Stream function-vorticity formulation of mixture mass flow

Journal Pre-proof Stream function-vorticity formulation of mixture mass flow Puskar R. Pokhrel, Shiva P. Pudasaini

PII: DOI: Reference:

S0020-7462(19)30406-8 https://doi.org/10.1016/j.ijnonlinmec.2019.103317 NLM 103317

To appear in:

International Journal of Non-Linear Mechanics

Received date : 8 June 2019 Revised date : 8 September 2019 Accepted date : 13 October 2019 Please cite this article as: P.R. Pokhrel and S.P. Pudasaini, Stream function-vorticity formulation of mixture mass flow, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103317. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier Ltd.

Journal Pre-proof

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Stream function - vorticity formulation of mixture mass flow

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Puskar R. Pokhrel1,2 , Shiva P. Pudasaini3 Pp1 Department of Mathematics, School of Science, Kathmandu University, Dhulikhel, Kavrepalanchok,

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Ppp Nepal

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PP2 Department of Mathematics, RR Campus, Tribhuvan University, Kathmandu, Nepal

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Pp 3 Institute of Geosciences, Geophysics Section, University of Bonn, Meckenheimer Allee 176, D-53115

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PpppBonn, Germany

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PPp Correspondence to: [email protected]

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Abstract: Employing a generalized quasi two-phase bulk mixture mass flow model derived from

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a general two-phase model (Pudasaini, 2012), here, we formulate a stream function - vorticity and

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vorticity-transport equation for a rapid flow of mixture of viscous fluid and solid particles down a

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channel. The original system of partial differential equations (PDEs) in velocity and pressure is

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converted into the reduced stream function - vorticity form as a close system of equations that is

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free of pressure which replaces the original system of three equations by a set of just two PDEs as

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an advantage of the new model. A novel pressure Poisson equation in terms of stream function,

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vorticity and the rate-dependent mixture viscosity is derived. For given mixture viscosity, our

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pressure Poisson equation can provide mixture pressure. The two equations are coupled through

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the Poisson equation to provide the full system in stream function, vorticity and mixture pressure

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to describe the dynamics of mixture flow. However, the pressure is decoupled and can also be

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computed separately using stream function and vorticity. We further reduce the new system to

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obtain exact expressions for stream function. One of the most important advancements here is

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the construction of a new pressure Poisson equation for shear mixture flows that includes yield

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strength of the mixture. We also discuss the importance of pressure Poisson equation induced by

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the flow field intensity, the yield strength, and free surface geometry. Our results show that the

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pressure Poisson equation is mainly characterized by the non-linear diffusion of the free surface.

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Furthermore, mixture pressures are derived analytically for thin and thick flows. Similarly, differ-

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ent flow scenarios such as pressure dominated flow; thick, low yield strength flow; and thin, high

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yield strength flows are analyzed. Several exact/analytical solutions are constructed for the pres-

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sure and flow depth distributions for incipient flow, shearing flow, free surface flow, propagating

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bore front and mass deposition. The novel models developed here and analytical results are con-

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sistent with the observed phenomena indicating their application potential in detailed description

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of the mixture flow dynamics more efficiently than the existing complex models.

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Keywords: Two-phase mass flow, Stream function, Vorticity, Vorticity-transport equation, Pres-

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sure Poisson equation, Analytical solutions.

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Introduction

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Debris flows, landslides and avalanches are some examples of mass transport phenomena in na-

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ture. Debris flows, generally occur in mountainous areas throughout the world, are extremely

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destructive and dangerous natural events. During these event, the mixture material undergoes

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rapid motion and large deformations (Kattel et al., 2018). Debris flows may also claim human

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lives and cause massive destruction of infrastructures (Pitman and Le, 2005; Pudasaini and Hut-

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ter, 2007; Mergili et al., 2017, 2018 a, 2018 b). A debris flow, as a typical example of geophysical

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mass flows, are gravity driven flows caused by intense rainfall or a sudden surge of water as a

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mixture of viscous fluid and granular solid particles down a slope (Takahashi, 1991; Hutter et al.,

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1996; Pitman and Le, 2005; Pudasaini, 2012). Such surges may occur due to the breaking off of

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a saturated landmass, or by a landslide impacting a river or lake, that causes overflow and may

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result into catastrophic dam collapse (Pudasaini 2012; Kattel et al., 2016; Mergili et al., 2017,

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2018 a).

