SPH Simulation of Molten-Fluid Flows with a Plastic Surface Skin: A Lava-Flow-Oriented Model Study

Abstract

Lava flows represent complex thermofluid phenomena in which surface cooling leads to the formation of a solidified surface layer. Understanding the influence of such a surface layer on fluid flow is an important issue in lava flow modeling. It also shares essential characteristics with a wide range of engineering problems involving surface solidification. However, the role of plastic surface skin in controlling flow deceleration and stopping behavior has not been sufficiently clarified in existing models. In this study, two-dimensional smoothed particle hydrodynamics (SPH) simulations were conducted to investigate the influence of surface skin formation on lava flow dynamics. The temperature dependence of viscosity was introduced to reproduce a plastic surface skin. The skin was represented as a low-temperature, high-viscosity region. Comparisons with simulations without surface skin formation demonstrated that the surface skin exhibits a suppressive effect on the flow. This behavior was consistent with qualitative observations of flowing lava. It was also found that this surface skin caused the successive deceleration characteristic in Bingham fluids. As a result, both the flow velocity and the flowing distance are affected. These results suggest that accurate lava flow simulations require models that incorporate both surface skin effects and non-Newtonian behavior.

1. Introduction

Lava flows represent complex thermofluid phenomena in which a solidified surface layer forms as a result of surface cooling during flow. Understanding the influence of a solidified surface layer on fluid flow is a fundamental problem. It is relevant in geophysical contexts and in a wide range of engineering applications. In mechanical engineering, similar phenomena are observed in processes such as continuous casting. In these processes, solidification shells and thin surface skins form, deform, and fracture. Related surface solidification phenomena are also observed in welding processes, where thin solidified layers appear on the surface of molten pools. These layers are associated with weld surface formation [1]. These geophysical and engineering phenomena share essential characteristics.

In lava flows, it is known that two types of solidified surface layers are formed depending on the surface cooling conditions [2]. When the cooling rate is high, the lava surface undergoes supercooling and becomes an amorphous glass. As a result, a thin solidified layer (skin) is formed, which deforms plastically. On the other hand, when the cooling rate is low, crystallization progresses and a rigid solidified layer (crust) is formed.

Based on observations of flowing lava, Hon et al. [3] described the advance of lava accompanied by the skin and the crust. The advance is interpreted as a repeated sequence of the processes outlined below. Initially, a plastically deformable skin with a thickness of approximately 1–2 mm forms on the lava surface. This skin retains the high-temperature lava inside and suppresses its motion. As the flow velocity decreases and cooling progresses further, a rigid crust begins to grow from the skin. When the crust thickness reaches several centimeters, it becomes strong enough to halt the incoming lava. Subsequently, when the crust is locally broken due to an increase in internal pressure, the interior lava flows out again, and the surface returns to a state without a solidified layer.

Furthermore, the rheological properties of lava flows are known to change from Newtonian to non-Newtonian behavior due to the presence of crystals and bubbles within the lava [4]. It has been reported that silicate melts and lavas with a low crystal fraction can be treated as Newtonian fluids over a wide range of temperatures and strain rates [5,6]. On the other hand, lavas with a high crystal fraction exhibit Bingham behavior with a finite yield stress [2]. This change in rheological properties has a significant influence on lava flow dynamics. In addition, lava is also a glass-forming liquid that forms an amorphous glass upon rapid cooling. Its viscosity shows a strong temperature dependence near the glass transition temperature [7].

For the mitigation of damage caused by lava flows, accurate prediction of lava flow behavior is an important social demand. Many previous studies have focused on lava properties and characteristic flow phenomena. These factors must be appropriately incorporated into predictive models. In the context, numerous studies have been conducted on lava flows using a variety of approaches, including field studies [3,5,6,7,8,9,10,11], experimental modeling [4,12,13,14], and numerical simulations [15,16,17,18,19,20,21,22,23,24,25].

Field studies include observations of lava flow behavior and cooling processes [3], as well as analyses of the rheological properties of remelted samples of solidified lava [7,8]. In addition, direct measurements of flowing lava have been conducted to evaluate rheological properties, surface temperature, and flow velocity [9]. However, due to the low frequency of eruptions accompanied by lava flows and the complexity of the phenomena, there are limitations to achieving a comprehensive understanding based only on field observations.

Therefore, analog experiments and numerical simulations play an important role in the study of lava flows. In analog experiments, polyethylene glycol (PEG) is often used to model lava. The materials can reproduce the formation of a rigid crust associated with surface cooling [12,13,14]. Based on these analog experiments, Lyman et al. [13,14] extended the similarity solution for viscosity-dominated gravity currents proposed by Huppert [26]. They incorporated non-Newtonian behavior and introduced the formation of a surface crust as a stopping condition for the flow. On the other hand, these models do not account for the influence of a plastic surface skin on flow behavior.

In numerical simulations, models based on the shallow water equation are widely used in operational forecasting of lava inflow areas [15]. These equations treat velocity and temperature distributions as depth-averaged quantities. This treatment makes it difficult to directly represent vertically structured phenomena, such as viscosity changes associated with surface cooling and the formation of a solidified surface layer.

Phase changes by cooling and deformation of solid–liquid interfaces complicate numerical simulations. For this reason, several previous studies have proposed simplified models for heat transport and phase change. For example, Starodubtsev et al. [18,19] represented solidification behavior indirectly by introducing a time-dependent viscosity model, without solving heat transport. In addition, mesh-based computational methods such as finite volume and finite element methods have the disadvantage of requiring remeshing to accurately treat the deformation of the solid–liquid interface [16,17]. In contrast, smoothed particle hydrodynamics (SPH), which is one of the mesh-free particle methods, is well suited to simulating flows with phase change and free surfaces. SPH-based approaches have also been applied to geohazard modeling, such as landslide runout analysis [27] and debris-flow simulations [28]. These phenomena share several kinematic features with lava flows, including free-surface evolution and complex rheological behavior. These studies [27,28] demonstrate the usefulness of SPH as a numerical framework for gravity-driven mass movements. Despite this advantage and usefulness, only a few studies have reported lava flow simulations that consider heat transport and solidification using particle methods [20,21,22,24,25,29].

Hérault et al. [24] conducted lava flow simulations for the actual topography of Mt. Etna, considering heat transport, temperature-dependent viscosity, and non-Newtonian rheology. Through qualitative comparisons with observed volcanic landforms and numerical results, they demonstrated the necessity of modeling lava flows as non-Newtonian fluids with temperature-dependent viscosity. However, this study did not provide a detailed discussion of the specific shapes or formation processes of characteristic lava flow structures, such as lava levees and lava accumulations. Tomita et al. [25] conducted SPH simulations of lava flows using molten metal as an analog material. Under simplified conditions, they successfully reproduced the morphology of characteristic lava flow structures and discussed their geometrical features. Nevertheless, the formation of a surface skin and its mechanical influence on flow behavior were not examined. More recently, Zago [29] developed a more advanced phase transition model for lava flow simulations, incorporating solidus and liquidus temperatures. However, surface solidification at the free surface is regarded as outside the scope of this framework. Accordingly, the formation of a surface skin is not considered in this model. Overall, previous studies [20,21,22,24,25,29] have not sufficiently considered the formation of a surface skin or its influence on flow behavior.

