Govind, Pooja Sharma, Ravi Chaudhary, B. K. Sharma, Preeti, Faisal Z. Duraihem
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{"title":"化学反应型和磁铁矿型Cu-Al2O3水基混合纳米流体在Riga板上的热扩散和质量扩散特性的数值研究","authors":"Govind, Pooja Sharma, Ravi Chaudhary, B. K. Sharma, Preeti, Faisal Z. Duraihem","doi":"10.1002/adts.202500352","DOIUrl":null,"url":null,"abstract":"The existing work intends to explore the thermal and mass diffusion properties in chemically reactive and magnetite hybrid nanofluids (consisting of <span data-altimg=\"/cms/asset/f0a514a9-3c88-4a53-bc8c-1ca644cdaa33/adts70009-math-0001.png\"></span><mjx-container ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0001.png\"><mjx-semantics><mjx-mrow><mjx-mrow data-semantic-children=\"12,8\" data-semantic-content=\"7\" data-semantic- data-semantic-role=\"division\" data-semantic-speech=\"normal upper A normal l 2 normal upper O 3 divided by upper C u\" data-semantic-type=\"infixop\"><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,3,6\" data-semantic-content=\"10,11\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"1,2\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"4,5\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"infixop,/\" data-semantic-parent=\"13\" data-semantic-role=\"division\" data-semantic-type=\"operator\" rspace=\"1\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mspace style=\"width: 0.28em;\"></mjx-mspace></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0001\" display=\"inline\" location=\"graphic/adts70009-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mrow data-semantic-=\"\" data-semantic-children=\"12,8\" data-semantic-content=\"7\" data-semantic-role=\"division\" data-semantic-speech=\"normal upper A normal l 2 normal upper O 3 divided by upper C u\" data-semantic-type=\"infixop\"><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,3,6\" data-semantic-content=\"10,11\" data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">A</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><msub data-semantic-=\"\" data-semantic-children=\"1,2\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">l</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"3\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><msub data-semantic-=\"\" data-semantic-children=\"4,5\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">O</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></msub></mrow><mo data-semantic-=\"\" data-semantic-operator=\"infixop,/\" data-semantic-parent=\"13\" data-semantic-role=\"division\" data-semantic-type=\"operator\">/</mo><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-parent=\"13\" data-semantic-role=\"unknown\" data-semantic-type=\"identifier\">Cu</mi></mrow><mspace width=\"0.28em\"></mspace></mrow>${\\mathrm{A}}{{\\mathrm{l}}_{2 }}{{\\mathrm{O}}_3}/{\\mathrm{Cu}}\\;$</annotation></semantics></math></mjx-assistive-mml></mjx-container> nanoparticles dispersed in <span data-altimg=\"/cms/asset/437e1cb3-62db-4991-a939-035e4b38558d/adts70009-math-0002.png\"></span><mjx-container ctxtmenu_counter=\"11\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0002.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,3\" data-semantic-content=\"4\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"normal upper H 2 normal upper O\" data-semantic-type=\"infixop\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0002\" display=\"inline\" location=\"graphic/adts70009-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,3\" data-semantic-content=\"4\" data-semantic-role=\"implicit\" data-semantic-speech=\"normal upper H 2 normal upper O\" data-semantic-type=\"infixop\"><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">H</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></msub><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">O</mi></mrow>$ {{\\mathrm{H}}_2}{\\mathrm{O}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container>) flow across a Riga plate, considering influential factors like radiation, exponential heat source, viscous dissipation, magnetic field, and activation energy. The governing equations are modified into ordinary differential equations by employing similarity transformation, The bvp4c solver in MATLAB is used to solve differential equations to obtain the numerical solution. The range of non-dimensional parameters 0.1 < Z < 1, 0 < Kp < 0.5, 1 < R < 5, 0.1 < Ec < 0.4, 0.1 < <span data-altimg=\"/cms/asset/61db4efd-251c-4051-87f8-25f3811f4ccb/adts70009-math-0003.png\"></span><mjx-container ctxtmenu_counter=\"12\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0003.png\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"phi 1\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0003\" display=\"inline\" location=\"graphic/adts70009-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"greekletter\" data-semantic-speech=\"phi 1\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">φ</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">1</mn></msub>${\\varphi _1}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < <span