Effects of geometry in the operation of coaxial electrosprays

Journal of Aerosol Science 167 (2023) 106075 Available online 6 September 2022 0021-8502/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Journal of Aerosol Science journal homepage: www.elsevier.com/locate/jaerosci Effects of geometry in the operation of coaxial electrosprays J.M. López-Herreraa,∗, M.A. Herradaa, M. Gamero-Castañoc, A.M. Gañán-Calvoa,b aDepartmento de Ing. Aerospacial y Mecánica de Fluidos, ETSI, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain bLaboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, 41092, Sevilla, Spain cDepartment of Mechanical and Aerospace Engineering, The Henry Samueli School of Engineering, University of California-Irvine, 92617, CA, USA ARTICLE INFO Editor: Dr. Chris Hogan Keywords: Coaxial electrospray Leaky dielectric model Taylor cone jet Electrospray ABSTRACT Coaxial electrospray is a well established procedure to generate coaxial microand nanostructures (microcapsules and coaxial fibers). In this work we show the capabilities of a high-precision numerical code using dynamic mapping, analytic Jacobian formulation and the Taylor–Melcher leaky dielectric model (LDM) to deal with real coaxial electrospray configurations. Since the number of parameters involved in the problem is large, we focus on an experimentally tested liquid pair configuration where the thickness, relative position, and end sharpness of the coaxial feeding tubes are geometrical parameters of interest. The effects of those parameters and the voltage applied at the tubes on the coaxial liquid menisci, liquid streamlines, size of jet and core, and electric current are discussed. 1. Introduction Coaxial electrospraying (CES) is a technique for generating micro/nano size compound structures that was first demonstrated by Loscertales et al. (2002). In the most extended configuration (see Fig. 1.a) two capillary tubes are placed concentrically and two fluids, immiscible or not, are made to flow separately through the inner tube and through the annular gap. An electric voltage is applied at the tubes with respect to a downstream grounded electrode. As a consequence, both fluids are funneled by electrical forces to a much smaller spatial scale than the size of the feeding capillaries. The most interesting regime is the so-called steady Taylor cone-jet mode (TCJ) in which a conical meniscus, anchored to the outer capillary, is focused into a steady jet. Typically, CES leads to compound drops by varicose Rayleigh–Plateau instabilities when low viscosity fluids are electrosprayed. On the other hand, if the fluid viscosities are large, the coaxial jet is thinned without limit, whips and spreads spatially by Coulombic repulsion. In this case, the method is named electrospinning (Reneker & Yarin,2008) and is used as a source of compound fibers (Sun et al.,2003). Electrospinning usually also involves rheological forces derived from the use of polymer solutions. The technique is often used with other physicochemical processes, such as solvent evaporation, to obtain solid structures in a single step. CES exhibits advantages over methods such as spray drying, emulsion solvent evaporation and nanoprecipitation to encapsulate drugs in polymers shells (Gañán-Calvo et al.,2013). It can also be employed to encapsulate sensitive bio macromolecules such as DNA, RNA, proteins and enzymes (Boda et al.,2018;Chen et al.,2019), although potential disadvantages such as electrochemical damage of active molecules, gas discharges or undesired impact of capsules on the collector could preclude its use. CES has also been used in environmental engineering, particularly for biosensing and catalysis purposes (Chang et al.,2010a,2010b;Yan et al., 2019). ∗Corresponding author. E-mail address: [email protected] (J.M. López-Herrera). https://doi.org/10.1016/j.jaerosci.2022.106075 Received 8 July 2022; Received in revised form 26 August 2022; Accepted 28 August 2022 Journal of Aerosol Science 167 (2023) 106075 2 J.M. López-Herrera et al. Fig. 1. (a) Sketch of the problem with definition of dimensionless parameters. The rectangle denotes the computational domain. (b) Computational subdomains and grids for the original and mapped variables. Despite the large number of publications reporting its use in different applications (see review papers by Gañán-Calvo et al., 2018;Rosell-Llompart et al.,2018;Zamani et al.,2013;Zhang et al.,2012), few studies have focused on the physical processes and the influence of operating conditions relevant to this technique. The core and shell sizes of the compound products depend mainly on the flow rates imposed and fluid properties, especially the electric conductivity and the surface and interfacial