Reconstruction of the 3D pressure field and energy dissipation of a Taylor droplet from a \(\upmu\)PIV measurement

This study processes the results of the experimental work of Mießner et al. (2020). In order to deliver experimental context, we very briefly provide some specifications of the experimental setup; an in-depth description of the experimental details of the \(\upmu\)PIV-study is given by Mießner et al. (2019) and Mießner et al. (2020).

Steady and pulsation-free volumetric flow rates of the disperse phase \(Q_\mathrm {d}\) (water-glycerin) and the continuous phase \(Q_\mathrm {c}\) (octanol) establish a regular Taylor droplet train behind the pin-hole of a flow-focussing (FF) device. The Taylor flow’s volume fraction is \(\varepsilon _\mathrm {d} = 0.5\). A thin octanol film (\(\delta \lesssim\) 1\(\upmu \hbox {m}\)) always separates the droplets from the smooth hydrophobic wall material (polydimethylsiloxane, abbreviated as PDMS). The refractive index (RI) of the droplet phase is matched to that of the continuous phase flow. The flow profile of the Taylor droplets is optically measured 5 mm downstream of the FF-junction using \(\upmu\)PIV. The side walls of the channel are not ideally parallel and enclose an angle of 4.7\(^\circ\). We neglect this small trapezoidal deviation and consider the microchannel cross section to be rectangular (\(W_\mathrm {ch}\) = 104\(\upmu\)m, \(H_\mathrm {ch}\) = 96\(\upmu\)m). Both phases are seeded with fluorescent tracer particles (Rhodamine B coating) of different seeding density (\(c_d \approx 0.7 c_c\)) to enable optical discrimination between the phases. A pulsed Nd:YAG laser excites the fluorescence of the particles. Images of the fluorescent tracer signal are recorded with a CCD-camera through a microscope. The z-position of the focal plane is controlled with a piezo-stepper (MIPOS500SG, Piezosystem Jena GmbH with a precision of 8 nm), which enables to scan through the measurement volume. Per measurement plane a minimal number of 120 valid images is necessary for the \(\upmu\)PIV evaluation. The symmetry of the microchannel allows to reduce the amount of data to one half of the measurement volume. After image preprocessing, plane-wise ensemble-PIV evaluations results in a 3D2C velocity field of a mean quasi-stationary Taylor droplet in the measurement domain. The out-of-plane velocity is reconstructed based on the conservation of mass. A detailed description of the method is given by Brücker (1995, 1997). An overview of the flow conditions is given in Table 1.

The interface shape of a Taylor droplet changes from static conditions (without flow) to dynamic conditions (Mießner et al. 2019; Helmers et al. 2019b). At steady state, a clean droplet interface dA between two immiscible phases is subject to the following balance of normal forces (Fig. 2).

Herein, \(F^j_{u,i}\) refers to the normal forces induced by the adjoining flow conditions, \(F^j_{p,i}\) indicates the normal forces exerted by the pressure and \(F^j_{\mathrm {LP,i}}\) represents the normal forces that arise from the energetic molecular interaction inside the individual phases. The latter macroscopically results in the surface tension forces and is related to the Laplace-pressure. The index j refers to the flow conditions with \(j=\mathrm {stat}\) denoting the static case (without flow) and \(j=\mathrm {dyn}\) representing the influence of velocity field. The index i refers to either the disperse phase \(i=\mathrm {d}\) or the continuous phase \(i=\mathrm {c}\). Tangential forces immediately induce flow on ideal clean interfaces and do not contribute to this balance. The experimental results of Mießner et al. (2020) clearly show interface mobility and support the assumption of a minor interface contamination and an ideal interface behavior.