data-altimg=\"/cms/asset/9e82d65f-9fea-47ac-9d4d-753b90f5b150/adts70009-math-0004.png\"></span><mjx-container ctxtmenu_counter=\"13\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0004.png\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"greekletter\" data-semantic-speech=\"phi 2\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0004\" display=\"inline\" location=\"graphic/adts70009-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"greekletter\" data-semantic-speech=\"phi 2\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"greekletter\" data-semantic-type=\"identifier\">φ</mi><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn></msub>${\\varphi _2}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < <span data-altimg=\"/cms/asset/680a4b69-6012-4b4b-bb47-f00bf6503fde/adts70009-math-0005.png\"></span><mjx-container ctxtmenu_counter=\"14\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0005.png\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript upper E\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0005\" display=\"inline\" location=\"graphic/adts70009-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript upper E\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">Q</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">E</mi></msub>${Q_E}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < Bi < 0.4, 1 < <span data-altimg=\"/cms/asset/8f886440-f812-438f-a39f-a31188dcbf32/adts70009-math-0006.png\"></span><mjx-container ctxtmenu_counter=\"15\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0006.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,3\" data-semantic-content=\"4\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"upper R e Subscript x\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\"1,2\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0006\" display=\"inline\" location=\"graphic/adts70009-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"0,3\" data-semantic-content=\"4\" data-semantic-role=\"implicit\" data-semantic-speech=\"upper R e Subscript x\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">R</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"5\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><msub data-semantic-=\"\" data-semantic-children=\"1,2\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">e</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">x</mi></msub></mrow>$R{e_x}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 5, and 0.1 < 𝛾 < 0.7. It is detected that temperature distribution enhances with the rise of Hartmann number and radiation, whereas velocity distribution increases with the rise of modified Hartmann number. As activation energy is raised, the concentration profile is also enhanced. Nusselt number increases with a rise of exponential heat source, for numerical values <span data-altimg=\"/cms/asset/36b9bd0e-5ca2-4f74-b4cf-7538af5dc89f/adts70009-math-0007.png\"></span><mjx-container ctxtmenu_counter=\"16\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0007.png\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript normal upper E\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0007\" display=\"inline\" location=\"graphic/adts70009-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript normal upper E\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">Q</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">E</mi></msub>${Q_{\\mathrm{E}}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> = 0.2 and <span data-altimg=\"/cms/asset/f1bbab0c-123d-43ce-8775-a9af6df95694/adts70009-math-0008.png\"></span><mjx-container ctxtmenu_counter=\"17\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0008.png\"><mjx-semantics><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript normal upper E\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0008\" display=\"inline\" location=\"graphic/adts70009-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msub data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper Q Subscript normal upper E\" data-semantic-type=\"subscript\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">Q</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" mathvariant=\"normal\">E</mi></msub>${Q_{\\mathrm{E}}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> = 0.3. The corresponding Nusselt numbers are 2.8232 <span data-altimg=\"/cms/asset/9c04f7bc-dfd7-4ed9-bbca-c7af5beda223/adts70009-math-0009.png\"></span><mjx-container ctxtmenu_counter=\"18\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0009.