1 Introduction

Studies of gas bubble departure in liquids, from a needle or an orifice, are conducted in the fields of chemical processes, pharmaceutical industries or environmental science [1]. A periodic bubble flow is investigated for the mining of hydrocarbon deposits. The appearance of a chaotic flow of the gaseous phase in the liquid during the extraction of hydrocarbons leads to disturbances in the amount of extracted raw materials and in extreme cases may lead to damage to the extraction equipment [2]. In medicine, ultrafine gas bubbles are used to transport of medicaments [3]. An occurrence of periodic bubble departures is necessary during the water aeration process, which is used to e.g. water purification [4, 5]. Moreover, a better understanding of the factors which lead to chaotic or periodic bubble departures is helpful for controlling underwater methane flow. Consequently, this knowledge can be applied to reduce the greenhouse effect [6].

Studies of liquid velocity above the needle after bubble departures are crucial for understanding bubble–liquid and bubble–bubble interactions. The hydrodynamic interaction between the departed bubble and needle can modify the time of bubble growth and bubble waiting time [1, 7]. Consequently, for bubbles departing from a single needle, changes of liquid velocity can cause chaotic bubble departures [8]. For bubbles departed from twin needles, chaotic changes of liquid velocity above the needle outlet can cause desynchronization of simultaneous or alternating bubble departures [9, 10].

There are a lot of papers reporting the non-linear behaviors of the bubbling process [11,12,13,14,15,16,17]. In these papers, two groups of phenomena responsible for the appearance of chaotic bubble behaviors are investigated. The first group is connected with bubble interface oscillations and liquid flow above the needle or orifice. The second group is connected with processes which appear in the gas supply system.

PIV techniques are generally used to identify the liquid flow around the bubbles. In paper [18], the phenomena responsible for the appearance of the bubble’s oscillatory trajectory was analysed using the PIV technique. It was shown that the transition to oscillatory trajectory of the bubble is connected with the appearance of two vortex tubes of stream-wise vorticity behind the bubble. It appears when the bubble aspect ratio exceeds a critical value (for large Reynolds number), and bubble growth becomes unstable and induces a horizontal force acting on the bubble. In paper [19], it was shown that the chaotic character of bubbles’ trajectories is caused by departed bubble shape and liquid flow generated by the moving bubble in the bubble column. Paper [20] shows that bubbles rise in approximately identical trajectories up to a distance of less than 40 mm from the needle outlet. The distance of bubble trajectories from the vertical axis increases as bubble departure frequency increases. In paper [21], it was shown (using the PIV technique) that the variation of bubble rising trajectory and bubble shape is closely associated with the liquid’s viscosity but less related to the frequency of bubble formation. The bubble departure diameter was about 6 mm. It was observed that bubble rising trajectory changes from a rectilinear path to a zigzag and spiral path as the liquid’s viscosity decreases. Paper [22] employs the PIV method to estimate liquid velocity above the needles. In this investigation, attempts were made to determine the velocity of the liquid above the needle, however, this method only allowed for determination of velocity in the macro scale. Consequently, in the present paper, investigations of changes of hydrodynamic boundary conditions above the needle during bubble growth and bubble waiting time stages were performed using pressure sensors located in the neighborhood of the needle.

In the present paper, the influence of bubble departure on liquid pressure in the needle’s neighbourhood was investigated. In the experiment, bubbles were generated from a needle (inner diameter was 1.1 mm). The air volume flow rate supplied to the needle was selected so that the bubbles departed periodically at two characteristic frequencies. Every 0.85 s, a bubble with a diameter of 7.1 mm departed from the needle, and then after 0.063 s, a smaller bubble departed (diameter of 4.6 mm). In order to investigate liquid pressure changes in the neighbourhood of the needle, a measuring needle was used. The measuring needle was placed at a distance of 1.5 mm from the main needle (the needle from which the bubbles were generated). In the measuring needle, the water–air interface moved as the velocity of the liquid above it changed. Changes of interface position cause the pressure changes in this needle (tip of the needle). These pressure changes were measured by a pressure sensor and recorded using the data acquisition station. Simultaneously, the pressure fluctuations in the main needle, caused by the bubble departures stages, were recorded. Bubble growth was recorded by a high-speed camera. The pressure changes and the frames of videos were synchronized. It was shown that subsequent stages of bubble growth modify the hydrodynamic conditions around the needle. During the bubble growth stage, the liquid velocity decreases, and the bubble growth time increases. When the liquid pressure in the needle’s neighbourhood decreases—the liquid’s velocity increases, the height of liquid penetration into the capillary is deeper and the bubble waiting time becomes lower. Moreover, the changes of liquid pressure and liquid velocity above the needle modify the shape of the growing bubble.