tension, see López-Herrera et al. (2003). This study also proposes the scaling laws for the size of cores and shells, and discusses in detail the concept of a driving liquid, first introduced by Loscertales et al. (2002), where electrostatic forces are concentrated. Furthermore Chen et al. (2005) and Mei and Chen (2008) provide a classification of operational modes; Mei and Chen (2007) study the effects of surface tension on the encapsulation efficiency; and Morais et al. (2020) study the effects of the concentration and type of solvent in the fabrication of polymeric compound particles. The geometry of the concentric capillaries may influence the flow pattern in the menisci and, eventually, in the transition region. Sofokleous et al. (2016) is the only study addressing this, specifically by experimentally investigating how the inner capillary tip displacement affects the morphology of the final products. These authors generated encapsulated particles and fibers by varying the concentration of a polymeric solution fed through the annular gap. For the physical properties of their liquid formulations, they concluded that the optimal configuration occurs when the inner tube is retracted 2 mm inside the outer one (the outer/inner diameters of outer and inner tubes are 1820/1350 μm and 560/305 μm respectively). However, the experimental characterization of such a complex system is extremely time consuming, and an alternative method based on a numerical model is specially useful to make these parametric studies. Toolboxes like multiphysics COMSOL facilitate the implementation of numerical schemes for this complex problem. In fact, Zhao et al. (2019) use this platform to simulate coaxial electrohydrodynamic jets. Their scheme uses a non-steady finite volume method in which the position of the interfaces is tracked solving the Cahn–Hilliard equation. Their results are consistent with the experimental data. Recently, López-Herrera et al. (2020) have developed a numerical scheme to model CES of two immiscible liquids. These simulations investigated the key transport processes in the conical meniscus, the transition region and the compound jet, where the numerical solution provides the stress, charge and velocity fields, as well as the free interfaces. López-Herrera et al. (2020) focused on the roles of the flow rates and conductivities of the liquids involved, revealing the power of a numerical simulation for providing a deep insight into CES. The present article extends the numerical scheme of López-Herrera et al. (2020) to study the influence of the geometry of the concentric tubes. In particular we study the effects of the thickness of the tubes, the protrusion of the inner capillary and the sharpness of the edges in CES. Journal of Aerosol Science 167 (2023) 106075 3 J.M. López-Herrera et al. 2. Formulation of the problem Our model solves the Navier–Stokes equations for two incompressible and immiscible fluids, and the Maxwell electric equations in the limit of the Taylor–Melcher Leaky Dielectric model (LDM). Thus, we assume that the charge accumulates on the free interfaces, while the bulk of the fluids is free of charge and acts as an Ohmic resistor (Saville,1997). The model is closed by imposing conservation equations for the surface charge. The positions of the interfaces are part of the solution. In what follows, the dimensional magnitudes are denoted with a tilde; otherwise, they are dimensionless. For example,  𝐿𝑜stands for the dimensional length of the outer tube, while 𝐿𝑜is dimensionless. The numerical scheme is based on an analytical transformation of coordinates from the physical space (𝑟, 𝑥)to the mapped computational space (𝜂, 𝜉)as depicted in Fig. 1.b. Three computational phases corresponding to each different fluid are indicated: Phases 𝑖,𝑜, and 𝑎are the fluid circulating through the inner tube, the fluid flowing through the annular gap, and the dynamically negligible surrounding atmosphere, respectively. The density of the inner fluid 𝜌𝑖, the outer surface tension 𝛾𝑜, the inner radius of the inner tube  𝑅𝑖, and the permittivity of air 𝜀𝑎are used as a set of magnitudes to nondimensionalize the equations. The bulk equations for the LDM are well established (Saville,1997); for each phase it has to be imposed a solenoidal velocity field, the momentum equation without volumetric electrical forces and the Laplace equation for the electric potential. The constraints at the free surface include the balance of stresses along the normal and tangential directions (stressed due to pressure, viscous, electric and interfacial tensions), the kinematic velocity condition, conservation of charge, and the condition for the jump of the normal components of the electric field across the