Emulsion Droplet Pair Coalescence under a Direct Current Electric Field

<div>(a) Change in the capillary pressure with a coalesced droplet stretching at different droplet pair radius ratios. Coalesced droplet becomes unstable if <span class="nowrap"><svg height="16.2017pt" id="M45" style="vertical-align:-3.943501pt" version="1.1" viewbox="-0.0498162 -12.2582 126.374 16.2017" width="126.374pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g><g transform="matrix(.013,0,0,-0.013,7.385,0)"><path d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM538 -11L509 26C488 -7 456 -31 418 -31C390 -31 345 -11 310 6C266 27 213 50 173 50C123 50 80 18 51 -24L82 -57C102 -25 135 0 173 0C199 0 246 -21 281 -38C325 -59 376 -81 418 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618C432 618 480 575 480 485Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,98.709,-5.741)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,98.644,3.784)"><path d="M452 282C452 397 389 451 302 451C268 451 227 441 188 422C95 377 24 281 24 154C24 65 70 -12 173 -12C240 -12 297 16 342 52C418 113 452 201 452 282ZM359 282C359 163 308 66 247 39C237 35 225 32 212 32C151 32 114 79 114 157C114 301 180 380 223 398C240 405 250 408 265 408C315 408 359 369 359 282Z"></path></g><g transform="matrix(.013,0,0,-0.013,103.655,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,108.153,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,112.651,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,117.083,-5.741)"><path d="M381 704H317L48 -163H112L381 704Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,120.991,-5.741)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g></svg>.</span> (b) Critical electric capillary number for the onset of instability of the coalesced droplet at different droplet pair radius ratios. Blue open diamond symbols show stable complete coalescence whereas red crossed diamond symbols show unstable coalescence. Here, a solid line is obtained from equation (<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/8420722/figures/#EEq8">9</a>) depicting the onset of unstable coalescence. Open and crossed symbols are simulation results for the castor oil droplet pair coalescence in the silicone oil. A reasonably good agreement is observed between theory and simulations.</div>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Emulsion Droplet Pair Coalescence under a Direct Current Electric Field 