<p>Hydromagnetic natural convection induced in a square cavity filled with a non-Newtonian fluid and confining a heating rhombic solid block submitted to multiple subdivisions is numerically analyzed. The heating blocks are maintained isothermal at <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">${T}_{H}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>H</mi> </msub> </math></EquationSource> </InlineEquation> and the cavity is cooled through its vertical walls at a constant temperature <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">${T}_{C}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>C</mi> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">${T}_{H}>{T}_{C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>H</mi> </msub> <mo>></mo> <msub> <mi>T</mi> <mi>C</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>). The multiple-relaxation-time lattice-Boltzmann-method (MRT-LBM) is used to simulate the momentum and energy equations by using D2Q9 and D2Q5 discretization schemes, respectively. The effect of the Rayleigh number (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">${10}^{4}\le Ra\le {10}^{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mn>4</mn> </msup> <mo>≤</mo> <mi>R</mi> <mi>a</mi> <mo>≤</mo> <msup> <mrow> <mn>10</mn> </mrow> <mn>6</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>), the Hartmann number (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="95" /> </InlineMediaObject> <EquationSource Format="TEX">$0\le Ha\le 50$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>H</mi> <mi>a</mi> <mo>≤</mo> <mn>50</mn> </mrow> </math></EquationSource> </InlineEquation>), the power-law index (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$0.7\le n\le 1.3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.7</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>1.3</mn> </mrow> </math></EquationSource> </InlineEquation>) and the fragmentation of the initial block (<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$16$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>16</mn> </mrow> </math></EquationSource> </InlineEquation>) on the fluid flow and heat transfer is examined. An adaptive Artificial Neural Networks (ANN) approach is proposed, trained and tested based on the numerical results provided by the MRT-LBM. The results obtained show that the variations of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$Ha$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Ha</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$Ra$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Ra</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Nb</mi> </mrow> </math></EquationSource> </InlineEquation> and the power-law index significantly influence the heat transfer rate and the flow intensity in the system considered. More specifically, heat transfer rate is improved by increasing the Rayleigh number and the number of subdivisions. For <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$Ra={10}^{6}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>a</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mn>6</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, increasing <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Nb</mi> </mrow> </math></EquationSource> </InlineEquation> from 1 to 16 enhances heat transfer by up to 55.8%, depending on fluid rheology, whereas increasing Hartmann number and power-law index reduce heat transfer. In the reference case (<InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), transitioning from a shear-thinning (<InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$n=0.7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </math></EquationSource> </InlineEquation>) fluid to a Newtonian (<InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq17.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </math></EquationSource> </InlineEquation>) or shear-thickening (<InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1.3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1.3</mn> </mrow> </math></EquationSource> </InlineEquation>) fluid decreases the heat transfer rate by approximately 51.9 and 72.6%, respectively. Furthermore, increasing <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$Ha$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Ha</mi> </mrow> </math></EquationSource> </InlineEquation> from 0 to 50 leads to heat transfer reductions of 51.4, 56, and 48.5% for <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$n=0.7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, 4 and 16; and 25.8, 24.4, and 16.8% for <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq22.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>; and 10, 6.5, and 2.7% for <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1000_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$n=1.3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1.3</mn> </mrow> </math></EquationSource> </InlineEquation>, demonstrating a stronger damping effect in shear thinning fluids compared to Newtonian or shear-thickening cases.</p>

MHD natural convection of non-Newtonian fluids in a square cavity with a subdivided rhombic-shaped heating element

期刊论文
发表日期：

2025年8月11日

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Khalid Chtaibi,
Mohammed Hasnaoui,
Abdelkhalek Amahmid,
Youssef Dahani,
Haïkel Ben Hamed,
Abdelghani Raji

摘要

Hydromagnetic natural convection induced in a square cavity filled with a non-Newtonian fluid and confining a heating rhombic solid block submitted to multiple subdivisions is numerically analyzed. The heating blocks are maintained isothermal at ${T}_{H}$ $T_{H}$ and the cavity is cooled through its vertical walls at a constant temperature ${T}_{C}$ $T_{C}$ ( ${T}_{H}>{T}_{C}$ $T_{H} > T_{C}$ ). The multiple-relaxation-time lattice-Boltzmann-method (MRT-LBM) is used to simulate the momentum and energy equations by using D2Q9 and D2Q5 discretization schemes, respectively. The effect of the Rayleigh number ( ${10}^{4}\le Ra\le {10}^{6}$ $10^{4} \leq R a \leq 10^{6}$ ), the Hartmann number ( $0\le Ha\le 50$ $0 \leq H a \leq 50$ ), the power-law index ( $0.7\le n\le 1.3$ $0.7 \leq n \leq 1.3$ ) and the fragmentation of the initial block ( $Nb=1$ $N b = 1$ , $4$ $4$ and $16$ $16$ ) on the fluid flow and heat transfer is examined. An adaptive Artificial Neural Networks (ANN) approach is proposed, trained and tested based on the numerical results provided by the MRT-LBM. The results obtained show that the variations of $Ha$ $Ha$ , $Ra$ $Ra$ , $Nb$ $Nb$ and the power-law index significantly influence the heat transfer rate and the flow intensity in the system considered. More specifically, heat transfer rate is improved by increasing the Rayleigh number and the number of subdivisions. For $Ra={10}^{6}$ $R a = 10^{6}$ , increasing $Nb$ $Nb$ from 1 to 16 enhances heat transfer by up to 55.8%, depending on fluid rheology, whereas increasing Hartmann number and power-law index reduce heat transfer. In the reference case ( $Nb=1$ $N b = 1$ ), transitioning from a shear-thinning ( $n=0.7$ $n = 0.7$ ) fluid to a Newtonian ( $n=1.0$ $n = 1.0$ ) or shear-thickening ( $n=1.3$ $n = 1.3$ ) fluid decreases the heat transfer rate by approximately 51.9 and 72.6%, respectively. Furthermore, increasing $Ha$ $Ha$ from 0 to 50 leads to heat transfer reductions of 51.4, 56, and 48.5% for $n=0.7$ $n = 0.7$ at $Nb=1$ $N b = 1$ , 4 and 16; and 25.8, 24.4, and 16.8% for $n=1$ $n = 1$ ; and 10, 6.5, and 2.7% for $n=1.3$ $n = 1.3$ , demonstrating a stronger damping effect in shear thinning fluids compared to Newtonian or shear-thickening cases.