To address this gap in previous studies [20,21,22,24,25,29], the objective of this study is to clarify the influence of the plastic surface skin on lava flow behavior. For this purpose, the cooling-induced surface skin is explicitly modeled as a low-temperature, high-viscosity layer to represent its plastic behavior. Two-dimensional SPH simulations are conducted to analyze the flow of a virtual fluid representing lava. The fluid flows down on a horizontal plate while forming a plastically deformable surface skin. Its influence on flow deceleration and stopping behavior is evaluated for both Newtonian and Bingham rheological conditions.

It should be noted that the plastic surface skin considered in this study is not intended to represent a rigid crust such as that introduced in previous lava flow models. In these gravity-current-based models [13,14], the crust is treated as a stopping condition that determines the final flow length. In contrast, this study focuses on the influence of a plastically deformable surface skin on flow dynamics during the advancing stage, and on its coupling with Bingham rheology.

2. Computational Method

2.1. Governing Equations

In this study, the computational object is the lava flow over a horizontal flat plate with cooling from the surroundings. The thermofluid behavior is described by the continuity equation, the Navier–Stokes equation, and the enthalpy transport equation. Although the governing equations and their discretizations are independent of the choice of unit system, all physical quantities in this study are expressed in the SI unit system. The continuity equation for a flow with constant density is given by

$$\nabla ·\overrightarrow{u}=0,$$

(1)

where $\overrightarrow{u}$ is the velocity. For a fluid with constant density, the Navier–Stokes equation in Lagrangian form is written as

$${\displaystyle \frac{D\overrightarrow{u}}{Dt}}=-{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{1}{\rho }}\nabla ·\left\{\mu \left[(\nabla \otimes \overrightarrow{u})+{(\nabla \otimes \overrightarrow{u})}^{\mathrm{T}}\right]\right\}+\overrightarrow{g},$$

(2)

where t is the time, $\rho$ is the density, p is the pressure, $\mu$ is the viscosity, and $\overrightarrow{g}$ denotes the gravitational acceleration. Here, the superscript ^T denotes the transpose of a tensor, and the operator ⊗ represents the tensor product. The Lagrangian form of the enthalpy transport equation is given by

$$\rho {\displaystyle \frac{Dh}{Dt}}=\nabla ·(\kappa \nabla T)-Q,$$

(3)

where h is the specific enthalpy, $\kappa$ is the thermal conductivity, T is the temperature, and Q denotes the heat loss term, which includes radiative heat loss and heat loss to the ambient air. Assuming that the specific enthalpy depends only on temperature and using specific heat at constant pressure $c_p=dh/dT$, Equation (3) can be rewritten as the following temperature evolution equation:

$${\displaystyle \frac{DT}{Dt}}={\displaystyle \frac{1}{\rho c_p}}\nabla ·(\kappa \nabla T)-{\displaystyle \frac{Q}{\rho c_p}}.$$

(4)

2.2. Discretization of Governing Equations in SPH Method

2.2.1. Treatment of Physical Quantities and Differential Operators in SPH

In the SPH method, a physical quantity $\varphi$ at the position $\overrightarrow{r}={\overrightarrow{r}}_a$ is evaluated by taking into account the contributions from the neighboring particles b. Defining the relative position vector from position a to position of particle b as ${\overrightarrow{r}}_{ba}={\overrightarrow{r}}_b-{\overrightarrow{r}}_a$, the $\varphi \left({\overrightarrow{r}}_a\right)$ can be expressed as follows [30]:

$$\varphi \left(\overrightarrow{r}={\overrightarrow{r}}_a\right)=\sum _b{m}_b{\displaystyle \frac{{\varphi }_b}{{\rho }_b}}W\left({\overrightarrow{r}}_{ba},H\right),$$

(5)

where m is the particle mass, and W is the kernel function that represents the spatial distribution of the physical quantity carried by each particle. The kernel value is determined by the distance $|{\overrightarrow{r}}_{ba}|$ between points a and b and the particle diameter H. In the following, the notation $W_{ba}=W({\overrightarrow{r}}_{ba},H)$ is used. In this study, the M4 spline function for a two-dimensional space was used as the kernel function. This kernel is a standard and widely used in SPH simulations [30,31].

Furthermore, the most fundamental model for the gradient of a physical quantity is obtained by differentiating the kernel function in Equation (5), and is expressed as follows [30]:

$$\nabla \varphi \left(\overrightarrow{r}={\overrightarrow{r}}_a\right)=\sum _{b\ne a}{m}_b{\displaystyle \frac{{\varphi }_b}{{\rho }_b}}\nabla W_{ba},$$

(6)

where $\nabla W_{ba}$ denotes the gradient of the kernel function, which is given by

$$\nabla W_{ba}=-{\displaystyle \frac{{\overrightarrow{r}}_{ba}}{\left|{\overrightarrow{r}}_{ba}\right|}}{\displaystyle \frac{d{W}_{ba}}{d\left|{\overrightarrow{r}}_{ba}\right|}}.$$

(7)

In addition, the Laplacian of a physical quantity is modeled using the gradient of the kernel function, $\nabla W_{ba}$, as follows [32]:

$${\nabla }^2\varphi \left(\overrightarrow{r}={\overrightarrow{r}}_a\right)=2\sum _{b\ne a}{\displaystyle \frac{m_b}{{\rho }_b}}{\displaystyle \frac{{\varphi }_b-{\varphi }_a}{|{\overrightarrow{r}}_{ba}|}}\left({\displaystyle \frac{{\overrightarrow{r}}_{ba}}{\left|{\overrightarrow{r}}_{ba}\right|}}·\nabla W_{ba}\right)=-2\sum _{b\ne a}{\displaystyle \frac{m_b}{{\rho }_b}}{\displaystyle \frac{{\varphi }_b-{\varphi }_a}{|{\overrightarrow{r}}_{ba}|}}{\displaystyle \frac{d{W}_{ba}}{d|{\overrightarrow{r}}_{ba}|}}.$$

(8)

2.2.2. Discretization of Navier–Stokes Equation

In this study, particle mass and volume and the total number of particles are kept constant. At each computational step, particle positions are corrected by the algorithm of incompressibility [33] to maintain a spatially uniform particle number density. This approach keeps the local density nearly constant in space and time. As a result, the continuity equation does not need to be explicitly solved. Details of the implementation of the algorithm of incompressibility are given in previous studies [25,33]. Therefore, only the Navier–Stokes equation and the enthalpy transport equation were solved.

By applying the SPH discretization to Equation (2), the following form is obtained:

$${\displaystyle \frac{D{\overrightarrow{u}}_a}{Dt}}=-\sum _{b\ne a}{m}_b\left({\displaystyle \frac{p_a}{{\rho }_a^2}}+{\displaystyle \frac{p_b}{{\rho }_b^2}}\right)\nabla W_{ba}+{\displaystyle \frac{2}{{\rho }_a}}\sum _{b\ne a}{\displaystyle \frac{2{\mu }_a{\mu }_b}{{\mu }_a+{\mu }_b}}{\displaystyle \frac{m_b}{{\rho }_b}}{\displaystyle \frac{{\overrightarrow{u}}_b-{\overrightarrow{u}}_a}{|{\overrightarrow{r}}_{ba}|}}\left({\displaystyle \frac{{\overrightarrow{r}}_{ba}}{\left|{\overrightarrow{r}}_{ba}\right|}}·\nabla W_{ba}\right)+{\overrightarrow{g}}_a.$$

(9)

The first term on the right-hand side of Equation (9) represents the pressure gradient, which is discretized in SPH to satisfy the action–reaction law for each interacting particle pair. This term was replaced by the incompressibility algorithm proposed by Shigeta et al. [33], in order to represent incompressible fluid behavior.