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"6\" data-semantic-content=\"0\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"times 10 Superscript negative 0.7\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mspace style=\"width: 0.33em;\"></mjx-mspace><mjx-msup data-semantic-children=\"2,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"superscript\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn><mjx-script style=\"vertical-align: 0.393em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\" size=\"s\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"float\" data-semantic-type=\"number\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0009\" display=\"inline\" location=\"graphic/adts70009-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"6\" data-semantic-content=\"0\" data-semantic-role=\"unknown\" data-semantic-speech=\"times 10 Superscript negative 0.7\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo><mspace width=\"0.33em\"></mspace><msup data-semantic-=\"\" data-semantic-children=\"2,5\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"superscript\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">10</mn><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\">−</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"float\" data-semantic-type=\"number\">0.7</mn></mrow></msup></mrow>$ \\times\\ {10^{ - 0.7}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and 5.5421 <span data-altimg=\"/cms/asset/0c3733e9-a252-4030-b6fc-8be41093d005/adts70009-math-0010.png\"></span><mjx-container ctxtmenu_counter=\"19\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/adts70009-math-0010.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"6\" data-semantic-content=\"0\" data-semantic- data-semantic-role=\"unknown\" data-semantic-speech=\"times 10 Superscript negative 0.7\" data-semantic-type=\"prefixop\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\" rspace=\"4\" space=\"4\"><mjx-c></mjx-c></mjx-mo><mjx-mspace style=\"width: 0.33em;\"></mjx-mspace><mjx-msup data-semantic-children=\"2,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"superscript\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn><mjx-script style=\"vertical-align: 0.393em;\"><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\" size=\"s\"><mjx-mo data-semantic- data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\" rspace=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"float\" data-semantic-type=\"number\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0010\" display=\"inline\" location=\"graphic/adts70009-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"6\" data-semantic-content=\"0\" data-semantic-role=\"unknown\" data-semantic-speech=\"times 10 Superscript negative 0.7\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,×\" data-semantic-parent=\"7\" data-semantic-role=\"unknown\" data-semantic-type=\"operator\">×</mo><mspace width=\"0.33em\"></mspace><msup data-semantic-=\"\" data-semantic-children=\"2,5\" data-semantic-parent=\"7\" data-semantic-role=\"integer\" data-semantic-type=\"superscript\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"6\" data-semantic-role=\"integer\" data-semantic-type=\"number\">10</mn><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4\" data-semantic-content=\"3\" data-semantic-parent=\"6\" data-semantic-role=\"negative\" data-semantic-type=\"prefixop\"><mo data-semantic-=\"\" data-semantic-operator=\"prefixop,−\" data-semantic-parent=\"5\" data-semantic-role=\"subtraction\" data-semantic-type=\"operator\">−</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"float\" data-semantic-type=\"number\">0.7</mn></mrow></msup></mrow>$ \\times\\ {10^{ - 0.7}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, respectively. The hybrid nanofluid flows over the Riga plate has numerous applications in mechanical engineering, including fuel systems, magnetohydrodynamic (MHD) pumps, energy systems, thermal reactors, etc.","PeriodicalId":7219,"journal":{"name":"Advanced Theory and Simulations","volume":"45 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Numerical Investigation of Thermal and Mass Diffusion Characteristics in Chemically Reactive and Magnetite Cu–Al2O3 Water-Based Hybrid Nanofluid Flow over Riga Plate\",\"authors\":\"Govind, Pooja Sharma, Ravi Chaudhary, B. K. Sharma, Preeti, Faisal Z. Duraihem\",\"doi\":\"10.1002/adts.202500352\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The existing work intends to explore the thermal and mass diffusion properties in chemically reactive and magnetite hybrid nanofluids (consisting of <span data-altimg=\\\"/cms/asset/f0a514a9-3c88-4a53-bc8c-1ca644cdaa33/adts70009-math-0001.png\\\"></span><mjx-container ctxtmenu_counter=\\\"10\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0001.png\\\"><mjx-semantics><mjx-mrow><mjx-mrow data-semantic-children=\\\"12,8\\\" data-semantic-content=\\\"7\\\" data-semantic- data-semantic-role=\\\"division\\\" data-semantic-speech=\\\"normal upper A normal l 2 normal upper O 3 divided by upper C u\\\" data-semantic-type=\\\"infixop\\\"><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,3,6\\\" data-semantic-content=\\\"10,11\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\\\"1,2\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\\\"4,5\\\" data-semantic- data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\" space=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi></mjx-mrow><mjx-mspace style=\\\"width: 0.28em;\\\"></mjx-mspace></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0001\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0001.