A generalized quasi two-phase bulk mixture model for mass flow

Accepted Manuscript A generalized quasi two-phase bulk mixture model for mass flow Puskar R. Pokhrel, Khim B. Khattri, Bhadra Man Tuladhar, Shiva P. Pudasaini

PII: DOI: Reference:

S0020-7462(17)30635-2 https://doi.org/10.1016/j.ijnonlinmec.2017.12.003 NLM 2946

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International Journal of Non-Linear Mechanics

Received date : 15 September 2017 Revised date : 18 September 2017 Accepted date : 4 December 2017 Please cite this article as: P.R. Pokhrel, K.B. Khattri, B.M. Tuladhar, S.P. Pudasaini, A generalized quasi two-phase bulk mixture model for mass flow, International Journal of Non-Linear Mechanics (2017), https://doi.org/10.1016/j.ijnonlinmec.2017.12.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A Generalized Quasi Two-phase Bulk Mixture Model for Mass Flow

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Puskar R. Pokhrel1,2 , Khim B. Khattri1 , Bhadra Man Tuladhar1 , Shiva P. Pudasaini3 1

3 2

4 3

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School of Science, Kathmandu University, Dhulikhel, Kavre, Nepal

Department of Mathematics, R. R. Campus, Tribhuvan University, Kathmandu, Nepal

Department of Geophysics, Steinmann Institute, University of Bonn, Meckenheimer Allee 176, D-53115 Bonn, Germany

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Correspondence to: Puskar R. Pokhrel, Email: [email protected]

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Abstract: We employ full dimensional two-phase mass flow equations (Pudasaini, 2012) to develop a gen-

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eralized quasi two-phase bulk model for a rapid flow of a debris mixture consisting of viscous fluid and solid

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particles down a channel. The emerging model, as a set of coupled partial differential equations, is charac-

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terized by some new mechanical and dynamical aspects of generalized bulk and shear viscosities, pressure,

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velocities and effective friction for the mixture where all these are evolving as functions of several dynamical

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variables, physical parameters, inertial and dynamical coefficients and drift factors. These coefficients and

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factors are uniquely constructed, contain the underlying physics of the system, and reveal strong coupling

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between the phases. The new model is mechanically described by an extended pressure and rate-dependent

15

Coulumb-viscoplastic rheology of mixture. The model is expected to simulate the velocities and pressure of

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the bulk mixture much faster than the two-phase mass flow model. Furthermore, the introduction of the ve-

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locity and pressure drifts makes it possible to reconstruct the two-phase dynamics. This shows the application

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potential of the new model presented here.

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Keywords: Mixture flow, Quasi-two-phase flow, Generalized bulk and shear viscosities, Generalized mixture

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velocities and pressure, Velocity and pressure drifts

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1

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Geophysical mass flows such as debris flows, avalanches and landslides generally contain more than one mate-

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rial and often form a mixture of soil and rock particles with a significant to dominant quantity of interstitial

24

fluid (Johnson, 1970; O’Brien et al., 1993; Iverson and Denlinger, 2001; Pitman and Le, 2005; Pudasaini et al.,

25

2005; Takahashi, 2007). Debris flows as a typical example of geophysical mass flows, are effectively two-phase

26

and gravity driven flows caused by intense rainfall or a sudden surge of water. Such surges may occur due to

Introduction

1

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the breaking off of a saturated landmass or dam, or by a landslide event that impacts a lake, ocean or river

28

causing over flow that may result in catastrophic dam collapse (Pitman et al., 2013; Mergili et al., 2017a).

29

As the flood surge rushes down the slope, it may transform into a debris flow by entraining the bed material

30

(Hungr et al., 2005; Iverson, 2012; Pudasaini and Fischer, 2016; Mergili et al., 2017c).

31

Debris flows, which generally occur in mountainous areas throughout the world, are extremely destructive

32

and dangerous natural events. A debris flow differs from other types of mass flows including rock and snow

33

avalanches because it contains solid particles and viscous fluid both of which generate fundamentally different

34

forces that significantly influence the motion. During these events, the mixture material undergoes rapid

35

motion and large deformations. The fast motion together with the evolving mixture density provides the

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debris flow with potentially huge energy and destructive power. Especially the interactions between solid and

37

fluid forces can cause exceptionally long run out (Iverson, 1997; Pitman and Le, 2005; Pudasaini and Miller,

38

2013; Pudasaini and Fischer, 2016).

39

In simulations, debris flows are often treated as a single phase mass flows (e.g., Chen, 1988; O’Brien et

40

al., 1993; Takahashi, 2007), mixture flows (Iverson and Denlinger, 2001; Bohorquez, 2008), a two-fluid flow

41

(Pitman and Le, 2005), and a two-layered model (Fernandez-Nieto et al., 2008). With a major advance in

42

two-phase debris flow modeling, Pudasaini (2012) proposed a comprehensive theory and simulation technique.

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The model accounts for strong interactions between the solid and the fluid. The model includes buoyancy

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and other three important physical aspects of the flow: enhanced non-Newtonian viscous stress, virtual mass

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and the generalized drag. The depth-averaged version of this general two-phase mass flow model has been

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widely applied in different flow scenarios: rock-ice avalanche dynamic simulation (Pudasaini and Krautblatter,

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2014), tsunami generated by debris impact on lakes and submarine sediment transport (Kafle et al., 2016),

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and glacial lake outburst flood (Kattel et al., 2016). Furthermore, employing the general two-phase mass

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flow model (Pudasaini, 2012) in the computational technique, r.avaflow, Mergili et al. (2017a) presented the

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simulation results for the interaction and propagation of mass flow in real mountain topographies. Using the

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same model and tool, Mergili et al. (2017b, 2017c) studied the complex hydro-geomorphic process chains

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in a multi-lake outburst flood in the Santa Cruz Valley (Cordillera Blanca, Peru). These are the first ever

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explicit simulations for the evolution of the solid and fluid phase in subaerial and submarine two-phase debris

2

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flows. Numerical results indicate that the model can adequately describe the complex dynamics of subaerial

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two-phase debris flows, particle-laden flows, sediment transport, submarine debris flows and associated phe-

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nomena. These results highlight the applicability of the two-phase model to a wide range of geophysical mass

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flows.

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Here, by reducing the general three-dimensional full two-phase model (Pudasaini, 2012), we generate a gener-

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alized quasi two-phase full (non-depth averaged) two-dimensional model for rapid bulk mixture flow down a

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channel. To do so, we combine the solid and fluid phase mass balances, and similarly the momentum balance

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equations to obtain a quasi-two phase bulk mixture model. This is achieved by introducing the drift factors

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for fluid velocities and the fluid pressure in terms of the solid velocities and solid pressure, respectively. We

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constitute some structurally new concepts of dynamically evolving drift induced generalized barycentric veloc-

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ities and pressure for the bulk mixture flow. Similarly, two further structures of bulk mixture viscosities, and

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the effective bulk and shear viscosities lead to the emergence of the representative complex mixture viscosity.

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The emerging model for bulk mixture flow consists of the full (non-depth averaged) two-dimensional quasi

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two-phase mass and momentum conservation equations for the generalized mixture velocities and pressure.

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The derived mass and momentum equations appear in non-conventional form due to the inertial coefficients,

69

and the form of the mixture viscosities. This model structure results in a novel, dynamically evolving effective

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mixture friction coefficient, which made it possible to extend the pressure- and rate-dependent Coulomb-

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viscoplastic granular flow rheology (Domnik and Pudasaini, 2012; Domnik et al., 2013) to the flow of debris

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mixtures. This is an important mechanical aspect of the derived mixture model.

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Several important research activities have been carried in the past in the mixture theory of continuum mechan-

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ics (Truesdell, 1957; Atkin and Craine, 1976a, 1976b; Bowen, 1976; Bedford and Drumheller, 1983; Truesdell,

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1984; Rajagopal and Tao, 1995; Massoudi, 2003, 2010). They have focused on the modeling of a mixture of

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two fluids and fluid mixed with solid particles so that each component is considered as a single continuum.

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Iverson (1997), Iverson and Denlinger (2001) and Pudasaini et al. (2005) progressed significantly in modelling

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effectively single-phase mixture flows taking into account the pore fluid pressure. Furthermore, Fernandez-

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Neito et al. (2008) presented two-layered model whereas Pitman and Le (2005) proposed a two-fluid model

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for debris flow. However, our modeling approach first develops the mixture pressure, velocity and mixture

3

81

shear and bulk viscosities to define the mixture stress tensor. The two-phase dynamics of solid and fluid

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can then be re-constructed from the definition of the mixture quantities and drift coefficients based on the

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known velocity and pressure of the mixture. This makes our model mechanically and structurally novel. On

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the one hand, the simulations based on the new model should be computationally substantially faster than

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the full two-phase simulations. On the other hand, such simulations are expected to be more accurate than

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the effectively single-phase models (Iverson and Denlinger, 2001; Pudasaini et al., 2005). This highlights the

87

application potential of the proposed model.

