Emanating Jets As Shaped by Surface Tension Forces.

We show that emanating jets can be regarded as growing liquid towers, which are shaped by the twofold action of surface tension: first the emanated fluid is being accelerated back by surface tension force, herewith creating the boundary conditions to solve the shape of the liquid tower as a solution of an equation mathematically related to the hydrostatic Young-Laplace equation, known to give solutions for the shape of pending and sessile droplets, and wherein the only relevant forces are gravity g and surface tension γ. We explain that for an emanating jet under specific constraints all mass parts with density ρ will experience a uniform time dependent acceleration a( t). An asymptotic solution is subsequently numerically derived by making the corresponding Young-Laplace type equation dimensionless and by dividing all lengths by a generalized time dependent capillary length λc( t) = [Formula: see text]. The time dependent surface tension γ( t) can be derived by measuring both time dependent acceleration a( t) and time dependent capillary length λc( t). Jetting experiments with water and coffee show that the dynamic surface tension behavior according to the emanating jet method and with the well-known maximum bubble pressure method are the same, herewith verifying the proposed model.

Introduction

Various methods exist for measuring the dynamic surface tension (DST) γ(t) to study surfactant adsorption and surface tension dynamics at small and large time scales.[1,2] The DST is a function of surface age and usually decreases in time when surfactants are transported, adsorbed, and reoriented at the interface before the equilibrium surface tension value is reached. Known methods for measuring the surface tension include force methods such as the Du Noüy ring[1] and Wilhelmy plate[1] technique, or shape methods such as the sessile and pendant-drop technique, or pressure methods such as the small bubble surfactometer technique.[1] Normally, the DST is a time-dependent function of the age of an initially prepared surfactant-depleted surface and decreases in time when surfactants are being adsorbed and reoriented at the air-liquid surface. When a fresh new surface is created, a mass transport of surfactants toward this surface will always occur by diffusion and convection. The contribution of convection depends on the method used. Examples of methods, in which convection is imposed deliberately but in a controlled and usually known manner, are the maximum bubble pressure method and the oscillating jet method.[3−5]

Here, we present a new method to determine the DST at small time scales between 1 and 50 ms by studying the dynamics of an emanating liquid–air interface, for example, when a coffee drop falls into a coffee bath and a short-lived emanating coffee jet is formed (see Figure ). In nature, one can encounter many different transient shapes of pendant drops, dripping threads, and emanating jets; see Figure .

Figure 5

Snapshots of a coffee jet. (a) Snapshots at 3.0, 5.0, 10.0, 15.0, and 20.0 ms after initial outburst of the jet. Blue dotted lines correspond to fitting the asymptotic shape function (Figure ), yielding for each jet a specific value for the capillary length λc in millimeter. Scale bar in the graph is 2.5 mm. (b) Time plot of the jet height H(t) measured from base to top of the jet. At 21.0 and 36.0 ms, a droplet is emitted from the tip resulting in a discontinuity in H(t). (c) Acceleration a(t) of the jet obtained from the second-time derivative (−∂2H(t)/∂t2) of the time plot of H and as obtained by fitting the asymptotic shape function to the snapshots plotted as (⧫).

Figure 1

Young–Laplace equation describing solutions that apply for the shape of various liquid bodies. (a) Sessile and pending drops; courtesy: Feikje Breimer and (b) pending honey thread; courtesy: Feikje Breimer. (c) Emanating jet or liquid tower from a water surface. It has a shape that can be derived from an equation (see eq ) mathematically related to the hydrostatic Young−Laplace equation.

Using the calculus of variations, Gauss[6] unified earlier mathematical results of Young and Laplace[7] to obtain equations and boundary conditions describing surface-tension-determined shapes.[8,9] It is known that solving the Young–Laplace equation is equivalent to minimizing the thermodynamic potential of a fluid–fluid system that may be in contact with a solid substrate.[10] This might imply that non-equilibrium liquid–air interfaces will always evolve toward a more stable solution of the Young–Laplace equation, such as that described by the shape of sessile and pending drops.[11−18]

In Figure b, a pending honey thread is depicted, which slowly retracts to the spoon. The shape of the slowly moving honey–air interface can be described by a quasi-static solution of the Young–Laplace equation.[19,20] We show here that the Young–Laplace equation may be extended to describe the relatively fast varying shape of an emanating jet interface and to derive the DST (Figure c). Note a geometric similarity in Figure b,c, that is, the pending thread and emanating jet are images, which seem mirrored around a horizontal plane.

