<p>This study explores multiple factors effects such as radiation, heat sources and sinks, activation energy, and an exponentially varying space-based heat source on bioconvection movement of a third-grade nanofluid (BTGNF) inside a stretched cylindrical surface incorporating Buongiorno nanofluid model to scrutinize the impacts of thermophoresis and Brownian motion. The analysis is carried out using an artificial neural network (ANN) based on a multilayer perceptron model. Numerical data for training, validating, and testing are generated using a robust numerical solver BVP4C method. The ANN model is used to predict key parameters such as the skin friction coefficient (SFC), local Nusselt number (LNN), local Sherwood number (LSN), and density of motile microorganisms (DMMO). The investigation is grounded in specific theoretical assumptions related to fluid flow behavior. Each physical characteristic is defined within a certain range: <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="144" /> </InlineMediaObject> <EquationSource Format="TEX">$0< {\beta }_{1}, {\beta }_{2}, {\beta }_{3} \le 0.8,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>3</mn> </msub> <mo>≤</mo> <mn>0.8</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le \lambda \le 1.0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="171" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le M, Nr, Nc \le 1.0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mi>r</mi> <mo>,</mo> <mi>N</mi> <mi>c</mi> <mo>≤</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$1< Pr \le 4.0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>P</mi> <mi>r</mi> <mo>≤</mo> <mn>4.0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="145" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le Nb, Nt \le 0.7,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>N</mi> <mi>b</mi> <mo>,</mo> <mi>N</mi> <mi>t</mi> <mo>≤</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$1.1 \le Le \le 2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.1</mn> <mo>≤</mo> <mi>L</mi> <mi>e</mi> <mo>≤</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="107" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le E< 1.0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>E</mi> <mo><</mo> <mn>1.0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le Lb \le 0.7,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>L</mi> <mi>b</mi> <mo>≤</mo> <mn>0.7</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq9.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">$2 \le Pe \le 4,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>P</mi> <mi>e</mi> <mo>≤</mo> <mn>4</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11740_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$0.1 \le {\lambda }_{1}, {\lambda }_{2}, {\lambda }_{3} \le 0.7.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.1</mn> <mo>≤</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>3</mn> </msub> <mo>≤</mo> <mn>0.7</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The temperature of the nanofluid rises significantly due to the influence of an exponentially varying space-based heat source and its related parameter. Additionally, both the solutal Biot number and the activation energy parameter contribute to an increase in the nanofluid’s concentration. On the other hand, an inverse relationship is observed between the bioconvection Lewis number and the distribution of microorganisms. The consequences display that the ANN models are highly effective at predicting the values of SFC, LNN, LSN, and DMMO with remarkably low error rates of − 0.12%, 0.03%, 0.01%, and 0.002, respectively. Similarly, the BTGNF-ANN model achieved very low mean squared error and high correlation (R) values, further confirming that the ANN models are capable of making highly accurate predictions.</p>

Artificial neural networks model prediction for thermal, heat source and activation energy effects in bioconvection magnetocross third grade nanofluid across an extended cylinder

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发表日期：

2025年9月12日

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Nehad Ali Shah,
Khalid Masood,
Zeeshan,
B. C. Prasannakumara

摘要

This study explores multiple factors effects such as radiation, heat sources and sinks, activation energy, and an exponentially varying space-based heat source on bioconvection movement of a third-grade nanofluid (BTGNF) inside a stretched cylindrical surface incorporating Buongiorno nanofluid model to scrutinize the impacts of thermophoresis and Brownian motion. The analysis is carried out using an artificial neural network (ANN) based on a multilayer perceptron model. Numerical data for training, validating, and testing are generated using a robust numerical solver BVP4C method. The ANN model is used to predict key parameters such as the skin friction coefficient (SFC), local Nusselt number (LNN), local Sherwood number (LSN), and density of motile microorganisms (DMMO). The investigation is grounded in specific theoretical assumptions related to fluid flow behavior. Each physical characteristic is defined within a certain range: $0< {\beta }_{1}, {\beta }_{2}, {\beta }_{3} \le 0.8,$ $0 < β_{1}, β_{2}, β_{3} \leq 0.8,$ $0.1 \le \lambda \le 1.0,$ $0.1 \leq λ \leq 1.0,$ $0.1 \le M, Nr, Nc \le 1.0,$ $0.1 \leq M, N r, N c \leq 1.0,$ $1< Pr \le 4.0,$ $1 < P r \leq 4.0,$ $0.1 \le Nb, Nt \le 0.7,$ $0.1 \leq N b, N t \leq 0.7,$ $1.1 \le Le \le 2,$ $1.1 \leq L e \leq 2,$ $0.1 \le E< 1.0,$ $0.1 \leq E < 1.0,$ $0.1 \le Lb \le 0.7,$ $0.1 \leq L b \leq 0.7,$ $2 \le Pe \le 4,$ $2 \leq P e \leq 4,$ and $0.1 \le {\lambda }_{1}, {\lambda }_{2}, {\lambda }_{3} \le 0.7.$ $0.1 \leq λ_{1}, λ_{2}, λ_{3} \leq 0.7 .$ The temperature of the nanofluid rises significantly due to the influence of an exponentially varying space-based heat source and its related parameter. Additionally, both the solutal Biot number and the activation energy parameter contribute to an increase in the nanofluid’s concentration. On the other hand, an inverse relationship is observed between the bioconvection Lewis number and the distribution of microorganisms. The consequences display that the ANN models are highly effective at predicting the values of SFC, LNN, LSN, and DMMO with remarkably low error rates of − 0.12%, 0.03%, 0.01%, and 0.002, respectively. Similarly, the BTGNF-ANN model achieved very low mean squared error and high correlation (R) values, further confirming that the ANN models are capable of making highly accurate predictions.