88

2

89

In a two-phase model, the phases have distinct material properties. In general, the fluid phase is characterized

90

by an isotropic stress distribution and a viscosity ηf and density ρf . Similarly, the solid phase is characterized

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by an anisotropic stress distribution, as a function of an internal frictional angle φ and the basal friction angle

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δ and the material density ρs (Savage and Hutter, 1989; Pitman and Le, 2005). The balance equations for

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the three-dimensional, two-phase debris flow are given by (Pudasaini, 2012):

Three Dimensional Model for Two-Phase Mass Flow

∂αs + ∇ · (αs us ) = 0, ∂t

(1)

∂αf + ∇.(αf uf ) = 0, ∂t

(2)

∂ (αs ρs us ) + ∇.(αs ρs us ⊗ us ) = αs ρs f − ∇.αs Ts + ps ∇αs + Ms , ∂t

(3)

∂ (αf ρf uf ) + ∇.(αf ρf uf ⊗ uf ) = αf ρf f − αf ∇pf + ∇.αf τ f + Mf , ∂t

(4)

94

where (1) and (2) are the mass balances for the solid and fluid, and (3) and (4) are the momentum balances

95

for the solid and the fluid fractions, respectively, with Ms =

τf

αs αf (ρs − ρf )g (uf − us )|uf − us |J−1 [UT {PF(Rep ) + (1 − P)G(Rep )}]J     ∂uf ∂us +αs ρf CV M + uf .∇uf − + us .∇us = −Mf , ∂t ∂t

  A(αf ) = ηf ∇uf + (∇uf )T − ηf [(∇αs )(uf − us ) + (uf − us )(∇αs )] . αf

(5) (6)

96

In these model equations, t is time, αs is the solid volume fraction, αf (= 1 − αs ) is the fluid volume fraction.

97

Similarly, us = (us , vs , ws ) is the velocity vector for solid and uf = (uf , vf , wf ) is that for fluid, ⊗ is the tensor 4

98

product, f is the body force density; Ts is the solid stress tensor; ps and pf are solid and the fluid pressures;

99

Ms and Mf (= −Ms ) are interfacial momentum transfers for solid and fluid respectively. Furthermore, τ f

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denotes the extra stress for fluid. The exponent J can have the values 1 (for linear drag) and 2 (for quadratic

101

drag). The first term of Ms is the drag force induced by the differences of the phase velocities, and the

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coefficient of drag CDG =

103

G and fluid-like F drag contributions to flow resistance. F and G are functions of densities, volume fractions,

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and the particle Reynolds number Rep given by F = (ρf /ρs )(αf /αs )3 Rep /180 and G = αf

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is a parameter depending on Rep (Richardson and Zaki, 1954; Pitman and Le, 2005). The behavior of drag

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depends on the interpolation parameter P ∈ [0, 1] which combines G and F. Setting P = 0 is more suitable

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when the solid particles are moving through a fluid and P = 1 for flows of fluids through dense packing of

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grains (Pitman and Le, 2005; Pudasaini, 2012). The drag coefficient depends on the solid volume fraction,

109

and increases with increasing values of αs and P. The second term of Ms is the virtual mass force (Ishii, 1975;

110

Ishii and Zuber, 1979; Drew, 1983; Drew and Lahey, 1987; Rivero et al., 1991; Maxey and Riley, 1993) induced

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by the difference in the acceleration between phases with CV M as virtual mass coefficient. With virtual mass

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force, the solid particles may bring more fluid mass with them. This results in loss of some inertia of solid

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mass that induces the kinetic energy of the fluid (Pudasaini, 2012). The first term of τ f is the Newtonian

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viscous stress with T as the transpose operator, and the second term of τ f is an enhanced Non-Newtonian

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viscous stress that depends on the solid volume-fraction gradient ∇αs (also see, Ishii, 1975; Drew, 1983; Ishii

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and Hibiki, 2006; Pudasaini, 2012). A is the mobility of fluid at the particle-fluid interface. There is a strong

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coupling between the solid and the fluid momentum transfer both through the interfacial momentum transfer,

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which includes the viscous drag, the virtual mass force; and the enhanced viscous fluid stress.

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Here, by reducing the three-dimensional, full two-phase model (1)-(4), we generate a generalized quasi two-

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phase, full two-dimensional model for rapid bulk mixture flow down a channel. For simplicity, we assume

122

that the difference between the phase accelerations is negligible, i.e., the virtual mass coefficient is close to

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zero, and the non-Newtonian term in the fluid extra stress tensor is insignificant. However, we note that in

αs αf (ρs − ρf )g is modeled by a linear combination of solid-like [UT {PF(Rep ) + (1 − P)G(Rep )}]J M (Rep )−1

, where M

A Quasi Two-phase Generalized Bulk Mixture Model

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Figure 1: Side view of the rapid flow of debris material along an inclined chute. ζ is the inclination angle, (usin , ufin ) and (hsin , hfin ) are solid and fluid phase inflow velocities and heights (modified from Domnik and Pudasaini, 2012).

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the following model formulations, these terms could be retained and the model can be extended for three-

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dimensional unconfined flows. Let us = (us , ws ), uf = (uf , wf ) be the respective velocity components for

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solid and fluid in the down-slope (x) and perpendicular to the channel surface (z) directions as shown in Fig.

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(1). Since αs + αf = 1, adding the mass balance equations (1) and (2), we get the mixture mass balance ∇.(αs us + αf uf ) = 0.

(7)

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In order to develop a reduced, but generalized quasi two-phase bulk mixture model we introduce the solid and

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fluid velocity drift coefficients (Zuber and Findlay, 1965; Wallis, 1969; Manninen et al., 1996; Krasnopolsky

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et al., 2016; Pudasaini and Fischer, 2016), (λu , λw ) and the pressure drift coefficient λp , as

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uf = λu us , wf = λw ws ; pf = λp ps .

(8)

i i ∂ h ∂ h (αs + λu αf )us + (αs + λw αf )ws = 0. ∂x ∂z

(9)

Then (7) reduces to

6

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This mass balance is explicitly written in terms of the solid phase (velocities). However, the fluid-phase is also

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represented through the velocity drift parameters (λu , λw ). Interestingly, it reduces to a traditional single-

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phase bulk mass balance when λu = λw = 1, which does not distinguish between the solid and fluid velocity,

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or the phase dynamics (Iverson and Denlinger, 2001; Pudasaini et al., 2005; Fernandez-Nieto et al., 2008).

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As the solid and fluid phases separately are considered to be incompressible (Mills, 1966), ρs and ρf are both

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constants. Dividing by ρs and ρf on both sides of (3) and (4) respectively, we get ∂ αs Ms αs , (αs us ) + ∇.(αs us ⊗ us ) = αs f − ∇ps + ∇. τ s + ∂t ρs ρs ρs

(10)

αf αf Mf ∂ (αf uf ) + ∇.(αf uf ⊗ uf ) = αf f − ∇pf + ∇. τ f + , ∂t ρf ρf ρf

(11)

138

where, the solid and fluid Cauchy stresses are now written in terms of the solid and fluid pressures (ps , pf )

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and deviatoric stresses (τ s , τ f ), respectively (Drew, 1983). Following Domnik and Pudasaini (2012) and

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Domnik et al. (2013), we will model the internal deformation of the solid component in debris bulk as a

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Coulomb-viscoplastic material with yield strength (Bingham, 1922; Jop et al., 2006; Ancy, 2007; Balmforth

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and Frigaard, 2007; Forterre and Pouliquen, 2008; Pailha and Pouliquen, 2009). The extra fluid stress tensor

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τ f for fluid, and τ s for solid are (Goddard, 1996; Tardos, 1997; Domnik et al., 2013; Schaefer, 2014)   Ds τ f = ηf ∇uf + (∇uf )T , τ s = 2ηs Ds + 2τys . ||Ds ||

(12)

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Here, τ s is the deviatoric stress tensor for a viscoplastic material (granular fluid). With the yield stress

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τys = τc + τp ps , where τc is cohesion, it becomes pressure- and rate-dependent Coulomb-viscoplastic law for

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solid-phase (Domnik and Pudasaini, 2012; Domnik et al., 2013; von Boetticher et al., 2016, 2017). As in the

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fluid, the symmetric part of the velocity gradient Ds =

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is the dynamic viscosity. The norm of Ds is defined by ||Ds || =

149

Pudasaini, 2012). The effective kinematic viscosity for solid takes the form: νse = νs +

τys , ||Ds ||

1 2 [∇us

+ (∇us )T ] is the strain-rate tensor, and ηs p

2tr(D2s ) (Jop et al., 2006; Domnik and

(13)

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where νs = ηs /ρs and τys := τys /ρs are the solid kinematic viscosity and yield stress normalized by the solid

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density. During the mass flow, with τys = 0, the material behaves as a Newtonian fluid for which νse = νs .