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"12,8\\\" data-semantic-content=\\\"7\\\" data-semantic-role=\\\"division\\\" data-semantic-speech=\\\"normal upper A normal l 2 normal upper O 3 divided by upper C u\\\" data-semantic-type=\\\"infixop\\\"><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,3,6\\\" data-semantic-content=\\\"10,11\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"implicit\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">A</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"1,2\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">l</mi><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></msub><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"4,5\\\" data-semantic-parent=\\\"12\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">O</mi><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">3</mn></msub></mrow><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"infixop,/\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"division\\\" data-semantic-type=\\\"operator\\\">/</mo><mi data-semantic-=\\\"\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"13\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"identifier\\\">Cu</mi></mrow><mspace width=\\\"0.28em\\\"></mspace></mrow>${\\\\mathrm{A}}{{\\\\mathrm{l}}_{2 }}{{\\\\mathrm{O}}_3}/{\\\\mathrm{Cu}}\\\\;$</annotation></semantics></math></mjx-assistive-mml></mjx-container> nanoparticles dispersed in <span data-altimg=\\\"/cms/asset/437e1cb3-62db-4991-a939-035e4b38558d/adts70009-math-0002.png\\\"></span><mjx-container ctxtmenu_counter=\\\"11\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0002.png\\\"><mjx-semantics><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"2,3\\\" data-semantic-content=\\\"4\\\" data-semantic- data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"normal upper H 2 normal upper O\\\" data-semantic-type=\\\"infixop\\\"><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0002\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0002.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"2,3\\\" data-semantic-content=\\\"4\\\" data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"normal upper H 2 normal upper O\\\" data-semantic-type=\\\"infixop\\\"><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">H</mi><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></msub><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">O</mi></mrow>$ {{\\\\mathrm{H}}_2}{\\\\mathrm{O}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container>) flow across a Riga plate, considering influential factors like radiation, exponential heat source, viscous dissipation, magnetic field, and activation energy. The governing equations are modified into ordinary differential equations by employing similarity transformation, The bvp4c solver in MATLAB is used to solve differential equations to obtain the numerical solution. The range of non-dimensional parameters 0.1 < Z < 1, 0 < Kp < 0.5, 1 < R < 5, 0.1 < Ec < 0.4, 0.1 < <span data-altimg=\\\"/cms/asset/61db4efd-251c-4051-87f8-25f3811f4ccb/adts70009-math-0003.png\\\"></span><mjx-container ctxtmenu_counter=\\\"12\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0003.png\\\"><mjx-semantics><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"greekletter\\\" data-semantic-speech=\\\"phi 1\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0003\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0003.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-speech=\\\"phi 1\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">φ</mi><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">1</mn></msub>${\\\\varphi _1}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < <span data-altimg=\\\"/cms/asset/9e82d65f-9fea-47ac-9d4d-753b90f5b150/adts70009-math-0004.png\\\"></span><mjx-container ctxtmenu_counter=\\\"13\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0004.png\\\"><mjx-semantics><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"greekletter\\\" data-semantic-speech=\\\"phi 2\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0004\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0004.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-speech=\\\"phi 2\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"greekletter\\\" data-semantic-type=\\\"identifier\\\">φ</mi><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">2</mn></msub>${\\\\varphi _2}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < <span data-altimg=\\\"/cms/asset/680a4b69-6012-4b4b-bb47-f00bf6503fde/adts70009-math-0005.png\\\"></span><mjx-container ctxtmenu_counter=\\\"14\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0005.png\\\"><mjx-semantics><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript upper E\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0005\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0005.