Geophysical mass flows such as mudflows and lahars commonly exhibit vertical heterogeneity, often inferred from observed concentration gradients associated with particle settling, consolidation, and fines migration. Standard Herschel–Bulkley (HB) rheology assumes spatially uniform material parameters and therefore cannot explicitly represent depth-dependent rheological structure. This work introduces a depth-dependent constitutive framework in which the HB consistency is promoted from a constant parameter to a prescribed spatial field K(z), enabling a continuous first-order representation of vertical rheological heterogeneity without additional concentration-transport equations or multilayer discretizations. Unlike modified or parametric HB formulations that adjust bulk rheological parameters through temperature, composition, regularization, or empirical coefficients while retaining a depth-uniform response, the proposed formulation embeds vertical stratification directly into the constitutive law. The framework is implemented in a three-dimensional weakly compressible smoothed particle hydrodynamics (WCSPH) solver and evaluated using dam-break configurations. Homogeneous HB, downward-stratified (DS), and upward-stratified (US) cases are compared in terms of front propagation, arrival time, and apparent front angle. The results show that the vertical location of high-consistency material modifies both the mobility and geometry of the advancing front: DS configurations produce basal braking with a relatively mobile upper layer, whereas US configurations generate a resistant upper layer that delays surface relaxation and maintains steeper fronts. Quantitatively, relative to an intermediate homogeneous HB reference, the depth-dependent formulation reduces the front position at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t^*=2.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>t</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>2.0</mn> </mrow> </math></EquationSource> </InlineEquation> by approximately <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$14\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>14</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$24\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>24</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> for DS cases and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$30\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>30</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$40\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>40</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> for US cases, while the arrival time to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x_f=1.0~\textrm{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>1.0</mn> <mspace width="3.33333pt" /> <mtext>m</mtext> </mrow> </math></EquationSource> </InlineEquation> increases by approximately <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$19\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>19</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$172\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>172</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> for DS and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$88\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>88</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation>–<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$780\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>780</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> for US, depending on <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$n$</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>. These results indicate that vertical rheological structure can act as a first-order control on mudflow dynamics and that the proposed HBK framework provides a physically interpretable single-phase approach for representing stratified non-Newtonian flows beyond depth-uniform HB rheology.

A depth-dependent constitutive framework for stratified non-newtonian flows: beyond homogeneous Herschel–Bulkley rheology

期刊论文
出版模式：

开放获取
发表日期：

2026年7月10日

查看全文

Diego Valencia-Enríquez,
Javier Revelo-Fuelagán,
John Erick Ortiz Guzmán,
Francisco Ricardo Mafla Chamorro,
Bianca Marcela Miranda Portilla,
Fausto Andrés Escobar Revelo,
Merylin Cristina Ortega Ortega

摘要

Geophysical mass flows such as mudflows and lahars commonly exhibit vertical heterogeneity, often inferred from observed concentration gradients associated with particle settling, consolidation, and fines migration. Standard Herschel–Bulkley (HB) rheology assumes spatially uniform material parameters and therefore cannot explicitly represent depth-dependent rheological structure. This work introduces a depth-dependent constitutive framework in which the HB consistency is promoted from a constant parameter to a prescribed spatial field K(z), enabling a continuous first-order representation of vertical rheological heterogeneity without additional concentration-transport equations or multilayer discretizations. Unlike modified or parametric HB formulations that adjust bulk rheological parameters through temperature, composition, regularization, or empirical coefficients while retaining a depth-uniform response, the proposed formulation embeds vertical stratification directly into the constitutive law. The framework is implemented in a three-dimensional weakly compressible smoothed particle hydrodynamics (WCSPH) solver and evaluated using dam-break configurations. Homogeneous HB, downward-stratified (DS), and upward-stratified (US) cases are compared in terms of front propagation, arrival time, and apparent front angle. The results show that the vertical location of high-consistency material modifies both the mobility and geometry of the advancing front: DS configurations produce basal braking with a relatively mobile upper layer, whereas US configurations generate a resistant upper layer that delays surface relaxation and maintains steeper fronts. Quantitatively, relative to an intermediate homogeneous HB reference, the depth-dependent formulation reduces the front position at $t^*=2.0$ $t^{*} = 2.0$ by approximately $14\%$ $14 %$ – $24\%$ $24 %$ for DS cases and $30\%$ $30 %$ – $40\%$ $40 %$ for US cases, while the arrival time to $x_f=1.0~\textrm{m}$ $x_{f} = 1.0 m$ increases by approximately $19\%$ $19 %$ – $172\%$ $172 %$ for DS and $88\%$ $88 %$ – $780\%$ $780 %$ for US, depending on $n$ $n$ . These results indicate that vertical rheological structure can act as a first-order control on mudflow dynamics and that the proposed HBK framework provides a physically interpretable single-phase approach for representing stratified non-Newtonian flows beyond depth-uniform HB rheology.