Equation 1 allows to compare the force balance at static conditions (\(j =\) stat) with the force balance at dynamic conditions (\(j =\) dyn). A consideration of the individual terms simplifies the equation. The forces exerted by the pressure on the respective sides of the interface cancel out (\(F^j_{p,\mathrm {d}} = F^j_{p,\mathrm {c}}\)). Without flow, the velocity forces of the static conditions vanish (\(F^\mathrm {stat}_{u,i} = 0\)). We combine the forces across the interface to a resulting force that can be calculated from the Laplace-pressure \(\varDelta F^j_{\mathrm {LP}} = +F^j_{\mathrm {LP,\mathrm {d}}} - F^j_{\mathrm {LP,\mathrm {c}}}\). A separation between the flow-induced forces and the interface tension forces links the shape deformation directly to the adjoining flow field:A division of the forces by the interface area dA results in a relation between the flow-induced pressure field and the Laplace-pressure difference between the static and the dynamic interface shape:In disregard of the involved material parameters of the Taylor flow (\(\sigma , \rho _\mathrm {d}, \rho _\mathrm {c}, \eta _\mathrm {d}, \eta _\mathrm {c}\)), the left-hand side of Eq. 3 consists of geometry information, while the right-hand side is based on the flow field. This allows a direct quantitative comparison between the curvature-dependent Laplace pressure of the interface geometry and the velocity field-dependent pressure difference at the interface.

Two approaches are available to compute the pressure distribution from the \(\upmu\)PIV measurements. One is based on the solution of the Poisson equation and requires a set of (Neumann/Dirichlet) boundary conditions to derive the pressure field (Gurka et al. 1999; Koschatzky et al. 2011; De Kat and van Oudheusden 2012). The second approach involves the direct integration of the momentum equation by means of finite differences. The main problem with the latter method is related to the accumulation of noise and integration error, which is successively incorporated into the derived pressure field (Baur 1999; Liu and Katz 2006; Jaw et al. 2009; Tronchin et al. 2015). Recently, Cai et al. (2020) have proposed a variational formulation for the pressure-from-velocity problem in two dimensions.

Charonko et al. (2010) reported that line integral methods perform better for the internal flow, while the pressure Poisson equation is superior for the external flow. De Kat and van Oudheusden (2012) proposed guidelines for the temporal and spatial resolution of the PIV-data. While the acquisition frequency has no relevance in the investigated quasi-stationary Taylor flow, the interrogation window size is supposed to be 5 times smaller than the flow structures to properly resolve the pressure features. In the presented case, the interrogation window size is about 14 times smaller than the vortex features.

In contrast to the applications of the above stated papers, the \(\upmu\)PIV-study incorporated into this paper deals with Taylor droplets at low \(\rm Ca_{\it c}\) and \(\rm Re_{\it c}\), i.e. the surface tension forces dominate the viscous forces and inertia plays a subordinate role. Thus, we decided to apply a direct integration scheme to obtain the pressure, since we address an internal flow and need to consider the second velocity derivatives of the viscous dissipation.

The Navier–Stokes equation for stationary incompressible viscous flows with negligible influence of body forces reads as follows:Based on the measured velocity field, the velocity gradient tensor is calculated together with the second derivatives. For the estimate of the pressure from the flow field, we directly integrate the Navier-Stokes equation. Since we cannot provide a reference pressure for the flow field, we omit the integration constant (\(c = 0\)).The direct integration procedure is done separately for the convective and the dissipative velocity contributions. Each integration step with respect to a coordinate axis is performed twice: Along and against the axis-direction. The two respective results are averaged to reduce the influence of error accumulation (Charonko et al. 2010). The interface approximation of Mießner et al. (2019) serves as a logical discriminator to attribute the material properties to the according phases. The convective contributions are added to the viscous terms to receive the entire 3D pressure field.

Macroscopically, work is added to the experimentally investigated flow by a syringe pump that builds up a pressure gradient to establish the flow (kinetic energy) in the microchannel. To maintain a flow, the overall work added to the flow system needs to compensate for directional changes, potential energy changes and frictional losses. In the considered case, the system is quasi-stationary, isothermal and incompressible, volume forces have no effect and Stokes-flow conditions apply for a Newtonian fluid (\(\rm Re_{\it c}=0.052\)). Thus, the work added by the pump to the stationary flow solely compensates for the total friction losses in the system.