The well-known hydrostatic Young–Laplace equation[10,12,19,20] expresses that the hydrostatic pressure ρgz with ρ mass density and g gravity constant at position z is compensated by the surface tension (γ)-based Laplace pressure 2γ/Rc(z)with Rc(z) ≡ 1/2R1 + 1/2R2 denoting the mean radius of curvature and is determined by two principal radii of curvature (R1, R2) corresponding to tangent circles in two perpendicular planes. Please note that the z = 0 level will be determined by physical considerations.

The asymptotic concave shape of a pending thread is a solution of the Young–Laplace equation, provided the surface-tension-based forces are in equilibrium with the gravitational force (Figure a).

Figure 2

(a) Force balance between gravitation and surface tension forces of a pending thread. When the formed liquid thread including the drop remains stable for some time, this implies that for every height value z, the gravitation and surface tension forces are in equilibrium. (b) For an emanating jet, the gravitation and surface tension forces are not in equilibrium and the jet will always accelerate in the downward direction with a value larger than g.

For a pending thread (Figure a), the net surface tension force Fsurface tension is directed upward along the axial direction of the liquid thread having a circumference 2πR(zo) at height zo with a magnitude F(zo) = 2πR(zo)·γ. At a height zo, the hydrostatic pressure will counter-exert a downward-directed force Fhydrostatic on the cross-section πR2(zo) of the liquid with a magnitude πR2(zo)·ρgzo. A downward gravitational force Fgravity on all of the thread mass between z = zo and z = ∞ will create an additional downward pulling force with magnitude g·mthread (zo) with mthread (zo) = ρ∫∞πR2(z)dz. When the formed liquid thread including the drop remains stable for some time, this implies that for every height value z, the gravitation and surface tension forces are in equilibrium.

For an emanating jet (Figure b), the net surface tension force Fsurface tension is directed downward along the axial direction of the liquid thread having a circumference 2πR(zo) at height zo with a magnitude F(zo) = 2πR(zo)·γ. The time-dependent surface-tension-based downward acceleration d(t) (≡a(t) – g) will create (Newton’s third law) a positive pressure gradient in the jet with a magnitude ρd(t)z, herewith forming the boundary conditions for shaping the jet. At height zo, an upward-directed force Fpress gradient on the cross-section πR2(zo) of the liquid will therefore be present with the magnitude πR2(zo)·ρdzo because of the surface-tension-driven deceleration d of the jet. A downward gravitational force Fgravity on all of the thread mass between z = zo and z = ∞ will create another downward-directed force with the magnitude g·mjet(zo) with mjet(zo) = ρ∫∞πR2(z)dz. The sum of all of these forces induce an acceleration with a magnitude a = g + d being larger than g toward the liquid bath. In the Supporting Information, a section “Self Consistent Solution” is included, showing that the downward acceleration based on the acting physical forces is/remains uniform throughout the whole jet, that is, d(z,t) = d(t). Also, in the section “Coaxial Cylinder Model for the Equation of Motion” in the Supporting Information, an alternative derivation for the motion and shape of the jet is presented based on the above considerations. A more elaborate theoretical model, including viscosity and other inertial contributions, will be presented next.

Theoretical Approach

Considering an emanating jet liquid–air interface, such as that depicted in Figures c, 4a, 5a, or 9a–c, clearly, we cannot neglect here the dynamic movement of the jet fluid toward the bath, neither through the action of gravity nor through the surface-tension-based attraction forces between the bath liquid surface and the jet surface. For an axisymmetric fluid element, the following momentum[21] equation applies in the upward z direction

Figure 4

Rise of a typical pure water jet at 24 °C. (a) Sequential snapshots after impact of a 2.5 mm-sized droplet. The tip of the jet appears at 0 ms just above the fluid surface. Scale bars are 2000 μm. (b) Height H of the tip grows to 9.6 mm in 24.0 ms. (c) Time-dependent deceleration a(t) as derived from the time plot of H(t). The initial velocity of the jet is about 2 m/s and declines rapidly.