The second term on the right-hand side is the viscosity term. In particle methods, the Laplacian operator appearing in the viscous term can be discretized using SPH and Moving Particle Semi-implicit (MPS) formulations. In this simulation, the viscosity is varied in both space and time. In such cases, the Navier–Stokes equation must be discretized without assuming a constant viscosity over the computational domain. However, this leads to more complicated discretization. Therefore, the SPH-type Laplacian formulation proposed by Morris et al. [31] was adopted, following previous studies that successfully simulated flows with spatio-temporal variations in viscosity [24,31].

In this scheme, the viscosity is initially assumed to be constant, and the term is discretized using the SPH Laplacian model presented in Equation (8),

$${\displaystyle \frac{{\mu }_a}{{\rho }_a}}{\nabla }^2{\overrightarrow{u}}_a={\displaystyle \frac{2\mu }{{\rho }_a}}\sum _{b\ne a}{\displaystyle \frac{m_b}{{\rho }_b}}{\displaystyle \frac{{\overrightarrow{u}}_b-{\overrightarrow{u}}_a}{|{\overrightarrow{r}}_{ba}|}}\left({\displaystyle \frac{{\overrightarrow{r}}_{ba}}{\left|{\overrightarrow{r}}_{ba}\right|}}·\nabla W_{ba}\right).$$

(10)

Subsequently, the viscosity $\mu$ in Equation (10) is redefined as the mean viscosity of the two interacting particles.

$${\displaystyle \frac{1}{{\rho }_a}}\nabla ·\left\{{\mu }_a\left[(\nabla \otimes {\overrightarrow{u}}_a)+{(\nabla \otimes {\overrightarrow{u}}_a)}^{\mathrm{T}}\right]\right\}={\displaystyle \frac{2}{{\rho }_a}}\sum _{b\ne a}{\displaystyle \frac{2{\mu }_a{\mu }_b}{{\mu }_a+{\mu }_b}}{\displaystyle \frac{m_b}{{\rho }_b}}{\displaystyle \frac{{\overrightarrow{u}}_b-{\overrightarrow{u}}_a}{|{\overrightarrow{r}}_{ba}|}}\left({\displaystyle \frac{{\overrightarrow{r}}_{ba}}{\left|{\overrightarrow{r}}_{ba}\right|}}·\nabla W_{ba}\right).$$

(11)

The mean value was evaluated using the harmonic mean, in order to treat the large spatial gradients of viscosity.

2.2.3. Discretization of Enthalpy Transport Equation

By applying the SPH discretization to Equation (4), the following form is obtained:

$${\displaystyle \frac{D{T}_a}{Dt}}={\displaystyle \frac{2\delta }{{\lambda }_a{n}_a{\rho }_a{c}_{p_a}}}\sum _b{\displaystyle \frac{{\kappa }_a+{\kappa }_b}{2}}\left(T_b-T_a\right)W_{ba}-{\displaystyle \frac{Q_a}{{\rho }_a{c}_{p_a}}},$$

(12)

where $\delta$ is the number of spatial dimensions, $\lambda$ is the weighted average of the squared values of distances to neighboring particles within the influence radius, and n denotes the particle number density. The first term on the right-hand side represents heat conduction, which is discretized using the Laplacian model of the MPS method [34]. In the SPH method, the Laplacian is represented as a summation of the gradients of kernel functions, which tends to underestimate its value near the surface. On the other hand, in the MPS formulation, the Laplacian is evaluated by taking a weighted average of the kernel functions as a weighting function, which is expected to reduce the underestimation near the surface. The second term on the right-hand side is the heat loss term. According to the previous study by Keszthelyi et al. [10], heat flux due to atmospheric convection and thermal radiation play a dominant role in the cooling of the lava surface. Therefore, likewise in this study, the heat flux to the ambient air $Q_a^{\mathrm{FLUX}}$ and the radiation on the surface $Q_a^{\mathrm{RAD}}$ are considered as the heat loss rate $Q_a$

$$Q_a=Q_a^{\mathrm{FLUX}}+Q_a^{\mathrm{RAD}}.$$

(13)

These effects were applied only to the surface particles. To identify the surface particles, the method proposed by Ito et al. [35] was used. Figure 1 shows a schematic diagram of this method. For a surface particle, neighboring particles are unevenly distributed within the influence radius. As a result, the number-density-based centers shift from the actual particle positions toward the interior of the fluid. The magnitude of this displacement is computed as

$$\Delta {\overrightarrow{r}}_a^{\mathrm{wc}}={\displaystyle \frac{{\displaystyle \sum _b{\overrightarrow{r}}_b{W}_{ba}}}{{\displaystyle \sum _b{W}_{ba}}}}-{\overrightarrow{r}}_a.$$

(14)

In this simulation, particles with $\left|\Delta {\overrightarrow{r}}_a^{\mathrm{wc}}\right|>0.3H$, were classified as surface particles in this simulation.

Figure 1. Schematic diagram of the surface-particle identification method proposed by Ito et al. [35]. Purple and red circles denote surface particles and bulk particles, respectively. The dashed circle indicates the influence radius of the particle of interest. Black dots indicate the actual particle positions, and green dots show the corresponding number-density-based centers. For surface particles, an uneven neighbor distribution causes larger displacement between these two positions. Particles whose displacement $\left|\Delta {\overrightarrow{r}}_a^{\mathrm{wc}}\right|>0.3H$ are classified as surface particles.

Figure 1. Schematic diagram of the surface-particle identification method proposed by Ito et al. [35]. Purple and red circles denote surface particles and bulk particles, respectively. The dashed circle indicates the influence radius of the particle of interest. Black dots indicate the actual particle positions, and green dots show the corresponding number-density-based centers. For surface particles, an uneven neighbor distribution causes larger displacement between these two positions. Particles whose displacement $\left|\Delta {\overrightarrow{r}}_a^{\mathrm{wc}}\right|>0.3H$ are classified as surface particles.

The heat flux to the ambient air is formulated as follows:

$$Q_a^{\mathrm{FLUX}}={\displaystyle \frac{S_a}{V_a}}{\displaystyle \frac{{\kappa }_a{\kappa }_{\mathrm{air}}\left(T_a-T_{\mathrm{air}}\right)}{{\kappa }_{a}d{z}^{\prime }+{\kappa }_{\mathrm{air}}dz}},$$

(15)

where S is the cross-sectional length of a particle, V is the area of the computational particle in the 2D simulation, $d{z}^{\prime }$ denotes the distance from the surface to air phase, and $dz$ is the radius of a computational particle. The subscript “$\mathrm{air}$” represents a physical quantity associated with the ambient air. Figure 2 illustrates the concept of this formulation. This equation is derived by assuming that the heat flux from the particle center (point C in Figure 2) to the particle surface (point B) is equal to the heat flux from the particle surface (point B) to the air phase (point A), and by formulating these conditions as simultaneous equations for the particle surface temperature.

The heat loss due to radiation is expressed by the Stefan–Boltzmann law as follows:

$$Q_a^{\mathrm{RAD}}={\displaystyle \frac{S_a}{V_a}}\epsilon \sigma \left(T_a^4-T_{\mathrm{air}}^4\right),$$

(16)

where $\epsilon$ is the emissivity and $\sigma$ is the Stefan–Boltzmann constant.

