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Mathematical Model for Axisymmetric Taylor Flows Inside a Drop

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Mathematical Model for Axisymmetric Taylor Flows Inside a Drop fluids Article Mathematical Model for Axisymmetric Taylor Flows Inside a Drop Ilya V. Makeev 1,* , Rufat Sh. Abiev 2 and Igor Yu. Popov 1 1 Faculty of Control Systems and Robotics, Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University), Kronverkskiy, 49, 197101 Saint Petersburg, Russia; [email protected] 2 Department of Optimization of Chemical and Biotechnological Equipment, Saint Petersburg State Institute of Technology (Technical University), Moscovskiy Prospect, 26, 190013 Saint Petersburg, Russia; [email protected] * Correspondence: [email protected] Abstract: Analytical solutions of the Stokes equations written as a differential equation for the Stokes stream function were obtained. These solutions describe three-dimensional axisymmetric flows of a viscous liquid inside a drop that has the shape of a spheroid of rotation and have a similar set of characteristics with Taylor flows inside bubbles that occur during the transfer of a two-component mixture through tubes. Keywords: Taylor flows; Stokes equations; Benchmark solution 1. Introduction The Taylor flow regime is the most favorable of the known regimes for microchannels, since it provides, on the one hand, an intensification of mass transfer due to Taylor vortices, and on the other hand, a sufficient residence time, in view of the rather moderate velocities of phase motion (typically 0.01–0.1 m/s, and not larger than 0.8–1 m/s), which are typical Citation: Makeev, I.V.; Abiev, R.S.; for this regime [1,2]. The velocity of bubbles or droplets is characterized by capillary Popov, I.Y. Mathematical Model for number Ca = µU/σ where µ is a liquid viscosity, U is adispersed phase (bubbles or Axisymmetric Taylor Flows Inside a droplets) velocity and σ is an interfacial tension. Usually Ca number does not exceed Drop. Fluids 2021, 6, 7. 0.01–0.1, and the Reynolds number for the flows in micro and mini-channels is less than https://dx.doi.org/10.3390/fluids 2000, corresponding thus to the laminar flow [3]. These features make it possible to achieve 6010007 a high mixing intensity without significantly complicating the equipment and generate great interest in this type of flow. Received: 2 December 2020 In the last two to three decades, many papers were devoted to both the hydrodynamics Accepted: 24 December 2020 of the Taylor flow and mass transfer in it [4–10]. Most of these works in fluid dynamics Published: 26 December 2020 are based on experimental or numerical experiments. At the same time, it is important to Publisher’s Note: MDPI stays neu- search for independent theoretical methods that allow one to obtain a general a priori idea tral with regard to jurisdictional claims of the dispersed flow inside droplets or bubbles, including the shape of streamlines. in published maps and institutional Such two-component liquid streams belong to the class of Taylor streams and are affiliations. widely used in various industries [1,2]. One of the first models describing the motion of Taylor bubbles in channels was published by F. Bretherton [11]. The extended Bretherton model for Taylor bubbles at moderate capillary numbers was proposed by E. Klaseboer [12]. Copyright: © 2020 by the authors. Li- A number of works [13–16] were devoted to the study of Taylor flows. In [17], censee MDPI, Basel, Switzerland. This the method of micro Particle Image Velocimetry is used to study Taylor vortices inside an article is an open access article distributed elongated bubble. In [18] the circulation patterns within the dispersed phase for regular under the terms and conditions of the Taylor two-phase flows were constructed using the FEATFLOW program. Creative Commons Attribution (CC BY) It is worth noting that the actual shape of the Taylor bubbles may be far from the ideal license (https://creativecommons.org/ geometric shape. Nevertheless, modeling, which considers the Taylor flow in a domain licenses/by/4.0/). Fluids 2021, 6, 7. https://dx.doi.org/10.3390/fluids6010007 https://www.mdpi.com/journal/fluids Fluids 2021, 6, x FORFluids PEER 2021 REVIEW, 6, x FOR PEER REVIEW 2 of 10 2 of 10 Fluids 2021, 6, 7 2 of 10 It is worth notingIt is worth that the noting actual that shape the actualof the shapeTaylor of bubbles the Taylor may bubblesbe far fro maym the be far from the ideal geometricideal shape. geometric Nevertheless, shape. Nevertheless, modeling, which modeling, considers which the considers Taylor flow the inTaylor a flow in a domain with an idealized form, is used in a number of studies [19]. In [20,21], the shape domain with an idealizedwith form, an is idealized used in a form, number is used of studies in a number [19]. In of [20,21 studies], the [19 