Figure 9

Rising water jet. Blue dotted lines represent experimental contours of the jet. (a) Time frame of 8 ms, (b) time frame of 16 ms, and (c) time frame of 24 ms. Corresponding graphs show (●) dimensional curvature values 2/Rc as a function of z derived from the experimental contours using eq ; (—) green solid line is a linear plot according to eq .

Here, the velocity v = v(z,t) describes the flow field in the axial z direction and has been averaged in the radial direction. R(z,t) describes the radius of the emanating jet at time t and axial position z, where R1, R2 = R1,2(z,t) are the corresponding principal radii of curvature.

If we assume that the inertial axial velocity gradient term is small with respect to the gravitational acceleration (v∂v ≪ g), then eq reduces towe assume that the time-dependent axial acceleration a(z,t) ≡ −∂v is independent of the place in the downward z-direction, and that the surface tension γ(z,t) is only dependent on time, therefore γ(z,t) = γ(t). Finally, we assume that the viscosity term is small with respect to gravity: . Then, eq may be written as

If we define the mean curvature 1/Rc with 1/Rc ≡ 1/2R1 + 1/2R2 and integrate eq by the variable z, we getwith C, a constant of integration; in this case, C = 0 because we define .

We can make eq with C = 0 dimensionless and independent of time by introducing a generalized time-dependent capillary length λc(t) = , and we further define Rc* ≡ Rc/λc and z*≡ z/λc.

We get

Possible functional shapes of R*(z*) are determined by the following nonlinear differential equation[21]R*′ and R*″ are the first and second derivative of R* with respect to z*. For large values of height z*, both R*′ and R*″ tend to zero; the asymptotic solution becomes R*(z*) = 1/z*. This approximate solution can be used as a starting point to solve eq numerically.

Using Mathematica NDSolve algorithm, a fully concave solution with a singularity at R* = 0 has been obtained by using boundary conditions R*(z* = 10) = 0.1 and R*′(z* = 10) = −0.01 and is shown in Figure . From eq at z* = 0, it follows that the sum of the inverse of the two principal radii of curvature R1* and R2* is zero and that the dimensionless radial value here is about Rbase* (z* = 0) = 1.39 (see Figure ). Table S1 plots numerical values for z* and R* and an approximate polynomial relation between z* and R* has also been derived and both are attached in the Supporting Information. Also, in the Supporting Information, it is derived that based on the actual physical forces, the downward acceleration remains uniform throughout the whole jet, thus when d(z,t) = d(t), the jet has attained the asymptotic shape according to Figure .

Figure 3

Dimensionless asymptotic solution of eq . Height z* and radius R* are given in the dimensionless units z/λc and R/λc with λc as the capillary length. Dimensionless plot (blue dotted line) of the concave jet shape according to the numerical solution R*(z*) of eq . Please note that at the base for z* = 0, the radius Rbase* has a dimensionless length of 1.39. For z* = −0.5, the radius R* goes to infinity.

From Figure and Table S1, it follows that for z* = −0.5, the value of R* ≡ R/λc → ∞. Obviously, this is the value of z* for which the jet starts developing and should be taken as the zero point corresponding to a deepened surface level (z = −0.5λc(t)) of the fluid bath (z = 0) close to the jet.

Whereas for a pending thread, the uniform gravitational (hydrostatic) pressure gradient ρg and the static surface tension γ solely determines its shape; in the case of the emanating jet, the surface-tension-based pressure gradient ρd(t) ≡ ρ(a(t) – g) and γ(t) will determine the shape of the jet. Effectively for the emanating jet all the liquid above the surface bath is subject to a free fall, so gravity will not contribute to the shape; therefore, we have to subtract g from a(t). Think about the case of free-falling droplets that are perfectly round because there is no hydrostatic pressure gradient inside the droplet present when it falls. Note also that we do not need any assumption about an initial radial velocity distribution of the fluid inside the emanating jet. Different radial or axial regions of the jet may be subject to different velocities; the shape-determining parameter and the main assumption are that the uniform but time-dependent surface-tension-based acceleration d(t) and γ(t) will apply to all different regions of the jet.