In this simulation, solidification was modeled by increasing the viscosity depending on the temperature, rather than by explicitly treating a solid–liquid interface. As a result, no additional boundary treatment associated with phase change is required. This viscosity-based treatment captures the plastic behavior of the cooling-induced amorphous glassy surface skin. The formulation and material parameters of the temperature-dependent viscosity are described in detail in Section Modeling of Lava Properties.

2.3. Computational Conditions

Figure 3 shows the computational domain. In this figure, brown and red areas represent the locations of solid and liquid particles, respectively. A rectangular container with outer dimensions of $8.0\times {10}^{-1}\mathrm{m}\left(x\right)\times 3.2\times {10}^{-1}\mathrm{m}\left(y\right)$ and a thickness of $4.0\times {10}^{-3}\mathrm{m}$, was prepared. The origin of the coordinate system was defined at a position $4.0\times {10}^{-3}\mathrm{m}$ in the x-direction and $4.0\times {10}^{-3}\mathrm{m}$ in the y-direction from the lower-left corner of the container. In addition, a horizontal plate was placed in the region $0.8\mathrm{m}\le x\le 4.0\mathrm{m}$ and $-4.0\times {10}^{-3}\mathrm{m}\le y\le 0.0\mathrm{m}$. Then, 129,600 liquid particles were arranged in a regular lattice within the region $0.0\mathrm{m}\le x\le 5.40\times {10}^{-1}\mathrm{m}$ and $0.0\mathrm{m}\le y\le 2.40\times {10}^{-1}\mathrm{m}$. Figure 3a shows a schematic of the particle configuration at this step. As shown here, the liquid particles are initially accumulated in the lower-left region of the container, which form a dam. Subsequently, a dam-break simulation in which the dam collapsed under gravity acting in the $-y$-direction was conducted as a preliminary computation. When the x-directional velocities of all fluid particles converged to $|u_x|<1.0\times {10}^{-3}\mathrm{m}/\mathrm{s}$, this state was regarded as the initial condition of main simulation. Figure 3b illustrates a schematic of the resulting particle configuration. Finally, in the main computation, the right wall of the container was removed, and the fluid particles flowed onto a horizontal plate, as shown in Figure 3c. Inside the container, the fluid was treated as an isothermal fluid at a temperature of $T=1500\mathrm{K}$. Heat transport was considered only for particles located on the horizontal plate ($x>0.8\mathrm{m}$).

In actual lava flows, the thickness of the surface skin is typically $1.0\times {10}^{-3}$–$2.0\times {10}^{-3}\mathrm{m}$ [3]. To quantitatively resolve the detailed formation mechanism of such a thin solidified layer, a sufficiently small particle diameter would be required. This is because the particle diameter determines the spatial resolution of surface heat loss, as the heat loss is averaged over the surface particle area (volume in three-dimensional simulations). However, adopting such a particle diameter would lead to a large number of computational particles and a significantly increased computational cost.

This study focuses on the macroscopic influence of the presence of a surface skin formed by surface cooling on the overall flow behavior. The detailed formation mechanism of the skin is beyond the scope of the present study. Based on this modeling strategy, the particle diameter was set to $1.0\times {10}^{-3}\mathrm{m}$ in this simulation.

Due to this setting, the spatial resolution is relatively coarse. As mentioned above, because heat fluxes due to atmospheric convection and radiative cooling are averaged over the particle area at the free surface, the resulting temperature decrease of surface particles cannot be resolved accurately. Accordingly, in order to enhance the temperature response of surface particles and to reproduce skin formation within a feasible computational time, the effects of atmospheric convection and radiation at the surface particles were artificially amplified by a factor of ten. The influence of the artificially amplified surface cooling was examined by a sensitivity analysis (see Appendix A).

Furthermore, sufficient temporal resolution was ensured under a feasible computational cost. A time step of $\Delta t=1.0\times {10}^{-4}\mathrm{s}$ was employed throughout the simulation.

In addition, the ambient air temperature was kept constant at $T_{\mathrm{air}}=300\mathrm{K}$, and its thermal conductivity was set to ${\kappa }_{\mathrm{air}}=2.63\times {10}^{-2}\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$ [36]. The distance from the lava surface to the air phase was set equal to the particle radius ($d{z}^{\prime }=0.5\times {10}^{-3}\mathrm{m}$).

Table 1. Computational parameters used in the simulations.

Parameter	Value	Unit
Dimension number $\delta$	2	-
Diameter of computational particle H	$1.0\times {10}^{-3}$	$\mathrm{m}$
Time step $\Delta t$	$1.0\times {10}^{-4}$	$\mathrm{s}$
Gravitational acceleration g	$9.8$	$\mathrm{m}/{\mathrm{s}}^2$
Thermal conductivity of soil ${\kappa }_{\mathrm{soil}}$	$0.52335$	$\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$
Specific heat of soil $c_{\mathrm{soil}}$	$1842.19$	$\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$
Density of soil ${\rho }_{\mathrm{soil}}$	2000	$\mathrm{kg}/{\mathrm{m}}^3$
Temperature of air $T_{\mathrm{air}}$	300	$\mathrm{K}$
Thermal conductivity of air ${\kappa }_{\mathrm{air}}$	$2.63\times {10}^{-2}$	$\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$
Distance from surface particles to air $d{z}^{\prime }$	$0.5\times {10}^{-3}$	$\mathrm{m}$

summarizes the computational conditions. The solid phase was assumed to be soil, and its physical properties were taken from the literature [37].

The bottom of the solid phase was treated as an adiabatic boundary. The validity of this assumption can be assessed using a thermal diffusion-length estimate. The characteristic thermal penetration depth ${\ell }_T$ is given by

$${\ell }_T\sim \sqrt{{\displaystyle \frac{\kappa }{\rho c_p}}t}.$$

(17)

Using the soil properties listed in Table 1 and the present simulation time scale ($\sim 10\mathrm{s}$), the penetration depth is estimated to be ${\ell }_T\approx 1.2\times {10}^{-3}\mathrm{m}$, which is smaller than the soil phase thickness ($4.0\times {10}^{-3}\mathrm{m}$). Therefore, the influence of the bottom boundary condition is expected to be limited in the present simulations. Based on this consideration, an adiabatic boundary condition was adopted as the simplest choice for the bottom boundary of the soil phase.

Modeling of Lava Properties

Because lava effusion is commonly observed for low-viscosity basaltic magma, the material properties of liquid particles in this simulation were based on data for basaltic lava. Comprehensive surveys of physical properties conducted within a single volcano are limited. Therefore, data reported in previous studies from various volcanoes were collected.

For the lava density, a constant value of $\rho =1370\mathrm{kg}/{\mathrm{m}}^3$ was adopted. It is based on measurements of flowing basaltic lava sampled at Piton de la Fournaise [9].

For the viscosity of lava, experimentally measured values obtained by remelting solidified basaltic lava samples from Mauna Ulu, Kīlauea volcano, were used [7]. In this study [7], the measured viscosity values were fitted using the Vogel–Fulcher–Tammann (VFT) equation and expressed as a function of temperature as follows:

$${log}_{10}{\mu }_{\mathrm{basalt}}\left(T\right)=-5.08+{\displaystyle \frac{6140.5}{T-558.8}},$$

(18)

where ${\mu }_{\mathrm{basalt}}\left(T\right)$ and T are in $\mathrm{Pa}\mathrm{s}$ and $\mathrm{K}$ units, respectively. Figure 4 shows the temperature dependence of the viscosity. The blue curve in this figure represents the viscosity calculated based on Equation (18).

