The Cattaneo–Christov double diffusion model generalizes classical diffusion theory by incorporating relaxation time effects enabling the characterization of heat and mass transport with finite propagation speeds. This advancement allows the model to effectively describes electrically induced magnetohydrodynamic (EMHD) flow and radiative heat transfer over a stretched surface for improved heat and mass transfer in Williamson nanofluids. In investigating these phenomena, the nonlinear partial differential equations are reduced to ordinary differential equations by employing a similarity transformation and solving them with the BP4C method. In addition, an artificial neural network trained with the Levenberg–Marquardt backpropagation algorithm is utilized to forecast flow regimes and heat transfer. For training, the dataset was divided into 70% training, 15% validation, and 15% test sets. Performance was assessed in terms of mean square error and regression analysis. Results from the ANN and computational approaches agree well with the earlier works. Notably, (LM-BP) provides an accurate analysis with a validation error of approximately 1.2%. It is also concluded that velocity decreases with a higher Weissenberg number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\left(\text{We}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mtext>We</mtext> </mfenced> </math></EquationSource> </InlineEquation> but increases with electric MHD strength. In contrast, the thermal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left({\lambda }_{\text{t}}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>λ</mi> <mtext>t</mtext> </msub> </mfenced> </math></EquationSource> </InlineEquation> and solute <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\left({\lambda }_{\text{c}}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mi>λ</mi> <mtext>c</mtext> </msub> </mfenced> </math></EquationSource> </InlineEquation> relaxation parameters are negatively correlated with temperature and concentration, respectively.

Numerical investigation of heat transfer in Williamson nanofluid flow with the Cattaneo–Christov double diffusion model using an artificial neural network approach

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2026年3月21日

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M. Waqas Ashraf,
Zhoushun Zheng,
Khurram Shabbir,
M. Israr Ur Rehman

摘要

The Cattaneo–Christov double diffusion model generalizes classical diffusion theory by incorporating relaxation time effects enabling the characterization of heat and mass transport with finite propagation speeds. This advancement allows the model to effectively describes electrically induced magnetohydrodynamic (EMHD) flow and radiative heat transfer over a stretched surface for improved heat and mass transfer in Williamson nanofluids. In investigating these phenomena, the nonlinear partial differential equations are reduced to ordinary differential equations by employing a similarity transformation and solving them with the BP4C method. In addition, an artificial neural network trained with the Levenberg–Marquardt backpropagation algorithm is utilized to forecast flow regimes and heat transfer. For training, the dataset was divided into 70% training, 15% validation, and 15% test sets. Performance was assessed in terms of mean square error and regression analysis. Results from the ANN and computational approaches agree well with the earlier works. Notably, (LM-BP) provides an accurate analysis with a validation error of approximately 1.2%. It is also concluded that velocity decreases with a higher Weissenberg number $\left(\text{We}\right)$ $(We)$ but increases with electric MHD strength. In contrast, the thermal $\left({\lambda }_{\text{t}}\right)$ $(λ_{t})$ and solute $\left({\lambda }_{\text{c}}\right)$ $(λ_{c})$ relaxation parameters are negatively correlated with temperature and concentration, respectively.