Experimental Section

First, we have performed experiments with pure water jets to derive the surface tension value of water from the data and to check the model.

In Figure a at 0 ms, the tip of the jet appears just above the fluid surface, a large part of the jet is not visible because the base of the jet is still at the bottom of the cavity below the fluid surface. The fully developed jet including the base becomes clearly visible in the last snapshot at 24.00 ms. A Movie at 6000 fps is provided in the Supporting Information. In Figure b, a time plot of the height of the pure water jet shows an asymmetric parabolic profile corresponding with the rise and fall of the jet. The downward acceleration being the second-time derivative of the height is presented in Figure c. The initial acceleration a is estimated to be more than a few 100 m/s2, but after 16 ms, the acceleration drops to values less than a few 10 m/s2. The presented model predicts that the time dependent acceleration a(t) is related to the time dependent capillary length λc(t) = , and this length can be derived from the relation Rbase ≡ 1.39λc. By substituting actual measured values of λc and a in the capillary length equation corresponding values for γ(t) can be calculated. Averaging the results for a number of experiments we verified that the dynamic surface tension of water[17] on a time scale of 1−20 ms is constant with value 72 ± 4 m/s2.

After the experimental validation of the model, further experiments have been performed with coffee, a liquid known to have nonconstant surface tension values in the domain of 1–50 ms.[22,23] Data from the emanating coffee jets have been analyzed to derive the DST γ(t) from the predicted concave jet shape with measured time-dependent parameter values for the capillary length λc(t) and inertial acceleration d(t). The physical process is illustrated in Figure a, where snapshots taken from a coffee jet are shown at times between 3 and 20 ms after the outburst of the jet (Movie is provided in the Supporting Information). The varying height H(t) of the tip of the jet is given in Figure b and the acceleration a(t), being the second-time derivative of the height plot (a(t) ≡ −∂2H(t)/∂t2), is depicted in Figure c. The jet acceleration a(t) drops quickly from 100 to 200 m/s2 to a value of about 20 m/s2 after 15–20 ms, a value still above the gravitational constant (g = 9.81 m/s2). Besides gravity, the jet will experience a surface-tension-based force accelerating the jet back to the liquid surface; the total acceleration a(t) can then be considered as a sum of two parts a(t) = d(t) + g with d(t), the DST-based acceleration.

In each of the snapshots of Figure a, the time-dependent value for the capillary length λc has been obtained by fitting the asymptotic shape function. Please note that the physical z = 0 axis in the snapshots is taken here as the z* = −0.5 axis. If we take the measured acceleration of the tip a(t) ≡ −∂2H(t)/∂t2 and substitute this value together with λc in the relation , we can calculate the DST, γ(t). Please note that the varying height H(t) of the tip of the jet is given in Figure b and that the total downward acceleration a(t), being the second time derivative of the height plot a(t) ≡ −∂2H(t)/∂t2, is depicted in Figure c.

Comparing the theoretical prediction of the asymptotic shape function with experimental results has been depicted in Figure a–c. To verify the shape function: the first step in the comparison procedure is to use the fact that the base of the jet Rbase ≡ 1.39λc. This gives a first rough experimental estimate of the capillary length λc. Next, we enlarge the experimental photographs until individual pixels become visible, determine the conversion factor between pixels and millimeters, and record the coordinates of the pixels of the interface.

Figure 6

Comparison of the predicted asymptotic shape (including absolute size) with the actual measured contour of the jet (a–c). Experimental contour plots (green solid line) of the coffee jet, as depicted in Figure , at 3.0, 5.0, and 10.0 ms after the initial outburst of the jet and theoretically obtained asymptotic concave jet shape (blue dotted lines) according to eq (see also Figure ). The vertical z-axis is related to the axial distance z using z ≡ λc(z* + 0.5).

