<p>The study of bioconvective flow of Casson-Williamson nanoliquid over a rough slender cylinder has many engineering and industrial applications. However, limited research exists on such type of flow under the combined effects of inclined magnetic field and liquid oxygen diffusion in conjunction with nanoparticles. The current research fill this gap by investigating the flow, heat and mass transfer characteristics by using a set of coupled, highly nonlinear partial differential equations. The nonsimilar transformations are used to reduce them into a set of dimensionless equations. The quasilinearization technique and an implicit finite difference scheme are used for mathematical simplification. The effects of many important parameters namely, mixed convection parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda \left( { - 2 \le \lambda \le 10} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mfenced close=")" open="("> <mrow> <mo>-</mo> <mn>2</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>10</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Williamson parameter <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">$W\left( {0 \le W \le 2} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mfenced close=")" open="("> <mrow> <mn>0</mn> <mo>≤</mo> <mi>W</mi> <mo>≤</mo> <mn>2</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Brownian motion parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$Nb\left( {0.1 \le Nb \le 0.6} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mi>b</mi> <mfenced close=")" open="("> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>N</mi> <mi>b</mi> <mo>≤</mo> <mn>0.6</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Magnetic field <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="135" /> </InlineMediaObject> <EquationSource Format="TEX">$M\left( {0 \le M \le 0.6} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mrow> <mn>0</mn> <mo>≤</mo> <mi>M</mi> <mo>≤</mo> <mn>0.6</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> roughness parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="138" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \left( {0.01 \le \varepsilon \le 0.1} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mfenced close=")" open="("> <mrow> <mn>0.01</mn> <mo>≤</mo> <mi>ε</mi> <mo>≤</mo> <mn>0.1</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> frequency parameter <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$n\left( {10 \le n \le 50} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mfenced close=")" open="("> <mrow> <mn>10</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>50</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> bio-convection’s Lewis number <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="149" /> </InlineMediaObject> <EquationSource Format="TEX">$Lb\left( {0.1 \le Lb \le 0.5} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>b</mi> <mfenced close=")" open="("> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>L</mi> <mi>b</mi> <mo>≤</mo> <mn>0.5</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Rayleigh number <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="152" /> </InlineMediaObject> <EquationSource Format="TEX">$Rb\left( {0.1 \le Rb \le 1.0} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>b</mi> <mfenced close=")" open="("> <mrow> <mn>0.1</mn> <mo>≤</mo> <mi>R</mi> <mi>b</mi> <mo>≤</mo> <mn>1.0</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Lewis number <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$Le\left( {1 \le Le \le 10} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>e</mi> <mfenced close=")" open="("> <mrow> <mn>1</mn> <mo>≤</mo> <mi>L</mi> <mi>e</mi> <mo>≤</mo> <mn>10</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> Schmidt number <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">$Sc\left( {1 \le Sc \le 3} \right),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>c</mi> <mfenced close=")" open="("> <mrow> <mn>1</mn> <mo>≤</mo> <mi>S</mi> <mi>c</mi> <mo>≤</mo> <mn>3</mn> </mrow> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and Peclet number <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_904_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$Pe\left( {1 \le Pe \le 5} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>e</mi> <mfenced close=")" open="("> <mrow> <mn>1</mn> <mo>≤</mo> <mi>P</mi> <mi>e</mi> <mo>≤</mo> <mn>5</mn> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> on the velocity, temperature, and concentration characteristics are investigated and illustrated through graphs. The skin friction coefficient increases approximately 14% and 75% when the roughness parameter escalates from 0.01 to 0.05 and the magnetic parameter changes from 0 to 0.3, respectively. The microbial density increases by about 40% as the Peclet number rises from 1 to 2. The mass transfer of liquid oxygen is significantly greater with an elevated values of the Peclet number and the bioconvection Lewis number. The coefficient of heat transfer upsurges for the roughness parameter, this is a vital role in electronic cooling techniques, nuclear reactor cooling, and polymer fibre coating. The present model may be used to solve various biological, bioengineering, biomedical, geophysical activities, architectural thermal insulation, and ecological problems.</p>

Computational study of Casson–Williamson nanoliquid flow over a slender cylinder in presence of oxytactic microbes: A new insights of inclined magnetic field and surface roughness

期刊论文
发表日期：

2025年6月1日

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P. M. Patil,
Sunil Benawadi

摘要

The study of bioconvective flow of Casson-Williamson nanoliquid over a rough slender cylinder has many engineering and industrial applications. However, limited research exists on such type of flow under the combined effects of inclined magnetic field and liquid oxygen diffusion in conjunction with nanoparticles. The current research fill this gap by investigating the flow, heat and mass transfer characteristics by using a set of coupled, highly nonlinear partial differential equations. The nonsimilar transformations are used to reduce them into a set of dimensionless equations. The quasilinearization technique and an implicit finite difference scheme are used for mathematical simplification. The effects of many important parameters namely, mixed convection parameter $\lambda \left( { - 2 \le \lambda \le 10} \right),$ $λ (- 2 \leq λ \leq 10),$ Williamson parameter $W\left( {0 \le W \le 2} \right),$ $W (0 \leq W \leq 2),$ Brownian motion parameter $Nb\left( {0.1 \le Nb \le 0.6} \right),$ $N b (0.1 \leq N b \leq 0.6),$ Magnetic field $M\left( {0 \le M \le 0.6} \right),$ $M (0 \leq M \leq 0.6),$ roughness parameter $\alpha \left( {0.01 \le \varepsilon \le 0.1} \right),$ $α (0.01 \leq ε \leq 0.1),$ frequency parameter $n\left( {10 \le n \le 50} \right),$ $n (10 \leq n \leq 50),$ bio-convection’s Lewis number $Lb\left( {0.1 \le Lb \le 0.5} \right),$ $L b (0.1 \leq L b \leq 0.5),$ Rayleigh number $Rb\left( {0.1 \le Rb \le 1.0} \right),$ $R b (0.1 \leq R b \leq 1.0),$ Lewis number $Le\left( {1 \le Le \le 10} \right),$ $L e (1 \leq L e \leq 10),$ Schmidt number $Sc\left( {1 \le Sc \le 3} \right),$ $S c (1 \leq S c \leq 3),$ and Peclet number $Pe\left( {1 \le Pe \le 5} \right)$ $P e (1 \leq P e \leq 5)$ on the velocity, temperature, and concentration characteristics are investigated and illustrated through graphs. The skin friction coefficient increases approximately 14% and 75% when the roughness parameter escalates from 0.01 to 0.05 and the magnetic parameter changes from 0 to 0.3, respectively. The microbial density increases by about 40% as the Peclet number rises from 1 to 2. The mass transfer of liquid oxygen is significantly greater with an elevated values of the Peclet number and the bioconvection Lewis number. The coefficient of heat transfer upsurges for the roughness parameter, this is a vital role in electronic cooling techniques, nuclear reactor cooling, and polymer fibre coating. The present model may be used to solve various biological, bioengineering, biomedical, geophysical activities, architectural thermal insulation, and ecological problems.