Turbulent impingement jet cleaning of thick viscoplastic layers

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I. INTRODUCTION
Cleaning is an essential unit operation in the fast-moving consumer goods (FMCG) industries, including the manufacture of pharmaceuticals, food, and personal care products [1,2].
Cleaning is performed to remove residual material at product changeover, to ensure batch integrity, and to remove fouling layers [3].The cost of cleaning can be considerable, comprising direct costs for equipment, cleaning agents, energy and treating the wastes generated as well as indirect costs associated with lost production [4].Many operations in these sectors employ automated cleaning methods, notably cleaning-in-place (CIP) operations, in which liquids are circulated through equipment in a sequence of steps, for example, to purge, clean, rinse out cleaning fluids, and prepare the surfaces for the next operation [5][6][7].CIP protocols are often overdesigned to ensure complete cleaning, and design is still approached in a semi-empirical manner [8].Improving cleaning processes requires a better understanding of the underlying science so that test results can be reliably related to process performance [9].
In this context, the material being removed is typically termed the "soil" or "soiling layer" (also adopted in this work), due to its spatially uniform composition and properties.Fouling layers, which accumulate over time, are rarely uniform as the local properties depend on the history of the material at a given depth [3].
CIP operations typically employ liquid flows to overcome cohesive forces within the soil layer and adhesive forces between the soil and the surface [10,11].Wilson and Landel [12] have characterized the flows as confined (filling the flow channel, as in heat exchangers, valves, and piping) or free-surface (e.g., flows generated by impinging jets, droplet impact, and films created by sprays, and falling films).This work focuses on one aspect of cleaning by impinging liquid jets, where a nozzle, which may be stationary or moving, directs a steady stream of liquid towards a surface coated by a soil.Very viscous soils, particularly those exhibiting viscoplasticity such as hair gels [13], some lotions [14], and toothpastes [15], require the CIP fluid to provide momentum in order to achieve cleaning over an acceptable timescale, and in practice this is often achieved by impinging liquid jets.
The thickness of the liquid film generated by an impinging jet, ĥ, is of order ' Dj /2', where Dj , is the diameter of the jet.The nozzle diameter ( Dn ), or its radius, therefore sets a characteristic length scale for fluid flow problems.The other characteristic dimension is the thickness of the soil layer, δ0 , and Fernandes et al. [16] identified three regimes for cleaning (see Table 1): (i) very thin layers ( δ0 < ĥ), (ii) thin layers ( δ0 ∼ ĥ), (iii) thick layers ( δ0 >> ĥ).Most of the studies of impinging jet cleaning to date (e.g., Table 1) have focused on very thin and thin layers.For instance, Yeckel and Middleman [17] conducted a classic investigation into the removal of a viscous, hydrophobic Newtonian oil layer by an impinging water jet, comparing experimental data with a model for the rate of removal.Fernandes and Wilson [18] recently presented a first-order model for the rate of removal of thin layers of an immiscible viscoplastic soil fluid, which explained the observed variation in cleaning rate with radial position, layer rheology, and jet flow rate.Their model was based on considerations of the viscous dissipation rate within a shallow wedge of soil at the cleaning front.Table 1 provides an overview of previous studies focused on soil layer cleaning using impinging jets.Note that, throughout this paper, we distinguish dimensional parameters by adding a circumflex (ˆ), while dimensionless parameters are presented without it.
Thick soil layers tend to be found in the lower regions of tanks and similar containers, particularly in cases where the container walls direct toward a drain.There has been relatively little work on the cleaning of thick soil layers by impinging jets until recently.Tuck et al. [26] reported experimental studies, using inclined jets, on the cleaning of thick viscoplastic soil layers (Carbopol ® layers, δ 0 = 1 − 8) and did not find good agreement with the Fernandes and Wilson model [18].They dyed the soil with a fluorescent tracer and imaged its removal from above: they observed a distinct delay between jet impingement and the appearance of a cleaned region, attributed to water entrapment under the thick viscoplastic layer, leading to the formation of a 'blister'.The observed cleaning rate was strongly affected by the scaled layer thickness, δ 0 = δ0 / Dn .Chee et al. [27] recently studied the removal of thick viscoplastic layers, with 0.33 ≤ δ 0 ≤ 4.65, by turbulent coherent impinging jets impinging normally onto the soil layer.They observed removal both from above, measuring the (radial) size of the blister or crater (denoted b, see Panel (c) in Table 1) and from below, considering the radius of the region clear of soil (denoted â).They estimated the shape of the cleaning front as a conical frustum crowned by a spherical cap, characterized by angle α.They reported a transition between cratering and blistering regimes when δ 0 ≳ 1 and explored the cleaning rate and behavior after a blister ruptured.They employed 4 different soil materials and characterized the rheology in terms of the inertial factor reported by Uth and Deshpande [28] in their study of water jets tunneling into viscous greases, viz.: Table 1.Overview of previous studies on the cleaning of viscous soil layers using impinging liquid jets.In the schematics (first column), the brown and blue colors denote soil and cleaning fluids, respectively.Here, Dn , ĥ, Vj,0 , τy , and δ0 denote nozzle diameter, liquid film thickness, injection velocity, yield stress of soil layer, and soil layer thickness, respectively.

Impinging jet cleaning Reference
where ρ, Vj,0 , and τy represent fluid density, injection velocity, and yield stress of soil layer, respectively.They studied opaque soil layers, so they were not able to confirm the shape of the cavity formed when the water jet penetrated the thick layer.
When a less viscous fluid is injected into or displaces a more viscous one, a well-known hydrodynamic instability, known as viscous fingering, can occur [29,30].Controlling viscous fingering in viscoplastic fluids relies on different key factors, including injection velocity, viscosity ratio, yield stress, and surface tension [31][32][33].In the context of cleaning soil layers with impinging jets, the radial flow after impingement displaces the soil fluid, forming a cleaned area with an approximately circular pattern [26,27,34].However, Chee and Wilson [24] showed that, in certain instances, the cleaned areas deviate from circular patterns, displaying non-circular cleaned areas, with obvious fingers at the interface between the cleaning and soil layer fluids.
Interest in the penetration of a low viscosity liquid jet into a nominally stationary viscoplastic fluid has increased, owing to its importance in the oil and gas industries.For instance, as oil and gas wells reach the end of their operational lifespan, plugging and abandonment (P&A) operations are conducted to prevent adverse environmental impacts [35].A vital step in P&A operations is the proper cleaning of the target region, removing undesirable materials that exhibit solid-gel features, comprising spent drilling mud, debris, formation cuttings, poor cement, etc. [36][37][38].Jet cleaning is known to be an efficient method for displacing/removing these materials [39].These operations differ from tank cleaning in the absence of a free surface, but there are evident overlaps with the initial stages of jet cleaning where the jet tunnels into the soil layer.While Hassanzadeh et al. [40] recently reported the application of a series of techniques to quantify the penetration of a liquid jet into large volumes of transparent viscoplastic fluids (Carbopol solutions), this paper reports the use of these techniques, and others, to investigate the cleaning of thick layers (2 of Carbopol solutions.The techniques allow the shape of the cavity generated by the jet to be monitored directly and features, such as the cavity radius and cleaning front angle, to be determined accurately.The use of transparent soil layers reveals, for the first time, the presence of air bubbles entrained by the jet, and a brief analysis of the bubbles' characteristics, including their trajectory, diameter, and velocity, is also presented.
The paper is organized as follows: Section II outlines our experiments, materials, and techniques employed to study the problem.In Section III, we begin with a phenomenological analysis, followed by an exploration of cleaning features, and subsequently, the presentation of bubble characteristics.The paper concludes with Section IV.

