Quantifying the influence of conduit inclination on Taylor Bubble behaviour in basaltic magmas

7 The ascent of single Taylor bubbles suspended in a range of Newtonian liquids, scaled to mimic 8 basaltic magmas, within vertical and inclined tubes has been studied experimentally over the range 0° 9 to 70° (where 0° is vertical and 90° is horizontal). Using measurements of Taylor bubble parameters 10 (ascent velocity and film thickness) and morphology, alongside dimensionless numbers, we show that 11 inclination has a clear effect on bubble behaviour and morphology. Notably, ascent velocity peaks at 12 inclinations of 40 to 50°, with proportional velocity increases of ~40-90% with respect to vertical 13 ascent values, before decreasing again. This work provides a basis for the expansion of existing 14 theoretical framework to accommodate for conduit inclination in volcanic scenarios. 15


Introduction
Developing the theoretical frameworks of gas-magma dynamics in volcanic conduits is a fundamental aspect to the understanding of basaltic activity dynamics worldwide (Seyfried and Freundt, 2000).
Such activity can range from quiescent passive activity through to the end member of lava fountaining from vents or fissures (Parfitt, 2004;Parfitt and Wilson, 1995). Key to understanding the drivers of these styles of activity are the gas flow regimes in the conduit, and how individual bubbles may interact and travel during ascent (Sparks, 2003(Sparks, , 1978, none more so than one of the most common classes of basaltic activity, the strombolian eruption (Blackburn et al., 1976;Parfitt, 2004;Taddeucci et al., 2015). This class of basaltic activity involves the discrete ejection of incandescent pyroclasts, often alongside an ashy component, driven by the rapid expansion of Taylor bubbles (also known as gas slugs) as they reach the surface.
The morphology and behaviour of Taylor bubbles in relation to basaltic style activity has been a major focus, e.g., (James et al., 2009;Kawaguchi and Nishimura, 2015;Pering et al., 2016;Pering and McGonigle, 2018;Seyfried and Freundt, 2000;Spina et al., 2013), however, the formation of Taylor bubbles is also key, with different mechanisms allowing for differences in ascent times and hence affecting activity development. The classic formation mechanisms invoked are the rise speed dependent model (Parfitt, 2004;Parfitt and Wilson, 1995;Wilson and Head, 1981) and the collapsing foam model (Jaupart and Vergniolle, 1988;Jaupart, 1990, 1986). However, more recent work explores system complexities through the influence of crystal packing (Woitischek et al., 2020), rheological changes in the magma via viscous caps (Barth et al., 2019;Capponi et al., 2016;Del Bello et al., 2015) or a combination of the two (Oppenheimer et al., 2020), and finally through cascade coalescence of trains of ascending Taylor bubbles (Pering et al., 2017;Pering and McGonigle, 2018).
The structure and characteristic behaviours, e.g., ascent velocities, film thickness, and stability of Taylor bubbles ascending in vertical conduits has been described in detail by several previous works (Bugg and Saad, 2002;Campos and Carvalho, 1988;de Azevedo et al., 2016;Llewellin et al., 2012;Viana et al., 2003). The bubble is split into four major components ( Figure 1): (1) the nose is the uppermost part of the bubble and its approximately hemispherical structure remains widely unchanged between bubble and liquid types, (2) the main body which is surrounded by a falling liquid film and remains relatively stable throughout bubble ascent (Llewellin et al., 2012), (3) the tail, and (4) the wake. The body is further subdivided into (2a) the upper portion, where the film thickness accelerates and thins as it develops, and (2b) the lower portion, where the film has a steady thickness because the forces acting on it are in equilibrium. Tail morphology varies depending on regime stability and the wake can be classified as open, closed, or turbulent (Campos and Carvalho, 1988;Nogueira et al., 2006;Pinto et al., 1998). Quantitative experimental data for Taylor bubble ascent velocity and thickness of the falling film rising around Taylor bubbles in inclined tubes are far scarcer. James et al., (2004) investigated the dependence of flow patterns on tube inclination within a volcanic context, showing the effect of inconduit pressure fluctuations on the formation of large individual slugs, highlighting, the promotion of Taylor bubble flow from bubbly as inclination moves from 0 to ~ 30°.
Engineering applications such as bubble column reactors and transport pipelines have driven a comprehensive literature for two-phase flow in inclined tubes filled for Newtonian fluids (Shosho and Ryan, 2001). Zukoski (1966) can be attributed as one of the first studies to investigate the effects of tube inclination on Taylor bubble rise velocity using an experimental approach.
In this study, using a low-cost experimental approach, we address this gap in the theoretical literature covering the ascent of Taylor bubbles in inclined conduits, scaled to basaltic magmas. This study is of importance given that non vertical conduits are likely commonplace at strombolian explosion exhibiting locations, e.g., (Chouet et al., 1997;Ilanko et al., 2015;James et al., 2004).