Since the rheological behavior of lava flows is consistent with that of a Bingham fluid. As described above, a model used in the simulation should possess the expression of yield stress. Accordingly, an apparent viscosity model [38] was introduced, in which the apparent viscosity ${\mu }_{\mathrm{app}}$ is defined as a function of the second invariant of the strain rate tensor ${ϵ}_{\mathrm{I}\mathrm{I}}$ given by

$${\displaystyle {ϵ}_{\mathrm{I}\mathrm{I}}=\frac{\partial u_x}{\partial x}\frac{\partial u_y}{\partial y}-{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}}\right)}^2,}$$

(19)

where $\overrightarrow{u}=(u_x,u_y)$. The apparent viscosity ${\mu }_{\mathrm{app}}$ is expressed as a function of ${ϵ}_{\mathrm{I}\mathrm{I}}$ as follows:

$${\mu }_{\mathrm{app}}={\mu }_0+{\tau }_0\left[{\displaystyle \frac{1-\exp \left(-\xi \sqrt{|{ϵ}_{\mathrm{I}\mathrm{I}}|}\right)}{\sqrt{|{ϵ}_{\mathrm{I}\mathrm{I}}|}}}\right],$$

(20)

where ${\mu }_0$ is the reference viscosity, ${\tau }_0$ is the yield stress, and $\xi$ is a fitting parameter. This model can express not only the non-Newtonian behavior, but also temperature dependence of viscosity by treating the reference viscosity as a function of temperature.

However, in SPH simulations of viscous fluids, the following stability condition associated with viscosity must be satisfied [32]:

$$C{\displaystyle \frac{4H^2\rho }{\mu }}\ge \Delta t,$$

(21)

where C denotes a safety factor, and a value of $C=0.25$ was adopted in this simulation. Under the particle diameter and time step conditions listed in Table 1, the range of viscosity is estimated as $\mu \le 13.7\mathrm{Pa}\mathrm{s}$ from Equation (21).

To satisfy this constraint, in this simulation, the apparent viscosity given by Equation (20) was scaled by a factor of $1/100$ as follows:

$${\mu }_{\mathrm{app}}^{\prime }={\displaystyle \frac{{\mu }_0\left(T\right)}{100}}+{\displaystyle \frac{{\tau }_0}{100}}\left[{\displaystyle \frac{1-\exp \left(-\xi \sqrt{|{ϵ}_{\mathrm{I}\mathrm{I}}|}\right)}{\sqrt{|{ϵ}_{\mathrm{I}\mathrm{I}}|}}}\right],$$

(22)

where ${\mu }_{\mathrm{app}}^{\prime }={\mu }_{\mathrm{app}}/100$. This scaling is introduced to preserve the relative temperature dependence of viscosity while satisfying the numerical stability condition of the SPH scheme. However, even after this scaling, the apparent viscosity can still exceed the stability threshold by temperature decrease. Therefore, an upper limit of ${\mu }_{\mathrm{app}}^{\prime }\le 13.7\mathrm{Pa}\mathrm{s}$ was imposed. In addition, large spatial gradients of viscosity between neighboring particles can lead to numerical instability. Therefore, a lower limit was also introduced, and the viscosity was restricted to the range $0.685\mathrm{Pa}\mathrm{s}\le {\mu }_{\mathrm{app}}^{\prime }\le 13.7\mathrm{Pa}\mathrm{s}$.

The reference viscosity ${\mu }_0\left(T\right)$ was represented by the temperature-dependent viscosity of basaltic lava ${\mu }_{\mathrm{basalt}}\left(T\right)$. In Figure 4, the purple curve shows the scaled viscosity, ${\mu }_{\mathrm{basalt}}\left(T\right)/100$. The scaled reference viscosity ${\mu }_0^{\prime }\left(T\right)={\mu }_0\left(T\right)/100$ was also restricted by the same upper and lower limits, $0.685\mathrm{Pa}\mathrm{s}\le {\mu }_0^{\prime }\left(T\right)\le 13.7\mathrm{Pa}\mathrm{s}$. In Figure 4, the red curve denotes the temperature-dependent reference viscosity ${\mu }_0^{\prime }\left(T\right)$ applied in the simulations with prescribed upper and lower limits. In this simulation, regions where the temperature-dependent viscosity ${\mu }_0^{\prime }\left(T\right)=13.7\mathrm{Pa}\mathrm{s}$ were treated as the surface skin. The corresponding temperature for this condition is $T\le 1306\mathrm{K}$.

In addition, the yield stress was set to a constant value of ${\tau }_0=1000\mathrm{Pa}$ as a representative value within the range reported for basaltic lava flows [39].

For the thermal conductivity and specific heat, temperature-dependent properties of basaltic lava were adopted based on previous studies [10,11].

$$\kappa =0.848+1.1\times {10}^{-3}T,$$

(23)

$$c_p=\left\{\begin{array}{cc}1100& \left(1010<T\right)\hfill \\ 1211-{\displaystyle \frac{1.12\times {10}^5}{T}}& \left(1010\ge T\right),\hfill \end{array}\right.$$

(24)

where $\kappa$, T, and $c_p$ are in $\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$, $\mathrm{K}$, and $\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$ units, respectively.

Based on the formulation of the heat flux to the ambient air described in Equation (15), the following equation corresponds to the heat transfer coefficient $\alpha$:

$$\alpha ={\displaystyle \frac{{\kappa }_{\mathrm{particle}}{\kappa }_{\mathrm{air}}}{{\kappa }_{\mathrm{particle}}d{z}^{\prime }+{\kappa }_{\mathrm{air}}dz}}.$$

(25)

By substituting the computational conditions and the physical properties of lava and air described above into this equation, the heat transfer coefficient is evaluated as $\alpha \approx 50\mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right)$. This value is within the range of $70\pm 25\mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right)$ experimentally obtained for forced convection in a previous study [10].

The emissivity of lava is generally considered to be $\epsilon =0.9$ [2], and this value was also used in the present study.

Table 2. Physical properties of basaltic lava.

Property	Value	Unit
Density $\rho$	1370	$\mathrm{kg}/{\mathrm{m}}^3$
Scaled apparent viscosity ${\mu }_{\mathrm{app}}^{\prime }$	$0.685$–$13.7$	$\mathrm{Pa}\mathrm{s}$
Fitting parameter for viscosity model $\xi$	$1.5$	-
Yield stress ${\tau }_0$	1000	$\mathrm{Pa}$
Thermal conductivity $\kappa$	$1.178$–$2.498$	$\mathrm{W}/\left(\mathrm{m}\mathrm{K}\right)$
Specific heat at constant pressure $c_p$	$0.837$–1100	$\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$
Emissivity $\epsilon$	$0.9$	-

summarizes the physical properties of basaltic lava used in this study.

In this simulation, four different rheological conditions were considered to evaluate the effects of non-Newtonian behavior and formation of the surface skin on lava flow dynamics. Specifically, four rheological conditions were examined, combining Newtonian or Bingham models with constant or temperature-dependent viscosity (N, TN, B, and TB).