interface. Furthermore, the velocities and electric potential are continuous at the two free surfaces. At the inner interface the conservation of charge it must be taken into account not only the charged added from inner fluid by electrical conduction but the one withdrawn by conduction through the outer fluid. The flow is fully developed upstream in the capillary tubes, resulting in a parabolic velocity profile for the inlet boundary condition and nonslip velocity condition at the wall of the tubes. The boundary conditions for the far field electric potential at the top and right of the computational domain shown in Fig. 1.a are taken from the analytical form derived by Gañán-Calvo et al. (1994) which modified the analytical potential distribution created by a semiinfinite line at voltage 𝛷at a distance 𝐻from a grounded plate, to obtain the field associated with a semiinfinite cylinder of finite radius. At the left boundary we set a logarithmic profile of potential matching the analytical potential at the top with the applied voltage of the emitter (𝛷at 𝑟=𝑅𝑜+𝜀𝑜). The mapped domains are discretized in the radial 𝜂direction with the Chebyshev spectral collocation method, while grid points are uniformly distributed in the axial 𝜉direction. After writing the equations in the {𝜉,𝜂}variables, the spatial derivatives are discretized with fourth order accuracy. The program has been built in the computational Matlab©environment, which facilitates the formulation of the equations in the mapped variables through symbolic calculus. The resulting non linear set of equations is solved with the Newton–Raphson method. The analytical calculation of the Jacobian reduces drastically the cost of its computation with respect to a scheme that evaluates it numerically. We refer the reader to López-Herrera et al. (2020) for a detailed account of the model and the numerical methods, and only describe in detail the new elements needed to simulate the geometry of the tubes. The main departure from López-Herrera et al. (2020) is the consideration of tubes of finite thickness, which is needed to study the geometrical effects of interest. Tube thickness is incorporated by modifying the radial transformation of the phases 𝑖,𝑜and 𝑎 as: Phase𝑖∶𝜂=𝑟 𝑓1(𝑥)with 𝑓1(𝑥) = {1𝑥≤𝐿𝑖 𝑓𝑖(𝑥)𝑥 > 𝐿𝑖 Phase𝑜∶𝜂=𝑟−𝑓2(𝑥) 𝑔1(𝑥)−𝑓2(𝑥)with 𝑓2(𝑥) = {1 + 𝜀𝑖[1 − e𝛼𝑖(𝑥−𝐿𝑖)]𝑥≤𝐿𝑖 𝑓𝑖(𝑥)𝑥>𝐿𝑖 and 𝑔1(𝑥) = {𝑅𝑜𝑥≤𝐿𝑜 𝑓𝑜(𝑥)𝑥>𝐿𝑜 Phase𝑎∶𝜂=𝑟−𝑔2(𝑥) 𝑅−𝑔2(𝑥)with 𝑔2(𝑥) = {𝑅𝑜+𝜀𝑜[1 − e𝛼𝑜(𝑥−𝐿𝑜)]𝑥≤𝐿𝑜 𝑓𝑜(𝑥)𝑥>𝐿𝑜 (1) where 𝐿𝑖,𝑜,𝑅𝑖,𝑜 and 𝜀𝑖,𝑜 are the dimensionless length, inner radius and thickness of the inner/outer (𝑖, 𝑜)tube, respectively. Note that with the adimensionalization used 𝑅𝑖= 1. The parameter 𝛼𝑖,𝑜 controls the steepness of the tubes’ tips. 𝑓𝑖,𝑜(𝑥)are the dimensionless radial position of the inner and outer fluid interfaces, respectively. 3. Results Our study is carried out using a realistic case studied in López-Herrera et al. (2020), corresponding to a inner flow rate of deionized water  𝑄𝑖=200 μL/h, and an outer flow rate  𝑄𝑜=60 μL/h of tributyl phosphate (TBP). The physical properties and flow rates of the fluids are kept fixed, and we study different tube geometries. The values of the physical properties can be found in López-Herrera et al. (2020) (see Table 1). The outer tube is made of stainless steel and has outer and inner diameters of 711.2 μm and 533.4 μm, while the inner tube was a fused silica tube with 360 μm and 200 μm outer and inner diameters, and whose surface was metalized. In the experiments Journal of Aerosol Science 167 (2023) 106075 4 J.M. López-Herrera et al. Table 1 Density, viscosity, surface/interfacial tension, electrical conductivity and relative permittivity of the liquids used. The value of 𝛾reported for the water (first row) corresponds to the interfacial tension between water and TBP. Magnitudes are given in the SI units. Liquid 𝜌 𝜇 𝛾 𝜅 𝛽 (kg/m3) (Pa s) (N/m) (S/m) Water (inner) 1000 1.00 10−38.00 10−36.42 10−481 TBP (outer) 976 3.93 10−327.55 10−38.82 10−58.91 of López-Herrera et al. (2020) the tubes fully overlapped (𝛥𝐿 =𝐿𝑖−𝐿𝑜= 0). A stable coaxial cone-jet was formed by applying a voltage  𝛷=4570 V between the concentric tubes respect a grounded electrode located about 10 mm downstream. The electrical current,  𝐼=90 nA, was measured with an electrometer connected to the grounded electrode. Using this case as a reference, we next explore how the flow field and the shape of the meniscus are affected by changes in the geometry of the capillaries. We expect the changes to be confined to the flow lines in the menisci, with the transition region remaining unchanged due to the disparity in the characteristic lengths. In the present calculations a computational domain box of 𝐿×𝑅= (16 × 30) is used. The distance from the tube to the plate is 𝐻≃ 40. The number of points in the axial direction is 𝑁𝑧= 1151 while in the radial direction we impose 𝑁𝑖 𝑟= 30,𝑁𝑖 𝑟= 20 and 𝑁𝑖 𝑟= 115 for the inner, outer and ambient phases. The position of the outer tube is fixed to 𝐿𝑜= 3.7while the inner one will be varied. 