Methods
This study examines the Taylor bubble ascent process in vertical and inclined tubes using an existing and repeatable experimental approach based on previous studies from volcanic, fluid dynamics and chemical engineering backgrounds (summarised in Table 1) et al., (2003) and James et al., (2004) also conducted a similar investigation at inclinations of roughly 30°. Each tube was then filled with Newtonian liquids (see Table 2) to approximately 90% capacity to prevent spillage once the Taylor bubble was introduced, especially as tubes were inclined closer to the horizontal. Single Taylor bubbles were injected into the tubes using a gas syringe which was attached to the base of the pipe via rubber tubing. Injected gas volumes were kept constant for each tube diameter, using a 10 ml and 50 ml gas syringe for the 0.016 m and 0.027 m diameter tubes, respectively, ensuring that slug length was >1.5D (D = tube diameter) where bubble velocity is independent of slug length (Shosho and Ryan, 2001). Taylor bubble ascent was recorded using a tripod-mounted GoPro which was positioned parallel to the centre point of each tube to minimise camera distortion. Separate recordings for each bubble ascent were made a total of ten times per inclination for each liquid other than with golden syrup, where five repeats per inclination were recorded as its high viscosity severely inhibited ascent times. To prevent fluid mixing in between runs, tubes were rinsed thoroughly with soap and water then dried using compressed air. The tubes were filled with a set of Newtonian liquids: water, sunflower oil, olive oil, glycerol and golden syrup, all of which are low cost, non-toxic and readily available fluids. The density and viscosity of each liquid (Table 2) were then calculated. Liquid density, ρ was determined by calculating the weight of 100 ml of each liquid then substituting the values into the equation: where m is the mass of the liquid and V is the volume of the liquid. The viscosity (µ) of each liquid was calculated by dropping a steel ball into a 1 litre measuring cylinder, then calculating the fall velocity over 20 cm, and applying the following: where ρ s is the density of the sphere, ρ is the liquid density, g is acceleration due to gravity, a is the radius of the sphere and v is the velocity of the sphere. Each displayed value for liquid density and viscosity ( Table 2 is the average of three measurements. Taylor bubble velocities were measured individually for each repeat of each inclination for all liquids, using ImageJ (https://imagej.nih.gov/ij/). Ascent velocity (vb) was calculated by measuring the time taken for the tip of the nose to cover a marked distance of 50 cm ( Figure 2): where vb is bubble velocity, d is distance travelled and t is time. Taylor bubble ascent was measured over the central part of the tube (from the 25 to 75 cm marks) as ascent velocity reaches a steady value within five to ten radii of gas injection then remains constant, assuming there is no outside interference (Llewellin et al., 2012). In the case that the bubble had not fully formed or stabilised before reaching 25 cm the ascent time was instead measured between 50 and 75 cm, i.e., for glycerol and golden syrup, as bubble length increased in liquids of a higher viscosity, while inclination had a further lengthening effect on slug morphology.
To measure the thickness of the falling film, λ, in inclined tubes, the bubble width was added to the width of the lower film thickness (i.e., in an inclined tube the bubble ascends closer to the upper tube wall) then the total was subtracted from the internal diameter ( Figure 1). Falling film thickness was measured in vertical tubes by subtracting bubble width from internal tube diameter then halving the resulting value because falling films present symmetry in all directions around the bubble when the condition of the surrounding tube is vertical (de Azevedo et al., 2015).