Subsequently, we try to find the closest value that matches with the z and R(z) coordinates derived from the pixels on the jet surface in each snapshot. A relative fit error for the radius R is obtained by comparing the difference of the radius R according to the experimental contour of the jet and of the theoretically obtained shape function with . Next, we use numerically obtained values of eq for the dimensionless graph and the conversion factor to find the dimensional graph for the precise value of λc. Fitting the shape function to the snapshots yield for each jet a specific value for the capillary length λc in millimeter (see, for example, the numbers for λc in Figure a).

Beverages, such as coffee, beer, and wine, are multicomponent and multiphase systems containing many constituents which show surface activity by themselves or in association with other compounds.[22,23] Naturally occurring surfactants can be essentially divided into two classes based on the molecular weight: low molecular weight compounds (small organic molecules with a molecular weight up to 5000 Da) and high molecular weight compounds (macromolecules and biopolymers). The coexistence of both classes of surfactants in beverages is more a rule than an exception. Coffee components are typically proteins (5–10 mg/mL), carbohydrates (5–10 mg/mL), lipids (2–5 mg/mL), and caffeine (2–5 mg/mL). We prepared drip coffee that was brewed with 50 g/L of R&G (roasted and ground) coffee in medium–low hardness tap water (180–200 mg/L CaCO3) by using a Philips electric dripfilter apparatus (1100 W), and a standard paper filter (Melitta) was used. To derive the DST values at small time scales between 0.1 and 100 ms, we used the following algorithm. First, we derive γ(t) from the capillary length definition and find γ(t) ≡ λc(t)2(ρ(a(t) – g)). The DST γ(t) can thus be obtained by using for each snapshot the calculated value for λc as stated above and the corresponding value for the DST-based acceleration d(t) ≡ a(t) – g, as obtained from Figure c. The result is plotted as blue squares in Figure b.

Figure 7

DST plots of coffee at 20 °C. (a) Plots obtained with dynamic maximum bubble pressure tensiometer (γ-Lab, Germany) for ristretto (●), regular (■), lungo (▲), soluble coffee (⧫), and drip coffee (▼).[23] (b) DST measurements of drip coffee as obtained with maximum bubble pressure method ▲ and as obtained with the new emanating jet (blue filled square) method.

DST plots of different coffee brands[23] measured with a dynamic maximum bubble pressure tensiometer at 20 °C are linear-log plotted in Figure a. The surface tension measurements show that it is possible to discriminate different preparations on the basis of their surface tension properties. The fast decrease of the surface tension already observed at a short time confirms the presence of low-molecular-weight surface-active solutes in addition to higher-molecular-weight surfactants. The kinetics of adsorption are interpreted from surface tension log time plots[24] which often display three distinct regimes: (I) diffusion and adsorption determine an initial period of a small tension reduction. (II) Continued rearrangement defines a second regime, where the resulting number of interfacial contacts per surfactant molecule causes a steep DST decline. (III) A final regime occurs after full coverage and is attributed to continued relaxation of the adsorbed layer and build-up of multilayers. Normally at very short times, the adsorption process is influenced by concentration and availability of surface-active molecules. In this case, because of the high solid content of the brewed coffee, there is a surplus of these molecules present, and diffusion regime I seems not observable. We see directly the start of regime II, where continued rearrangement of interfacial contacts per surfactant molecule causes a steep tension decline.