3.1. Analysis of the retracted inner capillary configuration We first consider the case of a retracted and sharp inner tube, 𝛥𝐿 =𝐿𝑖−𝐿𝑜= −2.2and 𝛼𝑖= 1.2, as shown in Fig. 2.A. A key output of the numerical solution is the electric current circulating through the fluids. The LDM considers two charge transport mechanisms contributing to the electric current: bulk conduction through both liquid subdomains (𝐼𝑏,𝑖 and 𝐼𝑏,𝑜), and convection of surface charge on both free surfaces (𝐼𝑠,𝑖 and 𝐼𝑠,𝑜). At any axial position of the cone-jet, each type of current is given by 𝐼𝑠,𝑘(𝑥)=2𝜋𝑓𝑘(𝑥)𝜎𝑘(𝑥)𝑣𝑠,𝑘(𝑥), 𝐼𝑏,𝑘(𝑥) = ∫𝑓𝑘 𝑓0𝑘 𝜅𝑟 𝑘𝐸𝑥(𝑥) 2𝜋𝑟d𝑟(2) where 𝜎𝑘is the surface charge density on the outer surface of the 𝑘th domain, and 𝑘= {𝑖, 𝑜}. Due to conservation of charge, the sum of the four components must be constant and equal to the emitted current at any axial position downstream of the tubes, 𝐼=𝐼𝑏,𝑖(𝑥) + 𝐼𝑠,𝑖(𝑥) + 𝐼𝑏,𝑜(𝑥) + 𝐼𝑠,𝑜(𝑥).𝐸𝑥is the axial component of the electric field, and 𝑓0𝑘is equal to 0 or 𝑓𝑖for the inner or outer liquid subdomains, respectively. The current has been made dimensionless with the reference value 𝐼𝑜= (𝛾2 𝑜𝜀𝑎∕𝜌𝑖)1∕2. The pattern of streamlines together with the geometry of the capillaries are shown in Fig. 2.A, while the different current profiles are shown in Fig. 2.B. Interestingly, the current builds up by the injection of charge from the surface of the tubes, specially near their ends (see the magenta line, 𝐼𝑡𝑜𝑡𝑎𝑙). This is consistent with the screening of the electric field inside the capillaries, and the resulting reduction of bulk conduction. Consequently, sufficiently upstream, e.g. in region I, the current circulating through the fluids is negligible (upstream the current circulates through the electrodes). The screening of the electric field is not as effective in region II, and therefore an extended length of the outer tube is able to inject charge in the fluid. In addition, in region II both the electric and the viscous stresses on the inner meniscus are weak (i.e. both the electric and velocity fields are weak) compared to the interfacial tension, resulting in a nearly spherical shape. More precisely, the interface is almost exactly a prolate spheroid with a 1.187 elongation ratio in the axial direction extending along region II and into half of region III. The velocity field in the inner meniscus is so weak in the vicinity of the exit of the outer tube that the streamlines of the outer fluid (blue lines) detach from the meniscus, forming a small and slow recirculation region between the inner and outer menisci (see Figs. 2 and 3(a)–(d), around 𝑥= 3.5). Once in region III, the strong electrical stresses on the outer surface produce an intense acceleration of the outer liquid which drags the inner meniscus into a conical shape, creating a thin coating. At the applied voltage at the tubes used in the simulation, the inner meniscus can be accurately reduced to a cone with a semiangle of 45◦, tangent to the prolate spheroid already mentioned with the line of tangency approximately with radius 1; both the prolate spheroid and the tangent cone are shown in 2by red dashed lines. However, the conditions for the formation of a thin coating are not of a mechanical origin exclusively: depending on the ratio of viscosities, electrical conductivities, permittivities, and the surface/interfacial tensions, one may have a continuous variety of situations (Gañán-Calvo & Montanero,2021). In general, a thin coating or thin-layer configuration can be observed whenever the inner to outer conductivity ratio is smaller than unity and the outer medium is a dielectric (see Figure 3(a) in Gañán-Calvo and Montanero (2021)), due to the dominant screening effect of the outer meniscus charge layer, which leaves the inner meniscus relatively free to fill the entire inner space in the outer meniscus. This is the case here presented, belonging to a relatively ample range of parametrical combinations. The total current cannot vary in region III because there are no electrodes that can inject charge into the fluid. The electric fields and forces are more intense in this region, and the menisci adopt the conical shape first explained by Taylor (1964). The main mechanism for the electrical current shifts from bulk conduction to convection of surface charge in the cone-to-jet transition region, i.e. in the outer edge of region III (Fernandez de la Mora & Loscertales,1994;Gañán-Calvo,1997;Gañán-Calvo et al.,1993). Beyond