A. Setup
The experimental setup, depicted schematically in Figure 1a, featured a transparent rectangular tank with a square cross-section (0.The jet liquid was injected via a pump through a cylindrical nozzle positioned 0.1 m above the center of the target plane to ensure a coherent jet at the impingement point (less than the critical length, see [41]).The varied parameters were the layer thickness and the rheology of the soil fluid.Considering fixed values for the nozzle diameter and flow rate (see Table 2) resulted in: where Re and W e are the Reynolds and Weber numbers, respectively.In most of the experiments, the jet liquid was dyed to aid visualization.A volume of soil fluid was initially added to the bottom of the tank to create a layer with the desired thickness.The same ratio of the volume of jet liquid injected to the volume of the initial soil layer, approximately 0.1, was employed in all experiments.The dimensional parameter ranges are summarized in Table 2.According to the literature, the surface tension of water is 0.072 kg/s 2 [42,43].
All tests were performed at ambient temperature of 23  )) conducted at room temperature using a rheometer (DHR-3, TA Instruments) with a parallel-plate geometry (40 mm diameter, 1 mm gap).To mitigate wall slip effects during measurement, fine sandpapers (functioning as rough surfaces) were fixed to the rheometer plates.Figure 1b presents the results for the two Carbopol solutions used in our study, with yield stress behavior evident at lower shear rates for both solutions.The curves show the fits to the Herschel-Bulkley model [47]: where τ , γ are the shear stress and shear rate, respectively.Also, τy , κ, and n represent the yield stress, consistency and power-law index, respectively.The inset in Figure 1b depicts the variation of the viscosity versus the shear rate, calculated from μapp = τ γ , and it shows shear-thinning behavior.The fitted parameters for the Carbopol solutions are reported in Table 3.For convenience, we use the labels Newtonian, low-yield stress fluid, and high-yield stress fluid, for fluids I, II, and III, respectively.
We also conducted oscillatory rheometry tests at a fixed frequency (1 Hz) to explore potential relations between the viscous and elastic behaviors of our experimental fluids.
Figure 1c depicts the variation of the storage modulus ( Ĝ′ , representing the elastic characteristic) and the loss modulus ( Ĝ′′ , representing the viscous characteristic) as a function of the applied shear stress for our Carbopol gel samples.Initially, at small shear stresses, the storage modulus surpasses the loss modulus, indicating the solid-like behavior of the Carbopol gels before yielding.However, as the shear stress increases beyond a critical value, a significant drop in the storage modulus is observed.Simultaneously, the loss modulus initially rises, crosses the storage modulus curve, and then sharply decreases.Thus, Ĝ′′ becomes larger than Ĝ′ , signifying a transition from elastic solid-like behavior (with weak viscous properties) to elasto-viscoplastic behavior (with weak elastic properties).
While the elastic properties of Carbopol gel can be of importance under specific conditions, their effects on the jet cleaning dynamics are presumed to be negligible in this study, due to their relatively minor role compared to the dominant viscous and yield stress properties.To evaluate this assumption (i.e., negligible effects of elasticity), let us compare the elastic time scale, estimated as λ ≈ ( κ Ĝ′ ) 1 n [40], with our experimental time scale.Our findings indicate that the elastic time scale ranges from 0.003 to 0.005 seconds for Ĝ′ at the critical strain (near the yield stress).On the other hand, our experiments, which involve injecting a constant ratio of the jet fluid into the layer fluid, take between 0.5 and 5.0 seconds to complete.This comparison suggests that elastic effects are unlikely to play a significant role, as the elastic time scales are considerably shorter than the experimental time scales.
As another potentially relevant parameter, let us briefly discuss the potential influence of wall slip in our experiments.During the cleaning process, the domain can be divided into two main regions: inside and outside the cavity.Within the cavity, minimal surface slippage may be expected due to its predominantly Newtonian fluid content.On the other hand, outside the cavity, where the yield stress fluid (i.e., Carbopol gel) interacts with the solid surface, slippage of Carbopol on the bottom surface could possibly occur.In our experiments, the scenario where a viscoplastic fluid is pushed by a Newtonian fluid at the wall resembles displacement flow in a confined geometry.Literature on displacement flows, where a Carbopol gel is displaced by a Newtonian fluid, indicates no slip conditions at walls, especially downstream of the displacing flow [48][49][50].In addition, in our experiments, given the considerable layer thickness and the finite extent of the bottom surface, we expect the wall slip to have minimal impact.Furthermore, by adding particles inside the layer, significant movement on the wall are not observed (see Supplementary materials).However, some studies on drop spreading [51][52][53] and fluid flow in micro-channels [54,55] highlight the some effects of wall slip on viscoplastic flow behavior.Therefore, investigating its precise effects on the cleaning mechanism remains a promising avenue for future research.

