Radiative Flow of Copper and Aluminum Nanoparticles with Heat Source Phenomenon: Dual Numerical Simulations and Stability Analysis

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="2"><span style="width: 18.8699ptpx;"><svg height="12.3916pt" id="M214" style="vertical-align:-3.42948pt" version="1.1" viewbox="-0.0498162 -8.96212 18.8699 12.3916" width="18.8699pt" 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="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.02,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.181,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g></svg></span></td><td class="align_center" rowspan="2"><span style="width: 6.21949ptpx;"><svg height="9.49473pt" id="M215" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt" 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="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"></path></g></svg></span></td><td class="align_center" colspan="2"><span style="width: 10.0698ptpx;"><svg height="9.14241pt" id="M216" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -5.96091 10.0698 9.14241" width="10.0698pt" 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="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.005,3.132)"><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></svg></span></td></tr><tr class="thead"><td class="align_center">Upper branch</td><td class="align_center">Lower branch</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left" rowspan="2">0.01</td><td class="align_center">−3</td><td class="align_center">2.2281</td><td class="align_center">−2.2827</td></tr><tr><td class="align_center">−2.5</td><td class="align_center">1.9924</td><td class="align_center">−1.9001</td></tr><tr><td class="align_left" rowspan="2">0.03</td><td class="align_center">−3</td><td class="align_center">2.6295</td><td class="align_center">−2.8547</td></tr><tr><td class="align_center">−2.5</td><td class="align_center">2.1824</td><td class="align_center">−2.4373</td></tr><tr><td class="align_left" rowspan="2">0.05</td><td class="align_center">−3</td><td class="align_center">2.8626</td><td class="align_center">−3.0626</td></tr><tr><td class="align_center">−2.5</td><td class="align_center">2.25463</td><td class="align_center">−2.4515</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>The values of <svg height="9.14241pt" id="M210" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -5.96091 10.0698 9.14241" width="10.0698pt" 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="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.005,3.132)"><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></svg> for various values of <svg height="12.3916pt" id="M211" style="vertical-align:-3.42948pt" version="1.1" viewbox="-0.0498162 -8.96212 18.8699 12.3916" width="18.8699pt" 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="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.02,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.181,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g></svg> and <svg height="9.49473pt" id="M212" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt" 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="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"></path></g></svg> when <span class="nowrap"><svg height="11.439pt" id="M213" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 262.203 11.439" width="262.203pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.806,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.068,0)"><path d="M535 230V280H52V230H535Z"></path></g><g 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Mathematical Problems in Engineering

tab4

Table 4

Table 4: Radiative Flow of Copper and Aluminum Nanoparticles with Heat Source Phenomenon: Dual Numerical Simulations and Stability Analysis 