3.1.Theoretical Framework
Calculating the flow parameters of the Taylor bubbles allows for the development of our understanding of flow structure and movement while specifically providing a way to describe observed behaviour numerically at both laboratory and volcanic scales. The ascent velocity of Taylor bubbles, vb, in vertical tubes has been shown to be dependent on the liquid density ρ, viscosity µ, gravitational acceleration g, the liquid-gas interfacial tension σ, and the internal diameter of the tube D (Llewellin et al., 2012). These values have been combined to form dimensionless quantities, which are used to describe the behaviour of fluids. These include the Froude number: the Eotvos number: and the Reynolds number: The Froude number is a dimensionless velocity describing flow pattern over an obstacle, or the ratio of inertial and gravitational (buoyancy) forces. The Eotvos number, describes the importance of gravitational (buoyancy) forces, compared to surface tension forces. The Reynolds number represents the ratio of inertial and viscous forces in the liquid to help predict flow patterns in various fluid flow situations, especially in relation to turbulence and its implications on bubble formation and stability. A fourth parameter has also been calculated using the liquid property values; this is the dimensionless inverse viscosity (Nf): which represents the ratio of external to internal forces and liquid flow pattern in the wake.
Taylor bubble parameters, from our experiments, have been described as seen in Figures 3 to 13, where vb, Fr, Eo, Re, and λ are plotted as functions of tube inclination angle θ. Each solid line corresponds to a particular tube diameter D, and each subplot shows data for a different Newtonian fluid. For the sake of comparison to a volcanic scale, the range of dimensionless values covered by this study are: 1.2 x 10 -3 ≤ Re ≤ 7.9 x 10 3 , 3.1 x 10 -3 ≤ Fr ≤ 5.4 x 10 -1 , and 0.5 ≤ Nf ≤ 1.5 x 10 4 . The ranges calculated in cover the known range of estimated dimensionless parameters at real volcanic conduits (Tables 2 and 3), with water and golden syrup acting as appropriate endmembers while the remaining liquids (19.3 ≤ Nf ≤ 101) likely encompass the magmatic range for inverse viscosity observed at volcanoes such as Stromboli and Yasur (Table 3). As a function of inclination (excluding falling film thickness), all dimensionless attributes follow the same trend: as inclination angle increases from a vertical (θ = 0°) position they increase to a maximum value, typically within the range of 40° to 60°, then decrease. All relationships are shown to be nonlinear, with trends relating to inclination that agree with results from previous chemical engineering literature (Bendiksen, 1984;Shosho and Ryan, 2001;Nigam et al., 1995;Pokusaev et al., 2011;Zukoski, 1966).

Reynolds Number
High Reynolds numbers coincide with high Nf values, and higher Reynolds numbers are indicative of turbulence, therefore smaller values for Re occur at low Nf, where the flow regime is observed to be laminar. According to Campos and Carvalho (1988) wake shape is axisymmetric when Re < 180; this also applies to the results for this study within the range 0.5 ≤ Nf ≤ 101, i.e., all liquids but water.
Existing studies (Amaya-Bower and Lee, 2010; Campos and Carvalho, 1988) observe oscillation in the wake to occur within the range 180 ≤ Re ≤ 304, which somewhat agrees with observations in this study, although it is difficult to determine the exact threshold for the onset of turbulent flow as the values within this range are not explicitly covered by this dataset.  (Figure 4), the Reynolds number is very sensitive to variations in viscosity; this is also reflected in equation (6), meaning that in vertical tubes Re is a function of Nf only (in agreement with literature), however, it also becomes a function of tube inclination for θ > 0.

Froude Number
Fr increases when θ increases from vertical until a maximum is reached within the range 45 ≤ θ ≤ 65, then decreases for all fluids ( Figure 5). These results agree with the qualitative behaviour for Froude number as a function of inclination observed (Shosho and Ryan, 2001;Zukoski, 1966). Tube diameter has a significant and complex effect on Froude number; as liquid viscosity increases (and Nf decreases) Fr values decrease; this occurs to a more significant extent for D = 0.016 m than for D = 0.027m with glycerol and golden syrup. Specifically, for D = 0.016 m containing water, Fr is higher than for D = 0.027 m; for the range 44 ≤ Nf ≤ 101 (encompasses both oils). Fr values from both diameters are similar; then for the range 0.5 ≤ Nf ≤ 42 (glycerol and golden syrup) Fr is higher in D = 0.027 m than in D = 0.016 m. Shosho and Ryan (2001) propose that this occurs for Nf ≤ 50 (approximately), which agrees with the results seen here (glycerol and golden syrup). For both tube sizes the fluid with 0.5 ≤ Nf ≤ 1.1 (golden syrup) exhibited Fr values well below those seen in other liquids. The literature (Shosho and Ryan, 2001;Bugg and Saad, 1998;Llewellin et al., 2012) (Bugg and Saad, 2002) and values in other literature (Seyfried and Freundt, 2000;Shosho and Ryan, 2001;Viana et al., 2003;Llewellin et al., 2012). However, overall values for Fr calculated in this study were in the range 3.1 x 10 -3 < Fr ≈ 0.351 when inclination angle was greater than 0°. Pokusaev et al., (2011) alludes to an explanation for this complex behaviour when it is stated that the constant is upheld within a system that is in inertial control, i.e., when liquid flow is laminar and the shape of the bubble nose is hemispherical, which is not the case in inclined tubes.