The work to drive a single average quasi-stationary Taylor droplet through a microchannel is calculated by volume integration of the work done on individual fluid elements \({\mathcal {W}}\). The work done on the fluid elements composes a scalar field. For an individual element, the work is calculated from the forces \(\mathbf {F}\) acting along its path \(\mathbf {s}\).The forces are derived from the total change of momentum \(\mathbf {I}\) over time, with m, \(\rho\) and V being the mass, density and volume of the fluid element, respectively.The above-stated flow conditions (stationary Stokes-flow, etc.) simplify the Navier-Stokes equation in (4) toThus, the only forces that cause a pressure change in the considered Taylor flow are the friction forces. The work done on a fluid element to compensate the friction forces is expressed asHerein, the vector of the element’s path \(\mathbf {s}\) is estimated with its velocity and a short period of reference time \(\mathbf {s} = \mathbf {u}t\). A non-dimensional representation of the work done per fluid element emerges after division by a reference work. As reference, we chose the work done on a laminar pressure-driven (\(\varDelta p_\mathrm {HP,c}\)) single-phase flow (Hagen-Poiseuille flow - HP) of continuous phase (\(Q = Q_\mathrm {tot}\)) through the same square cross section \(A_\mathrm {ch}\).The scalar 3D field of dimensionless work \({\mathcal {W}}^*\) reads as follows:Negative values of the work indicate deceleration of the flow, while positive values represent acceleration of fluid elements.

The duration of the reference time can be chosen arbitrarily, since the considered Taylor flow is quasi-stationary in the relative frame of reference. This leads to the conclusion, that the dimensionless work \({\mathcal {W}}^*\) calculated above is equal to the dimensionless power loss \(P^*\) and the dimensionless pressure drop \(\varDelta {p}^*\).Despite the equivalence of the dimensionless quantities, we proceed the presentation of this study based on the above introduced concept of work that needs to be done on the flow to overcome the viscous losses in order to maintain the flow stationary.

The energetic minimization of the adhesion and cohesion forces between the molecules along and across the clean interface dA of the contacted phases (Fig. 2) determine its curvature at static flow conditions. The Laplace-pressure describes the macroscopic effect that relates the interface tension with the curvature of dA to the pressure in the disperse phase.When the flow field close to the interface exerts normal forces onto the interface dA, the curvature is deformed from the static shape (\(j = \mathrm {stat}\)) and the Laplace-pressure is altered (\(j = \mathrm {dyn}\)).

The pressure difference between both geometric states equals the pressure contribution exerted by the flow (Eq. 3). Thus, there are two methods to derive the pressure distribution on a moving Taylor droplet: i) With knowledge of the geometry of the interface shape, we calculate the curvature distribution and determine the Laplace-pressure distribution. ii) The evaluation of the 3D velocity field of a moving Taylor droplet delivers the pressure difference across the interface at the location of the interface.

An approximation of the Taylor droplet interface (Mießner et al. 2019) provides primary and secondary geometry information. Primary information means e.g. location, volume, interface area, while secondary information refers to e.g. the curvature distribution. The static and the dynamic shape of the droplet deliver the respective curvature distributions to calculate the curvature difference distribution \(\varDelta \kappa\). For the latter step, we apply the Matlab script “Surfature” (Claxton 2006):The expression in Eq. 14 is a direct geometric measure for the pressure exerted onto the interface. This scalar quantity is projected onto the interface and easily compared to the pressure derived from the velocity field.

To receive the flow-related pressure on the moving Taylor droplet interface, we subtract the pressure of the disperse phase \(p^\mathrm {dyn}_{u,\mathrm {d}}\) from pressure of the continuous phase \(p^\mathrm {dyn}_{u,\mathrm {c}}\) (Eq. 3). The equation is valid only at the position of the interface, because here the Laplace-pressure emerges from the material property changes. We use the interface approximation (Mießner et al. 2019) to determine the interface location of the moving droplet. The resulting pressure difference is a scalar quantity that is also projected onto the interface and compared to the curvature-based Laplace-pressure distribution.

We quantify the overall energy loss of the Taylor flow by determination of the work \({\mathcal {W}}^*_i\) that is necessary to keep the observed flow section stationary (Sec. 2.4). To obtain a dimensionless quantity, the work done to keep up the Taylor flow is divided by the work that is necessary to maintain a Hagen-Poiseuille flow of continuous phase with the same total volume flow through the same cross section. The experimental data set offers three different derivation possibilities for the overall energy loss of a moving Taylor droplet:

We divide the pressure field into a wall-proximate part and a core-flow to support the description of the Taylor flow phenomenology. The simple flow field of a laminar single-phase flow through a straight pipe solely evolves due to the wall contact of the fluid. In contrast, the flow in and around a Taylor droplet is more complex due to the additional presence of the droplet interfaces.