region III, the current is converted from dominant bulk conduction to surface charge convection in the outer interface, that Journal of Aerosol Science 167 (2023) 106075 5 J.M. López-Herrera et al. Fig. 2. The inner tube end is retracted to 𝛥𝐿 = −2.20. In the upper graph (panel A), the geometry of tubes and interfaces as well as the streamlines pattern of inner and outer fluids are shown. The shape of the inner interface approximately corresponds to a prolate spheroid tangent to a conical tip at 90o, as indicated by the red dashed lines. In the lower panel B, the axial profiles of dimensionless electric current 𝐼are plotted. Upper indexes 𝑠and 𝐵stand for surface and bulk current while subindexes 𝑜and 𝑖correspond to outer and inner phase, respectively (See Eq. (2)). The current is made dimensionless with the reference current  𝐼𝑜= (𝛾2 𝑜𝜀𝑎∕𝜌𝑖)1∕2 and the lengths with the inner radius of the inner tube,  𝑅𝑖. The dimensional total current in this simulation is  𝐼𝑡𝑜𝑡𝑎𝑙 =71.05 nA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 3. Streamlines for different relative positions of the inner tube respect to the outer one. In the top row, subplots (a,b,c,d), the inner tube is retracted to 𝛥𝐿 = −2.20. In the intermediate row, subplots (e,f,g,h), both tubes are overlapped (𝛥𝐿 = 0.). In the bottom row, graphs (i,j,k,l), the inner tube protrudes up to 𝛥𝐿 = 2.15. (a,e,i,b,f,j) correspond to a blunt outer tube end (𝛼𝑖= 10), while (c,g,k,d,h,l) are for a sharp end (𝛼𝑖= 1.2). Besides, (a,e,i,c,g,k) are for an inner tube blunt end, while (b,f,j,d,h,l) are for a sharp end. The dimensional current is in a narrow range between the maximum value obtained for the configuration shown in panel (j) (  𝐼𝑡𝑜𝑡𝑎𝑙 =71.88 nA) and the minimum one of panel (e) (  𝐼𝑡𝑜𝑡𝑎𝑙 =70.94 nA). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) eventually prevails. Interestingly, the bulk current through the outer fluid exhibits distinct peaks at the borders between regions. The first two cusps (corresponding to borders I–II at 𝑥= 1.48, and II–III at 𝑥= 3.69) essentially reproduce the maxima (peaks) of the outer annular cross section (see Fig. 2) in these transitions, due to the homogeneity of the inner electric field in the axial direction in those regions. The third smoother cusp (𝑥= 5.33) is caused by the local rise of the inner electric field close to the apex in region III, immediately followed by a decay in the jet consistent with the decay of bulk conduction at the expense of interfacial convection. Another interesting issue is the negative polarity of the inner surface convection current beyond region III. Although this is not a Journal of Aerosol Science 167 (2023) 106075 6 J.M. López-Herrera et al. Fig. 4. (color online) Axial distribution of the different terms of the interfacial normal stress balance at the inner interface 𝑓𝑖, Eq. (3) for the tubes geometry arrangements shown in Fig. 3 panels a,b,i and j. Labels {𝐓𝑝;𝐓𝑠𝑡;𝐓𝑣} refers to {‖𝑝‖;𝛾𝑖∇⋅𝐧𝑖;𝐧𝑖⋅‖𝜏𝑣‖⋅𝐧𝑖}. The normal (subscripts 𝑎and 𝑏for inner and outer sides) and tangential (subscript 𝑐) contributions to the electric stress term 𝐓𝑒, namely {𝐓𝑒 𝑎,𝐓𝑒 𝑏,𝐓𝑒 𝑐}={𝛽𝑜(𝐸𝑜 𝑛,𝑖)2∕2,𝛽𝑖(𝐸𝑖 𝑛,𝑖)2∕2,(𝛽𝑖−𝛽𝑜)𝐸2 𝑡,𝑖∕2}, are distinguished. readily explainable result, it could be attributed to the imbalance caused by the strong extraction of charges from the inner liquid that the rapidly accelerating outer surface generates. Despite the inner domain is the smaller conductivity one, it supplies the vast majority of dominant bulk current owing to the thinness of the outer layer. In addition, the viscous diffusion and mass continuity immediately lead to a nearly uniform inner and outer velocity profiles beyond region III in the coaxial jets. It appears that these effects together with the normal stress balance at the inner and outer jets would lead to a total excess of positive polarity charges moving in the downstream direction unless the inner surface charge layer would be negative. 3.2. Comparison between protruding, flushed and retracted tube configurations, and edge sharpness Fig. 3 shows the streamline pattern for different relative positions of the capillary tubes and the sharpness of their ends. In particular we simulate twelve configurations resulting from the permutation of three relative positions (retracted inner tube, 𝛥𝐿 = −2.2, top row; flushed inner tube, 𝛥𝐿 = 0, middle row; and protruding inner tube, 𝛥𝐿 = 2.15, bottom row), and two sharpness values for each tube (blunt 𝛼𝑘= 10, and sharp 𝛼𝑘= 1.2). The variation of the total current in the twelve cases is within ±1% of the average, which is to be expected because the cone-to-jet transition region is not substantially affected by the relative positions of the tubes or the sharpness or their ends. The shape of the