<p>This article addresses the onset of penetrative convection in a horizontal dusty ferrofluid layer with temperature-dependent viscosity, heated from below and bounded by permeable boundaries. The study considers three temperature-dependent viscosity variation laws viz: linear, exponential, and inverse viscosity variation as well as four distinct internal heat supply functions that induce either uniform or non-uniform heating within the dusty ferrofluid layer. The inclusion of permeable boundaries introduces a novel aspect to the analysis, enhancing its relevance across various scientific and engineering applications. Using linear stability theory and normal mode analysis, the eigenvalue problem is derived and solved for stationary convection modes through the application of a single-term Galerkin method. The impact of key parameters on the onset of convection is analyzed for different combinations of hydrodynamic boundary conditions, providing significant insights into the system’s stability characteristics. The study reveals that the combined effect of heat sources and sinks increases the system’s instability compared to using only one type of thermal input. The linear and exponential temperature-dependent viscosity variation laws have a destabilizing effect, whereas the inverse variation law has a stabilizing influence across all four heat supply functions. Additionally, the nonlinearity of the magnetization parameter (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_993_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(M_3\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>) and the dust particle parameter (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_993_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(h^{'}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>h</mi> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation>) destabilizes the system, while the permeability parameters (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_993_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(k_0^{'}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>k</mi> <mn>0</mn> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_993_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(k_1^{'}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>k</mi> <mn>1</mn> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation>) are found to have a stabilizing effect. Results from previous research are recovered as special cases from the findings of the present study. These findings offer new insights into the complex interplay between thermal, magnetic, and particulate effects in ferrofluid convection, with potential applications in aerospace engineering, energy systems, geophysical flows, and magnetic fluid technologies.</p>

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Penetrative convection in dusty ferrofluids via internal heating with temperature-dependent viscosity and permeable boundaries

  • Awneesh Kumar,
  • Pankaj Kumar

摘要

This article addresses the onset of penetrative convection in a horizontal dusty ferrofluid layer with temperature-dependent viscosity, heated from below and bounded by permeable boundaries. The study considers three temperature-dependent viscosity variation laws viz: linear, exponential, and inverse viscosity variation as well as four distinct internal heat supply functions that induce either uniform or non-uniform heating within the dusty ferrofluid layer. The inclusion of permeable boundaries introduces a novel aspect to the analysis, enhancing its relevance across various scientific and engineering applications. Using linear stability theory and normal mode analysis, the eigenvalue problem is derived and solved for stationary convection modes through the application of a single-term Galerkin method. The impact of key parameters on the onset of convection is analyzed for different combinations of hydrodynamic boundary conditions, providing significant insights into the system’s stability characteristics. The study reveals that the combined effect of heat sources and sinks increases the system’s instability compared to using only one type of thermal input. The linear and exponential temperature-dependent viscosity variation laws have a destabilizing effect, whereas the inverse variation law has a stabilizing influence across all four heat supply functions. Additionally, the nonlinearity of the magnetization parameter ( \(M_3\) M 3 ) and the dust particle parameter ( \(h^{'}\) h ) destabilizes the system, while the permeability parameters ( \(k_0^{'}\) k 0 and \(k_1^{'}\) k 1 ) are found to have a stabilizing effect. Results from previous research are recovered as special cases from the findings of the present study. These findings offer new insights into the complex interplay between thermal, magnetic, and particulate effects in ferrofluid convection, with potential applications in aerospace engineering, energy systems, geophysical flows, and magnetic fluid technologies.