Thus, we calculate the momentum thickness \(\delta _{2}\) of a reference Poiseuille flow in a circular microchannel (Fig. 4) to geometrically discriminate between an immediate wall influenced region and the remaining core of the flow. Details on the momentum thickness are provided by (Schlichting and Gersten 2016).

The calculation of momentum thickness \(\delta _{2}\) returns a defined length scale at a fixed geometric fraction of the channel heightThree cases are used to set the momentum thickness \(\delta _{2}\) into perspective: The normalized analytical 2D velocity profile of the single-phase Poiseuille flow through the cross-section of a circular microchannel (Fig. 5a), through a square microchannel (Fig. 5b) and a measured stream-wise velocity profile at the central cross-section of a Taylor droplet’s main vortex (Fig. 5c). For the latter case, half of the microchannel is depicted, since the measurement data cover only half of the flow volume.

For better comparability, the 2D shear distributions are normalized with the wall shear stress of a laminar flow in a circular microchannel \(\tau _{\mathrm {ref}}=\frac{8{\bar{U}}}{H_{ch}}\). Values close to one show regions where the shear profile of the respective case equals the reference value (Fig. 5d–e).

A dashed line represents the border of the wall-influenced flow layer. As expected, the reference shear and the wall-shear agree at the wall in the circular channel (Fig. 5d, dotted line). The normalized shear distribution in the laminar single-phase flow of a square channel shows increased shear at the center of the side walls compared to the circular channel (Fig. 5e, dotted line). The equality of the shear profile to the reference-shear moves inwards to the position of the momentum thickness \(\delta _{2}\). The same observation holds true for the measured case of the Taylor droplet (Fig. 5f). The solid black line indicates the interface position. In the gutter, additional shear is present outside the droplet due to the by-pass flow and the high viscosity ratio between droplet and bulk (\(\lambda = 2.625\)). The magnitude of the shear distribution is not ideally symmetric in the y-direction at the top wall, because the reconstruction of the velocity z-component is subject to integration error and integration of noise error. A comparison of the well-resolved measurement in the y-direction with the reconstructed results in the z-direction shows a similar trend of an increased shear in proximity of the momentum thickness. Therefore, we consider the layer thickness \(\delta _{2}\) to be a valid measure to discriminate between wall-dominated regions and the core of a Taylor flow.

The flow field in and around a moving Taylor droplet deforms the interface from its shape at rest. Abiev (2017) suggests to make use of the flow related interface deformation to estimate the pressure distribution. We use the geometry-based model data and a flow-based approach based on experimental data to quantify and compare the pressure difference on the droplet interface in Fig. 6. The pressure differences are normalized with the driving pressure for a single-phase Poiseuille-flow of continuous phase material (\(Q_\mathrm {HP},c = Q_\mathrm {tot}\)) through an equivalent channel cross section of droplet length \(L_\mathrm {d}\). A positive pressure refers to forces that act on a area element dA against the outwards pointing surface normals (Fig. 3b): As a result, the pressure pushes the interface inwards. The interface is moved outwards at regions of negative pressure. The green arcs mark the geometric entrances and exits of the gutters.

Figure 6a shows the Laplace-pressure difference (Eq. 14) derived from the geometric interface approximation. The interface approximation produces artifacts when used as the source for second-order information (e.g. the interface curvature). The 2D Laplace-pressure profile on the interface exhibits discontinuities at the joints of different interface parts: (i) at the entrances and exits of the gutter and the wall-film and (ii) at the joints of the wall-films. In addition, the droplet interface at the wall-film does not show any sign of deformation which is in contradiction to the findings of Kreutzer et al. (2018). Both, the artifacts and the absence of pressure in wall-films are the result of the simplification of the geometric boundary conditions and the assumptions that allowed to retrieve geometric first-order information, like the location of the Taylor droplet interface.

Despite these drawbacks, qualitative information on the droplet deformation from its static shape can be observed. The elongation of the front cap is caused by a suction region at the droplet’s tip (blue) and followed by a region of positive pressure that moves the interface inwards. At the droplet’s back cap the interface is initially pulled outwards behind the gutter exits and wall-film regions and subsequently compressed at the tip. Both qualitative results correspond to the cap deformation described by the experimentally derived correlation of Helmers et al. (2019a).