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Now, τ s in (12) can be written as τ s = 2νse Ds , where τ s := τ s /ρs . In what follows, for simplicity, we replace

153

νes by νs . 7

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Next, we construct a generalized single momentum balance equation for a bulk mixture flow. Adding (10)

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and (11), we get: ∂ Ms Mf (αs us + αf uf ) + ∇ · [(αs us ⊗ us ) + (αf uf ⊗ uf )] = f + ∇ · (αs τ s + αf τ f ) − (αs ∇ps + αf ∇pf ) + + , (14) ∂t ρs ρf

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where, pressures (ps , pf ) and stresses (τ s , τ f ) are normalized with respective densities, and the viscosities in

157

(τ s , τ f ) are kinematic viscosities. With uf = λu us and wf = λw ws , the local time derivative of the ‘mixture

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velocity’ in (14) becomes i ∂ ∂h (αs us + αf uf ) = (αs + λu αf )us , (αs + λw αf )ws . ∂t ∂t

159

(15)

The convective terms in (14) can be written as ∂ ∂ [(αs + λ2u αf )u2s ] + [(αs + λu λw αf )us ws ], ∂x ∂z  ∂ ∂ [(αs + λu λw αf )us ws ] + [(αs + λ2w αf )ws2 ] . ∂x ∂z

∇ · [(αs us ⊗ us ) + (αf uf ⊗ uf )] =

160

(16)

The divergence of the rate-of-strain tensors in (14) become ∇ · [αf τ f + αs τ s ]          h∂  ∂uf ∂uf ∂wf ∂ ∂ ∂us ∂ ∂us ∂ws 2νf αf + νf αf + + 2νs αs + νs αs + , ∂x ∂x ∂z ∂z ∂x ∂x ∂x ∂z ∂z ∂x           ∂uf ∂wf ∂wf ∂ ∂ ∂ ∂us ∂ws ∂ ∂ws i P P pppp νf αf + + 2νf αf + νs αs + + 2νs αs ∂x ∂z ∂x ∂z ∂z ∂x ∂z ∂x ∂z ∂z     h ∂ ∂us ∂ ∂us ∂ws PPp = 2 (νs αs + λu νf αf ) + (νs αs + λu νf αf ) + (νs αs + λw νf αf ) , ∂x ∂x ∂z ∂z ∂x     ∂ ∂us ∂ws ∂ ∂ws i P P pppp (νs αs + λu νf αf ) + (νs αs + λw νf αf ) +2 (νs αs + λw νf αf ) . (17) ∂x ∂z ∂x ∂z ∂z PPp =

161

With the definition pf = λp ps , the bulk mixture pressure gradients in (14) become   ∂ps ∂ps − (αs ∇ps + αf ∇pf ) = − (αs + λp αf ) , (αs + λp αf . ∂x ∂z

(18)

162

For simplicity, in the above formulations, we have assumed that variations of λu , λw , λp with x, and z are

163

negligible. Finally, we deal with the total interfacial momentum exchange in (14): αs αf g Ms Mf + = (1 − γ) (uf − us )|uf − us |J−1 ρs ρf [UT {PF(Rep ) + (1 − P)G(Rep )}]J   αs αf g 1 P ppppppP P − −1 (uf − us )|uf − us |J−1 . γ [UT {PF(Rep ) + (1 − P)G(Rep )}]J 8

164

165

  Since uf − us = (λu − 1)us , (λw − 1)ws , we have h i αs αf g (λu − 1)us , (λw − 1)ws Ms Mf 1 + = − (1 − γ)2 ρs ρf γ [UT {PF(Rep ) + (1 − P)G(Rep )}]J
which become zero if γ → 1 or λu → 1, λw → 1 .

J−1  (λ − 1)u , (λ − 1)w  , u s w s

(19)

166

So, with f = (fx , fz ) from (15)-(19), (14) yields the following x- and z- directional momentum balances for

167

the bulk mixture:  ∂ ∂  ∂ ∂ps [(αs + λu αf )us ] + (αs + λ2u αf )u2s + [(αs + λu λw αf )us ws ] = fx − (αs + λp αf ) ∂t ∂x ∂z ∂x     ∂us ∂ ∂us ∂ ∂ws (νs αs + λu νf αf ) + (νs αs + λu νf αf ) , ppppP P + 2 + (νs αs + λw νf αf ) ∂x ∂x ∂z ∂z ∂x

(20)

∂ ∂ ∂ ∂ps [(αs + λw αf )ws )] + [(αs + λu λw αf )us ws ] + [(αs + λ2w αf )ws2 )] = fz − (αs + λp αf ) ∂t ∂x ∂z ∂z     ∂us ∂ws ∂ ∂ws ∂ (νs αs + λu νf αf ) + (νs αs + λw νf αf ) +2 (νs αs + λw νf αf ) . (21) ppppP P + ∂x ∂z ∂x ∂z ∂z 168

With the definition of the bulk velocities, and the pressure of the mixture flow um = (αs + λu αf )us , wm = (αs + λw αf )ws , pm = (αs + λp αf )ps ,

169

the inertial coefficients Λuu =

170

(22)

αs + λ2u αf αs + αf λ2w αs + λu λw αf 1 1 , Λ = , Λuw = , Λu = , Λw = , (23) ww (αs + λu αf )2 (αs + λw αf )2 (αs + λu αf )(αs + λw αf ) αs + λu αf αs + λw αf

and the dynamical coefficients for mixture viscosities and pressure Ληu = νs αs + λu νf αf , Ληw = νs αs + λw νf αf , Λp = αs + λp αf ,

(24)

171

from (9), (20) and (21), the new dynamical model for bulk mixture flow consist of the non depth-averaged

172

two-dimensional quasi two-phase mass and momentum conservation equations: ∂um ∂wm + = 0, ∂x ∂z

(25)

∂um ∂ ∂ ∂pm + (Λuu u2m ) + (Λuw um wm ) = fx − ∂t ∂x ∂z ∂x     ∂ ∂(Λu um ) ∂ ∂(Λu um ) ∂(Λw wm ) PPPPPPPPP + 2 Ληu + Ληu + Ληw , ∂x ∂x ∂z ∂z ∂x ∂wm ∂ ∂ ∂pm 2 + (Λuw um wm ) + (Λww wm ) = fz − ∂t ∂x ∂z ∂z     ∂ ∂(Λu um ) ∂(Λw wm ) ∂ ∂(Λw wm ) PPPPPPPPP + Ληu + Λ ηw +2 Λ ηw , ∂x ∂z ∂x ∂z ∂z 9

(26)

(27)

173

where, for simplicity, the relations Λp ∂ps /∂x =: ∂pm /∂x, and Λp ∂ps /∂z =: ∂pm /∂z, are defined. The simple

174

situation being the one for which variation of Λp is negligible. pm is called the “kinematic pressure” for the

175

bulk. It appears that the dynamics of the bulk pressure, pm is complex. We will discuss on this in more detail

176

later. The inertial and dynamical coefficients in (26) and (27) are complex and coupled, and explicitly include

177

the physics and dynamics of mixture, such as the solid volume fraction, solid and fluid material densities

178

and viscosities; solid friction and yield strength, rate of deformation of solid, and velocity and pressure drift

179

parameters. Moreover, the mass and momentum equations in (25)-(27) appear in non-conventional form due

180

to the inertial coefficients (Λu , Λw ), and the non-equality of the mixture viscosities Ληu and Ληw . In the form,

181

the model equations (25)-(27) may look similar to the existing (effectively single-phase) mixture mass flow

182

models (Iverson, 1997; Iverson and Denlinger, 2001; Pudasaini et al., 2005). However, there are fundamentally

183

new aspects in the new model than those found in literature. This has been discussed in Section 5 and also

184

in the following sections.

185

From the application and computational point of view, the contraction (25)-(27) is useful, because it reduces

186

the full two-phase model to a quasi two-phase bulk model. This paves the way to apply the simple single-

187

phase computational tools of Domnik and Pudasaini (2012) and Domnik et al. (2013), with some amendments

188

to numerically solve the emerged model equations, that to a large extent, describe a mixture flow of solid

189

particles and viscous fluid. However, at the same time, the drift parameters (λu , λw , λp ) contain the basic

190

and important information relating the solid and fluid velocities, and the dynamic pressures. Further models

191

can be developed to describe these parameters. Alternatively, laboratory or field data can be utilized either

192

to obtain solid velocities (us , ws ) and pressure ps , or fluid velocities (uf , wf ) and pressure pf , and simulate

193

the dynamics for the other field variables. This makes the resulting model a generalized quasi two-phase bulk

194

mixture model. For λu = λw = λp = 1 the emerging model reduces to traditional effectively single-phase bulk

195

mixture model (Iverson and Denlinger, 2001; Pudasaini et al., 2005). The new model has a potential for a

196

wider application through constructing the functional relations for (λu , λw , λp ), or via parameterizations; the

197

advantage is that computationally it is faster than the full two-phase simulations, and, at the same time, more

198

accurate than the single-phase models.

