In this paper, we analyze the behavior of the ternary hybrid nanofluids composed by Alumina <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\left({\text{Al}}_{2}{\text{O}}_{3}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <msub> <mtext>Al</mtext> <mn>2</mn> </msub> <msub> <mtext>O</mtext> <mn>3</mn> </msub> </mfenced> </math></EquationSource> </InlineEquation>, Copper <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$(\text{Cu})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mtext>Cu</mtext> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and Titania <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$({\text{TiO}}_{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mtext>TiO</mtext> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> embedded in water- (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">${\text{H}}_{2}\text{O})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>H</mtext> <mn>2</mn> </msub> <mrow> <mtext>O</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>) based fluid. Using appropriate transformations, the partial differential equations are reduced to ordinary (similar) differential equations, which were numerically solved using the bvp4c function in MATLAB software. It is found that dual (upper and lower branch) solutions exist for shrinking parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda <0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and some values of the boundary layer thickness <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">${\eta }_{\infty },$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>η</mi> <mi>∞</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> so that the boundary conditions to be satisfied. To see which solution is stable and which is nonstable, a stability analysis has been performed. Thus, it is established that the upper branch solution is stable and physically realizable in practice and the lower solution is unstable and, therefore not physically realizable in practice.The results are presented in graphical and tabular form for different values of volume fraction, Prandtl number and shrinking parameter. The results show that the reduced skin friction and the reduced Nusselt number dynamically affect by growing the nanoparticle volume fraction, the type of nanoparticles and the shrinking parameter. The practical applications of hybrid nanofluids in main areas of research, includes solar energy, refrigeration as well as heating, ventilation, and air-conditioning applications, heat exchanger, heat pipes, coolant in machining and manufacturing, electronic cooling, automotive industry, generator cooling, transformer cooling, nuclear system cooling, biomedical, space, ships and defense are discussed in detail. As far as the authors know, no reported studies have solved the stagnation point past a stretching/shrinking sheet utilizing ternary hybrid nanofluids (Al2O3–CuO–TiO2/H2O) <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14195_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">$($</EquationSource> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> </math></EquationSource> </InlineEquation> water. The comparisons of the reduced skin friction coefficient and the reduced local Nusselt number are compared with papers from the open literature. The agreement is very good, so that we are confident that the present results are accurate and correct.

Dual solutions of ternary hybrid nanofluids stagnation point over a linearly stretching/shrinking sheet

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2025年6月21日

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C. Revnic,
T. Grosan,
I. Pop

摘要

In this paper, we analyze the behavior of the ternary hybrid nanofluids composed by Alumina $\left({\text{Al}}_{2}{\text{O}}_{3}\right)$ $({Al}_{2}, O_{3})$ , Copper $(\text{Cu})$ $(Cu)$ and Titania $({\text{TiO}}_{2})$ $({TiO}_{2})$ embedded in water- ( ${\text{H}}_{2}\text{O})$ $H_{2} O)$ ) based fluid. Using appropriate transformations, the partial differential equations are reduced to ordinary (similar) differential equations, which were numerically solved using the bvp4c function in MATLAB software. It is found that dual (upper and lower branch) solutions exist for shrinking parameter $\lambda <0$ $λ < 0$ and some values of the boundary layer thickness ${\eta }_{\infty },$ $η_{\infty},$ so that the boundary conditions to be satisfied. To see which solution is stable and which is nonstable, a stability analysis has been performed. Thus, it is established that the upper branch solution is stable and physically realizable in practice and the lower solution is unstable and, therefore not physically realizable in practice.

The results are presented in graphical and tabular form for different values of volume fraction, Prandtl number and shrinking parameter. The results show that the reduced skin friction and the reduced Nusselt number dynamically affect by growing the nanoparticle volume fraction, the type of nanoparticles and the shrinking parameter. The practical applications of hybrid nanofluids in main areas of research, includes solar energy, refrigeration as well as heating, ventilation, and air-conditioning applications, heat exchanger, heat pipes, coolant in machining and manufacturing, electronic cooling, automotive industry, generator cooling, transformer cooling, nuclear system cooling, biomedical, space, ships and defense are discussed in detail. As far as the authors know, no reported studies have solved the stagnation point past a stretching/shrinking sheet utilizing ternary hybrid nanofluids (Al₂O₃–CuO–TiO₂/H₂O) $($ $($ water. The comparisons of the reduced skin friction coefficient and the reduced local Nusselt number are compared with papers from the open literature. The agreement is very good, so that we are confident that the present results are accurate and correct.