C. Measurement techniques
We utilized high-speed imaging as the primary technique to capture the layer fluid flow behavior.This involved illuminating the entire domain from the top, side, and back using light-emitting diodes (LEDs), with a diffuser plate placed between the LEDs and the experimental setup to ensure uniform illumination.For visualization, we dyed the jet liquid using black ink (Higgins Fountain Pen India Ink).The experiments were captured using a high-speed digital camera (Canon EOS R5 with a Canon RF 24-70 mm lens) featuring a 45megapixel full-frame sensor.The experimental images were typically recorded at intervals of 8.3 milliseconds.Bottom-view images of the flow, obtained through a mirror positioned below the setup (as illustrated in Figure 1a), were also captured using the same camera.
The recorded light intensities were transformed into concentration maps using an in-house MATLAB code, enabling the detection of the boundary between the transparent and dyed fluids.
Particle image velocimetry (PIV) was used to estimate the velocity fields.Traditional PIV faces challenges in flows containing several bubbles, owing to the uncontrollable scattering of laser light.The use of bubbles (ranging from micrometer to centimeter sizes) as tracers has been introduced as a novel technique for calculating velocity fields (see [56][57][58][59]) and this approach was adopted to map the velocity field of the bubbles generated by the impinging jet.
In effect, the bubbles were considered as natural tracer particles.However, it is important to note that the main goal of the current study is to track the velocity behavior of the bubbles themselves, not the velocity field of the entire flow.To do so, the central vertical plane was illuminated sideways via a thin green laser sheet (∼ 0.002 m) generated by a DPSS laser system (Laserglow Technologies, wavelength = 532 nm, see Figure 1a).The high-speed digital camera (Canon EOS R5 with a Canon RF 24-70 mm lens), set perpendicular to the laser light sheet, allowed bubble motion to be traced.Image analysis by cross-correlation of two successive images at a high frame rate (short time interval) was conducted using the open-source software PIVlab [60].

III. RESULTS
This section highlights our main experimental findings, showing the substantial effect of yield stress and initial layer thickness on flow characteristics.First, we outline the typical flow behavior during the cleaning of thick viscoplastic layers using impinging jets.Following that, we characterize the flow behavior and reveal how various parameters can modify these characteristics.

A. General observation
Figure 2 illustrates the cleaning a thick layer (δ 0 ≈ 5.5) of different fluids by impinging jets (example videos can be found as Supplementary materials).For visualization purposes, the first row features dyed injection fluid, while the second row uses undyed water.In the case of the Newtonian layer, as shown in Figure 2a with water as the layer fluid, the jet penetrates the layer and forms a crater which subsequently expands along the base.In the early stages, splashing of the layer fluid and oscillation of the free surface are evident.Similar behavior is observed with the low-yield stress fluid, as seen in Figure 2b.However, there is less splashing of the layer fluid, less oscillation of the free surface, and the dyed jet liquid does not extend as far from the impingement point.With the high-yield stress fluid, as depicted in Figure 2c, the jet penetrates to the base and creates a cavity rather than a crater, with minimal fluid splashing.As the jet liquid spreads radially outward near the base, an observable bulge develops on the free surface above the cavity, and both features become more pronounced with time.This corresponds to the blister formation reported by [18,26].In our experiments, blister formation is exclusively observed in the presence of high-yield stress fluid (fluid III) across all thicknesses, as shown in Table 4.For other fluids and varying thicknesses, blister formation does not occur, and the cleaning exhibits distinct behaviors, such as cratering, mixing, etc.The range of blister formation observed in our experiments aligns with the trend reported by Chee et al. [27], who identified blister formation occurring within the range of IF −1 = (3.0− 20) × 10 −3 for thick soil layers (i.e., δ 0 > 1.5).
Inspection of the images in the bottom row of Figure 2a  in plunging jet aerators [61,62] and reactors [63], our experiments bring to light, for the first time, the presence of bubbles during the soil layer cleaning process.In Figure 2a  Dimensionless experimental times listed under each column.
resulting in an increase in the dark region.In the low-yield stress layer, a cratering regime seems to be established; however, the jet cannot mix well with the layer fluid, leading to a decrease in area.In the high-yield stress layer, the jet expands primarily below the blister region, resulting in the smallest area.

B. Geometrical features
In the jet cleaning process, the jet penetrates the layer, tunnels down to the base, and forms a cavity, pushing the layer fluid away along the bottom surface.A blister can form when the resistance to flow in the layer fluid favors sideways expansion rather than the breakup of the cavity's roof.It grows, eventually rupturing and forming a crater surrounded by a berm of layer fluid.When the jet liquid escapes from the cavity, it entrains soil layer fluid.In this work, the evolution of cavity growth is characterized in terms of the four dimensions depicted in Figure 3a, namely: • â, the radius of the base surface region cleaned by the jet; • ĉ, the maximum radius of the cavity; • α, the angle between the jet liquid and the base, analogous to a contact angle; • b, the radius of the blister region, on the top of the layer.
Parameters â and b are equivalent to those measured by Chee et al. [27]: they employed opaque viscoplastic fluids so they were unable to measure ĉ and they also estimated α from their measurements of â and b. Figure 3b provides an example of the bottom view of the cavity, with the red dashed line marking the extent of the cavity, exhibiting a roughly circular pattern (example videos can be found in the Supplementary material).This circularity is consistent with previous studies in the literature [26,27,64].There are noticeable fluctuations at the boundary of the cavity, suggesting a form of viscous fingering (due to the large viscosity ratio [65,66]) and/or the presence of bubbles.Chee et al. [24] also observed the fingering instability at the boundary of the cleaned area.This particular aspect warrants further attention.Here, we examine how each geometrical feature responds to variations in the main parameters.If we assume the force as a fraction of the momentum within the liquid film at a radius of â, the removal rate can be written as (assuming that the gravity effects are negligible): where t and M represent the time and momentum within the liquid film, respectively.f (ζ) denotes the strength of the viscoplastic layer, which can depend on parameters such as the layer thickness and yield stress (i.e., ζ = τy , δ0 , etc.).In the cleaned area, a film with a local thickness of ĥ1 − ĥ0 and an average velocity of Ûm is formed, characterized by an assumed velocity profile: Here, ŷ denotes the distance from the base, r is the radial position, and Û corresponds to the liquid velocity along the surface at position r and ŷ.Note that ĥ0 and ĥ1 denote the minimum and maximum heights of the cavity, respectively, at each time.The assumed velocity profile in the liquid film could be valid for any thick liquid film layer generated by an impinging jet, where ĥ ≳ Dj /2.If there is no fluid escaping from the cavity, we can write the conservation of mass as follows: where ĥ1 ĥ0 F dŷ = ĥ1 − ĥ0 .Then, M , which represents the momentum flux per unit of periphery, can be defined as follows: where ρ is the density of the jet liquid, with the note that in our work, ρ = ρj = ρa .If , therefore: Using Equation (7), Equation (8), Equation (9), we can arrive at the following form for M : Then, we consider a radial element (inside the liquid film) from r to r + dr and conduct a momentum balance on it, under the crude assumption that the pressure within the layer remains almost constant: where τw is the wall shear stress (τ w = μj ).Then, using Equation ( 6) and Equation (10), along with the assumptions of ĥ0 = 0 and rj ≤ r ≤ â, the following is obtained: Then, Ûm can be found as follows: Combining Equation ( 5), Equation (7), Equation ( 8), Equation ( 9) and Equation ( 13) at r = â leads to: Lastly, integration leads us to the final expression for â in the form of: where K = 5 f (ζ) 0.06 π 3 ρμ j m3 , termed cleaning rate constant.
In light of the concept of L c , one can reasonably assume that if L c < δ 0 , the jet loses its ability to effectively clean the target plate, resulting in a = 0.This suggests the dependence of jet cleaning behavior on both IF and δ 0 .For example, Chee et al. [27] demonstrated that within a constant range of IF −1 , i.e., IF −1 = (3 − 20) × 10 −3 , blister formation occurs only when δ 0 > 1.5.On the other hand, according to our findings (refer to Table 4), within a constant layer thickness range, i.e., δ 0 = 2.6 − 22.2, blister formation is observed only when We provide more details in Appendix A. In Figure 4a, we observe the variation of a versus time for various layer thicknesses, for the high-yield stress fluid.As seen, as the layer thickness increases, the cleaning rate reduces, leading to a corresponding decrease in K value.Also, our results confirm that the cleaning exhibits a growth rate proportional to t 0.2 , as obtained by our model.Figure 4b shows how a varies over time in different fluids, at a constant thickness of δ 0 ≈ 11.1.One can observe that as the yield stress increases, there is a corresponding reduction in the cleaning rate, indicated by the K values in Figure 4b.These results are in agreement with the findings presented by Tuck et al. [26].However, they noted an initial delay in the cleaning process for the thick layers.As mentioned by Chee et al. [27], this delay seems to be due to their measurement approach, which captured the flow behavior from the top, making the cleaning area not visible until a certain point.However, in Chee et al. [27] and the current study, where the cleaned area was captured from the bottom and side views, respectively, no such delay was observed.Moreover, the early values corresponding to the Newtonian layer cannot closely follow the trend, potentially due to the mixing of jet and layer fluids, which directly affects the cleaning behavior.
Figure 4c presents the variation of the cleaning rate constant against layer thickness for various layer fluids, with the errorbars representing the fitting error at 95% confidence bounds.As can be observed, increasing both δ 0 and the yields stress of the fluid results in a decrease in K, in agreement with the findings in the literature [26].Therefore, it can be concluded that increasing the yield stress and thickness of the layer contributes to an increase in the layer resistance, which, in turn, intensifies the cleaning complexity and, subsequently, leads to a reduction in the cleaning rate.