10

199

4

200

Here, we present the final set of model equations that is written in a convenient form for applications and

201

numerical simulations (Domnik et al., 2013).

202

4.1

203

To calculate the pressure pm , we construct the pressure Poisson equation for the bulk mixture. The x- and

204

z- directional momentum equations (26) and (27) can be written as

205

The New Model Equations

Pressure Poisson Equations

∂pm ∂um =− + F, ∂t ∂x

(28)

∂wm ∂pm =− + G, ∂t ∂z

(29)



(30)

where    ∂(Λu um ) ∂ ∂(Λu um ) ∂(Λw wm ) + Ληu +Ληw +fx , ∂x ∂z ∂z ∂x     ∂ ∂ ∂(Λw wm ) ∂ ∂(Λu um ) ∂ ∂(Λw wm ) 2 )+ +Ληw G = − (Λuw um wm )− (Λww wm Λ ηu +2 Λ ηw +fz . ∂x ∂z ∂x ∂z ∂x ∂z ∂z F =−

206

∂ ∂ ∂ (Λuu u2m )− (Λuw um wm )+2 ∂x ∂z ∂x

(31)

Partially differentiating (28) with respect to x, and (29) with respect to z, and adding them, we get ∂ ∂x

207

Ληu



∂um ∂t



+

∂ ∂z



∂wm ∂t



=

∂ ∂t



∂um ∂wm + ∂x ∂z



=−

∂ 2 pm ∂ 2 pm ∂F ∂G − + + . ∂x2 ∂z 2 ∂x ∂z

This, with the continuity equation (25) results in the following pressure Poisson equation for the mixture flow ∂ 2 pm ∂ 2 pm ∂F ∂G + = + . ∂x2 ∂z 2 ∂x ∂z

(32)

208

4.2

The Bulk Mixture Model

209

With the definitions of F and G from (30) and (31), the following set of x- and z-directional momentum, and

210

pressure Poisson equation together constitute a new generalized quasi two-phase mixture mass flow model: ∂pm ∂um =− + F, ∂t ∂x

(33)

∂wm ∂pm =− + G, ∂t ∂z

(34)

∂ 2 pm ∂ 2 pm ∂F ∂G + = + . ∂x2 ∂z 2 ∂x ∂z

(35)

11

211

The set (33)-(35) is a system of three highly non-linear partial differential equations with three unknowns,

212

namely, the generalized mixture velocities um , wm , and pressure pm . So, the system is closed and the numerical

213

integration is possible provided that the coefficients Λuu =

αs + λ2u αf αs + αf λ2w αs + λu λw αf 1 1 , Λww = , Λuw = , Λu = , Λw = , 2 (αs + λu αf ) (αs + λw αf )2 (αs + λu αf )(αs + λw αf ) αs + λu αf αs + λ w αf

Ληu = νs αs + λu νf αf , Ληw = νs αs + λw νf αf , Λp = αs + λp αf ,

(36)

214

are known. Equations (33)-(35) require appropriate boundary conditions for the velocities and pressure at

215

the free and the basal surface, and also the initial conditions for these variables. This has been discussed in

216

Section 8.

217

4.3

218

There are several ways to model the drift coefficients (Zuber and Findlay, 1965; Wallis, 1969; Manninen et

219

al., 1996; Krasnopolsky et al., 2016; Pudasaini and Fischer, 2016). Simplified flow situations can be utilized

220

to reduce the drift parameters λu and λw into a single one. Assuming locally small variations of αs and λu ,

221

the mixture mass balance (25), and the steady state solid mass balance (1) leads to

Modelling the Drift Coefficients

wf ≈ λ0 + λu ws ,

(37)

222

where λ0 is an integration constant whose value can be fixed from the boundary and physical conditions.

223

Since wf = λw ws , by replacing λw = λ0 /ws + λu , the model equations contain effectively a single velocity drift

224

parameter, λu . Note that this does not produce singularity, because while substituting λw in the respective

225

terms in the model equations (33)-(36), ws in the denominator cancels out. In particular, this shows that

226

when λ0 ≈ 0, λw ≈ λu . This is advantageous, because explicit descriptions are available for λu . GhoshHajra

227

et al. (2015) analytically derived an expression for λu , as a ratio of the fluid and solid pressure parameters

228

βf and βs , λu =

229

averaged equations (Kafle et al., 2016; Kattel et al., 2016). Then, we may dynamically obtain λu = uf /us .

230

Furthermore, we may also parameterize λw as a function of λu , e.g., a linear or a quadratic relation.

p βf /βs . Alternatively, us and uf can be obtained from the fast simulations of the depth

12

231

5

Discussions on Physical Aspects of the New Model

232

The model equations (33)-(35) are written as well structured partial differential equations in conservative form.

233

There are several important features of the new model. Some physical aspects, implications, and applicabilities

234

are discussed here. We have presented two new structures (concepts) of bulk mixture viscosities and pressures

235

as functions of the solid volume fraction, solid and fluid material densities and viscosities, solid friction and

236

yield strength, rate of deformation of solid, and velocity and pressure drift parameters. One of the advantages

237

of the coupled and generalized model (33) - (35) is that structurally it is the same as the single phase

238

granular or debris flow model (Domnik and Pudasaini, 2012; Domnik et al., 2013). So, in principle, the same

239

computational strategy can be applied to solve the new system. However, the complexity inherited by the

240

two-phase bulk mixture flow is contained in the parameters Λu , Λw , Λuu , Λuw , Λww , Ληu , Ληw , Λp . Importantly,

241

we can reconstruct some fundamental properties and dynamics of two-phase flows.

242

5.1

243

The two-phase dynamics is largely lost in the classically derived effectively single-phase mixture mass flow

244

model (Iverson and Denlinger, 2001; Pudasaini et al., 2005). But, in our derivation, once um , wm and pm are

245

obtained from (33)-(35), the two-phase dynamics of solid (us , ws , ps ) and fluid (uf , wf , pf ) can be calculated

246

from (22) and (8). That is, (us , ws , ps ) are obtained from (22), and then (uf , wf , pf ) are systematically re-

247

constructed from (8). However, how much reality these solutions represent depends on the physics contained

248

in (λu , λw , λp ), the viscosities (Ληu , Ληw ) and the description of αs .

249

5.2

250

The viscous terms appear in the momentum equations (20) and (21). From the third term on the right hand

251

side of (20), the bulk viscosity can be obtained

Reconstruction of the Two-phase Dynamics

Effective Bulk Viscosity

    ∂us ∂(λu us ) ∂ 1 ∂ λu ∂um + νf αf = νs αs um +νf αf um ≈ (νs αs Λu + νf αf λu Λu ) . (38) νs αs ∂x ∂x ∂x αs + λu αf ∂x αs + λu αf ∂x 252

253

Similarly, from the fourth term on the right hand side of (21), another expression for the bulk viscosity can be obtained νs αs

    ∂ws ∂(λw ws ) ∂ 1 ∂ λw ∂wm + νf αf = νs αs wm +νf αf wm ≈ (νs αs Λw + νf αf λw Λw ) . (39) ∂z ∂z ∂z αs + λw αf ∂z αs + λw αf ∂z

13

254

These expressions are valid if Λu and λu vary linearly with x, and Λw and λw vary linearly with z. These

255

approximations separately concern the x- and z-directional bulk viscosities. The effective (representative)

256

‘bulk viscosity’ of the mixture is obtained by averaging over (38) and (39): 1 1 ΛB ηm := νs αs (Λu + Λw ) + νf αf (λu Λu + λw Λw ). 2 2

(40)

257

If λu and λw are increasing, then (Λu + Λw ) and (λu Λu + λw Λw ) are decreasing. Thus, the viscosity of the

258

mixture is also decreasing. This means the higher are the fluid drifts, the faster is the fluid motion, and thus,

259

the higher is the deformation. This results in the lower bulk viscosity, as in a shear-thinning fluid (Pudasaini,

260

2011). This is physically meaningful.