inner meniscus depends strongly on the relative positions of the tube and very little on the sharpness of their ends. As already mentioned, since the screening of the electric field is larger as the inner tube is pulled backward, a nearly spheroidal shape meniscus results in subplots (a) to (d). In contrast, when the inner tube protrudes (subplots (i) to (l)), the inner meniscus is exposed to the electric field and adopts the classical conical shape. The screening effect is also responsible for the weak dependency of the streamlines on the tube sharpness (except in the vicinity of the end of the tube). This is not the case for the protruding configuration, subplots (i) to (l). As subplot (l) shows, the sharp tubes allow the outer fluid to form a thin layer fully surrounding the inner meniscus, i.e. from its anchor point to the inner tube. In this way the strong electrical stresses on the outer meniscus are more effectively transmitted by viscous diffusion to the inner fluid, which we expect to enhance the global stability of the system. That said, a rigorous global stability analysis is needed to provide definitive and quantitative answers to any stability problem. However, we find that the numerical scheme shows an expedite convergence to solutions that can be observed in reality, while real unstable configurations typically exhibit difficulties in convergence, or no convergence at all, similar to what is found in the real system that our model simulates. In the present system, the most serious stability problem where the above statement could fail would come from asymmetric instabilities (e.g. those that may arise at the higher voltages) that our strictly axisymmetric configuration excludes. 3.3. Comparison of stresses We next analyze the balance of normal stresses in the inner interface for four configurations in Fig. 3, namely for the cases in panels a, b, i and j. Fig. 4 plots the different components of the normal stress along the cone-to-jet transition region (see LópezHerrera et al.,2020;Saville,1997). The complementary Fig. 5 shows the position 𝑓𝑖of the inner interface and the thickness of outer fluid, (𝑓𝑜−𝑓𝑖). In all cases the origin of the axial coordinate is set at the center of the transition region, defined as the point 𝑥𝑜 where the second derivative of the inner interface is maximum. The balance of normal stress at the interface 𝑓𝑘is given by ‖𝑝‖ ⏟⏟⏟ 𝐓𝑝 +‖𝐧𝑘⋅𝝉𝑣⋅𝐧𝑘‖ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ 𝐓𝑣 −𝛾𝑘𝛁⋅𝐧𝑘 ⏟⏞⏟⏞⏟ 𝐓𝑠𝑡 = − 𝛽𝑜 (𝐸𝑜 𝑛,𝑘)2 2 ⏟⏞⏞⏟⏞⏞⏟ 𝐓𝑒 𝑎 +𝛽𝑖 (𝐸𝑖 𝑛,𝑘)2 2 ⏟⏞⏞⏟⏞⏞⏟ 𝐓𝑒 𝑏 − (𝛽𝑜−𝛽𝑖) 𝐸2 𝑡,𝑘 2 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ 𝐓𝑒 𝑐 (3) where the normal of the interfaces 𝐧𝑘points out outward see Fig. 1 and ∥ ∥𝑘stands for the jump of the quantity (pressure or viscous normal stresses) across the interface 𝑘(∥ ∥𝑖=𝑜−𝑖for the inner interface and ‖‖𝑜=𝑎−𝑜for the outer one). 𝐸𝑜∕𝑖 𝑛∕𝑡,𝑘 is the Journal of Aerosol Science 167 (2023) 106075 7 J.M. López-Herrera et al. Fig. 5. The shape of the inner interface  𝑓𝑖∕𝑟𝐺(blue line) is shown in the left 𝑦-axis, while the width of the outer layer, ( 𝑓𝑜− 𝑓𝑖)∕𝑟𝐺is shown on the right 𝑦-axis for the capillary arrangements shown in Fig. 3 panels a,b,i and j. The shapes are made dimensionless with the characteristic length for the jet radius, 𝑟𝐺=(𝜌𝑖𝜀𝑎 𝑄3 𝑖∕(𝛾𝑜𝜅𝑖))1∕6. The inset show a magnification of the jet region. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 6. (color online) Effect of the applied voltage at the tubes on the shape of the coaxial cone-jet, for protruding (upper panel) and flushed (lower panel) configurations. Note that the inner cone occupies nearly the entire cone volume from the apex up to the point where its radius equals that of the inner tube. The dimensional voltage applied at the tubes are 𝛷= 4570 V (𝑉= 8.19) and 3236 V (𝑉= 5.80). For the upper panel, the dimensional electrical currents are  𝐼𝑡𝑜𝑡𝑎𝑙 =69.89 nA and 71.88 nA for the lower and the higher applied voltage respectively. Similarly, for the lower panel the dimensional current would be  𝐼𝑡𝑜𝑡𝑎𝑙 =69.72 nA and 70.95 nA. (normal/tangential) component of the electric field evaluated in the outer/inner side of the interface 𝑘.Fig. 4 shows that Eq. (3) mostly reduces to a balance between the pressure term (𝐓𝑝) and the electrostatic suction caused by the tangential stresses (𝐓𝑐), with a smaller contribution of the capillary stress (𝐓𝑠𝑡). Other normal stresses acting on the inner interface are negligible. The geometry of the tubes has a significant effect on the tangential electric field which, in turn, determines the height of the peak of the 𝐓𝑐profile at the beginning of the jet. The relative position of the tubes is the most relevant factor: the 𝐓𝑐profile when the inner tube is protruding, 𝛥𝐿 = 2.15, is in average 