<p>Our goal is to investigate entropy generation in a two-dimensional steady incompressible flow of a Casson fluid over a nonlinear vertically curved stretching sheet. The study considers convective heat transfer along with natural convection influenced by gravity within the boundary layer flow of the Casson fluid over this nonlinear vertical curved surface. The governing equations are formulated in a curvilinear coordinate system to accurately represent the flow, which is affected by magnetohydrodynamics (MHD), an absorbent medium, suction/injection, Joule heating, viscous dissipation, and thermal radiation. To simplify the analysis, suitable similarity variables are introduced, transforming the nonlinear partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs), which are then solved numerically using the bvp4c technique. The numerical results obtained show good agreement with previously published findings. The outcomes are presented graphically, illustrating the influence of various physical flow parameters on the velocity and temperature profiles. Additionally, the study calculates and analyzes the skin friction coefficient <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{\prime\prime}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the local Nusselt number <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$-\theta ^{\prime}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It is observed that the curvature parameter δ has a similar effect on both <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$f^{\prime}\left(\xi \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mfenced close=")" open="("> <mi>ξ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta \left(\xi \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mfenced close=")" open="("> <mi>ξ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> with both profiles increasing as δ increases. The temperature profile <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\theta \left(\xi \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mfenced close=")" open="("> <mi>ξ</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> rises with increasing values of the radiation parameter <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$Rd$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Rd</mi> </mrow> </math></EquationSource> </InlineEquation>, magnetic <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$M$</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${\gamma }_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$Ec$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Ec</mi> </mrow> </math></EquationSource> </InlineEquation>, but decreases with a rise in the Prandtl number <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14259_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$Pr$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Pr</mi> </mrow> </math></EquationSource> </InlineEquation>. An increase in the magnetic parameter leads to a reduction in both the entropy generation rate and the Bejan number. This research offers novel insights into the interplay between entropy generation and flow dynamics, contributing significantly to the understanding of Casson fluid behavior in engineering applications.</p>

Comparative analysis of entropy optimization in Casson fluid flow over a nonlinear curved stretching sheet

期刊论文
发表日期：

2025年5月22日

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Musharafa Saleem,
Majid Hussain,
Umair Khan,
Syed Modassir Hussain

摘要

Our goal is to investigate entropy generation in a two-dimensional steady incompressible flow of a Casson fluid over a nonlinear vertically curved stretching sheet. The study considers convective heat transfer along with natural convection influenced by gravity within the boundary layer flow of the Casson fluid over this nonlinear vertical curved surface. The governing equations are formulated in a curvilinear coordinate system to accurately represent the flow, which is affected by magnetohydrodynamics (MHD), an absorbent medium, suction/injection, Joule heating, viscous dissipation, and thermal radiation. To simplify the analysis, suitable similarity variables are introduced, transforming the nonlinear partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs), which are then solved numerically using the bvp4c technique. The numerical results obtained show good agreement with previously published findings. The outcomes are presented graphically, illustrating the influence of various physical flow parameters on the velocity and temperature profiles. Additionally, the study calculates and analyzes the skin friction coefficient $f^{\prime\prime}(0)$ $f^{″} (0)$ and the local Nusselt number $-\theta ^{\prime}(0)$ $- θ^{'} (0)$ . It is observed that the curvature parameter δ has a similar effect on both $f^{\prime}\left(\xi \right)$ $f^{'} (ξ)$ and $\theta \left(\xi \right)$ $θ (ξ)$ with both profiles increasing as δ increases. The temperature profile $\theta \left(\xi \right)$ $θ (ξ)$ rises with increasing values of the radiation parameter $Rd$ $Rd$ , magnetic $M$ $M$ , ${\gamma }_{1}$ $γ_{1}$ , and $Ec$ $Ec$ , but decreases with a rise in the Prandtl number $Pr$ $\Pr$ . An increase in the magnetic parameter leads to a reduction in both the entropy generation rate and the Bejan number. This research offers novel insights into the interplay between entropy generation and flow dynamics, contributing significantly to the understanding of Casson fluid behavior in engineering applications.