261

5.3

262

The shear viscosities are contained in the fourth term on the right hand side of (20), and similarly, the third

263

term on the right hand side of (21):

264

265

Effective Shear Viscosity

(νs αs + λu νf αf )

∂us ∂ws ∂um ∂wm + (νs αs + λw νf αf ) ≈ (νs αs + λu νf αf )Λu + (νs αs + λw νf αf )Λw , ∂z ∂x ∂z ∂x

(41)

(νs αs + λu νf αf )

∂us ∂ws ∂um ∂wm + (νs αs + λw νf αf ) ≈ (νs αs + λu νf αf )Λu + (νs αs + λw νf αf )Λw . ∂z ∂x ∂z ∂x

(42)

This shows that there are two contributions also for the shear viscosity. Averaging these, we obtain a representative ‘shear viscosity’ of the mixture: 1 1 ΛSηm := νs αs (Λu + Λw ) + νf αf (λu Λu + λw Λw ). 2 2

(43)

266

S So, since ΛB ηm and Ληm are identical, (43) can be considered as the “representative mixture viscosity”, Ληm .

267

However, these are only simplifications, and that the better representations of the viscosities are as they

268

appear in (20), (21). In simple situations, (43) can be further reduced. If the x- and z-directional velocity

269

drifts are identical, i.e., λu ≈ λw (isotropic drifts), then Ληm ≈ (νs αs + νf αf λu ) Λu =

νs αs + νf αf λu . αs + λ u αf

(44)

270

This is a very important structure, because the mixture viscosity Ληm can be measured in a rheometer.

271

Figure 2 reveals that the mixture viscosity, as represented by the simplified form (44), varies strongly with

272

the velocity drift. The chosen parameters are νs = 4.99 m2 /s, νf = 10−3 m2 /s (Domnik et al., 2013; von 14

273

Boetticher et al., 2016, 2017), and for different values of αs . For higher solid fraction (dense flow), the drop in

274

the mixture viscosity with the drift is slower, however, for the lower solid fraction (dilute flow), the mixture

275

viscosity drops exponentially with the velocity drift. This indicates that the mixture flow behaves as a type

276

of shear-thinning material: weakly for higher solid fraction but strongly with the lower solid fraction. Such

277

a rheological behaviour of the mixture is often observed in debris flows (von Boetticher et al., 2016; 2017).

278

Figure 2 shows that Ληm is an inverse exponential function of λu . So, it can be approximated by a hyperbolic

279

tangent function:

Ληm = Λ0ηm (1 − ξ tanh(λχu )) .

(45)

αs = 0.75 αs = 0.50

4

αs = 0.25

m

Mixture viscosity: Λη (m2s−1)

5

3

αs = 0.15

2 1 0 0

0.5

1

1.5

2 2.5 3 Velocity drift: λu

3.5

4

4.5

5

Figure 2: Variation of the mixture viscosity with velocity drift: mixture viscosity depends strongly on the velocity drift and the solid volume fraction in the debris mixture and shows a type of shear-thinning behaviour.

Theoretical derivation Approx. with tanh function

4

m

Mixture viscosity: Λη ( m2 s−1)

5

3 2 1 0 0

0.5

1

1.5

2 2.5 3 Velocity drift: λu

3.5

4

4.5

5

Figure 3: Approximation of the mixture viscosity by a hyperbolic tangent function of drift.

15

12 αs = 0.25 Velocity drift: λu

10

αs = 0.20 αs = 0.15

8

αs = 0.10

6 4 2 0.2

0.3

0.4

0.5 0.6 0.7 Viscosity: Λη (m2 s−1)

0.8

0.9

1

m

Figure 4: Variation of the velocity drift with mixture viscosity.

280

As Λ0ηm is the reference value of Ληm at λu = 0, this can be determined relatively easily. However ξ and χ may

281

be determined from an experiment which is not within the scope here, but can be numerically determined for

282

results presented in Fig. 2. Figure 3 shows that the Ληm in (44) can very well be represented by a hyperbolic

283

tangent function of λu with the parameters Λ0ηm = 4.99 m2 /s, χ = 0.5 and ξ = 0.97. This is important from

284

an application point of view as a relatively simple function (45) can replace (44).

285

Furthermore, (44) shows that λu can be obtained analytically, and explicitly: λu =

(νs − Ληm ) αs . (Ληm − νf ) αf

(46)

286

Since usually νs − Ληm > 0 and Ληm − νf > 0, αs , αf > 0, λu > 0. Such an explicit form of the drift is novel,

287

and technically very important. Because, as mentioned earlier, Ληm may be measured in the lab, then with the

288

knowledge of νs and νf , (46) provides values for the drift, λu , as a function of Ληm . This is a great advantage

289

because this can be used to reconstruct the two-phase dynamics of the mixture. See later, Fig. 5 and Fig. 6,

290

and the corresponding descriptions on the phase reconstructions. Figure 4 plots (46) with νs = 4.99 m2 /s,

291

νf = 10−3 m2 /s, αs = 0.25, 0.20, 0.15, 0.10, and shows that higher is the solid volume fraction, higher are the

292

velocity drifts which decrease with higher viscosity. Figure 2 and Fig. 4 show that the viscosity and velocity

293

drift of the mixture flow vary inversely to each other. The same analysis can be performed for more complex

294

mixture viscosity represented by (43).

295

In connection to (22)1 and (22)2 , (46) can be utilized to reconstruct the solid and fluid velocities in terms of

296

the mixture velocity um and the drift λu . Figure 5 shows an incipient motion of a mixture material. The 16

297

chosen parameters are νs = 4.99m2 /s, νf = 10−3 m2 /s, αs = 0.25. This figure explains how the solid and fluid

298

velocities can be analytically reconstructed as a function of the mixture velocity. As the mass is released,

299

the fluid accelerates faster than the solid phase, but after a while the fluid velocity tends to saturate, which

300

is realistic (Pudasaini, 2011), as the fluid-phase is a viscous material that can reach to a terminal velocity.

301

However, the solid-phase continues to accelerate, this is also realistic because the solid phase is described by

302

a frictional granular material. So, such an incipient behaviour can be described by our modelling approach.

303

Figure 6 shows the depositional behaviour. Assume a situation, as described by Fig. 5 that after traveling

304

a certain distance, the mixture enters the depositional area. At the entry into the depositional area, the

305

solid-phase has higher velocity than the fluid-phase (as seen in Fig. 5). However, as the flow enters into this

306

area, the velocity of both the phases decrease steadily. Nevertheless, the depositional behaviour of the solid

307

and the fluid phase are completely different. The solid decelerates much faster than the fluid. This is so,

308

because the solid material is frictional which dissipates the energy more quickly than the fluid phase which

309

is viscous, that is relatively weaker (because of the viscous dissipation) than the solid. These are realistic

310

scenarios of mixture flow in depositional region.

311

Figure 5 and Fig. 6 show several important features of solid and fluid velocities in the mixture flows. First,

312

these are strongly non-linear. Second, both the solid and fluid velocities behave fundamentally differently.

313

The rates at which the solid and fluid velocities accelerate after the mass release, and the rates of their

Phase velocities: us, uf (ms−1)

2.5 2

Solid−phase velocity Fluid−phase velocity

1.5 1 0.5 0 0

0.2

0.4 0.6 0.8 1 The mixture velocity: um (ms−1)

1.2

1.4

Figure 5: Evolution of the incipient motion of mixture material that explains how the solid and fluid velocities can be reconstructed as a function of the mixture velocity.

17

Phase velocities: us, uf (ms−1)

2.5 Solid−phase velocity Fluid−phase velocity

2 1.5 1 0.5 0

1.2

1

0.8 0.6 0.4 The mixture velocity: um (ms−1)

0.2

0

Figure 6: Depositional behaviour of the solid and the fluid phases velocities as a function of the mixture velocity.

314

decelerations as they enter the depositional region are completely different. These are natural phenomena,

315

but simulated here for the first time in terms of the mixture velocity. This clearly shows the application

316

potential of the new model presented here.

317

If, further, the drifts are close to unity, then Ληm ≈ νs αs + νf αf ,

(47)

318

which is the simplest description of the viscosity of the bulk mixture, or the concentration-weighted average

319

viscosity (von Boetticher et al., 2016). The complex mixture viscosity as it appears in (40) or (43) is newly

320

constructed here, which is rich in its physics, and evolves mechanically as a complex and coupled function of

321

several physical and mechanical parameters and dynamical variables including the solid volume fraction αs ;

322

solid and fluid material densities and viscosities (because the viscosities are kinematic), solid friction, yield

323

strength and rate of deformation of solid (that appears in νs ), and velocity and pressure (because νs containing

324

yield strength that presents on solid pressure) drift parameters, etc.