Maximum radius of cavity (ĉ):
When dealing with the cleaning of a thick layer using impinging jets, a cavity is formed within the layer due to the entrapment of the injection fluid, and this cavity gradually grows with time.To quantify the cavity formation, we introduce the parameter ĉ, representing the maximum radius of the cavity.Figure 5a presents a schematic of cavity formation.As the cavity develops within the layer, a blister can form at the free surface due to the displaced fluid (see Figure 3a and Table 4).Based on the literature [27], the blister formation can be characterized by the parameter b, signifying the distance between the jet centerline and the edge of the blister (i.e., the radius of the deformed region on the surface of the soil layer).
Figure 5b shows that the maximum radius of the cavity closely matches that of the blistered region, a consistent trend observed across all blistering cases.This finding supports the assumption made by Chee et al. [27], although they were unable to confirm it due to the opacity of their layer fluids.Here, our focus is on the behavior of ĉ, and as depicted in Figure 5b, it can also provide insights into the variation of b.
To estimate the growth of cavity, we adopt the assumption that no jet fluid can escape from the formed cavity.This allows us to model the cavity formation as taking on the shape of a spherical frustum, as schematically depicted in Figure 5a.To proceed, we establish the following parameters: ĉ denotes the hemisphere's radius, and Ĥ represents the height of the virtual spherical cap.Now, using the volume conservation, the following can be obtained: where B represents the spherical frustum volume.To determine B, we subtract the volume of virtual spherical cap [67] from the hemisphere volume: where ĉ = Ĥ + ĥ.If we assume a linear relation between ĥ and ĉ (i.e., ĥ ĉ = λ), we can substitute Ĥ with (1 − λ)ĉ, where λ is a constant, and find: Now, we define ĉ = Ûc t, where Ûc represents the rate of radial increase.Then, the following equation can be expected to govern the growth of the maximum cavity radius: where ω is the constant coefficient.It is important to note that, based on Equation (20), ω depends on Vj,0 , Dn and λ.Given that the injection velocity and the nozzle diameter remain constant across all experiments, here, the variation of ω is solely controlled by λ.This observation potentially implies that our assumptions regarding cavity shape and the absence of liquid escape may be better supported in the high-yield stress fluid.Figure 5d presents the variation of ω values, obtained from Equation (20), versus δ 0 for different fluids.
As seen, the trend indicates a general decrease in ω with an increase in layer thickness.
Furthermore, enhancing the yield stress of the soil layer results in a decrease in ω.Therefore, these findings demonstrate that increasing both the yield stress and layer thickness enhances the layer resistance, leading to a subsequent decrease in the growth rate of the cavity.

Angle between jet boundary and surface (α):
Within the context of cleaning thick layers, α is known as another important geometric parameter of interest, representing the angle between the jet boundary at the end of â and the bottom surface (see Figure 3a and Figure 5a).In the recent study by Chee et al. [27], α was calculated using values of â and b: However, our experiments allow us to directly determine α through image analysis, thanks to the transparency of our fluids, regardless of the values of â and b.
Now, let us attempt to develop a simple model for determining α.To do so, we crudely assume that for r > â, τw ≈ τy at ŷ = ĥ0 (r); therefore, Equation ( 11) can be written as: and finally, we arrive at: where ĥ1 represents the height of the cavity in each time.Therefore, α (degree) can be calculated as follows: Figure 6a presents the temporal evolution of α for a typical experiment (fluid II & δ 0 = 5.6).The diamond, circle and square symbols represent α values obtained through direct measurement, Equation (24) and Equation ( 21), respectively.Firstly, α reaches almost steady values, which is consistent with the typical trend observed in the literature [27].
In addition, although the calculation of α based on â and b (i.e., Equation ( 21)) tends to significantly overpredict the experimental data, Equation ( 24) shows good agreement with the observed values.However, it is important to acknowledge that, since Equation ( 21) was originally formulated based on the cleaning thin layers [22], it might not be able to properly predict the variations in α observed in thick layers.Note that in the direct measurement approach, α for each time step is calculated by taking the average of the left and right angles (i.e., α = α L +α R
To have a better understanding, we investigate the variation of the steady values of α (α s ) versus the layer thickness across different layer fluids, as illustrated in Figure 6b.The filled and hollow symbols represent α s obtained using direct measurement and Equation (24), respectively.As seen, there is a trend of decreasing α s as the layer thickness increases.
Furthermore, transitioning from the Newtonian fluid to viscoplastic fluids is associated with a decrease in α s .In addition, predicted values by Equation ( 24) show a reasonable agreement with the experimental data.