• N: Newtonian fluid with a constant viscosity of ${\mu }_a=0.685\mathrm{Pa}\mathrm{s}$
• TN: Newtonian fluid with temperature-dependent viscosity (${\tau }_0=0\mathrm{Pa}$)
• B: Bingham fluid with a constant reference viscosity of ${\mu }_0^{\prime }=0.685\mathrm{Pa}\mathrm{s}$
• TB: Bingham fluid with temperature-dependent viscosity

In these thermal conditions (TN and TB), surface cooling leads to the formation of a low-temperature, high-viscosity surface layer (surface skin).

3. Results and Discussion

3.1. Validation and Physical Interpretation of Numerical Simulations

3.1.1. Validation for Newtonian Fluid via Similarity Solution

In this study, numerical simulations were considered for a Newtonian fluid with constant viscosity (condition N) that is instantaneously released onto a horizontal plate. The simulation results are compared with known theoretical scaling laws formulated for viscous-dominated flows [26], as follows:

$$L\left(t\right)=\beta {\left({\displaystyle \frac{1}{3}}{\displaystyle \frac{g\Delta \rho q^3}{\mu }}\right)}^{1/5}{t}^{1/5},$$

(26)

where L is the flow length, g is the gravitational acceleration, q represents the cross-sectional area of the released fluid. The constant $\beta$ is a dimensionless coefficient determined by the similarity solution [26]. In the case of a two-dimensional flow, $\beta =1.41$, $\Delta \rho$ denotes the density difference between the fluid and the ambient medium. In the present simulations, the ambient medium is not modeled. Therefore, $\Delta \rho$ is taken as $\Delta \rho ={\rho }_{\mathrm{fluid}}-{\rho }_{\mathrm{air}}\approx {\rho }_{\mathrm{fluid}}$. This assumption is reasonable for lava flows, since ${\rho }_{\mathrm{lava}}\gg {\rho }_{\mathrm{air}}$.

Figure 5 compares the simulated flow length of condition N with the theoretical solution for viscous-dominated flow proposed by Huppert [26]. To compare the simulation results with the theoretical scaling, it is necessary to identify the time range in which the flow is dominated by viscous effects. In actual physical processes, the early stage flow behavior is influenced by inertial slumping, which precedes the viscous-dominated flow [13]. Therefore, this inertial slumping stage should be excluded from the quantitative comparison.

According to the scaling relation given by Equation (26), the flow length follows $L\propto t^{1/5}$ in the viscous-dominated flow. When the flow follows the scaling, the quantity $L/t^{1/5}$ becomes constant. Therefore, the compensated quantity $\tilde{L}=L/t^{1/5}$ is introduced and evaluated to diagnose the transition to the viscous-dominated flow. Figure 6 shows the time evolution of the relative change of the compensated quantity, described as $\left|{\tilde{L}}_{n+1}-{\tilde{L}}_n\right|/{\tilde{L}}_n$. As time progresses, the magnitude of the relative change decreases. After approximately $t=5.0\mathrm{s}$, the relative change remains at a low level of order ${10}^{-4}$. Based on Figure 6, $t\ge 5.0\mathrm{s}$ is adopted as the viscous-dominated regime.

For the data within the viscous-dominated regime, the relative root-mean-square error (rRMSE) between the simulation result of condition N and the theoretical solution proposed by Huppert [26] is evaluated. The resulting rRMSE is $5.36\times {10}^{-2}$. This result indicates that the simulation results agree with the theoretical scaling within a relative error of approximately $5\%$.

In the following analysis, the time exponent is fixed to $1/5$, and only the constant $\tilde{L}=\beta {\left[g\Delta \rho q^3/\left(3\mu \right)\right]}^{1/5}$ is treated as a fitting parameter. The value of $\tilde{L}$ is determined by minimizing the rRMSE between the simulation result and the theoretical scaling. As a result, the optimal value is found to be $\tilde{L}=2.53$, while the theoretical value is $\tilde{L}=2.39$. With this optimized prefactor, the rRMSE is reduced to $1.25\times {10}^{-2}$. This result indicates that the temporal scaling predicted by the theory is well reproduced by the simulation.

These results indicate that the numerical simulations show reasonable agreement with the known theoretical scaling laws. This agreement supports the validity of the simulations for the Newtonian condition.

3.1.2. Validation for Bingham Fluid via Theoretical Stopping Length

For Bingham fluids, validation based on time evolution is not feasible because no analytical solution exists. Therefore, this study validates the Bingham model by comparison with the theoretical stopping length proposed by Lyman et al. [13]. This theoretical model is derived from the static balance between the yield stress and the gravitational driving force. The theoretical stopping length $L_y$ is given by

$$L_y={\left({\displaystyle \frac{9g\Delta \rho q^2}{8{\tau }_0}}\right)}^{1/3}.$$

(27)

In this study, the numerical simulation results of condition B are compared with this theoretical prediction, with a focus on the final flow length at stopping.

Figure 7 shows the temporal evolution of the flow front position obtained from the numerical simulation for condition B. The final stopping position predicted by the theoretical model of Lyman [13] is also shown for comparison.

As shown here, the stopping length measured from the origin in the simulation is approximately $L\approx 2.6\mathrm{m}$. In contrast, the theoretical model predicts a stopping length of $L=2.94\mathrm{m}$. The stopping length obtained from the numerical simulation corresponds to approximately $88\%$ of the theoretical prediction. Lyman [13] reported that, for Bingham fluids, the experimentally observed stopping lengths range from $79\%$ to $93\%$ of the theoretical values. In Figure 7, the thin dashed lines indicate this experimentally observed variability range. The result of simulation is within the variability range reported in previous experimental validations.

Based on the comparison described above, these results indicate that the numerical simulations show reasonable agreement with the theoretical model for Bingham fluids. This agreement supports the validity of the simulations in capturing the stopping behavior both qualitatively and quantitatively.

For temperature-dependent conditions, no analytical or experimental benchmark that includes both thermal evolution and surface skin formation is available. Accordingly, these conditions are evaluated as extensions of the validated isothermal Newtonian and Bingham models.

3.1.3. Evaluation Based on Mechanical Energy Framework

Based on the framework proposed by Ge et al. [40] for rock avalanches, the evolution of the mechanical energy budget is analyzed in this study. Although the material system is different, both phenomena are gravity-driven mass movements. Therefore, the same framework can be applied to interpret the evolution of the mechanical energy budget of the present flow. Following this framework, the potential energy $PE$, kinetic energy $KE$, mechanical energy $ME$, and dissipated energy $DE$ are defined as

$$PE\left(x\right)=\sum _a{m}_{a}g{y}_a,$$

(28)

$$KE\left(x\right)=\sum _a{\displaystyle \frac{1}{2}}{m}_a{\left|{\overrightarrow{v}}_a\right|}^2,$$

(29)

$$ME\left(x\right)=PE\left(x\right)+KE\left(x\right),$$

(30)

$$DE\left(x\right)=ME\left(x_0\right)-ME\left(x\right).$$

(31)

• Here, x denotes the position of the flow front, and $ME\left(x_0\right)$ is the reference mechanical energy. The energies are evaluated by summing the contributions of all lava particles at the time when the flow front reaches the position x. It should be noted that $DE$ is defined here as the loss of mechanical energy from the reference state. It includes not only physical dissipation but also other energy losses associated with numerical diffusion. In this study, $DE$ is used as an effective measure of energy loss to analyze the relative contribution of dissipation among different conditions.