325

5.4

326

The pressure of the bulk mixture is defined in (22). The solid and fluid pressures can be expressed as

Mixture Pressure

αs ρs ps =

pm αs ρs , αs + λp αf

18

αf ρf pf =

(pm αf ρf )λp . αs + λp αf

(48)

327

With this, the mixture pressure can also be written as pm =

Λp (αs ρs ps + αf ρf pf ) . αs ρs + λp αs ρf

(49)

328

In terms of generalized mixture density (ρpm = αs ρs + λp αf ρf ) with respect to pressure drift λp , the mixture

329

pressure is pm =

Λp (αs ρs ps + αf ρf pf ) . ρpm

330

This constitutes a general concept of dynamically evolving (bulk) mixture pressure. From (22) and (49), we

331

have two equivalent representations of the mixture pressure. For dilute flows (if the extra pressure generated

332

by the grain contacts is very small, i.e., λp ≈ 1), then pm reduces to pm =

1 (αs ρs ps + αf ρf pf ), ρm

(50)

333

where, ρm = αs ρs + αf ρf . pm in (50) is the classical definition of the mixture (barycentric) pressure. So,

334

(49) is called the drift induced generalized barycentric pressure for the mixture flow. This shows that λp

335

may play a crucial role in the pressure dynamics. If the fluid pressure increases (say strongly, i.e., more than

336

hydrostatic), then pm becomes much greater than ps . In this situation, pm ≈ λp αf ps . If λp is very small, from

337

(48), we obtain pm = αs ps . The pressures ps and pf are dynamic. So, both can be positive, or negative, or

338

one of them positive and the other negative. Assume that ps > 0, and the fluid pressure is negative (λp < 0).

339

But, whether pm > 0 or < 0 depends on αs + λp αf . If αs + λp αf > 0, pm > 0, otherwise pm < 0. So,

340

one of the phases may have locally negative pressure, however the bulk pressure can still be positive. It may

341

depend on, how the negative fluid pressure develops and the amount of the locally available fluid. Many

342

other interesting possibilities can be described. For example, large shearing of the solid matrix can result in

343

dilatation leading to the decreased fluid pressure. This implies that bulk pressure can be smaller than the solid

344

pressure, because αs + λp αf < 1. On the contrary, during compaction of the solid phase, the fluid pressure

345

increases substantially, leading to the higher bulk pressure than that of the solid. Therefore, the dynamics of

346

the bulk pressure pm is much more complex than just the dynamics of the solid and fluid pressures separately.

19

347

5.5

Mixture Velocity

348

As for the bulk pressure, the bulk mixture velocities um = (αs + λu αf )us , wm = (αs + λw αf )ws can be

349

expressed as um =

350

αs ρs us + αf ρf uf αs ρs ws + αf ρf wf , wm = . Λu (αs ρs + λu αf ρf ) Λw (αs ρs + λw αf ρf )

With the generalized mixture densities with respect to velocity drifts λu and λw , ρum = αs ρs + λu αf ρf ; ρw m = αs ρs + λw αf ρf ,

351

the mixture velocities can be written in alternative forms as um =

1 1 (αs ρs us + αf ρf uf ) , wm = (αs ρs ws + αf ρf wf ) . Λu ρum Λw ρw m

(51)

352

These are called the drift induced generalized barycentric velocities, or simply the generalized mixture veloc-

353

ities. In special situation when the velocity drifts are close to unity, these expressions reduce to the classical

354

barycentric velocities: um =

1 1 (αs ρs us + αf ρf uf ) , wm = (αs ρs ws + αf ρf wf ) . ρm ρm

(52)

355

As for the pressure, the properties of the generalized velocities can be discussed.

356

6

357

The models obtained in (25)-(27) can be further simplified. Here, we derive reduced models as characterized

358

either by the volume fraction, or by the drift coefficients.

359

6.1

360

Up to now, no assumption has been made on the solid (or, fluid) volume fraction αs . As it stands now, αs

361

appears to be an internal variable. There are two possibilities. (i) αs can be considered as a state variable

362

which is the general situation. This requires an extra closure or, a transport equation namely, the advection-

363

diffusion (Iverson and Denlinger, 2001; Pudasaini et al., 2005):

Reduced Models

Characterization with the Volume Fraction

∂αs ∂αs ∂αs ∂ 2 αs ∂ 2 αs + us + ws − Dx 2 − Dz 2 = 0, ∂t ∂x ∂z ∂ x ∂ z 20

(53)

364

which says that αs advects with the velocity (us , ws ) and diffuses with (Dx , Dz ). Exact solutions can be

365

constructed in computation assuming that (us , ws ) and (Dx , Dz ) are known for some particular time step.

366

This can be updated as required. This way, the model equations (25)-(27) and (53) are dynamically coupled.

367

However, much advanced sub-advection, sub-diffusion models (Pudasaini, 2016) can also be utilized. (ii)

368

In simple situations, either αs can be parameterized, or be considered as constant, which is the scenario in

369

homogeneous flow.

370

6.2

371

6.2.1

372

In situations when the solid and fluid pressure are close to each other, the pressure drift coefficient is close

373

to unity, i.e., λp ≈ 1. When instantaneous pressures are in equilibrium, this can be a valid assumption, for

374

example, if the curvature and/or the surface tension is negligible. This holds only for substantially dilute

375

flows with very low particle contacts. When the contact area is a small fraction of the total interfacial area

376

(e.g., loose/intermittent contacts), interfacial solid and fluid pressures can be approximated by fluid pressure

377

(Kuo et al., 1976; Gough, 1980; Drew, 1983). Then, the model equations (9), (20) and (21) reduce to

Characterization with Drifts Pressure Drift Factor Close to Unity

i i ∂ h ∂ h (αs + λu αf )us + (αs + λw αf )ws = 0, ∂x ∂z

378

 ∂ ∂  ∂ ∂ps [(αs + λu αf )us ] + (αs + λ2u αf )u2s + [(αs + λu λw αf )us ws ] = fx − ∂t ∂x ∂z ∂x     ∂ ∂us ∂ ∂us ∂ws (νs αs + λu νf αf ) + (νs αs + λu νf αf ) , (54) ppppP P + 2 + (νs αs + λw νf αf ) ∂x ∂x ∂z ∂z ∂x ∂ ∂ ∂ ∂ps [(αs + λw αf )ws )] + [(αs + λu λw αf )us ws ] + [(αs + λ2w αf )ws2 )] = fz − ∂t ∂x ∂z ∂z     ∂ ∂us ∂ws ∂ ∂ws ppppP P + (νs αs + λu νf αf ) + (νs αs + λw νf αf ) +2 (νs αs + λw νf αf ) . (55) ∂x ∂z ∂x ∂z ∂z

379

There are only two velocity drifts remaining in these equations.

21

380

6.2.2

Identical Velocity Drift Factors

381

The velocity drift factors may have the identical values in the x- and z-directions: λu = λw 6= 1. Then, (9),

382

(54) and (55) reduce to i i ∂ h ∂ h (αs + λu αf )us + (αs + λu αf )ws = 0, ∂x ∂z

  ∂  ∂  ∂ps ∂ [(αs + λu αf )us ] + (αs + λ2u αf )u2s + (αs + λ2u αf )us ws = fx − (αs + λp αf ) ∂t ∂x ∂z ∂x      ∂ ∂us ∂ ∂us ∂ws ppppP P + 2 (νs αs + λu νf αf ) + (νs αs + λu νf αf ) , + ∂x ∂x ∂z ∂z ∂x

383

∂ ∂ ∂ ∂ps [(αs + λu αf )ws )] + [(αs + λ2u αf )us ws ] + [(αs + λ2u αf )ws2 )] = fz − (αs + λp αf ) ∂t ∂x ∂z ∂z      ∂us ∂ws ∂ ∂ws ∂ (νs αs + λu νf αf ) +2 (νs αs + λu νf αf ) . ppppP P + + ∂x ∂z ∂x ∂z ∂z

(56)

(57)

(58)

384

As compared to the general situation, the advantage now is that the bulk mixture viscosity is simply given by

385

the single expression νs αs + λu νf αf , which, however, still evolves as a function of the velocity drift λu , and

386

other physical and dynamical variables included in νs and αs .

387

6.2.3

388

In the most simple situation, when the velocity drifts are close to unity (λu , λw ≈ 1), (56)-(58) further reduce

389

to

Velocity Drift Factors Close to Unity

∂us ∂ws + = 0, ∂x ∂z

(59)

     ∂us ∂ ∂us ∂ws Λm + Λm + , ∂x ∂z ∂z ∂x      ∂ws ∂(us ws ) ∂(ws2 ) ∂ps ∂ ∂us ∂ws ∂ ∂ws + + = fz − + Λm + +2 Λm , ∂t ∂x ∂z ∂z ∂x ∂z ∂x ∂z ∂z ∂us ∂(u2s ) ∂(us ws ) ∂ps ∂ + + = fx − +2 ∂t ∂x ∂z ∂x ∂x

(60) (61)

390

where Λm = νs αs + νf αf is now the traditional simple bulk mixture viscosity which is independent of the

391

velocity drift. And, since the velocity drifts are close to unity, the pressure drift may also be close to unity,

392

which is legitimate. However, for similar solid and fluid velocities, the solid and fluid pressures can be different.