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript upper E\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">Q</mi><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">E</mi></msub>${Q_E}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 0.4, 0.1 < Bi < 0.4, 1 < <span data-altimg=\\\"/cms/asset/8f886440-f812-438f-a39f-a31188dcbf32/adts70009-math-0006.png\\\"></span><mjx-container ctxtmenu_counter=\\\"15\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0006.png\\\"><mjx-semantics><mjx-mrow data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,3\\\" data-semantic-content=\\\"4\\\" data-semantic- data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"upper R e Subscript x\\\" data-semantic-type=\\\"infixop\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\\\"true\\\" data-semantic- data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\" style=\\\"margin-left: 0.056em; margin-right: 0.056em;\\\"><mjx-c></mjx-c></mjx-mo><mjx-msub data-semantic-children=\\\"1,2\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0006\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0006.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:unit\\\" data-semantic-children=\\\"0,3\\\" data-semantic-content=\\\"4\\\" data-semantic-role=\\\"implicit\\\" data-semantic-speech=\\\"upper R e Subscript x\\\" data-semantic-type=\\\"infixop\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">R</mi><mo data-semantic-=\\\"\\\" data-semantic-added=\\\"true\\\" data-semantic-operator=\\\"infixop,\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"multiplication\\\" data-semantic-type=\\\"operator\\\"></mo><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"1,2\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">e</mi><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"3\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">x</mi></msub></mrow>$R{e_x}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> < 5, and 0.1 < 𝛾 < 0.7. It is detected that temperature distribution enhances with the rise of Hartmann number and radiation, whereas velocity distribution increases with the rise of modified Hartmann number. As activation energy is raised, the concentration profile is also enhanced. Nusselt number increases with a rise of exponential heat source, for numerical values <span data-altimg=\\\"/cms/asset/36b9bd0e-5ca2-4f74-b4cf-7538af5dc89f/adts70009-math-0007.png\\\"></span><mjx-container ctxtmenu_counter=\\\"16\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0007.png\\\"><mjx-semantics><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript normal upper E\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0007\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0007.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript normal upper E\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">Q</mi><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">E</mi></msub>${Q_{\\\\mathrm{E}}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> = 0.2 and <span data-altimg=\\\"/cms/asset/f1bbab0c-123d-43ce-8775-a9af6df95694/adts70009-math-0008.png\\\"></span><mjx-container ctxtmenu_counter=\\\"17\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0008.png\\\"><mjx-semantics><mjx-msub data-semantic-children=\\\"0,1\\\" data-semantic- data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript normal upper E\\\" data-semantic-type=\\\"subscript\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\\\"vertical-align: -0.15em;\\\"><mjx-mi data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" size=\\\"s\\\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0008\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0008.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><msub data-semantic-=\\\"\\\" data-semantic-children=\\\"0,1\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-speech=\\\"upper Q Subscript normal upper E\\\" data-semantic-type=\\\"subscript\\\"><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"italic\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\">Q</mi><mi data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"2\\\" data-semantic-role=\\\"latinletter\\\" data-semantic-type=\\\"identifier\\\" mathvariant=\\\"normal\\\">E</mi></msub>${Q_{\\\\mathrm{E}}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> = 0.3. The corresponding Nusselt numbers are 2.8232 <span data-altimg=\\\"/cms/asset/9c04f7bc-dfd7-4ed9-bbca-c7af5beda223/adts70009-math-0009.png\\\"></span><mjx-container ctxtmenu_counter=\\\"18\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0009.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"6\\\" data-semantic-content=\\\"0\\\" data-semantic- data-semantic-role=\\\"unknown\\\" data-semantic-speech=\\\"times 10 Superscript negative 0.7\\\" data-semantic-type=\\\"prefixop\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"4\\\" space=\\\"4\\\"><mjx-c></mjx-c></mjx-mo><mjx-mspace style=\\\"width: 0.33em;\\\"></mjx-mspace><mjx-msup data-semantic-children=\\\"2,5\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"superscript\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn><mjx-script style=\\\"vertical-align: 0.393em;\\\"><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\" size=\\\"s\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"float\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0009\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0009.