There are numerous real-world applications of phenomena caused by abruptly started or stopped object motion. One such illustration is an application of airbags in motor vehicles. The primary purpose of this research is to investigate impulsive mixed convective Williamson ternary nanofluid flow over a rotating rough sphere in the presence of periodic magnetic effects. The external stream is primarily responsible for the time-dependent flow. A sinusoidal waveform mathematically models the rough surface of the sphere with small amplitude and high frequency. Thus, surface gradient and skin-friction exhibit wavy effects in the boundary layer regime. Under suitable initial and boundary conditions, the governing equations of the Williamson fluid flow, which in the current flow problem include the effects of heat diffusion and rotation, are highly coupled nonlinear PDEs. These are converted to non-dimensional forms by applying the semi-similar transformations, for which numerical semi-similar solutions are produced using the quasi-linearization technique followed by implicit finite difference approximation. The ranges of some important parameters considered are 2 ≤ Ri ≤ 10 (Richardson number), 0 ≤ M ≤ 4 (magnetic), 0 ≤ ϕi ≤ 0.04, i = 1, 2, 3 (nanoparticles volume fraction), 0 ≤ Wp ≤ 1 (Williamson parameter), 0 ≤ λ ≤ 5 (rotation parameter). The streamwise velocity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14173_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {F\left( {\xi ,\eta } \right)} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mi>F</mi> <mfenced close=")" open="("> <mrow> <mi>ξ</mi> <mo>,</mo> <mi>η</mi> </mrow> </mfenced> </mrow> </mfenced> </math></EquationSource> </InlineEquation>, skin-friction <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14173_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{x}}} } \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mn>1</mn> <mrow> <mfenced open="/"> <mphantom> <mpadded width="0pt"> <mn>12</mn> </mpadded> </mphantom> </mfenced> </mrow> <mn>2</mn> </mrow> </msup> <mi>C</mi> <msub> <mi>f</mi> <mtext>x</mtext> </msub> </mrow> </mfenced> </math></EquationSource> </InlineEquation> and rotational skin-friction <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14173_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{y}}} } \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mn>1</mn> <mrow> <mfenced open="/"> <mphantom> <mpadded width="0pt"> <mn>12</mn> </mpadded> </mphantom> </mfenced> </mrow> <mn>2</mn> </mrow> </msup> <mi>C</mi> <msub> <mi>f</mi> <mtext>y</mtext> </msub> </mrow> </mfenced> </math></EquationSource> </InlineEquation> are all enhanced by increasing Ri and λ values. The upsurging λ values from 0 to 4 amend the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14173_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{\text{x}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mn>1</mn> <mrow> <mfenced open="/"> <mphantom> <mpadded width="0pt"> <mn>12</mn> </mpadded> </mphantom> </mfenced> </mrow> <mn>2</mn> </mrow> </msup> <mi>C</mi> <msub> <mi>f</mi> <mtext>x</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation> approximately by 17% at ξ = 1 and Ri = 10. The amplitude of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10973_2025_14173_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{x}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <msup> <mi>e</mi> <mrow> <mn>1</mn> <mrow> <mfenced open="/"> <mphantom> <mpadded width="0pt"> <mn>12</mn> </mpadded> </mphantom> </mfenced> </mrow> <mn>2</mn> </mrow> </msup> <mi>C</mi> <msub> <mi>f</mi> <mtext>x</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation> for n = 50 is enhanced by about 98% at ξ = 0.5 when the roughness ε upsurge from 0.001 to 0.005. Changing the shape of the nanoparticles from spherical to laminar results in an increase in approximately 46% at ξ = 0.5. Furthermore, the current results strongly correlate with similar outcomes reported in the literature.

Impulsive mixed convection of Williamson ternary nanofluid over a spinning rough sphere: influence of periodic magnetic field

期刊论文
发表日期：

2025年4月28日

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P. M. Patil,
Bharath Goudar,
Mikhail A. Sheremet

摘要

There are numerous real-world applications of phenomena caused by abruptly started or stopped object motion. One such illustration is an application of airbags in motor vehicles. The primary purpose of this research is to investigate impulsive mixed convective Williamson ternary nanofluid flow over a rotating rough sphere in the presence of periodic magnetic effects. The external stream is primarily responsible for the time-dependent flow. A sinusoidal waveform mathematically models the rough surface of the sphere with small amplitude and high frequency. Thus, surface gradient and skin-friction exhibit wavy effects in the boundary layer regime. Under suitable initial and boundary conditions, the governing equations of the Williamson fluid flow, which in the current flow problem include the effects of heat diffusion and rotation, are highly coupled nonlinear PDEs. These are converted to non-dimensional forms by applying the semi-similar transformations, for which numerical semi-similar solutions are produced using the quasi-linearization technique followed by implicit finite difference approximation. The ranges of some important parameters considered are 2 ≤ Ri ≤ 10 (Richardson number), 0 ≤ M ≤ 4 (magnetic), 0 ≤ ϕ_i ≤ 0.04, i = 1, 2, 3 (nanoparticles volume fraction), 0 ≤ Wp ≤ 1 (Williamson parameter), 0 ≤ λ ≤ 5 (rotation parameter). The streamwise velocity $\left( {F\left( {\xi ,\eta } \right)} \right)$ $(F (ξ, η))$ , skin-friction $\left( {Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{x}}} } \right)$ $(R e^{1 /) 2} C f_{x})$ and rotational skin-friction $\left( {Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{y}}} } \right)$ $(R e^{1 /) 2} C f_{y})$ are all enhanced by increasing Ri and λ values. The upsurging λ values from 0 to 4 amend the $Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{\text{x}}$ $R e^{1 /) 2} C f_{x}$ approximately by 17% at ξ = 1 and Ri = 10. The amplitude of $Re^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Cf_{{\text{x}}}$ $R e^{1 /) 2} C f_{x}$ for n = 50 is enhanced by about 98% at ξ = 0.5 when the roughness ε upsurge from 0.001 to 0.005. Changing the shape of the nanoparticles from spherical to laminar results in an increase in approximately 46% at ξ = 0.5. Furthermore, the current results strongly correlate with similar outcomes reported in the literature.