Film Thickness and Interfacial Tension
The data collected for film thickness in this study refers to the 'upper falling film thickness', i.e., that located on the upper side of the Taylor bubble where it comes into close contact with the upper wall of the tube. This deviation away from the central axis of the tube occurs because of the buoyancy or level effects acting on the bubble. As a result of this, the 'lower film thickness', which refers to the film thickness on the opposite side of the bubble, increases to a maximum with increasing tube angle and decreasing Nf . As with Llewellin et al., (2012), in this study it is not assumed that the falling film is uniformly thin, so λ is used to denote film thickness instead of the conventional use of δ. Falling film thickness has an inverse relationship with all other measured parameters ( Figure 6).
Except for water, film thickness decreases with increasing inclination until a minimum value is reached within the range of 45 ≤ θ ≤ 55 (in line with the maxima of all other parameters) before increasing again as inclination approaches 70° for the range 0.5 ≤ Nf ≤ 101. Llewellin et al., (2012) states that surface tension may play a different role in controlling film thickness (and therefore rise velocity) for transitional and turbulent films. This can be observed in Figure 7 where the relationship of Eo and film thickness as a function of inclination appears sigmoidal in nature for high Nf values (water) as inclination increases from the vertical. The significance of this was not investigated further due to a lack of available data for high Nf but may be a result of a breakdown of the thin film assumption (Pokusaev et al., 2011), which states that when film thickness is thin local tube curvature can be neglected. Film thickness decreases as the role of interfacial tension becomes more important, i.e., as Eo decreases, film thickness also decreases ( Figure 8) and these occur alongside a decrease in Nf.
High Eo values indicate that the system is relatively unaffected by interfacial tension, whereas a low value indicates that interfacial tension dominates (typically for Eo < 1). Therefore, the decreasing value of Eo with increasing inclination angle indicates an increasing dependence on inclination angle as buoyancy plays a larger role and pushes the bubble up against the upper wall of the tube as inclination angle approaches 70°. Thinner falling films occur for liquids of low viscosities and film thickness increases with increasing viscosity, whereas large Nf and Re values give a thin falling film compared to pipe radius in vertical tubes. This observation is also consistent with inclined tubes, with the addition of film thickness also being a function of inclination. According to Bugg et al., (2002) film thickness is independent of Re and hence of ascent velocity for 10 -5 ≤ Re ≤ 1, meaning that for Re > 2 film thickness decreases as Reynolds number increases, which agrees with observations made in this study. Seyfried and Freundt (2000) states that film thickness is strongly dependent on Eo number for Eo < 40, and this dependence is more pronounced for smaller Nf values, i.e., more viscous liquids. As a result of this interfacial tension is negligible for Eo > 40 (Viana et al., 2003), which applies for most of this dataset as such only data for water ( Figure 7) and olive oil ( Figure 8) were plotted to show variation. Taylor bubbles cannot form when Eo < 4 because capillary forces restrict bubble movement (Bugg et al., 1998). While this may be relevant to studies on a laboratory scale, this cannot occur at a volcanic scale where conduit diameters are several meters wide.

Taylor Bubble Shape
A qualitative change in bubble geometry has been observed to occur in response to increasing inclination. Within tubes of vertical geometry bubble morphology is axisymmetric and the Taylor bubble ascends the pipe centre (Zukoski, 1966) and consistently with previous works (Table 1), the shape of the nose and body remain qualitatively the same for both diameters in vertical tubes, i.e., when θ = 0, independent of bubble length. In Taylor bubbles the ratio of bubble to tube diameter is > 0.6, so tube diameter is the controlling length governing nose shape (de Azevedo et al., 2015), and as liquid viscosity increases, the shape of the bubble nose becomes blunter and velocity decreases. These observations also apply to inclined tubes in addition to a gradual change in bubble shape that is observed with increasing inclination. For all inclinations, including vertical, the bubble nose has a rounded shape, however, nose shape becomes more oblate, and the lower side of the bubble becomes flatter as the bubble lengthens and thins in response to an increased inclination angle. These observations are displayed in Figure 9 and agree with data from previous studies (Zukoski, 1966;James et al., 2004Pokusaev et al., 2011. The influence of the contact angle of the slug on the tube wall affects bubble morphology; for θ = 0 (vertical) a standard stable bubble structure is observed across all Nf values, although turbulence can be observed in the wake for the range 7100 ≤ Nf ≤ 15560 (water). This changes when θ > 0, where an asymmetric slug configuration above the tube's central axis is observed to occur as a function of inclination angle. Tail morphology also changes for θ > 0, especially for lower Nf values ( Figure 10) where flow is less turbulent. Tail shape is thought to remain stable for Nf ≤ 600 (Campos and de Carvalho, 1988;Viana et al., 2003;Llewellin et al., 2012), and results of this study show tail stability to be maintained up to Nf = 101, which possibly aligns with previous findings as the next highest data point for this dataset is Nf = 7100, meaning the value for stability lies within the range 101 ≤ Nf ≤ 7100. Tail morphology is flat where viscous effects are unimportant and takes on the shape of an oblate spheroid where viscous effects become important in vertical tubes (Bugg et al., 1998), this also occurs with inclined tubes to an extent, however, tail shape becomes more wedge shaped or pointed at greater inclinations ( Figure 10) for low and high viscosity liquids respectively.