2 Experimental setup and measurement technique

In the experiment, bubbles were generated in a glass tank (200 × 150 × 500 mm), which was filled with distilled water. The bubbles were generated from a brass needle (in the present paper this needle is called the main needle) with an inner diameter equal to 1.1 mm. The neighbouring needle was used to record the changes of liquid pressure (this needle is called measuring needle) during bubble departure. A diagram of the experimental setup is shown in Fig. 1.

Fig. 1
figure 1

Experimental setup: 1-Glass tank, 2 l-pressure-measuring brass needle s, 2r-brass needle generating bubbles, 3-pressure sensors, 4-air valve, 5-flow meter, 6-air tank, 7-proportional air valve, 8-air pump, 9-camera, 10-light source, 11-semi-transparent glass

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The needles were placed at the bottom of the tank. Air pressure changes were measured using an MPX12DP silicon pressure sensor, whose sensitivity was 5.5 mV/kPa. The pressure changes were measured in both needles. The air pressure sensors were connected to the data acquisition system (DT9800 series USB Function Modules for Data Acquisition system), which allows us to record signals from two or more channels without time delays. The air pressure changes were recorded with a sampling frequency of 2 kHz. The air volume flow rate was equal to q = 0.011 l/min and was set using the air valve and controlled using a BRONKORST flow meter. The accuracy of the flow meter was equal to 5%. The air flow rate was selected so that the bubbles departed at two characteristic frequencies. The first bubble (with diameter equal to 7.1 mm) departed after 0.85 s. The next bubble (with diameter 4.6 mm) departed 0.063 s after the first. The error of bubble diameter measurement was equal to one pixel (0.1 mm).

During the experiment, the videos containing the bubbles' departure process were recorded with a CASIO EX FX 1 high-speed camera. The videos, recorded in grayscale (1200 fps), were divided into frames. The time series of pressure changes and the frames of videos were synchronized. In this method laser–phototransistor system and semi-transparent glass were used. In measurements the part of the laser beam light was going through the semi-transparent glass and fell on the phototransistor and the part of laser beam light was reflected. In the frames of films we observed the laser beam light reflected from the glass plate and data acquisition system recorded the laser beam presence on the phototransistor—in the form of a voltage time series. During the experiment the laser beam was covering by sheet of paper and we observed vanishing of the laser beam in the frames of films and decrease of voltage generated by phototransistor—recorded by data acquisition system. The point of laser beam decay on the film frame and the first point of the lowest voltage value was the data synchronization point. In order to assign the frames contain the subsequent bubble stages above the needle to the time series of pressure fluctuations, the time series of pressure fluctuations were resampled to 1200 Hz using a computer program. The synchronization error is related to the difference in video sampling frequency (1200 Hz) and frequency recording by data acquisition station (2000 Hz) and it is about 1.3 ms. More details of the method of data synchronization has been described in paper [23].

At the start of the experiment, the position of the air–liquid interface in the measuring needle was placed about 1 mm above the needle’s tip. In this case, the bubble’s departure from the main needle was not observed. The initial position of the interface is correlated with the hydrostatic pressure of the liquid above the needle’s outlet. Bubble departure from the main needle caused the velocity of the liquid above the needles to change, and consequently, caused changes of the interface’s position in the measuring needle. Changes of the air–liquid interface cause changes of air pressure in the needle, which were measured by pressure sensors. In contrast to the widely used PIV methods and acoustic methods, the use of a measuring needle next to the main needle is helpful in measuring the liquid pressure differences caused by successive stages of bubble growth. As a consequence, it also allowed us to study the pressure conditions above the needle during the bubble growth stage. Changes in the pressure of the liquid are correlated with the changes in the velocity of the liquid.

3 Experimental results

The time series of pressure fluctuations in the main and measuring needles are presented in Fig. 2.

Fig. 2
figure 2

Time series of pressure fluctuations in the main (dashed line) and measuring (continuous line) needles

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It was observed that the pressure fluctuations in the main needle are synchronized with the pressure changes in the main needle (Fig. 2). This means that the pressure fluctuations in the measuring needle are caused by the successive stages of bubble departures.

The pressure fluctuations in the needles for a single cycle of bubble departure and synchronized frames of videos showing the successive bubble growth stages are presented in Fig. 3.