C. Comparison between impingement submerged and plunging jets
Plunging and submerged jets are two distinct types of jet flows, each exhibiting unique behavioral features.The key differentiation between them is the distance between the nozzle and ambient fluid: in plunging jets, there is a significant gap between the nozzle tip and the ambient fluid, whereas submerged jets feature a configuration where there is no gap, often with the nozzle tip immersed within the ambient fluid.Hence, it is reasonable to consider the jet impinging into a thick layer as resembling a plunging jet, given the presence of a gap between the jet and the soil layer, and the fact that the jets hit the horizontal free surface of the liquid.The impingement of the jet onto the free surface of ambient fluid forms a cavity at the point of impact, accompanied by the entrainment of air [62].respectively.In each panel of Figure 7, the first and second rows depict the sequence of experimental snapshots of submerged and plunging jets, respectively.Also, the third row presents the sequence of snapshots for the plunging jet experiments, wherein both the jet and ambient fluids are transparent, thereby providing a clearer perspective on the air entrainment phenomena observed in our experiments.In the second and third rows of Figure 7a, the air entrainment process can be clearly observed.Based on these snapshots, we can divide the air entrainment process into two main phases: (i) the first generation of bubbles (FGB) and (ii) the second generation of bubbles (SGB).It can also be observed that the penetrating jet initially drags a cylindrical air void beneath the water surface (see the first frame).Once this void reaches a specific depth below the free surface, a significant portion of it detaches from the main body and transforms into a toroidal air bubble.This toroidal bubble then continues its descent until it hits the bottom surface, typically splitting into two large bubbles, thereby constituting FGB.In this stage, the jet entrains continuous small bubbles under the water, resulting in the formation of SGB.Then, FGB rapidly ascends towards the free surface and eventually disappears by the final frame.Another important observation when comparing the first and second rows of Figure 7a is that the mixing in the plunging jet seems to be more pronounced compared to the submerged jet, as evidenced by the larger black region in the second row than in the first row in the final frame.
While the same general scenario repeats for other fluids, as seen in Figure 7b for the low-  7c for the high-yield stress fluid (fluid III), it is crucial to highlight the significant influence of the yield stress on these dynamics.Increasing the yield stress leads to notable changes; for instance, the initial cylindrical void under the free surface becomes smaller.In the case of the low-yield stress fluid (Figure 7b), the toroidal bubble shape modifies and decreases in size compared to the Newtonian fluid.Furthermore, FGB moves upward but, before reaching the free surface, becomes entrapped within the main jet body.Moreover, SGB loses their ability to freely disperse in the ambient fluid, and the yield stress of the fluid keeps them in a confined area.In the high-yield stress fluid (Figure 7c), the toroidal bubble shape is completely changed, and instead of reaching the bottom surface intact, it collapses into numerous smaller bubbles with various sizes and shapes, forming what we refer to as SGB; consequently, FGB fails to form.In addition, SGB is trapped in a more confined area compared to the previous cases.It is important to note that, in all cases, most of the bubbles exhibit nearly spherical shapes.
By taking a closer look at Figure 7, an interesting behavior exhibited by bubbles comes to light.In Figure 7a, both FGB and SGB successfully escape from the jet fluid (i.e., cavity), becoming visible outside the black region.On the other hand, Figure 7b shows different behavior: only FGB tries to escape, moving near the edge of the jet but beyond its boundaries.In this case, there are almost no SGB visible beyond the black region.However, the most notable difference occurs in Figure 7c, where neither FGB nor SGB succeed in escaping from the jet fluid, resulting in the absence of observable bubbles beyond the confines of the black region.Now, let us attempt to understand the reason behind this behavior by examining the governing forces and effects.According to theoretical considerations [68], the movement of bubbles within a viscoplastic fluid is expected to be determined by the balance between three key forces: the yield stress, buoyancy, and surface tension forces.Considering the almost constant value of σ across all cases, it seems that the surface tension force may not be the primary driver behind the variations in bubble behavior within our experiments.Therefore, simplified relations can be obtained by writing the balance between the main forces (i.e., the yield stress and buoyancy forces): where Db is the bubble diameter and Y is known as the yield number [69].The aforementioned relation suggests that the motion of bubbles is predominantly governed by the yield number, with the existence of a critical yield number serving as a pivotal threshold.
Below this critical value, bubbles can navigate through the ambient fluid, whereas beyond it, bubbles are unable to pass through.For example, in Figure 7a, the yield number is zero, indicating the Newtonian fluid, hence, all bubbles can easily navigate within the ambient fluid.However, in Figure 7c, where the yield stress is large, no bubbles can penetrate the ambient fluid.In Figure 7b, the yield number seems to be about the critical value, where the large bubbles (i.e., FGB) can migrate into the ambient fluid, while the smaller bubbles (i.e., SGB) are unable to do so.).In all panels, the filled and hollow symbols correspond to the plunging jet (δ 0 ≈ 22.2) and the submerged jet (δ 0 ≈ 27.8), respectively.
Figure 8 compares the temporal evolution of the geometrical features for plunging and submerged jets.To minimize the effects of the fluid height on the flow behavior, we compare cases with closest heights, i.e., for plunging jet δ 0 ≈ 22.2 and for submerged jet δ 0 ≈ 27.8.
Note that this small height difference is considered to have a negligible effect on parameters over the long timeframe.For additional details, please see Appendix B. As seen in Figure 8a, as expected, the growth in a is significantly lower in the yield stress fluid compared to the Newtonian fluid, for both types of jets.In addition, it can be generally stated that a exhibits a greater increase in submerged jets compared to plunging jets, for fluids I and II.This variation could potentially be attributed to the influence of bubbles present in the fluid, affecting the overall density (density of jet+air), leading to a more upward direction in the plunging jets; as a consequence, a decreases.However, in the high-yield stress fluid (fluid III), the growth of a in the plunging jet surpasses that in the submerged one.The potential reason lies in the influence of the yield stress, which restricts the motion of the jet and overcomes the buoyancy force.As a result, the movement within the cavity increases, due to the presence of SGB, contributing to an increase in a In a simplified explanation, with an increase in the yield stress, it can be observed that c of the plunging jet attempts to surpass that of the submerged jet, and it finally accomplishes this in the high-yield stress fluid.This phenomenon may be attributed to the movements of bubbles in a more confined region, but a more comprehensive study is needed for a deeper understanding.
The evolution of α over time for submerged and plunging jets is plotted in Figure 8c.
As observed, α exhibits a non-monotonic behavior with increasing the yield stress, whereby transitioning from the Newtonian fluid to the low-yield stress fluid results in a decrease in α, and further transitioning to high-yield stress fluid leads to an increase in α.Also, α in the submerged jet almost displays higher values in comparison to the plunging jets, across all of the fluids.Similarly, in the context of droplet collisions on solid surfaces, it has been reported that entrapped air bubbles within droplets can modify the contact angle at the liquid-solid interface during such collisions [70].