shape]. In [20,21], the shape of the of the bubblesof is the approximated bubblesbubbles is approximated by is a approximated part of a by cylinder a part by aofwith part a cylinder hemispherical of a cylinder with hemispherical caps with or hemispherical a pro- caps or capsa pro- or a prolate late ellipsoid lateat ends. ellipsoid ellipsoid at ends. at ends. In the type ofIn flow the thattype weInof thefloware type considering, that of we flow are that aconsidering, mixture we are considering, of twoa mixture liquids a mixtureof that two have liquids of two dif- liquidsthat have that dif- have different ferent viscosityferent coefficients viscosityviscosity is coefficients transferred coefficients is through transferred is transferred the channelthrough through and the theonechannel channel liquid and is and onedistrib- one liquid liquid is isdistrib- distributed in the uted in the formuted of in bubbles the formform inside of of bubbles bubbles the other inside inside one the( theFigure other other 1). one oneThe (Figure (Figureliquid 1). interacting1). The The liquid liquid with interacting interacting the with with the the channel channel wallschannel moves alongwallswalls movesits movesaxis alongand along drives its itsaxis axisthe and gas and drives bubbles drives the theor gas the gas bubbles droplets bubbles or orofthe thethe droplets dropletssec- of of the the sec- second liquid. ond liquid. Atond a constant liquid. At two aa constant-constantphase velocity two-phasetwo-phase of a velocitymixturevelocity of ofof a liquidsa mixture mixture through of of liquids liquids the throughchannel, through the the channel, channel, this flow can this flow can thisbe considered flow canbe be consideredst consideredationary and stationary st ationaryaxisymmetric and and axisymmetric axisymmetric (for the case (for when(for the casethe the case whencapillary when the capillarythe capillary forces dominate forces dominateforces over dominate gravitationalover gravitationalover gravitationalforces, forces, i.e., for i.e.,forces, liquids for liquidsi.e. ,similar for similar liquids to tothe thesimilar water water toat at the thethe channelwater at diameter the smaller channel diameterchannel smaller diameterthan than≈ smaller 33 mm).mm). than 3 mm). Figure 1. Two-component liquid in circular channel (r,z). FigureFigure 1. Two 1.-componentTwo-component liquid liquidin circular in circular channel channel (r,z). (r,z). Due to the presenceDue to thofe friction presenceDue to force the of presence frictionat the boundary offorce friction at the forceof boundarytwo at fluids the boundary ofand two friction fluids of two be-and fluids friction and friction be- between tween the fluidtween and thethe fluidchannelthe fluid and walls, andthe channel the a vortex channel walls, flow walls, a occurs vortex a vortex in flow the flow occursinner occurs part in inthe of the innerthe inner bub- part part of of the the bub- bubbles during bles during thebles movement during thethe of movementmovement the mixture ofof through thethe mixturemixture the channel throughthrough ( theFigurethe channelchannel 2). (Figure(Figure2 ).2). Figure 2. Streamlines in the elongated bubble. Arrows characterize the velocity vectorsin the frame Figure 2. Streamlines in the elongated bubble. Arrows characterize the velocity vectorsin the frame Figure 2. Streamlines in the ofelongated the moving bubble. bubble. Arrows Reproduced characterize from the [14 velocity] with permission vectorsin the of Elsevier. frame of the moving bubble.of the moving Reproduced bubble. from Reproduced [14] with frompermission [14] with of Elsevier.permission of Elsevier. In our previous work [22], we obtained solutions for two-dimensional flows inside an In our previousIn ourwork ellipticalprevious [22], we region,work obtained [22], which solutionswe demonstrated obtained for solutionstwo the-dimensional presence for two of -flowsdimensional both the inside main flows toroidal inside vortex (similar an elliptical region,an elliptical whichto region, thatdemonstrated shown which in demonstrated Figurethe presence2) and twotheof both presence satellite the vortices.main of both toroidal Thesethe main vortex solutions toroidal generally vortex reflect the (similar to that(similar shown to in thatnature Figure shown of 2) the andin droplet/bubble Figure two satellite 2) and vortices.two flows satellite in theThese Taylorvortices. solutions two-phase These generally solutions flow. generally reflect the naturereflect of the naturedropletThe of / bubblethe system droplet flows of Stokes / bubblein the equations Taylor flows intwo is the often-phase Taylor used flow. two in - mathematicalphase flow. models to describe the The system ofThe Stokes systemflow equations of Stokes a liquid is often equations with used a set inis of oftenmathematical characteristics
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