<p>The principal objective of the current work is to investigate the combined effects of thermophoretic diffusion and chemical reactions on the dynamics of Williamson fluid, as well as heat and mass transmission, over a continuously magnetized Riga surface. In this analysis, variable thermal properties are also considered. To solve the model dimensionless nonlinear system of equations, an innovative scheme is introduced, the Morlet wavelet neural network algorithm. This scheme validates unparalleled performance and accuracy. Moreover, the statistical procedures such as mean, standard deviation, MSE, MAD, and ENSE validate the exceptional precision of the proposed scheme. The MSE ranges from <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{ - 10}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>; MAD is within the interval of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{ - 01}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>01</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{ - 10}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>; ENSE ranges from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{ - 15}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>15</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, while fitness curves fall within <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14191_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$10^{ - 10}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> for all four cases across multiple runs. Additionally, using visual illustrations of the outcome improves the overall understanding of fundamental phenomena. This research work emphasizes the potential practical applications in fluid dynamics and encourages further research to expose new insights and advancements in the field. Combining ANN techniques with numerical simulations opens new research avenues and promotes progress in engineering and fluid dynamics.</p>

Machine-assisted simulation of heat and mass transfer through a porous Riga surface with chemical reactions and radiation effects

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发表日期：

2025年4月21日

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Shazia Habib,
Saleem Nasir,
Zeeshan Khan,
Abdallah S. Berrouk,
Saeed Islam,
Asim Aamir

摘要

The principal objective of the current work is to investigate the combined effects of thermophoretic diffusion and chemical reactions on the dynamics of Williamson fluid, as well as heat and mass transmission, over a continuously magnetized Riga surface. In this analysis, variable thermal properties are also considered. To solve the model dimensionless nonlinear system of equations, an innovative scheme is introduced, the Morlet wavelet neural network algorithm. This scheme validates unparalleled performance and accuracy. Moreover, the statistical procedures such as mean, standard deviation, MSE, MAD, and ENSE validate the exceptional precision of the proposed scheme. The MSE ranges from $10^{0}$ $10^{0}$ to $10^{ - 10}$ $10^{- 10}$ ; MAD is within the interval of $10^{ - 01}$ $10^{- 01}$ and $10^{ - 10}$ $10^{- 10}$ ; ENSE ranges from $10^{0}$ $10^{0}$ to $10^{ - 15}$ $10^{- 15}$ , while fitness curves fall within $10^{0}$ $10^{0}$ and $10^{ - 10}$ $10^{- 10}$ for all four cases across multiple runs. Additionally, using visual illustrations of the outcome improves the overall understanding of fundamental phenomena. This research work emphasizes the potential practical applications in fluid dynamics and encourages further research to expose new insights and advancements in the field. Combining ANN techniques with numerical simulations opens new research avenues and promotes progress in engineering and fluid dynamics.