D. Presence of bubbles
Section II C showed that the presence of bubbles can effect certain geometrical features of our interest.Thus, it is insightful to characterize the bubble behavior and understand its evolvement with respect to different parameters.To do so, we define the certain characteristics, including the bubble trajectory, the bubble diameter, and the bubble velocity, and conduct a brief investigation into how these features vary in response to the main parameters.Note that, in this section, we consistently use the same time and length scales as in the earlier part of the paper to make our parameters/variables dimensionless, employing Dn for length, Dn / Vj,0 for time, and Vj,0 for velocity.

Bubble trajectory
In Figure 9a, we illustrate an example of the bubbles trajectory during the cleaning a thick soil layer.In this particular scenario, the layer consists of the low-yield stress fluid (i.e., fluid II), and the initial thickness is δ 0 ≈ 22.2.As mentioned in the previous section, the air entrainment process results in the formation of two types of bubbles, namely FGB and SGB.
In a typical experiment, during the initial phase when the jet interacts with the free surface, it carries a pocket of air shaped like a toroid into the layer.This toroidal bubble then impinges on the bottom surface, leading to its division into a few randomly distributed large bubbles.On the other hand, SGB forms through the continuous entrainment of air into the layer by the jet flow, resulting in a significantly smaller bubble size and a higher quantity compared to FGB.Moreover, in certain cases, FGB and toroidal bubbles may collapse, transforming into several smaller bubbles, which are then considered as SGB.As can be observed in Figure 9a, firstly, FGB is notably larger than SGB.Also, FGB appears earlier in the process compared to SGB.Interestingly, both bubble generations maintain relatively consistent sizes over time.Moreover, FGB tends to travel closer to the jet centerline, while SGB exhibits more prominent movement in the x direction and less in the y direction when compared to FGB.Furthermore, the asymmetric behavior of both FGB and SGB seems to result from the random generation of bubbles and their interactions with each other, the jet flow, and the bottom surface.• FGB fails to form in high-yield stress fluids (i.e., fluid III).
• When the initial layer thickness is δ 0 ≈ 2.8, the proximity of the free and bottom surfaces restricts the formation of FGB, resulting in their absence regardless of the fluid type.
• In the yield stress fluids, most of the bubbles gather near the cleaning front.However, as the yield stress increases, the confined region (i.e., cavity size) decreases, consequently causing the bubbles to migrate closer to the centerline.
• In the Newtonian fluid, a reduction in the initial layer thickness typically causes SGB to migrate further away from the jet centerline.Conversely, in the case of yield stress fluids, a decrease in the initial thickness tends to draw the bubbles nearer to the jet centerline.
• In general, increasing the fluid yield stress and reducing the initial layer thickness results in the SGB traveling less in the y direction.
• SGB exhibits their closest and farthest presence from the jet centerline in the highyield stress and the Newtonian fluids, respectively, both scenarios occurring at an initial thickness of δ 0 ≈ 2.8.

Bubble diameter
The bubble diameter is another important parameter, and delving into its analysis provides valuable insights that enhance our comprehension of the flow dynamics.Figure 10a displays the time-dependent evolution of median bubble diameter of SGB for various fluids at constant initial thickness (δ 0 ≈ 11.1).Also, to gain more insights, the time-averaged normal probability density function of bubble diameter ( P DF ) is plotted in Figure 10b for different fluids.Note that the normal probability density function ( P DF ) is given by [71]: where d and σ represent the mean diameter and the standard deviation of the distribution, respectively.As can be seen, over time, there is no significant change in the bubble diameter across all cases.Interestingly, an increase in the yield stress leads to larger bubble diameters.In other words, within viscoplastic fluids, the yield stress exerts a resistance force, hindering bubble divisions, and as a result, larger bubbles can be found in the highyield stress fluid.It should be noted that the number of bubbles analyzed depends on the layer fluid, layer thickness, and capture time.On average, the number of bubbles at each timestep is 410 for fluid I, 230 for fluid II, and 160 for fluid III.This trend seems reasonable because as the yield stress increases, bubble division decreases and less air is entrained into the layer.Considering the average number of bubbles, one potential reason that the normal distribution may not adequately capture the dispersion of the data is the sample size limitation.In addition, the inherent complexity of bubbles generations, interactions, divisions, and coalescences may also contribute to this discrepancy.The influence of the initial layer thickness on the time-averaged median bubble size is depicted in Figure 10c for different fluids.Our observation reveals that variations in the initial layer thickness do not have a significant impact on the time-averaged median bubble size.It is important to note that throughout our experiments, our jet flow remained coherent and continuous.This consistency is crucial because any instabilities or disturbances in the jet, such as breakup, oscillation, or vibration, can significantly impact bubble dynamics.For instance, research by Guyot et al. [72] demonstrated that oscillations in jet flow generally result in a reduction in bubble penetration depth.Similarly, Boudina et al. [73] showed that jet breakup prior to reaching the free surface can also lead to a decrease in bubble penetration depth.Now, let us attempt to determine the possible critical value of the yield number in our experiments.In our experiments, bubbles of differing sizes are entrained in both the jet and Carbopol gels.Close to the jet, as the fluids flow outwards, it is likely that the fluids are yielded.Further away, bubbles may become trapped in Carbopol that is slowly moving or static.This is governed by the yield number Y , which depends on the bubble shape and orientation [68,69].Given that the average sphericity (or circularity) of bubbles in our experiments, calculated following the methods of [74][75][76], exceeds 0.8 across all cases, it is reasonable to assume that the shape of bubbles in our experiment is nearly spherical.As explained in the previous section, when dealing with the Newtonian fluid (fluid I), both FGB and SGB experience free movements, easily escaping from the jet fluid (i.e., the dark region) and traveling through the ambient fluid.This free motion is attributed to the yield number being zero for both generations, resulting in the absence of any competing major resistance forces against buoyancy.When dealing with the high-yield stress fluid, the escape of bubbles from the jet fluid seems impossible, primarily attributed to the large yield number (Y > 1.0).However, the scenario varies for the low-yield stress fluid.In this case, although SGB remains trapped within the jet fluid, FGB successfully escapes the jet fluid and navigates within the ambient fluid, remaining close to the jet interface.Given the constant yield stress, the distinguishing factor between the behavior of FGB and SGB lies in the buoyancy forces, calculated directly based on the bubble diameters.Based on our calculations, the yield number for FGB and SGB approximates Y ≈ 0.07 and Y ≈ 0.6, respectively.Interestingly, our findings generally align well with the reported values in the literature.The theoretical critical values for spherical bubbles reported in [77,78] are Y ≈ 0.58 & 0.71, suggesting that, beyond these thresholds, bubbles do not move in viscoplastic fluids.Moreover, Sikorski et al. [79] proposed Y ≈ 0.5 as the critical yield number in an experimental study.

