Incompressible Fluids Simulation by Relaxing the Density-Invariant Condition in a Marine Simulator

<div>The 2D model of diffuse particles generated by a fluid particle f. <svg height="14.8173pt" id="M133" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 26.249 14.8173" width="26.249pt" 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="M494 0V32C450 38 426 53 393 105L296 260C330 305 354 338 375 361C411 396 421 401 476 410V442H293V410C332 402 337 394 323 371C311 351 295 324 274 293L227 371C213 393 218 402 252 410V442H16V410C70 402 80 391 108 344L196 199C166 159 140 123 115 95C81 54 61 43 7 32V0H196V32C156 41 150 53 167 81C182 105 199 138 219 166L269 79C287 49 283 38 245 32V0H494Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.646,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.144,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.578,0)"><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></svg> is the current position, <svg height="14.8173pt" id="M134" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 52.5334 14.8173" width="52.5334pt" 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="M494 0V32C450 38 426 53 393 105L296 260C330 305 354 338 375 361C411 396 421 401 476 410V442H293V410C332 402 337 394 323 371C311 351 295 324 274 293L227 371C213 393 218 402 252 410V442H16V410C70 402 80 391 108 344L196 199C166 159 140 123 115 95C81 54 61 43 7 32V0H196V32C156 41 150 53 167 81C182 105 199 138 219 166L269 79C287 49 283 38 245 32V0H494Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.646,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.144,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.483,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.02,0)"><path d="M600 0V24L339 665L311 657L43 24V0H600ZM497 50H107L303 539L497 50Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.347,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,47.78,0)"><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></svg> is the position, and <svg height="11.3463pt" id="M135" style="vertical-align:-5.52893pt" version="1.1" viewbox="-0.0498162 -5.81737 12.5478 11.3463" width="12.5478pt" 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="M483 442H303V410C358 401 362 388 351 355C329 286 296 195 278 149C262 190 226 288 200 358C188 393 193 400 239 410V442H-8V410C40 400 53 388 71 344C120 229 165 113 215 -11H254C315 141 364 261 400 342C421 389 433 400 483 410V442Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.132)"><path d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"></path></g></svg> is the velocity of a fluid particle.</div>

Mathematical Problems in Engineering

fig1

Figure 1

Figure 1: Incompressible Fluids Simulation by Relaxing the Density-Invariant Condition in a Marine Simulator 