Figure 6b depicts the pressure difference derived from the velocity field (Eq. 5 used in Eq. 3) at the interface of the Taylor droplet. In addition to the pressure information, the streamlines of the interface motion are shown (Mießner et al. 2020).

The white area at the top wall-film artificially closes the droplet. The interface gap exists due to a lack of measurement data. However, the flow symmetry allows access to the pressure conditions in the films at the side wall-film instead.

The effect of the front cap elongation and the back cap compression is clearly present in the measurement-based pressure. The general magnitude range of the pressure distribution is comparable between the two approaches. The cap deformations arise from the relative motion of the main vortices in the slugs. The ring shape stagnation region at the droplet front indicates the location where the slug’s wall-driven vortex attaches to the interface, while it detaches again at the droplet front tip. The pressure rises due to the viscous displacement that takes place between attachment and detachment. The influence of the detachment is smaller than predicted by the geometric representation (Fig. 6a). At the droplet back, inverted flow conditions with respect to the interface cause the back cap compression.

The pressure evolution from the wall-film entrance at the droplet front towards its rear exit shows a series of pressure changes. At the frontal stagnation region, the liquid of the continuous phase is either redirected to the droplet tip or in the direction of the wall-film. At the onset of the film the outer liquid is forced into the film and the pressure rises. After the transport of fluid through the film, the film-thinning increases the pressure. At the exit of the film, the liquid flows towards the ring-shaped stagnation region at the back, where it slows down again. This series of pressure changes feed back to the interface and coincide with the location where the onset of the bullet-shape for increased \(\rm Ca_{\it c}\) is situated (Taha and Cui 2006).

The pressure distribution along the gutter appears to be different compared to the geometric approach. While the magnitudes of the pressure at the gutter entrances are similar, the distribution in gutters deviates. The geometric approach delivers positive Laplace-pressures, while the flow-based pressure carries a negative sign. However, in both cases the pressure decreases from the front to the back of the gutter. Thus, the pressure gradient points into the same direction—against the flow direction of the Taylor droplets. This confirms the postulated pressure gradient inversion with respect to the flow direction by Abiev (2017).

An estimation of the pressure gradient along the gutter allows the determination of the volume flow through the gutter and calculation of the relative velocity (Helmers et al. 2019b). Based on the droplet interface geometry, two possibilities to analytically obtain the pressure along the gutter are feasible: an evaluation of (i) the curvature difference distribution between the static and the dynamic Taylor droplet shape (Fig. 6a, Eq. 14) and (ii) the Laplace-pressure difference between the mean gutter radius at the front and back of the gutter. The gutter radii are calculated as a by-product of the interface approximation (Mießner et al. 2019).

The measurement-based pressure difference is used as a reference to asses the accuracy of the theoretical approaches (Fig. 7). For this purpose, the distribution in the gutter is averaged in transverse direction (blue triangles). The measurement resolution in the x-direction of the source velocity field serves as sampling grid distance in flow direction. The schematic inset of Fig. 7 visualizes the sampling method for the gutter gradient. The offset of pressure at the back end of the gutter is removed. The result is normalized with the driving pressure of a single-phase Poiseuille-flow of continuous phase material (\(Q_\mathrm {HP},c = Q_\mathrm {tot}\)) through an equivalent channel of gutter length \(L_\mathrm {g}\) to quantify the influence of the droplets presence.

A linear fit to the averaged pressure difference (blue solid line) allows to estimate the overall gutter pressure gradient of the experiment. The confidence interval of the slope (\(\pm 9.1\%\)) is calculated to quantify the influence of the standard deviation of the measurement from the linear fit on the gradient (blue area).

The Laplace-pressure difference due to the geometric droplet deformation is averaged in the same manner (\(\circ\)), and a linear fit allows to estimate the pressure gradient in the gutter. Clearly, the slope of the model surface fit is half as steep as the pressure gradient of the measurement is (\(-55.4\%\)). We attribute the deviation between Laplace pressure difference of the entrances gutter (black dashed line) and the model’s mean curvature distribution (black dotted line) to the simplifications introduced to calculate the grid points of the interface shape. Thus, the curvature difference of the approximate interface cannot be used to estimate the gutter gradient.