Taylor Bubble Ascent Velocity
Overall, Taylor bubble ascent velocity in vertical and inclined tubes is a function of the combination of liquid properties and tube geometry, meaning the propagation rate of volcanic Taylor bubbles is a result of magmatic properties and internal conduit geometry. For the full range of Nf velocity rates increase when inclination increases up to a maximum within the range 40 ≤ θ ≤ 50 for D = 0.016m and 50 ≤ θ ≤ 60 for D = 0.027 m before decreasing again ( Figure   11). For this study, only five different fluids were used, in two different pipe diameters, meaning this study covers ten Nf values of the range 0.5 < Nf < 15560, however, only Nf values for the range 0.5 < Nf < 101 were plotted given their volcanic relevance. The angle at which the maxima occurs for Taylor bubble ascent velocity is lower for low Nf than for high Nf (Figure 12), consistent with findings by Zukoski (1966) and Nigam et al., (1995). The effect of inclination angle on bubble rise velocity is complex and non-linear as a result of the observable change in bubble morphology that occurs in response to changes in inclination. As long as ascent velocity is independent of bubble length it is a function of the shape of the bubble head (Pokusaev et al., 2011), namely the curvature of the nose in the vicinity to its critical point. This occurs when bubble length is > 1.5D (Shosho and Ryan, 2001;James et al., 2004).
For less viscous liquids, inclination has negligible influence on bubble length generated from the same air volume, but a tendency of increasing bubble length is observed in liquids of high viscosity (or low Nf) when inclination increases. Changes in Taylor bubble structure, which is a function of Nf and inclination, alters the overall drag coefficient and shape of the falling film surrounding the bubble and as a result influences the rate of ascent (Bugg et al., 1998). A system in laminar flow (low Nf, smaller diameter) increases the contribution of this viscous drag; this is displayed in Figure 13a where the maximum variation in proportional velocity change is observed to occur for D = 0.016 m at high viscosity (0.5 ≤ Nf ≤ 19.3), i.e., for golden syrup and glycerol. As flow regime becomes more turbulent (occurs with D = 0.027 m), the effect on proportional velocity change is smaller (Figure 13b).

Volcanic Context
These findings have a significant impact on the current knowledge of Taylor bubble dynamics in volcanic conduits, which are commonly assumed to contain vertical conduits in previous studies. Although this assumption is made on a purely theoretical basis, it is not a realistic approach to the study of bubble dynamics. Due to the nature of the trends displayed throughout this work. in comparison to data in Table 3, it can be assumed that the current ranges seen in literature for Taylor bubble flow on a volcanic scale would increase, specifically in reference to Taylor bubble ascent velocity rates and dimensionless parameters because the maximum value for each parameter always occurs at an inclination greater than 0° (vertical). Fr and Re are especially affected here, as they are direct functions of vb (see equations [4] and [6]) which is a function of changing tube and therefore conduit scale inclination. These results also have specific implications for surface eruptive behaviour and controls on volcanic explosivity.

Conclusions
Here, we illustrated a simple and effective experimental approach that allows for the measurement of single Taylor bubbles ascending within both vertical and inclined conduits in two-phase flow. We derived and calculated dimensionless numbers and Taylor bubble flow characteristics in comparison to a natural volcanic scale. We highlight that Taylor bubble morphology is found to be a strong function of inclination, while the nose shape is a major controlling function for vb for the full range of Nf studied here, especially when conduit inclination is within the range of 40 ≤ θ ≤ 60. We show a wider and higher range of vb, Fr and Re (as function of inclination) than is presently seen in the current volcanic literature, which predominantly uses vertical conduit geometry to estimate Taylor bubble flow characteristics within volcanic conduits.