20% taller than when the inner tube is retracted, 𝛥𝐿 = −2.20. The sharpness of the end of the inner capillary has a smaller effect (the sharper the tube the stronger the stress), only observable when the inner tube is protruding. The impact of these differences on the shape of the interfaces is almost negligible (see Fig. 5). The distances in the figure are made dimensionless with the characteristic length derived by Gañán-Calvo (2004) and Gañán-Calvo et al. (1997,2018), given by 𝑟𝐺=(𝜌𝑖𝜀𝑎 𝑄3 𝑖∕(𝛾𝑜𝜅𝑖))1∕6. The differences in shape induced by changes in the relative positions of the tubes are restricted to the cone Journal of Aerosol Science 167 (2023) 106075 8 J.M. López-Herrera et al. region, 𝑥<𝑥𝑜. The cone of the inner fluid has a larger angle and the region occupied by the outer fluid is thicker when the inner capillary is protruding 𝛥𝐿 = 2.15. However, once the jet region develops, 𝑥>𝑥𝑜, both the shape of the inner interface and the thickness of the outer fluid almost collapse into single curves independently of the geometry of the tubes. There is only a slight decrease in thickness of the outer fluid in the jet, due to the more intense tangential electric field associated with a protruding and sharp inner tube. However, as observed in panels j and l of Fig. 3 for sharp tubes in the protruding configuration, the larger electric tangential field also induces a larger tangential stress on the outer meniscus, revealed by a more intense recirculation pattern in the cone region, and this would contribute to a more stable operation of the CES. Finally, Fig. 6 illustrates the effect of the applied voltage at the tubes on the shape of the menisci, for flushed and protruding configurations. In either case increasing the applied voltage at the tubes reduces the volume of either menisci, bringing the transition region closer to the face of the tubes. We expect the protruding configuration to be more stable because the volume of the meniscus subject to potentially larger deviations is drastically reduced. In fact, the outer edge of the inner tube virtually divides the outer meniscus in two (rear and forward menisci) and serves as a supporting surface that should reduce the growth of perturbations (see panel on top). 4. Conclusion A useful model is presented here to investigate the influence of tubes’ geometry on the coaxial electrospray. As in a previous work (López-Herrera et al.,2020), we use the Taylor–Melcher leaky dielectric model (LDM) and the numerical scheme of Herrada and Montanero (2016). We analyze in detail two effects for a given set of liquid properties: (i) the protruding and retracted position of the inner tube at the exit of the outer tube, and (ii) the geometry of the tubes’ edges (blunt or sharp). We found significant differences in the tangential stresses at the inner interface between the retracted and protruding configurations, where the latter exposes the inner interface to stronger tangential stresses. In addition, we also find that the electric current carried by the liquid bulk (conduction) essentially reproduces the cross-sectional areas of the inner and outer domains at each axial position up to near the end of the outer meniscus in the retracted configuration. Except in those extreme cases where electrokinetic processes, gas ionization or ion evaporation become important (i.e, in cases not compliant with the LDM), the degree of accuracy achieved ensures that the findings revealed by the numerical simulations reflect the physics of the system studied. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This research has been supported by the Spanish Agencia Estatal de Investigación (grant numbers PID2019-108278RB and PCI2018-093040), and by the Junta de Andalucía, Spain (grant number P18-FR-3375). MGC was supported by the Air Force Office of Scientific Research, Award No. FA9550-21-1-0200. References Boda, S. K., Li, X., & Xie, J. (2018). Electrospraying an enabling technology for pharmaceutical and biomedical applications: A review. Journal of Aerosol Science, 125, 164–181. Chang, M.-W., Stride, E., & Edirisinghe, M. (2010a). Controlling the thickness of hollow polymeric microspheres prepared by electrohydrodynamic atomization. Journal of the Royal Society Interface,7(SUPPL. 4), S451–S460, cited By 48. Chang, M.