In contrast, the calculation of the gutter gradient with the Laplace-pressure difference between the mean gutter radius at the front and back of the gutter leads to a suitable estimation (Fig. 7, dashed line). The result lies well inside the variance of the measured slope (\(+2.7\%\)). The gray area indicates the sensitivity to a \(\pm 25\%\) slope change of the gutter radius approach. Thus, the approach of Helmers et al. (2019b) to estimate the driving pressure gradient of the gutter from the gutter radii is confirmed by the results presented here.

In disregard of a reference pressure and of the Laplace-pressure at rest, the presented 3D field solely shows the pressure that drives the flow inside and outside of the Taylor droplet. The flow is directed from left to right. Streamlines in a frame of reference relative to the droplet motion are given on half of the central symmetry plane (Fig. 8a). Bernoulli’s principle does not apply, since the viscous forces are 20 times stronger than the inertia forces (\(\rm Re_{\it c} = 0.05\), Stokes-flow conditions).

The pressure field of the Taylor droplet generally shows two different regions for its evolution: The pressure field in a layer close to the wall \(\delta _{2}\) and in the remaining core of the field. Inside the wall-proximate layer, the pressure decreases as expected in stream-wise direction. However, the direction of the pressure gradient reverses in the core of the flow.

In the core flow, the pressure builds up at the droplet front due to viscous displacement. The outer main vortex in the slug is forced to change direction due to the interface at the droplet front. This elevated pressure in front of the droplet is the actual pressure source that drives the by-pass flow through the gutters (Abiev 2017) and gives rise to the relative velocity.

The wall-driven main vortex B inside the droplet strongly changes the flow direction to evade the secondary vortices (A, C) that connect the inner to the outer field. Two regions of the droplet’s main vortex inner region B stand out: the converging rear develops a positive pressure, while the diverging front shows a negative sign.

The fundamental change of pressure conditions between the outer layer \(\delta _{2}\) and the core of the flow is attributed to the presence of shear at the wall (Fig. 8a). In the case of the experimental conditions applied here, the thickness of the wall-proximate layer amounts to \(\delta _{2}\). Perpendicular to the center plane (Fig. 8a), four sections are chosen to investigate the pressure evolution from the wall (Fig. 8b) towards the core of the flow (Fig. 8e). The black line indicates the position of the Taylor droplet interface. The pressure distribution at the wall (Fig. 8b) correlates with the droplet motion: increased pressure on the left causes the droplet motion downstream towards the right, while showing a decreasing pressure level.

To emphasize the change of the pressure gradient from the wall towards the core of the flow (Fig. 8b–e), the pressure is averaged in the z-direction to receive a pressure profile in downstream direction (Fig. 8f–i). A linear fit illustrates the change of pressure gradient (red lines). An inversed pressure gradient with respect to the flow direction is present at a wall distance of \({\tilde{y}} = \frac{\delta _{2}}{H_\mathrm {ch}}=1/15\). It seems that the core flow is moved through the microchannel and causes the shear-related pressure loss in the wall-layer. Inside the core flow, the viscous displacement of the recirculating main ring-vortices give rise to an elevated pressure.

In Fig. 9, the consideration of mean pressure profiles along the downstream direction provides an overview for the discussion of the pressure. For this purpose, we discriminate between the wall-proximate layer of thickness \(\delta _{2}\) and the core of the flow as well as between droplet and bulk. The color-coding indicates the disperse phase with red and the continuous phase with blue, while the core of the flow is represented with a thick line and the wall-layer with a thin line. The constant contribution of the Laplace-pressure is added to the droplet’s pressure profiles to visualize the pressure jump at the interface.

Inside the wall-layer of both phases (Fig. 9, thin lines) the mean pressure decreases as expected in downstream direction. The elevated droplet viscosity \(\lambda = \eta _\mathrm {d}/\eta _\mathrm {c} = 2.625\) causes a higher pressure drop in the momentum layer of the droplet in comparison with the layer of the continuous phase.

In the core flow of the droplet, a clear reversed pressure gradient is present in either phase (Fig. 9, thick lines). An expected maximum of the pressure is found at the rear stagnation point of the droplet (red thick line). The pressure fluctuation induced by the rear secondary vortex is followed by an almost linear pressure increase. The pressure fluctuations of the frontal secondary ring-vortex end in a local minimum at the frontal singularity of the droplet.