DST plots of drip coffee at 20 °C have been measured, both with a dynamic maximum bubble pressure tensiometer and according to the new emanating jet method, and are plotted in Figure b. The results show that the slope of the surface tension reduction according to the emanating jet method and with the maximum bubble pressure method is the same. Please note that in the maximum bubble method, the surface age is defined as the time interval from the start of the surface expansion to the point where the radius of the bubble equals the capillary radius.[1] The surface age is therefore always related to a specific measuring technique of the experimenter who must subjectively define the “start” of the prepared new surface. In the maximum bubble method, the surface age is the measured time interval from the start of the surface expansion to the point where the radius of the bubble equals the capillary radius, that is, a twofold expansion of the air-liquid interface from a value of πRcapillary2 to 2πRcapillary2. For the emanating jet method, the measured time interval is from the visible start of the jet, resulting in a much larger relative expansion from almost zero to the full jet surface area at t = 20 ms. We estimate that the total surface area of the jet at t = 8–10 ms is about half the size of the one at t = 20 ms (Figure a). To compare both DST measuring methods, it is reasonable to redefine the “start” of the prepared new surface for the emanating jet method at t = 8–10 ms instead of at t = 0 ms. It can be noted that the DST curves in Figure b will then overlap significantly.

Effect of Viscosity and Inertia

When will viscosity alter jet dynamics and the shape of the rising jet? The main contribution from viscosity is determined by the term 3η∂((∂v)R2)/R2 in eq . The validity of the model is hence co-determined by the contribution of the internal axial velocity gradient ∂v. Consider a drop forming on the tip of the jet, which is continuously supplied with the liquid flowing through the tip. For the jet of Figure , the velocity v through the tip with the radius 1 mm at height zo = 15 mm filling the drop is about 2 cm/s. The initial velocity of the jet coming out of the bath is however much higher, typically 20 mm in 20 ms, implying a jet velocity of 1 m/s. The total upward velocity vo is thus about 1.02 m/s. When there is an influx of liquid in the jet at a constant flow rate for a period of time, the ballistic transport within the jet at each moment implies that vo(z = 0)πRo(z = 0)2 = v(z)πR(z)2 throughout the whole jet and using R(z) ≈ λc2/z, we find ∂v = 2voz/zo2.

The viscous contribution 3η∂((∂v)R2)/R2 for all values z > λc then becomes 3η∂((∂v)R2)/R2 = 3η∂((2voz/zo2)λc4/z2)/λc4/z2 = 6ηvo/zo2 = 25. This value is much smaller than the contribution of ρd(t) in eq or eq , which is typically in the range 1000–100 000; see Figure c. Therefore, for a low viscous jet (η = 1 mPa s) with a moderate emanating velocity (v = 1 m/s), we see that the viscosity will not influence much on the shape of the emanating jet.

Ghabache et al.[25] have proposed, based on mass conservation, a shape solution in monomial form for R(z,t); and reads R(z,t) ∝ zatb with a and b arbitrary numbers fulfilling the relation a = −(2b + 1)/2. From their experiments[25] it was found that a = −1 and b = 1/2, herewith verifying mass conservation with the said relation. From our theoretical derivation based on the momentum equation we predict that there is only one solution with an asymptotic behaviour with a = −1 valid for large z values. Our experimental findings and further literature study indeed confirm that a = −1 for large 3 values. Applying mass conservation will then automatically yield b = 1/2 using a = −(2b + 1)/2. The mass conservation equation and the momentum equation are therefore both needed to describe the evolving shape of an emanating jet: The momentum equation predicts the radial distribution parameter a = −1 for large z values and the mass conservation equation yields then the temporal distribution parameter b = 1/2. Please note that our shape solution has always a finite volume (see Table S1), whereas the monomial solution R(z,t) ∝ zatb for a = −1 has an unbound volume for finite t values.

We have verified mass conservation (b = 1/2 when a = −1) for the first 20 ms for the coffee jet of Figure in Figure , thus that e.g. R(z = 0,t) ∝ t1/2. Please note that a linear dependence between Rbase2 and t implies that the mass flow rate φ in the jet from the bath is not constant but increases in time according to φ ∝ t1/2.

Figure 8

Verification of mass conservation for the coffee jet experiment; see Figure ; the fitted results for the coffee jet (blue Δ) verifies that the time exponent b = 1/2. The green line is the predicted linear dependence between Rbase2 and t, as proposed by Ghabache et al.[25]

We know proceed with the contribution of the second inertial term ρv∂zv in eq . This term will be much smaller than the first inertial term whenever ρa ≫ ρv∂v and will thus depend on the value of the initial jet velocity v and the gradient ∂v. The term ∂v has been analyzed in refs (30) and (31) and based on boundary-integral simulations with analytical modeling, it was concluded that the fluid inside the jet moves with a nearly constant speed (ballistically) upward, implying that the ∂v term is small. From our analysis and experiments with a small velocity, jetting up to 1−2 m/s, we conclude that the contribution of the second inertial term to the jet shape is small with respect to the first inertial term.

