dd} \right)\rho {{v}^{2}}{\text{/}}2 + 2D\gamma .$$
(2.2)
Here, the first term is the hydrodynamic drag force Fh and the second term is the doubled surface tension force Fc (Fig. 8a). When setting M ~ ρπD3/6 and Δt ~ D/\(v\) and taking into account (2.2), from (2.1) it follows that
$$\frac{{{{v}_{y}}}}{v}\sim 1 - \frac{{3d}}{{\pi D}} - \frac{{12}}{{\pi {\text{We}}}} \approx 1 - \frac{d}{D} - \frac{4}{{{\text{We}}}}.$$
(2.3)
Fig. 8.
On the estimation of the velocity of secondary jets. (a) Impact diagram: jet (1), secondary jet (2). (b) Comparison with experiment: straight line corresponds to dependence (2.3), points 1 correspond to the experimental data for d = 9 μm, points 2 to the experimental data for d = 25 μm, and dashed curve to the theoretical boundary (2.4) between jet deceleration and capture with the fiber.
Using (2.3), we can obtain the criterion for the capture of a jet by a fiber (\({{v}_{y}}\) = 0)
$${\text{We}} < \frac{4}{{1 - \frac{d}{D}}}\quad {\text{or}}\quad \frac{d}{D} > 1 - \frac{4}{{{\text{We}}}}.$$
(2.4)
Inequality (2.4) makes it possible to estimate the fiber diameter \({{d}_{*}}\), which will completely stop the jet with given characteristics (diameter, velocity, density, surface tension). Thus, for the case under study (\(v\) ~ 10 m/s and D ~ 50 μm), estimate (2.4) gives \({{d}_{*}}\) ~ 47 μm. The boundary (2.4) between the regimes of deceleration and jet capture is shown in Fig. 8b with a dotted line.
Estimate (2.3) is in agreement with the measurement results in the order of magnitude (Fig. 8b). At the same time, the predicted dependence of the degree of jet deceleration \({{v}_{y}}{\text{/}}v\) on the fiber diameter is not fully confirmed by experiment. A possible explanation is that the hydrodynamic resistance force Fh exerts only on a layer of liquid with a thickness of the order of d, while the rest of the jet passes through the fiber without noticeable resistance. It should also be noted that there is a large spread in the measured values of the degree of deceleration caused by the influence of factors that are difficult to control. It is possible that the improvement of the experimental technique and the data processing procedure (for example, the use of computer image analysis to determine the shape of secondary jets) will improve the repeatability of the results and will make it possible to discover new laws in the process of microjet impact on the fiber.
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3 SUMMARY
It was found that under the experimental parameters studied, the frontal impact of a water microjet on a microfiber leads to its splitting into two parts (during the tangential impact, the jet passes through the fiber almost without changing its shape). The longitudinal velocity of these parts (secondary jets) is approximately by two times, and the transverse velocity of their expansion is by 7–15 times less than the initial impact velocity. This indicates significant deceleration of the jet by the fiber. Estimates of the contribution of various forces to the impact dynamics indicate its predominantly inertial nature. The developed experimental technique makes it possible to study the processes of collision of microjets with fibers and opens up new possibilities for studying the high-speed interaction of liquid micro-objects with various obstacles.
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Translated by E.A. Pushkar