393

Even the simplest set of equations (59)-(61) is important because this includes the first order basic dynamics

394

of evolving viscosity that governs the dynamics of the mixture flow.

22

395

7

The Flow Rheology

396

The mixture motion and the settlement are described by constitutive law. Geophysical mass flows can be mod-

397

elled by viscoplastic rheology (Jop et al., 2006; Ancey, 2007; Balmforth et al., 2007; Forterre and Pouliquen,

398

2008). Domnik et al. (2013) proposed the pressure and rate-dependent Coulomb viscoplastic rheological

399

model to describe the full dynamics of the rapid flows of granular materials down the channels impinging on

400

rigid walls. Unlike Herschel-Bulkley, or Bingham plasticity, Domnik et al. (2013) model does not include

401

any fit parameter, and is fully described by the phenomenological parameters of the materials. With several

402

flow simulations, from the material collapse and silo outlet to the final depositions, they have demonstrated

403

the applicability of their new rheological model with very high computational performances. Their model

404

is capable of describing several important observation in wide range of geophysical mass flows as recently

405

demonstrated by von Boetticher et al. (2016, 2017).

406

Iverson (1997), Iverson and Denlinger (2001), Massoudi (2003, 2010) and Pudasaini et al. (2005) used the

407

total stress in a mixture as the sum of solid and fluid-phases stress tensors. On contrary, our approach is

408

completely different. We employ the mixture pressure, velocity and mixture shear viscosity (as constructed

409

in Section 5, and mixture strain-rate tensor, see below), to construct the mixture stress tensor. So, flow

410

rheology intrinsic to the model (25)-(27) is rather complex and non-conventional. It simultaneously includes

411

the physical and dynamical properties of both the solid and fluid components. From (25)-(27), the Cauchy

412

stress tensor for the mixture takes the form: σ m = −pm I + 2Ληm Dm ,

413

(62)

where, pm is the normalized pressure (49), Ληm is the mixture viscosity (43) 1 1 Ληm = νse αs (Λu + Λw ) + νf αf (λu Λu + λw Λw ). 2 2

414

Λ T Λ Dm = 12 [∇uΛ m + (∇um ) ] is the strain-rate tensor for mixture, where um = (Λu um , Λw wm ). So, the velocity

415

gradient terms in Dm are in extended and non-conventional form, and reduce to the standard form if Λu =

416

Λw = 1.

23

417

The effective kinematic viscosity for solid νse = νs +

τys ||Dm ||

(63)

418

tends to infinity as ||Dm || → 0. To overcome this problem in computation, Domnik et al. (2013) and von

419

Boetticher et al. (2016, 2017) introduced the exponential factor my as νse = νs +

 τys  1 − e−my ||Dm || . ||Dm ||

(64)

420

To capture typical flow behavior of a geophysical mass flows, Domnik et al. (2013) proposed a pressure

421

dependent yield stress τys = τc + τp pm /Λp , where τc is cohesion. Without cohesion, this forms a Drucker-

422

Prager yield criterion (Prager and Drucker, 1952) q ΠσDm > τp pm ,

(65)

423

where, ΠσDm is the second invariant of the deviatoric stress tensor (Prager and Drucker, 1952). The relation

424

(65) simply tells us that the mixture material undergoes plastic yielding when deviatoric stress is greater than

425

the yield stress. With several complex flow simulations, from silo outlet to the flow obstacle interactions, and

426

depositions with solid-fluid and fluid-solid transitions, Domnik et al. (2013) and von Boetticher et al. (2016,

427

2017) demonstrated that the pressure-dependent yield stress discussed above is a very efficient rheological

428

model.

429

8

430

To solve the equations (25), (28)-(31) numerically, we need boundary conditions. Usually, the free surface of

431

the debris bulk is assumed to be traction free. The friction is induced by the movement of debris mixture

432

on the sliding plane. Consider the normal vector n and a tangential vector t on sliding plane. In rapid

433

flows of debris material down the slopes, even the lowest particle layer in contact with the bottom boundary

434

moves with a non-zero and non-trivial velocity. Therefore, the generally used no-slip boundary condition does

435

not represent the flow physics (Domnik and Pudasaini, 2012). A non-zero slip velocity is determined by the

436

frictional strength, which depends on the load the material exerts on the rigid boundary. So, as in Domnik

437

b = σ b n · t to the normal pressure N b = σ b n · n at the sliding et al. (2013), we relate the shear-stress Tm m m m

Boundary Conditions

24

438

surface (b). This can be achieved by using the Coulomb sliding law for bulk: b Tm =

ubm b tan δm Nm , |ubm |

(66)

439

where the tangent of the effective bed friction angle δm for the mixture defines the “effective proportionality

440

constant”, and ubm is the velocity of the bulk at the base. Consequently, δm ≤ δs implies the emergence of a

441

new effective mixture friction coefficient µm = tan δm . This is uniquely defined via (66), and reveals that µm

442

or, δm , can evolve dynamically. Higher values of tan δm go along with a higher shearing and, therefore, with

443

a higher frictional force at the rigid boundary. The relation (66) asserts that the material yields plastically if

444

the shear stress attains the critical value. The flowing material is assumed to be non penetrating through the

445

boundary. So, n · ubm = 0 and also [(t · ∇)(n · um )]b = 0. Then, from (62), the normal and shear stresses at

446

the bottom are given by b Nm = pbm + 2Λbηm [(t · ∇)(t · um )]b ,

b and Tm = Λbηm [(n · ∇)(t · um )]b ,

(67)

447

which has been expressed in terms of the tangential velocity and the pressure at the boundary, and leads to

448

a pressure and rate-dependent Coulomb-viscoplastic sliding law. Using (67) in (66), one obtains [(n · ∇)(t · um )]b − 2cFm [(t · ∇)(t · um )]b =

449

450

cFm b p = Ferm pbm , Λbηm m

where the ratio between the debris friction and the viscous friction is given by Ferm =

(68) cFm (the effective Λbηm

friction ratio), and the friction factor cFm is defined by  b   um b    |ub | tan δm , um 6= 0, m cF =      0, ubm = 0.

451

The pressure and rate-dependent Coulomb-viscoplastic sliding law for generalized bulk mixture (68), dynami-

452

cally defines the bottom boundary velocity [t · um ]b (Domnik and Pudasaini, 2012). In case, if the bed friction

453

angle δm = 0, then there is no solid material friction, and cFm = 0. This results in the free slip condition

454

[(n · ∇)(t · um )]b = 0, so the basal shear-stress vanishes. For very high shear viscosity, Ferm ≈ 0, and there is

455

no more pressure dependency. So, (68) becomes 2cFm =

[(n · ∇)(t · um )]b , [(t · ∇)(t · um )]b 25

(69)

456

i.e., the friction factor is the ratio of the normal to the tangential velocity at the bottom.

457

We ponder the more general case, Ferm 6= 0. If we consider the flow in a narrow rectangular inclined channel

458

with the main flow direction parallel to the x-axis, the shearing mainly occurs at the bottom boundary with

459

the basal normal vector parallel to the z-direction. The shearing at the sidewalls may not have a significant

460

influence on the flow (Pudasaini et al., 2007; Domnik et al., 2013), and so will be neglected. Then, the

461

Coulomb shear-stress at the bottom (66) can be written as b Tm

=

cFm

"

pbm

+

2Λbηm



∂um ∂x

b #

.

(70)

462

So, we have extended the pressure- and rate-dependent Coulomb-viscoplastic granular flow rheology (Domnik

463

and Pudasaini, 2012; Domnik et al., 2013) to debris mixture flow. Moreover, the extended sliding law (68),

464

for the rectangular inclined channel is given by 

∂um ∂z

b

−

2cFm



∂um ∂x

b

=

cFm b p · Λbηm m

(71)

465

The equation (71) is for the pressure in terms of velocity gradient or vice-versa (Domnik and Pudasaini, 2012).

466

So, the pressure dependent velocity is employed as the boundary condition at the sliding surface and the top

467

surface is traction free. Furthermore the bottom shear stress in (71) depends only on the normal stress, i.e.,

468

on the velocity near the sliding surface.

469

We further employ the von Neumann boundary condition for the pressure by applying the momentum con-

470

servation in the direction of the normal at the rigid boundaries, i.e., n · 4pm = n · F·

(72)

471

This closes the derivation of the model equations and the boundary conditions for a generalized quasi two-

472

phase mixture mass flow.