Figure 8 shows the evolution of the mechanical energy budget as a function of the flow-front position for the four simulation conditions. In all conditions, a short initial transient is observed immediately after the onset of motion. This initial transient is considered to originate from numerical effects of the SPH method. It disappears when the flow front reaches approximately $x=1.2\mathrm{m}$. Therefore, the region $x<1.2\mathrm{m}$ is defined as the initial transient region. In addition, the mechanical energy $ME$ at $x=1.2\mathrm{m}$ is used as the reference mechanical energy $ME\left(x_0\right)$ for evaluating the dissipated energy $DE$.

In addition, Figure 9 shows the evolution of the mechanical energy $ME$ as a function of the flow-front position in the initial transient region. No significant difference is observed between the two Newtonian conditions (N and TN), nor between the two Bingham conditions (B and TB). Furthermore, at the end of the initial transient region ($x=1.2\mathrm{m}$), the difference among the four conditions is negligible. Therefore, the initial transient does not influence the comparative discussion among different conditions presented below.

As shown in Figure 8, the energy conversion process after the initial transient ($x>1.2\mathrm{m}$) can be described in the same way for all conditions. In the early stage, potential energy $PE$ decreases while kinetic energy $KE$ increases. This is the conversion from potential energy to kinetic energy. At the same time, the dissipated energy $DE$ also increases continuously. As the flow develops, the kinetic energy $KE$ decreases and becomes very small eventually. This corresponds to a state in which the forward propagation is significantly slowed, and the energy budget is dominated by $DE$. A similar qualitative trend in energy conversion was reported in a previous study [40] for gravity-driven flows.

3.2. Effects of Bingham Behavior on Flow Deceleration and Distance

Figure 10 shows the time evolution of the flow front position for each condition. The blue, green, orange, and red lines correspond to the conditions N, TN, B, and TB, respectively.

As shown in Figure 10, a clear difference is observed in the approach to the stopped state between Newtonian fluids (N and TN) and Bingham fluids (B and TB). For Newtonian fluids, the flow velocity decreases gradually. In contrast, for Bingham fluids, the flow undergoes a rapid decrease in velocity before reaching the stopped state.

Figure 11 shows the velocity distributions in the x-direction for each condition when the flow front reaches $x=1.5\mathrm{m}$. It should be noted that the aspect ratio is changed for visibility in the figures. A difference is observed in the spatial extent of the region with velocities higher than $1.8\mathrm{m}/\mathrm{s}$ between Newtonian fluids (Figure 11a,b) and Bingham fluids (Figure 11c,d). On the other hand, no significant difference is found in this high-velocity region between the two Newtonian fluids (N and TN). A similar result is obtained between the two Bingham fluids (B and TB).

Figure 12 shows the velocity distributions in the x-direction for each condition when the flow front reaches $x=2.0\mathrm{m}$. The Bingham fluids (Figure 12c,d) exhibit overall lower velocities than the Newtonian fluids (Figure 12a,b). The region with velocities higher than $1.0\mathrm{m}/\mathrm{s}$ disappears in the Bingham fluids.

Figure 13 shows the velocity distributions in the x-direction for each condition when the flow front reaches $x=3.0\mathrm{m}$ for Newtonian fluids (N and TN) and $x=2.4\mathrm{m}$ for Bingham fluids (B and TB). In addition, Figure 14 shows the velocity distributions at $t=10.83\mathrm{s}$ when the flow under condition N reaches the domain boundary. A comparison between Figure 13d and Figure 14d for condition TB shows that the flow front exhibits almost no further displacement. This observation allows the Bingham flows under condition TB to be regarded as stopped at this stage. In particular, as shown in Figure 13d, the flow under condition TB has already lost most of its velocity at $x=2.4\mathrm{m}$.

Figure 15 shows the apparent viscosity distributions for Bingham fluids (B and TB) when the flow reaches at $x=2.0\mathrm{m}$ and $x=2.4\mathrm{m}$. Here, the apparent viscosity distribution under condition TB at $x=2.4\mathrm{m}$ (Figure 15d) is compared with that at $x=2.0\mathrm{m}$ (Figure 15c), where the flow is still advancing. This comparison shows that the apparent viscosity increases over the entire flow region during this region. A similar comparison is conducted for condition B. The apparent viscosity distribution at $x=2.4\mathrm{m}$ (Figure 15b) is compared with that at $x=2.0\mathrm{m}$ (Figure 15a). The increase in the apparent viscosity is found to start from the upstream region.

The apparent viscosity distribution for condition B (Figure 15a,b) are compared with the velocity distributions at the same times (Figure 12c and Figure 13c). The upstream region exhibits lower velocities than the downstream region. This observation indicates that the apparent viscosity increases in regions where the flow velocity decreases.

Therefore, a feedback process within the Bingham fluid model is suggested. A decrease in flow velocity reduces the velocity gradient. This reduction leads to an increase in the apparent viscosity. The increased apparent viscosity further decreases the flow velocity. The repetition of this process can cause a rapid decrease in velocity and eventually lead to a stopping of the flow.

Through this feedback process, the simulation captures behavior that is consistent with the characteristics of a Bingham fluid. In actual Bingham fluids, a decrease in the velocity gradient causes the shear stress to fall below the yield stress. As a result, the fluid exhibits solid-like behavior.

To further examine the influence of Bingham behavior from a viewpoint of mechanical energy, the evolution of the mechanical energy budget is analyzed. As shown in Figure 8, the overall trend of the mechanical energy budget is qualitatively similar among the different conditions. Figure 16 presents the evolution of the normalized dissipated energy $DE/ME\left(x_0\right)$ as a function of the flow-front position. A clear difference is observed between the Newtonian (N and TN) and Bingham fluids (B and TB). In particular, the Bingham fluids exhibit a larger increase in dissipation over a shorter flowing distance. This enhanced dissipation can be interpreted as energy loss associated with the yield stress.

In this simulation, the Bingham fluid behavior was confirmed to have an influence on the flow deceleration and on the flowing distance. These results are consistent with the understanding [24] that non-Newtonian fluid models are required for accurate prediction of lava flow behavior.

3.3. Effects of Skin Formation on Flow Deceleration and Distance

As shown in Figure 10, the time evolution of the flow front position indicates that both Newtonian fluids (N and TN) exhibit similar gradual deceleration trends. However, at the same elapsed time, condition TN exhibits a shorter flowing distance than condition N. This difference in flowing distance is found to increase as time progresses.

Similarly, both Bingham fluids (B and TB) exhibit similar abrupt deceleration trends. However, at the same elapsed time, condition TB exhibits a shorter flowing distance than condition B. As a result, the final stopping distance under condition TB is smaller than that under condition B.

Figure 11, Figure 12, Figure 13 and Figure 14 show the velocity distributions in the x-direction for four conditions. First, the velocity distributions of the two Newtonian fluids (N and TN) are compared. When the flow front reaches $x=1.5\mathrm{m}$ (Figure 11a,b), the velocity distributions of both Newtonian fluids (N and TN) are similar. When the flow front reaches $x=2.0\mathrm{m}$ (Figure 12a,b), the spatial extent of the region with velocities between $1.4\mathrm{m}/\mathrm{s}$ and $1.6\mathrm{m}/\mathrm{s}$ is smaller under condition TN than under condition N. When the flow front reaches $x=3.0\mathrm{m}$ (Figure 13a,b), the difference in velocity between conditions N and TN becomes more pronounced. At $t=10.83\mathrm{s}$, the front position under condition TN (Figure 14b) is located further upstream than that for condition N (Figure 14a).