Fig. 3
figure 3

Example of pressure changes in the main (dashed line) and measuring (continuous line) needles and frames of videos showing bubble departure stages a–e

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The time series of pressure changes in the gas supply system (main needle) is shown as a continuous line in Fig. 3. The pressure changes in the measuring needle were marked with a dashed line (Fig. 3). The frames of videos showing the subsequent stages of bubble departure are presented above the time series in Fig. 3a–e. In the experiment, bubbles depart periodically at two frequencies. Every 0.85 s, a bubble with a diameter of 7.1 mm departed from the needle, and then after 0.063 s, a smaller bubble departed (diameter of 4.6 mm). The bubble waiting time was equal to 0.72 s. Displacement of the liquid from the needle causes a decrease in liquid pressure in the needle’s neighborhood (Fig. 3b–c). The pressure decrease continues throughout the growth of the greater bubble (Fig. 3c–f). Just after bubble departure, the liquid pressure increases (Fig. 3g). During growth of the smaller bubble, the pressure stabilizes. Then, when the liquid penetrates into the needle, the liquid pressure increases slightly (Fig. 3a).

The changes of liquid pressure and liquid velocity above the needle modify the shape of the growing bubble. The first bubble grew over a shorter time than the second bubble in the same cycle of bubble departures. During the growth of this bubble, a decrease of pressure value was observed in the measuring needle. The shape of the first bubble is close to a sphere, and its diameter is equal to 7.1 mm. The second bubble grows when the pressure of the liquid above the needle increases. The elevated pressure of the liquid surrounding the growing bubble causes a decrease in the drag force. The shape of this bubble is close to an ellipsoid, and its diameter is equal to 4.6 mm.

4 Discussion

Based on experimental and numerical results, the liquid’s flow above the needle during the successive stages of bubble departure is proposed. Schematic changes of liquid flow above the needle are presented in Fig. 4.

Fig. 4
figure 4

Schematic changes of liquid flow above the needle during the bubble waiting time and successive stages of bubble departures. a bubble waiting time—liquid penetrates into the needle, b bubble waiting time—maximum position of meniscus, c bubble waiting time—discharge of liquid from the needle, d the beginning of bubble growth, e bubble growth, f growth of the subsequent bubble

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During undisturbed (by bubble growth) liquid flow in the needle’s neighbourhood, the liquid pressure is steady at the maximum measured value. The position of the air – liquid interface in the measuring needle was placed about 1 mm behind the needle’s tip. The first stage of bubble waiting time—liquid penetrates into the needle (Fig. 4a), causes a pressure increase in the main needle e (Fig. 3a, − pc). The liquid flows above the measuring needle, which causes a slight decrease in liquid pressure (Fig. 3a, − pcn). In the next stage, the liquid stops flowing into the needle and vl = vg = 0 (Fig. 4b), then the liquid pressure above the measuring needle temporarily increases. Next, the liquid is displaced from the needle (Fig. 4c), and the liquid pressure in the main needle’s neighbourhood decreases (Fig. 3b, c). Right after the needle is emptied of the liquid, the bubble growth stage is observed (Fig. 4d). The greater bubble’s growth causes the formation of the liquid flow around the bubble and an increase in liquid velocity (vpp1) in the needle’s neighbourhood. Consequently, the bubble growth stage causes a decrease in liquid pressure in the needle’s neighbourhood (Figs. 3c–f, 4e). Just after the bubble’s departure, and as the bubble moves away from the needle, the liquid pressure decreases (Figs. 3g, 4f), because the bubble–needle hydrodynamic interaction becomes weak. During growth of the smaller bubble, the pressure stabilizes. Then, during the liquid’s penetration into the needle, the liquid pressure increases slightly (Figs. 3a, 4a).

5 Conclusions

In contrast to the widely used acoustic method, the proposed method allows for investigating pressure changes in the neighbourhood of the needle from which the bubbles are generated. Pressure changes can be recorded simultaneously in the measuring needle and in the needle used to generate bubbles. Synchronization of data from the acquisition station and data from the high-speed camera allowed us to compare the results of pressure changes in the control needle with the shape and size of the generated bubbles. The applied control needle does not interfere with the flows generated by growing bubbles.

Subsequent stages of bubble growth change the hydrodynamic conditions around the needle. The greatest pressure changes are observed during the bubble growing time. During this time in firs moment the pressure increases slightly and after that it decreases significantly until the first bubble breaks off (diameter is equal to 7.1 mm). Moreover the changes of liquid pressure are connected with the bubble diameter, the growth of second bubble (diameter is equal to 4.6 mm) causes the increase of pressure to neutral value. During the bubble waiting time we observe slight pressure fluctuations.

The changes of liquid pressure and liquid velocity above the needle modify the shape of the growing bubble. The shape of the first bubble, which grew during the decrease of liquid pressure in the needle’s neighbourhood, is close to a sphere, and its diameter is equal to 7.1 mm. The shape of the second bubble, which grew during the increase of liquid pressure in the needle’s neighbourhood, is close to an ellipsoid, and its diameter is equal to 4.6 mm.

Presented in the paper method of the measured of liquid pressure above the needle can be used in investigations of bubble interaction and controlling of bubble departure process. In addition, the results presented in the paper can be used to numerical investigation of the interaction between bubbles detaching from two or more needles.