Bubble velocity
Our PIV tool delivers two-dimensional velocity fields depicting the motion of bubbles over time at the central plane of the system (z = 0).Note that, here, our objective is to obtain the velocity fields for SGB.The obtained instantaneous velocity can be decomposed into the mean and fluctuation velocities: where u b and v b represent the velocity component in the x and y directions, respectively.
Also, the symbols "¯" and " ′ " signify the mean and fluctuation of the velocity components, respectively.Note that all velocities are made dimensionless by the injection velocity, Vj,0 .To better compare the velocity behaviors across various fluids, we conclude this subsection by presenting two-dimensional contours of the mean velocity profile in the x − y plane, with the mean velocity calculated as follows: Considering the bubble velocity patterns illustrated in Figure 11b-d, we can derive the following insights: • The dimensionless mean bubble velocities in fluid I (Figure 11b), fluid II (Figure 11c), and fluid III (Figure 11d) fall within the ranges of (0, 0.36) m s , (0, 0.08) m s , and (0, 0.07) m s , respectively.
• The bubble rise velocity within a Newtonian fluid can be determined using the proposed correlation in the literature [80,81]: using which the bubble velocity in water (i.e., fluid I) in our experiment can be predicted as Vb ≈ 0.38 m s .This prediction is almost consistent with the range of our PIV results, particularly in regions far from the bottom surface where the influence of bottom impingement diminishes, allowing the bubble to exhibit its own velocity.It is important to highlight that we rely on two assumptions when using Equation (29) for our Newtonian case: (i) negligible viscous drag and (ii) the nearly spherical shape of the bubble.
• In the Newtonian fluid (i.e., Figure 11b), the bubble velocity exhibits a relatively axisymmetric behavior, whereas in the yield stress fluids (i.e., Figure 11c and Fig- ure 11d), the velocity deviates from this axisymmetric pattern.
• When transitioning from the Newtonian fluid to the low-yield stress fluid, there is a significant reduction in the velocity magnitude.However, when moving from the lowyield stress fluid to the high-yield stress fluid, the velocity magnitude experiences a slight decrease.This observation generally aligns with the related literature on bubble motion in yield stress fluids [79,82], wherein an increase in effective viscosity (or yield stress) typically corresponds to a decrease in the bubble velocity.
• As the yield stress increases, the region with negligible velocity ( Vb ≈ 0) develops, indicating that bubbles are constrained to a more confined area with a higher yield stress.
• In the Newtonian fluid, the velocity magnitude typically rises with an increase in the fluid height (with the maximum velocity observed near the free surface), which aligns with the findings in the literature [83].However, in the yield stress fluids, the maximum velocity is observed near the bottom surface.

IV. SUMMARY
Our experimental study delved into the complex flow dynamics of turbulent jets used for cleaning viscoplastic layers.Employing non-intrusive experimental techniques, we analyzed the impact of the initial layer thickness and the rheological characteristics, in particular the yield stress, on the cleaning behavior.To quantify the cleaning process, we considered various geometrical features, including â, ĉ, and α.Our findings demonstrated that, as the yield stress and initial layer thickness increase, there is a decrease in the cleaning rate, owing to the increased resistance of the layer.Also, our proposed relation for the cavity growth reasonably predicted the trend of our experimental results in terms of the temporal evolution of ĉ.Moreover, our developed model for predicting α revealed a good agreement with the direct measurement.
The results of our research revealed that, apart from the jet liquid, air entrainment, producing bubbles, play a role in interacting with the jet to clean the viscoplastic layer.Our observation confirmed that the entrapment of bubbles within the layer contributes to the formation of blister shapes.We also characterized the behavior of bubbles in the cleaning layers, including their trajectory, size distribution, and velocity.For instance, we observed that a higher yield stress leads to larger bubble sizes, shorter travel distances, and decreased bubble velocity.These insights into the flow dynamics and bubble interactions contribute to a better understanding of the thick viscoplastic layer cleaning using turbulent jets.
Our study has a potential to guide further research and applications in this area, paving the way for advancements in cleaning technologies.Future research directions may involve exploring instabilities observed in the cleaned area radius and the cavity radius.Also, conducting a detailed analysis of the bubbles behavior and their interactions within the thick cleaning layer can provide valuable insights.Investigating the effects of varying parameters (e.g., Re and W e) presents another avenue for exploration.Moreover, examining the effects of surface parameters such as wall slip and roughness, as well as fluid elasticity, would deepen our understanding of cleaning behavior in realistic conditions.Finally, obtaining exact velocity profiles within the cavity would offer further insights into the underlying cleaning mechanism.

2 FIG. 1 :
FIG. 1: (a) Experimental setup (dimensions not to scale) for our viscoplastic fluid layer (VPF) cleaning study, using high-speed imaging and PIV.(b) Shear stress (τ ) vs. shear rate ( γ) for fluid II (red triangles) and III (blue circles), with Herschel-Bulkley model fit (Equation (4)) and inset showing apparent viscosity data.(c) The results of the oscillatory shear tests: filled symbols represent storage (G ′ ) moduli, while hollow symbols represent loss (G ′′ ) moduli.The measurements for fluid II and fluid III are denoted by red and blue colors, respectively.
Figure2c, the bubbles are confined to a smaller area (around the base of the blister cavity) than in the other two cases.While the entrainment of air into a liquid has been reported

17 ,Figure 2b 17 FIG. 2 : 12 FIG. 2 :
Figure 2b 17 and Figure 2c 17 , we observe that the flow boundary changes over time for fluids with varying yield stress levels: none, low, and high, respectively.Furthermore, Figure 2a 27 , Figure 2b 27 , and Figure 2c 27 display long-exposure images from experiments conducted without ink, revealing the bubble movement zones.As can be seen, as the yield stress increases (transitioning from Figure 2a to Figure 2c), this area diminishes.Given the constant injection volume, the decrease in area may stem from the presence of different regimes.In the Newtonian layer, the jet fluid entrains and mixes with the layer fluid,

Figure
Figure 3cshows an example of the evolution of the cleaned area radius over time: â increases consistently with time as liquid is injected.The removal rate seems to be dictated by the force applied on the viscoplastic layer at the interface of layer, bottom surface, and jet liquid, resulting in the formation of an almost circular shape with a radius of â[26,27,64].