Similarly, the velocity distributions of the two Bingham fluids (B and TB) are compared. When the flow front reaches $x=1.5\mathrm{m}$ (Figure 11c,d), the velocity distributions of both Bingham fluids (B and TB) are similar, as in the Newtonian fluids. When the flow front reaches $x=2.0\mathrm{m}$ (Figure 12c,d), the velocity under condition TB is lower overall than that under condition B. When the flow front reaches $x=2.4\mathrm{m}$ (Figure 13d), the flow under condition TB has lost most of its velocity, and the front position shows almost no further change, even at $t=10.83\mathrm{s}$ (Figure 14d). On the other hand, at $x=2.4\mathrm{m}$ (Figure 13c), the flow under condition B continues to advance. At $t=10.83\mathrm{s}$, the front position under condition B (Figure 14c) is located further downstream than that under condition TB (Figure 14d).

Figure 17 shows the apparent viscosity distributions at the time when flows reach at $x=2.0\mathrm{m}$ for each condition. Under the temperature-dependent conditions TN (Figure 17b) and TB (Figure 17d), a high-viscosity region with ${\mu }_{\mathrm{app}}^{\prime }=13.7\mathrm{Pa}\mathrm{s}$ is observed at the flow surface.

Figure 18 shows the temperature distributions at the time when flows reach at $x=2.0\mathrm{m}$ for thermal conditions (TN and TB). For the condition TN, a comparison between the viscosity (Figure 17b) and the temperature (Figure 18a) indicates that this surface region satisfies ${\mu }_{\mathrm{app}}^{\prime }=13.7\mathrm{Pa}\mathrm{s}$ and $T\le 1306\mathrm{K}$. A similar relationship between the apparent viscosity (Figure 17d) and temperature (Figure 18b) is also observed in the condition TB. This result shows that the surface layer is represented as a low-temperature, high-viscosity skin.

On the other hand, the viscosity field of condition N (Figure 17a) and the that of condition TN (Figure 17b) show that the scaled viscosity ${\mu }_{\mathrm{app}}^{\prime }$ inside the flow is mostly lower than $0.7\mathrm{Pa}\mathrm{s}$. This observation indicates that cooling-induced skin formation occurs only at the flow surface.

Similarly, the apparent viscosity field of condition B (Figure 17c) and the that of condition TB (Figure 17d) show comparable viscosity distributions inside the flow. This result indicates that skin formation due to cooling is confined to the flow surface, even for the Bingham fluids.

On the other hand, Figure 15b,d show the apparent viscosity distributions when the flow fronts reach $x=2.4\mathrm{m}$. At this stage, Figure 15b,d indicate that the apparent viscosity inside the flow is overall higher under condition TB than under condition B. This result suggests that the velocity reduction process associated with Bingham behavior occurs at a shorter flowing distance under condition TB than under condition B.

To further examine the influence of surface skin from a viewpoint of mechanical energy, the evolution of the mechanical energy budget is analyzed. As shown in Figure 8, the overall trend of the mechanical energy budget is qualitatively similar among the different conditions. Figure 16 presents the evolution of the normalized dissipated energy $DE/ME\left(x_0\right)$ as a function of the flow-front position. The conditions with surface skin exhibit a larger increase in dissipation over a shorter flowing distance than the corresponding conditions without skin. This result indicates that the surface skin acts as an additional dissipative mechanism that promotes mechanical energy loss.

Based on the above comparisons of the velocity distributions, their temporal evolution, and the mechanical energy analysis, the effects of the surface skin are evaluated for both the Newtonian fluids (N and TN) and the Bingham fluids (B and TB).

These results indicate that the cooling-induced surface skin suppresses the flow and reduces the flow velocity. This interpretation is consistent with qualitative observations of lava flows. In such flows, a plastically deformable surface skin formed at the flow surface retains the internal lava near the flow front [3].

Under condition TN, the suppression of the flow by the surface skin causes the flow to exhibit longer arrival times at corresponding positions than under condition N. Similarly, under condition TB, the surface skin causes the Bingham related velocity reduction to occur at a shorter flowing distance than under condition B. As a result, this behavior leads to a difference in the final stopping position of the flow.

These results suggest that surface skin formation plays a role in the flow characteristics of lava flows, and that accurate simulation of lava flow behavior requires viscosity models that appropriately account for both non-Newtonian behavior and skin formation.

4. Conclusions

The objective of this study was to clarify the influence of the plastic surface skin on lava flow behavior. To this end, two-dimensional SPH simulations were conducted to analyze the flow of a lava-inspired virtual fluid on a horizontal plate. The surface skin was represented as a low-temperature, high-viscosity region through the introduction of a temperature-dependent viscosity. The flow behavior was evaluated by comparing conditions with and without non-Newtonian (Bingham) behavior and with and without surface skin formation.

The main results of this study are summarized as follows:

1.

The numerical results obtained for the isothermal Newtonian condition showed reasonable agreement with established theoretical solution [26] in terms of the time evolution of the flow. In addition, the results for the isothermal Bingham condition were consistent with theoretical prediction [13] regarding the final stopping position of the flow. These agreements support the validity of the present simulation framework for describing viscous-dominated and yield-stress-controlled flow behavior.

2.

By introducing the apparent viscosity model, it was confirmed that the apparent viscosity increases as the flow velocity decreases. This increase in apparent viscosity further reduces the flow velocity and eventually leads to the stopping of the lava flow. This behavior qualitatively reproduces the characteristics of a Bingham fluid. These results are consistent with the prevailing understanding that non-Newtonian fluid models are required for accurate prediction of lava flowing distances.

3.

By considering the temperature dependence of viscosity, a plastic surface skin was successfully reproduced in the numerical simulations. Comparison with simulations without surface skin formation revealed that the surface skin reduces the flow velocity. This behavior is consistent with qualitative observations [3]. In particular, when Bingham behavior is taken into account, the suppressive effect of the surface skin causes the successive deceleration characteristic in Bingham fluids, as described above, to emerge at a shorter flowing distance.

These results suggest the coupling effect of non-Newtonian behavior and surface skin formation governs lava flow deceleration and the final flowing distance. Accordingly, accurate prediction of lava flowing distances requires models that incorporate both effects. These present findings highlight the importance of surface skin formation in lava flow dynamics. Similar physical considerations are also relevant to engineering and other thermofluid systems involving surface solidification.

Although this study is based on a two-dimensional model, the suppressive effect associated with surface skin formation is considered to be qualitatively robust. This effect originates from cooling localized near the free surface and the resulting increase in near-surface viscosity, which are not inherently dependent on the flow dimensionality.

However, in actual three-dimensional lava flows, lateral skin formation is expected to occur. It remains to be clarified how three-dimensional effect influences the quantitative evaluation of lava flow velocity, flow width, and flowing distance.

For such quantitative evaluation at natural flow scales, it is necessary to incorporate realistic thermophysical properties of actual lava. However, in practice, the time step size in SPH simulations is constrained by a viscosity-related stability condition as expressed in Equation (21). When treating high-viscosity fluids such as lava, this stability condition severely restricts the allowable time step. To maintain a feasible computational cost, the particle diameter must be increased, which leads to a reduction in spatial resolution. This limitation makes it difficult to accurately resolve surface phenomena, such as the detailed structure and growth process of a thin surface skin.

To overcome this limitation, it is necessary to introduce a semi-implicit treatment [22,41] of the viscous term to relax the severe time step restriction in future work. Such developments are expected to contribute to more accurate quantitative prediction of lava flow behavior.