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To properly describe the flow dynamics, Pudasaini (2012) has developed a generalized two-phase

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mass flow model by unifying the dry granular avalanche model of Savage and Hutter (1989),

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debris mixture flow model of Iverson and Denlinger (2001), and the two-fluid debris flow model

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of Pitman and Le (2005). This model reveals strong coupling between solid and fluid particles

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through inter-facial momentum transfer. The model includes several important and dominant

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physical aspects of flow such as the generalized drag force, virtual mass force, buoyancy forces

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and enhanced non-Newtonian viscous stress.

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Using the general two-phase mass flow model of Pudasaini (2012), Pokhrel et al. (2018) de-

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veloped a new generalized two-dimensional quasi-two-phase bulk mixture model which includes

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effective mixtures viscosity that evolves mechanically as a coupled function of several physical

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and mechanical parameters including dynamical variables. The new model consists of generalized

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mixture velocities and pressure. The model appears in non-conventional form due to the inertial

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coefficients, and the complex mixture viscosities. This structure resulted in the emergence of a

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new and dynamically evolving effective mixture friction coefficient, and general mixture viscosity,

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and reveals important mechanical aspects of the model. Later, by applying the same model formu-

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lation, Khattri and Pudasaini (2018) produced an extended model so as to include further physics

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of two-phase flow such as virtual mass force, generalized drag and non-Newtonian viscous stress

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of mixture mass flow. In this contribution, we present a stream function - vorticity formulation

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of the mixture mass flow model in Pokhrel et al. (2018).

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In fluid dynamics, stream function is used in flow visualization for incompressible flow. The ve-

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locity components are determined as the derivatives of some stream function, and the vorticity

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vector is defined by the flow field. It is an important concept and related to the average angu-

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lar momentum of a fluid particle, and the flow with circular streamlines (Chen and Xie, 2016).

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Significant research has been carried out on the formulation of the problem in terms of stream

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function, vorticity, velocity and pressure fields, for example, the groundwater flow (Slichter, 1897;

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Senger and Fogg, 1990). The solutions of two-dimensional variable-density ground water flow

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problems have been achieved using stream function (Anderson and Woessner, 1992; Evans and

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Raffensperger, 1992). Stokes, Navier-Stokes, and Stokes-Darcy problems have been formulated

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in terms of stream function, vorticity, velocity and pressure fields with their numerical solutions

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using different techniques (Ern et al., 1999; Cockburn and Cui, 2012; Alvarez et al., 2016). Anaya

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et al. (2015, 2016) have constructed the vorticity, velocity, and pressure field of the Brinkman

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problem.

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Following Pokhrel et al. (2018), our modeling approach first obtains a kinematic equation cou-

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pling the stream function and the vorticity for a mixture mass flow. This resulted in a general

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vorticity-transport equation that depends on the dynamics of the stream function and mixture

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viscosity. Importantly, we have derived a novel pressure Poisson equation for the mixture flow in

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terms of the mixture viscosity, stream function and vorticity, so that it can provide the mixture

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pressure for a given mixture viscosity. The importance of the construction of the explicit form of

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the pressure lies in its practical application as it is much harder to obtain or measure real pressure

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of the flow. Furthermore, the generalized pressure Poisson equation here exclusively includes shear

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viscosity for the mixture flow. This makes our model mechanically and structurally novel. We

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have developed a full system of equation in stream function, vorticity and mixture pressure that

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describe the dynamics of mixture motion. The stream function retains the flow properties of the

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mixture. The various reduced model equations are constructed by employing constant vorticity

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and isotropic stream function. A vorticity-transport equation is reduced into the model equation

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in terms of mixture viscosity and stream function for which exact solutions are constructed for

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uniform flow. A general form of stream function and velocity field of the reduced system are also

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constructed. The pressure Poisson equation has been reduced for different flow situations, and it

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has been analyzed for different physical scenarios. We also establish an exact analytical solution

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for non-hydrostatic pressure distribution in mixture flow. The pressure is high for high strength

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flow. Moreover, the pressure field is quadratic with flow depth, but with a complex non-linear

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coefficient that varies with the flow geometry or dynamics of free surface. Such a special pressure

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profile is developed here for the first time. We also construct a complex, and exact/analytical

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solution for free surface of the flow, and modeled the flow depth along the slope. The pressure

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solutions for mixture flow are constructed from the pressure Poisson model for different physical

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situations. The dynamics of the source terms in flow dynamics of the new pressure Poisson model

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is exclusively presented. Further analysis of the source terms is carried that resulted in the con-

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struction of analytic/exact solutions. Furthermore, various reduced models of the new pressure

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Poisson equation for shear mixture flow are developed and analyzed by transforming the model

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into non-dimensional form. This allowed in constructing several exact/analytical solutions of pres-

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sure Poisson equation with different physical scenarios and flow situations. We obtain a quadratic

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pressure profile which is an important extension of classical hydrostatic pressure distribution. For

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the dynamic pressure dominated flow, the total pressure distribution must be quadratic. Con-

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struction of all these novel and exact/analytical solutions for mixture mass flow problems were

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possible due to the new stream function - vorticity formulation of the quasi two-phase bulk mix-

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ture mass flow model (Pokhrel et al., 2018). The proposed models and their different solutions

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presented here indicate the application potential of our work.