Asymptotic Shape Solution and Other Solutions

When the actual shape deviates from the predicted asymptotic shape the physical picture will change; if the liquid jet is somewhere slightly thicker or thinner than the predicted shape (cf. Figure ), and thus having locally more or less mass, the downward acceleration will no longer be uniform throughout the whole jet. Also, we have not incorporated any possible growing instabilities according to the Rayleigh–Plateau instability mechanism yielding the formation of one or more drops.[21,26] Pitts[9] showed that the profile curve of a stable pendant drop or multiple curved jet structure never contains more than one inflection point. In our case, working with viscous liquids, we either do not see any drop formation or that only one single drop forms at the end of the liquid jet. Surface-tension-based instabilities may, as discussed above, subsequently take over, leading to pinch-off of the drop or the whole liquid jet/drop system evolving toward a fluid system with nonuniform d(z,t) values.

Many different studies related to jet formation after the impact of a drop or flat object on a liquid surface have been published.[26−31] In most jet formation analyses, the surface tension has not been considered as a parameter influencing the jet shape except for the tip region of the jet,[30,31] where drop formation takes place. This study is the first one to acknowledge that the evolving shape of small emanating jets can be understood and described by surface tension forces using a mathematical framework based on the Young–Laplace equation.

A more detailed verification of other existing Young–Laplace-type solutions than that of the solution with an asymptotic shape is presented in Figure . In this experiment, a water droplet with a diameter of 2.5 mm is released 26 cm above the surface of a deep-water bath; a 6000 fps Movie is provided in the Supporting Information. Snapshots made at time frames 8, 16, and 24 ms after the first appearance of the jet above the water surface are depicted, and their contours are highlighted with blue dots. The values for the curvature 2/Rc of these contours are derived by calculating the corresponding principal radii of curvature of R1, R2 = R1,2(z,t) from the graphs. The green solid line is subsequently derived by plotting 2/Rc(z) = ρzd/γ (eq ) with the obtained d values of, respectively, 22 ± 4, 18 ± 3, and 13 ± 2 m/s2.

In the graphs in Figure , the plotted curvature values (•) follow the slope of the green line implying that the surface-tension-based acceleration d applies for the fluid motion in the z-direction. This verifies the main assumption of the model that the inertial pressure gradient is indeed uniform in the whole emanating jet and has the magnitude ρd. The uniform pressure gradient also extends toward the tip region of the jet. Although a downward step in the curvature (cf. Laplace pressure) is noticed at time frames 16 and 24 ms (indicating a growing droplet), the uniform inertial pressure gradient is also present in the tip of the jet because the slope of the green solid line remains constant inside the prolate-shaped droplet.

Summary

We conclude that moderate emanating jets can best be regarded as growing liquid towers, which are shaped by the twofold action of surface tension: first that the emanated fluid is being accelerated back to the bath by surface tension, herewith creating a uniform time-dependent pressure gradient ρd(t) in the liquid tower, and that secondly this pressure gradient defines the boundary conditions to solve the shape of the liquid tower as a time-dependent solution of eq . For the first time, an asymptotic solution for the shape function has been numerically obtained and compared with experimental data on jet shapes. The shape function solution has been verified for small nonviscous jets with a moderate initial velocity up to 1–2 m/s. Dynamic surface tension measurements according to the well-known bubble pressure method and the new emanating jet method have been compared for drip coffee at 20 °C. The results show that the slope of the surface tension reduction according to the emanating jet method and with the maximum bubble pressure method is the same. A sound synchronization of the start of the surface age is necessary to compare both methods in more detail. Finally, there is presented experimental proof that other liquid tower shapes than the asymptotic shape solution can also be described with the presented model as expressed by eqs −7.