473

9

474

The full dimensional (non-depth-averaged) simulation of the debris mixture flow is a new topic. However,

475

here, we mention some similarities and major differences between our new model and some other seemingly

476

similar or, alternative models existing in the literature. This will make it clear the usefulness of the proposed

Essence of the New Model

26

477

model and its application potential.

478

Since, in general, all the coefficients in (36) are away from unity, even in reduced form, the convective and

479

viscous terms in (26) and (27) are different than in Domnik and Pudasaini (2012) and Domnik et al. (2013).

480

Differences emerge from the fact that all Λ’s may vary as functions of αs ’s and λ’s. Furthermore, (um , wm )

481

and pm are state variables for the mixture, from which, with the knowledge of αs and λ, the state variables

482

for the solid (us , ws ; ps ), and for the fluid (uf , wf ; pf ) can be reconstructed. From the technical point of view,

483

this is a major advantage of the proposed model that could not have been achieved from classical formulations

484

of the mixture models (Iverson and Denlinger, 2001; Pudasaini et al., 2005).

485

As an alternative model for the debris mixture flows, we consider von Boetticher et al. (2016) in which the

486

bulk density is computed as a linear combination of the phase densities from the knowledge of the volume

487

fractions as transport quantities. In our formulation, we do not require to dynamically compute the bulk

488

density. So, we do not have to deal with the difficulties with boundary conditions in simulations with the

489

mixture density gradients and its evolutions as in von Boetticher et al. (2016). This stems from the fact

490

that in von Boetticher et al. (2016), the bulk transport equations are considered whereas we begin with two

491

explicit mass and momentum balances for the solid and fluid phases. Then, we constructed the bulk mixture

492

model consisting of the mass and momentum balances for the mixture. This resulted in the emergence of the

493

velocity and pressure drift factors, inertial and dynamical coefficients for mixture velocities, viscosities and

494

pressures. Our simple formulation provides possibilities for the construction of the full dynamics of the solid

495

and fluid phases. This also allowed to construct several technically important reduced models that can be

496

applied from simple to complex flow situations. The inertial, pressure and viscous terms in (33)-(35) appear

497

in non-classical forms. Effective viscosity takes a very general form. The bulk and shear viscosities, that are

498

in extended and general form, are different. However, the reduced bulk and shear viscosities for the mixture

499

appear to be equivalent. This representative mixture viscosity has further been reduced to simple mixture

500

viscosity as in von Boetticher et al. (2016) that also includes the power law rheology together with the yield

501

strength for the slurry fluid, whereas we model that as a non-Newtonian viscous fluid. In von Boetticher et

502

al. (2016, 2017), the effective mixture viscosity is a complex linear combination between the phase viscosities,

503

whereas in our formulation the mixture viscosities are complex and non-linear combinations between the solid

27

504

and fluid-phase viscosities. The same applies for the velocities and pressures in our model while in the von

505

Boetticher et al. (2016) model the velocities and pressure are simply the bulk velocities and pressure.

506

10

507

The depth-averaged version of the two-phase mass flow model (Pudasaini, 2012) has been successfully applied

508

to different flows of mixture materials. Here, we focused on some particular aspects of the full-dimensional

509

mixture flows down a channel. First, we considered the general three-dimensional and two-phase particle-fluid

510

mixture mass flow model. Then, in order to develop a reduced but, generalized quasi two-phase bulk mixture

511

model, we introduced the solid and fluid velocity drift coefficients, and also the pressure drift coefficient. From

512

the application and computational point of view, this is useful, as it reduces the full two-phase model to quasi

513

two-phase bulk model. This paves the way to apply the simple single-phase computational tools to numerically

514

solve the new model equations. At the same time, the drifts contain important information relating the solid

515

and fluid velocities, and the dynamic pressures. This makes the resulting model a generalized quasi two-phase

516

bulk mixture model.

517

We developed two different mixture viscosities and pressures. The drift induced generalized mixture velocities

518

are obtained. The effective bulk and shear viscosities are obtained by respectively averaging two alternative

519

bulk and shear viscosities appearing in the model derivation. We noticed that the bulk and shear viscosities

520

are identical. This can be considered as the mixture viscosity. The constructed mixture viscosity is rich in

521

its physics and evolves mechanically as a coupled function of several physical and mechanical parameters and

522

dynamical variables including the solid volume fraction, solid and fluid material densities and viscosities, solid

523

friction and yield strength, rate of deformation of solid, and velocity and pressure drift parameters. This

524

shows that the higher are the fluid drifts, faster is the fluid motion, and thus, higher is the deformation. This

525

results in the lower bulk viscosity. In simple situations, the viscosity can be further reduced to the simplest

526

description of the viscosity of the bulk mixture. We have also introduced a general concept of dynamically

527

evolving drift induced generalized pressure for the bulk mixture flow. This shows that the pressure drift

528

coefficient plays a crucial role in the pressure dynamics. We reveal that, during large shearing, dilatation and

529

compaction, the mixture pressure may respond completely differently as compared to the solid or fluid pressure

Discussions and Summary

28

530

separately. Therefore, the dynamics of the bulk pressure is much more complex than just the dynamics of

531

the solid and fluid pressures separately. For dilute flows, it reduces to the classical definition of the mixture

532

pressure. The emergence of an effective mixture friction coefficient, that evolve dynamically, reveals one of the

533

most important mechanical aspects of our mixture model. We have extended the pressure- and rate-dependent

534

Coulomb-viscoplastic granular flow rheology to debris mixture flow. Such a sliding law for generalized bulk

535

mixture dynamically defines the bottom boundary velocity.

536

With the constructed bulk mixture velocities and the pressure, the inertial coefficients, and the coefficients

537

of dynamical mixture viscosities and pressure, our model for bulk mixture flow consists of the full two-

538

dimensional quasi two-phase mass and momentum conservation equations. It appears that the dynamics of

539

the bulk pressure, the inertial and dynamical coefficients are complex and coupled, and explicitly include the

540

physics and dynamics of mixture. The final model equations are written as a well structured system of three

541

non-linear partial differential equations in conservative form with three unknowns, namely, the generalized

542

mixture velocities and pressure. This system is written in a convenient form for numerical simulations.

543

Moreover, the newly derived mass and momentum equations appear in non-conventional form due to the

544

inertial coefficients, and the non-equality of the mixture viscosities.

545

We outlined some important physical aspects, implications, and applicabilities of the new model. The model

546

has a potential for a wider applications, it will be computationally faster than the full two-phase simulations,

547

and, at the same time, more accurate than the single-phase models. The great advantages of the coupled

548

and generalized model is that structurally it is the same as the existing single phase granular or debris flow

549

model. So, in principle, the same computational strategy can be applied to solve this new system. However,

550

the complexity and generality inherited by the two-phase bulk mixture flow are contained in the inertial,

551

dynamical and pressure coefficients. Once mixture velocity and pressure are obtained, the two-phase dynamics

552

of solid and fluid can be re-constructed from the definition of the mixture quantities and drift coefficients.

553

We derived reduced models as characterized either by the volume fraction, or by the drift coefficients. Solid

554

volume fraction appears to be an internal variable for which an evolution equation can be constructed, or

555

simply parameterized. In the most simplified situation, the model equations consist of a single velocity drift

556

parameter. This is advantageous, because there are analytical expressions available for this. Alternatively, 29

557

this factor can be obtained from the fast simulations of the depth averaged equations. This allows us to

558

dynamically obtain the drift coefficient. For simple flow configuration with unit drift coefficients, the emerging

559

model reduces to traditional effectively single-phase bulk mixture model. In special situation when the velocity

560

and drifts are close to unity, the generalized mixture velocities, and pressure reduce to the classical barycentric

561

velocities and pressure. This further reduces the complexity of the model system. For the identical velocity

562

drift coefficients, the generalized viscosities reduce to a single classical mixture viscosity. In the most simple

563

flow configuration, if further the velocity drifts are close to unity, then the mixture viscosity becomes the

564

traditional simple bulk mixture viscosity. Even the simplest set of equations is important because this includes

565

the first order basic dynamics of evolving mixture viscosity that governs the dynamics of the mixture flow.

566

Acknowledgments

567

We thank the reviewers, and the editor Giuseppe Saccomandi for their constructive reviews and suggestions

568

that resulted in the substantially improved paper. The idea of the model development and the model reduction

569

was generated by Shiva P. Pudasaini. We gratefully acknowledge the financial support provided by the German

570

Research Foundation (DFG) through the research project, PU 386/3-1: “Development of a GIS-based Open

571

Source Simulation Tool for Modelling General Avalanche and Debris Flows over Natural Topography”. Puskar

572

R. Pokhrel acknowledges University Grants Commission (UGC), Nepal for the financial support provided as

573

a PhD fellowship.

574

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