Figure 25 = 5 . 6 (
Figure 3c displays the variation of a versus t 0.2 for an experiment (δ 0 ≈ 11.1 in the highyield stress fluid).The plot also includes various values of K superimposed on it.As seen, the results for K = 1.21 demonstrate a good consistency with the data points.It should be noted that we initially assume M y (with M y representing the momentum/force required to cause yielding of the yield stress fluids) to be negligible compared to M , and the deviation of data points in the initial time steps from the predicted trend can be attributed to this assumption.Before delving into the influence of the main parameters on â behavior, it is important to highlight recent findings by Hassanzadeh et al. [40], which identify a critical length where jet penetration experiences a sudden slowdown, accompanied by a significant reduction in jet momentum.They have proposed a correlation for this critical length (L c ) as follows: L c = Lc Dn = 5.6 τy ρ V 2 j,0

Figure 3 FIG. 5 :
Figure5cshows how c evolves over time for different fluids (at the same layer thickness, δ 0 = 11.1).As observed, an increase in yield stress leads to a decrease in c.This implies that within the Newtonian layer, the cavity exhibits faster growth, owing to the absence

60 FIG. 6 : 2 .
FIG. 6: Variation of α (degree) versus time for an experiment with δ 0 = 5.6 and fluid II.The data points obtained by: direct measurement (blue diamond), Equation (24) (yellow circle), Equation (21) (black square).In each time step, α L and α R denote the left and right angles, respectively, and α is their average, calculated as α = α L +α R 2 .(b) Variation of steady values of α (α s ) versus δ 0 .The cases with fluid I, fluid II, and fluid III as layer fluids are respectively depicted by the black square, red circle, and blue diamond.The filled and hollow symbols correspond to α s obtained by direct measurement and

Figure 7a ,
Figure 7a, Figure 7b, and 7c display the injection of jet fluid into the Newtonian fluid (fluid I), the low-yield stress fluid (fluid II), and the high-yield stress fluid (fluid III),

FIG. 7 :
FIG. 7: Experimental snapshots of Newtonian jet into fluids I, II, and III.Panels a, b, c: first and second rows show submerged and plunging jet sequences, respectively.Third row: transparent jet and ambient fluid in plunging experiments.Plunging jets have δ 0 ≈ 22.2 and submerged jets have δ 0 ≈ 27.8.Each scenario's final snapshot includes magnified bubble sample.Dimensionless experimental times listed under each column.

3 FIG. 8 :
FIG. 8: Time evolution of different geometrical features: (a) a, (b) c, and (c) α.For all of the experiments, the jet is water, and soil layer fluid is denoted by distinct symbols: fluid I (black circle), fluid II (red square), and fluid III (blue triangle).(d) Variation of the time-averaged a, ā (blue circle), and time-averaged c, c (red diamond), versus the reverse of inertial factor (IF −1).In all panels, the filled and hollow symbols correspond to the

Figure 8b plots the•
Figure 8b plots the temporal evolution of c for submerged and plunging jets at different fluids.Interestingly, we can observe three different behaviors in terms of c values:

Figure 8d presents how
Figure 8d presents how the time-averaged cleaned area radius (ā) and the time-averaged maximum cavity radius (c) vary with IF −1 , for plunging and submerged jets.The trend shows that as IF −1 increases, both ā and c decrease for both jet types, meaning that the increasing IF −1 (or decreasing IF ) makes the cleaning process more challenging.Moreover, only for the high-yield stress fluid (i.e., fluid III), the submerged jet exhibits lower values for both ā and c compared to the plunging jet.

Figure 9b and
Figure 9b and Figure 9c depict the behavior of FGB and SGB, respectively, under various conditions, including different fluid types and initial layer thicknesses.The following insights can be derived from Figure 9a-c:

20 FIG. 9 :
FIG. 9: (a) Bubble trajectory for the case with the low-yield stress fluid (i.e., fluid II) and δ 0 ≈ 22.2.The size and the color of symbols represent the bubble size and the dimensionless elapsed time, respectively.The circle and triangle symbols correspond to FGB and SGB, respectively.The trajectory of (b) FGB and (c) SGB at different conditions.In (b) and (c), the symbol shapes represent the initial layer thickness: diamond (δ 0 ≈ 2.8), circle (δ 0 ≈ 5.6), square (δ 0 ≈ 11.1), star (δ 0 ≈ 22.2), and symbol color represents the different fluids: black (fluid I), red (fluid II), blue (fluid III).The dotted lines represent the centerline of the jet impingement and the different gray horizontal dashed lines show the free surface for different initial thickness layers.

2 FIG. 10 :
FIG. 10: (a) Variation of median bubble size ( Db,50 ) versus time ( t) at a constant initial layer thickness (i.e., δ 0 ≈ 11.1) for different fluids: fluid I (black star), fluid II (red circle), fluid III (blue diamond).(b) The time-averaged diameter size distribution of the bubbles in different fluids.The intensity of the colors represents the intensity of the yield stress (i.e., a darker color shows a higher yield stress fluid).The (standard deviation, variance) of the data are as follows: fluid I (0.56,0.31), fluid II (0.76,0.57), fluid III (0.42,0.18).(c) Variation of time-averaged Db,50 versus the initial layer thickness, for different fluids: fluid I (black square), fluid II (red circle), fluid III (blue diamond).

Figure 11a illustrates how
Figure 11a illustrates how the velocity components vary over time at a specific spatial point within the domain of interest, for the Newtonian fluid.This plot present the temporal behavior of both u b and v b , along with their mean and fluctuation velocities.As evident from the results, the mean velocity in the x direction (i.e., u b ) approaches nearly zero, highlighting the predominant upward motion of bubbles in the y direction, driven by buoyancy.

Table 2 .
Dimensional parameters ranges in experiments.A hat symbol (ˆ) indicates dimensional parameters; absence signifies dimensionless parameters.Jet and soil layer densities are effectively identical, ρ = ρj = ρa .Subscripts j and a represent jet liquid and ambient soil layer fluid, respectively.

Table 3 .
Rheological characterization of the soil layer fluids.Here, m = μa μj is the viscosity ratio,