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"6\\\" data-semantic-content=\\\"0\\\" data-semantic-role=\\\"unknown\\\" data-semantic-speech=\\\"times 10 Superscript negative 0.7\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\">×</mo><mspace width=\\\"0.33em\\\"></mspace><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"2,5\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"superscript\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">10</mn><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\">−</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"float\\\" data-semantic-type=\\\"number\\\">0.7</mn></mrow></msup></mrow>$ \\\\times\\\\ {10^{ - 0.7}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container> and 5.5421 <span data-altimg=\\\"/cms/asset/0c3733e9-a252-4030-b6fc-8be41093d005/adts70009-math-0010.png\\\"></span><mjx-container ctxtmenu_counter=\\\"19\\\" ctxtmenu_oldtabindex=\\\"1\\\" jax=\\\"CHTML\\\" role=\\\"application\\\" sre-explorer- style=\\\"font-size: 103%; position: relative;\\\" tabindex=\\\"0\\\"><mjx-math aria-hidden=\\\"true\\\" location=\\\"graphic/adts70009-math-0010.png\\\"><mjx-semantics><mjx-mrow data-semantic-children=\\\"6\\\" data-semantic-content=\\\"0\\\" data-semantic- data-semantic-role=\\\"unknown\\\" data-semantic-speech=\\\"times 10 Superscript negative 0.7\\\" data-semantic-type=\\\"prefixop\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"4\\\" space=\\\"4\\\"><mjx-c></mjx-c></mjx-mo><mjx-mspace style=\\\"width: 0.33em;\\\"></mjx-mspace><mjx-msup data-semantic-children=\\\"2,5\\\" data-semantic- data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"superscript\\\"><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn><mjx-script style=\\\"vertical-align: 0.393em;\\\"><mjx-mrow data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic- data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\" size=\\\"s\\\"><mjx-mo data-semantic- data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\" rspace=\\\"1\\\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic- data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"float\\\" data-semantic-type=\\\"number\\\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mn></mjx-mrow></mjx-script></mjx-msup></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml display=\\\"inline\\\" unselectable=\\\"on\\\"><math altimg=\\\"urn:x-wiley:25130390:media:adts70009:adts70009-math-0010\\\" display=\\\"inline\\\" location=\\\"graphic/adts70009-math-0010.png\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><semantics><mrow data-semantic-=\\\"\\\" data-semantic-children=\\\"6\\\" data-semantic-content=\\\"0\\\" data-semantic-role=\\\"unknown\\\" data-semantic-speech=\\\"times 10 Superscript negative 0.7\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,×\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"unknown\\\" data-semantic-type=\\\"operator\\\">×</mo><mspace width=\\\"0.33em\\\"></mspace><msup data-semantic-=\\\"\\\" data-semantic-children=\\\"2,5\\\" data-semantic-parent=\\\"7\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"superscript\\\"><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"integer\\\" data-semantic-type=\\\"number\\\">10</mn><mrow data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-children=\\\"4\\\" data-semantic-content=\\\"3\\\" data-semantic-parent=\\\"6\\\" data-semantic-role=\\\"negative\\\" data-semantic-type=\\\"prefixop\\\"><mo data-semantic-=\\\"\\\" data-semantic-operator=\\\"prefixop,−\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"subtraction\\\" data-semantic-type=\\\"operator\\\">−</mo><mn data-semantic-=\\\"\\\" data-semantic-annotation=\\\"clearspeak:simple\\\" data-semantic-font=\\\"normal\\\" data-semantic-parent=\\\"5\\\" data-semantic-role=\\\"float\\\" data-semantic-type=\\\"number\\\">0.7</mn></mrow></msup></mrow>$ \\\\times\\\\ {10^{ - 0.7}}$</annotation></semantics></math></mjx-assistive-mml></mjx-container>, respectively. The hybrid nanofluid flows over the Riga plate has numerous applications in mechanical engineering, including fuel systems, magnetohydrodynamic (MHD) pumps, energy systems, thermal reactors, etc.\",\"PeriodicalId\":7219,\"journal\":{\"name\":\"Advanced Theory and Simulations\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2025-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advanced Theory and Simulations\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1002/adts.202500352\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advanced Theory and Simulations","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1002/adts.202500352","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
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