The evolution of the mean pressure in the core flow of the continuous phase (Fig. 9 thick blue line) along the droplet also shows the reversed pressure gradient. Most of the profile is almost constant from the back to the front of the droplet. The reversed pressure increase that drives gutter flow is located between the frontal gutter entrance and the frontal onset of the wall-film. Further downstream, the pressure is almost constant again. This again confirms the postulated pressure gradient inversion with respect to the flow direction by Abiev (2017).

The distribution of the work done on the flow allows to locate the major contributions to the energy loss of the flow. For visualization reasons, the distribution of the work done \({\mathcal {W}}^*\) is scaled with the number of sampling points \(\varPi _i N_i\) of the velocity field (Fig. 10). The droplet moves in the positive x-direction.

The work distribution in the central symmetry plane is shown in Fig. 10a. The interface shape is indicated with a thick black line. The streamlines relate the flow field to the distribution of work done on the flow.

Deceleration work (blue) is mainly performed due to the directional change of downstream the velocity component of the flow field. Consequentially, the peak deceleration is located at the annular stagnation regions, where the fluid of the main vortices is forced to reverse direction.

Acceleration work is performed (red) in the caps of the Taylor droplet and inside the wall-influenced layer \(\delta _{2}\). The counter rotating secondary vortices (A, C) in the caps are driven by and receive their energy from the adjoining wall-induced main vortices. Except for the strong directional change of the main vortices, the fluid close to the wall is accelerated due to the no-slip condition at the wall. The layer-thickness of apparent acceleration amounts roughly to \(\delta _{2}\) and coincides with the momentum layer thickness where also the wall-related pressure gradient is situated (Fig. 8a).

Inside this layer, wall-proximate sections through the field of work are presented in Fig. 10b–e. It is noteworthy that in transverse direction the acceleration regions are spread mainly over the width of the wall-film of the Taylor droplet. With increasing wall distance \({\tilde{y}}\) the deceleration of the flow grows where the droplet interface redirects the flow.

The respective z-averaged profiles of work done on the flow \({\mathcal {W}}^*\) along the flow direction are given in Fig. 10f–i. Moving step-wise inwards from the wall, the acceleration intensity increases between f–g, and decreases when reaching \(\delta _{2}\) (Fig. 10i). In the case of this experiment, we consider the momentum layer thickness \(\delta _{2}\) to be a valid measure to distinguish the wall-proximate flow from the core of the Taylor flow.

We show and discuss the cumulated energy dissipation of a Taylor droplet, which is in its dimensionless form equivalent to the overall pressure drop of the Taylor droplet (Eq. 12). Thus, we quantify the cumulated work done on the flow, because the energy loss of a Taylor flow due to viscous dissipation equals the work done on the flow to maintain the flow stationary.

Since we provide indirect and derived data on the loss of energy, we apply and compare the results of three different methods. These calculations are based on (i) the resistance forces at the interface of the droplet and the microchannel wall \({\mathcal {W}}^*_F\), (ii) the actual work done on the flow field \({\mathcal {W}}^*_u\) and (iii) the pressure loss at the channel wall \({\mathcal {W}}^*_{\varDelta p,\mathrm {wall}}\). As reference quantity, we use the work done on a comparable single-phase flow \({\mathcal {W}}_\mathrm {ref}\) (Eq. 10). The discussion of the results always begins at the tip of the rear cap (\(x/H_\mathrm {ch} = -L_d/H_\mathrm {ch} = -1.58\)) and proceeds in flow direction (positive x-direction).

All presented approaches to quantify the energy dissipation of the two-phase flow are derivations of high order. They indirectly depend on the reconstructed 3D flow field to a variable degree: While pressure-based work \({\mathcal {W}}^*_{\varDelta p,\mathrm {wall}}\) involves double differentiation and an integration, the interface force-related work \({\mathcal {W}}^*_{F}\) is subject to additional conversion and integration steps. Since the work done on the flow \({\mathcal {W}}^*_u\) includes only double derivatives, we consider the result to be more reliable. From this perspective, the tendency towards lower magnitudes of total work done on the Taylor flow correlates with a decreasing accuracy due to the indirectness of the quantification approaches. In addition, the force-based approach \({\mathcal {W}}^*_{F}\) does solely include forces in flow direction and omits contributions of the transverse flow.