-W., Stride, E., & Edirisinghe, M. (2010b). A new method for the preparation of monoporous hollow microspheres. Langmuir,26(7), 5115–5121, cited By 99. Chen, X., Jia, L., Yin, X., Cheng, J., & Lu, J. (2005). Spraying modes in coaxial jet electrospray with outer driving liquid. Physics of Fluids,17(3), Article 032101. Chen, C., Liu, W., Jiang, P., & Hong, T. (2019). Coaxial electrohydrodynamic atomization for the production of drug-loaded micro/nanoparticles. Micromachines, 10(2). Fernandez de la Mora, J., & Loscertales, I. (1994). The current emitted by highly conducting Taylor cones. Journal of Fluid Mechanics,260, 155–184. Gañán-Calvo, A., & Montanero, J. (2021). Self-similar electrohydrodynamic solutions in multiple coaxial taylor cones. Journal of Fluid Mechanics,915, R1. Gañán-Calvo, A., Montanero, J., Martín-Banderas, L., & Flores-Mosquera, M. (2013). Building functional materials for health care and pharmacy from microfluidic principles and flow focusing. Advanced Drug Delivery Reviews,65(11–12), 1447–1469. Gañán-Calvo, A. M. (1997). Cone-jet analytical extension of taylor’s electrostatic solution and the asymptotic universal scaling laws in electrospraying. Physical Review Letters,79(2), 217–220. Gañán-Calvo, A. (2004). On the general scaling theory for electrospraying. Journal of Fluid Mechanics,507, 203–212. Gañán-Calvo, A. M., Barrero, A., & Pantano, C. (1993). The electrohydrodynamics of electrified conical menisci. Journal of Aerosol Science,24, S19–20. Gañán-Calvo, A., Dávila, J., & Barrero, A. (1997). Current and droplet size in the electrospraying of liquids. Scaling laws. Journal of Aerosol Science,28(2), 249–275. Gañán-Calvo, A., Lasheras, J., Dávila, J., & Barrero, A. (1994). The electrostatic spray emitted from an electrified conical meniscus. Journal of Aerosol Science, 25(6), 1121–1142. Gañán-Calvo, A., López-Herrera, J., Herrada, M., Ramos, A., & Montanero, J. (2018). Review on the physics of electrospray: From electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray. Journal of Aerosol Science,125, 32–56. Herrada, M. A., & Montanero, J. M. (2016). A numerical method to study the dynamics of capillary fluid systems. Journal of Computational Physics,306, 137–147. Journal of Aerosol Science 167 (2023) 106075 9 J.M. López-Herrera et al. López-Herrera, J., Barrero, A., López, A., Loscertales, I., & Márquez, M. (2003). Coaxial jets generated from electrified taylor cones. Scaling laws. Journal of Aerosol Science,34(5), 535–552. López-Herrera, J. M., Herrada, M. A., Gamero-Castaño, M., & Gañán-Calvo, A. M. (2020). A numerical simulation of coaxial electrosprays. Journal of Fluid Mechanics,885, A15. Loscertales, I. G., Barrero, A., Guerrero, I., Cortijo, R., Marquez, M., & Gañán-Calvo, A. M. (2002). Micro/nano encapsulation via electrified coaxial liquid jets. Science,295(5560), 1695–1698. Mei, F., & Chen, D.-R. (2007). Investigation of compound jet electrospray: Particle encapsulation. Physics of Fluids,19(10), Article 103303. Mei, F., & Chen, D.-R. (2008). Operational modes of dual-capillary electrospraying and the formation of the stable compound cone-jet mod. Aerosol and Air Quality Research,8(2), 218–232. Morais, A. Í., Vieira, E. G., Afewerki, S., Sousa, R. B., Honorio, L. M., Cambrussi, A. N., Santos, J. A., Bezerra, R. D., Furtini, J. A., Silva-Filho, E. C., Webster, T. J., & Lobo, A. O. (2020). Fabrication of polymeric microparticles by electrospray: The impact of experimental parameters. Journal of Functional Biomaterials, 11(1). Reneker, D., & Yarin, A. (2008). Electrospinning jets and polymer nanofibers. Polymer,49(10), 2387–2425. Rosell-Llompart, J., Grifoll, J., & Loscertales, I. G. (2018). Electrosprays in the cone-jet mode: From taylor cone formation to spray development. Journal of Aerosol Science,125, 2–31. Saville, D. A. (1997). Electrohydrodynamics:The Taylor-Melcher leaky dielectric model. Annual Review of Fluid Mechanics,29(1), 27–64. Sofokleous, P., Lau, W. K., Edirisinghe, M., & Stride, E. (2016). The effect of needle tip displacement in co-axial electrohydrodynamic processing. RSC Advances, 6(79), 75258–75268. Sun, Z., Zussman, E., Yarin, A., Wendorff, J., & Greiner, A. (2003). Compound core–shell polymer nanofibers by co-electrospinning. Advanced Materials,15(22), 1929–1932. Taylor, G. (1964). Disintegration of water drops in an electric field. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 280(1382), 383–397. Yan, T., Ren, H., Liu, J., Ou, Y., & Jiao, Q. (2019). Nested core–shell electrosprayed microspheres: A potential assembly with synergetic interface for energy release. Materials Letters,241, 92–95. Zamani, M., Prabhakaran, M., & Ramakrishna, S. (2013). Advances in drug delivery via electrospun and electrosprayed nanomaterials. International Journal of Nanomedicine,8, 2997–3017. Zhang, L., Huang, J., Si, T., & Xu, R. X. (2012). Coaxial electrospray of microparticles and nanoparticles for biomedical applications. Expert Review of Medical Devices,9(6), 595–612. Zhao, X., Wang, D., Lin, Y., Sun, Y., Ren, T., Liang, J., & Madou, M. (2019). Numerical simulation of coaxial electrohydrodynamic jet and printing nanoscale structures. Microsystem Technologies,25, 4651–4661.