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For incompressible two-dimensional flows, quasi two phase mass flow model equations (Pokhrel et

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al., 2018) can be simplified by introducing the stream function ψ and vorticity ω as new dependent

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variables. For this, first we present a pressure Poisson equation for stream function and vorticity.

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Let v = (um , wm ) be the velocity components in the down-slope (x) and perpendicular to the

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channel surface (z), respectively (see, Fig. 1). And v can be expressed as

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Stream function - vorticity Poisson equation

v = um i + wm j = (um , wm ).

(1)

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For the irrotational motion, there exists a scalar potential function φ(x, z) such that the vector

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field can be written in terms of their gradients,

∂φ ∂φ i+ j. ∂x ∂z

(2)

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v = ∇φ(x, z) =

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From (1) and (2), one obtains um = ∂φ/∂x and wm = ∂φ/∂z. The function φ(x, z) is called the

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velocity potential, and the curves φ(x, z) = cφ give equipotential lines (curves) for different values

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of cφ . For incompressible fluid, divergence of the vector field v is zero, i.e.,   ∂ ∂ ∂2φ ∂2φ ∂um ∂wm ∇·v= i +j · (um i + wm j) = + 2 = + = 0. 2 ∂x ∂z ∂x ∂z ∂x ∂z

(3)

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This shows that the velocity potential is harmonic. Therefore, there exists a conjugate harmonic

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function ψ(x, z) such that a complex function w(z) = φ(x, z) + iψ(x, z) is analytic. Thus, φ and 4

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Figure 1: Side view of the rapid flow of debris material along an inclined chute. ζ is the inclination

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angle, umin , hin are inflow velocities and heights of the mixture flow (modified from Domnik and Pudasaini, 2012).

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ψ satisfy the Cauchy-Riemann equations:

∂φ ∂ψ =− . ∂z ∂x

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∂φ ∂ψ = , ∂x ∂z

(4)

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The function ψ(x, z) is called the stream function, and the curves ψ(x, z) = cψ , for different values

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of cψ , are called stream lines. Moreover, the slope at any point at the curve ψ(x, z) = cψ is dz ∂ψ/∂x ∂φ/∂z wm =− = = . dx ∂ψ/∂z ∂φ/∂x um

(5)

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This shows that the velocity of the flow is tangent to the curve ψ(x, z) = cψ . The vorticity ω is

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defined mathematically as the scalar product of unit normal with the curl of velocity vector: ω = (∇ × v) · k =

m

∂x

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∂um  ∂wm ∂um k·k= − . ∂z ∂x ∂z

(6)

Using um = ∂φ/∂x = ∂ψ/∂z and wm = ∂φ/∂z = −∂ψ/∂x in (6), we get

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 ∂w

∂2ψ ∂2ψ + = −ω. ∂x2 ∂z 2

(7)

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This is a kinematic equation connecting the stream function ψ and the vorticity ω. So, if we can

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model, parameterize, or, construct an equation for ω, we will have obtained a formulation that

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automatically produces divergence-free velocity field. We call (7) the stream function vorticity

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Poisson equation. 5

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Stream function - vorticity formulation of mixture mass flow

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3.1

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The components of velocities um and wm for the mixture flow in the down-slope and the normal

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to the flow direction, respectively, and the pressure for the mixture flow pm describe the rapid

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motion of a (debris) mass flow in a channel. The suffix m stands for the mixture. A generalized

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quasi two-phase bulk mixture model for mass flow (Pokhrel et al., 2018) are as follows:

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A generalized quasi two-phase bulk mixture model

∂um ∂wm + = 0, ∂x ∂z

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

(9)

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∂wm ∂ ∂ ∂pm 2 + (Λuw um wm ) + (Λww wm ) = fz − ∂t ∂x ∂z ∂z     ∂(Λu um ) ∂(Λw wm ) ∂ ∂(Λw wm ) ∂ Ληu + Ληw +2 Ληw . PPPPPPPPP + ∂x ∂z ∂x ∂z ∂z

(8)

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In the above equations, t is time, pm = (αs + λp αf )ps , um = (αs + λu αf )us , wm = (αs + λw αf )ws ,

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where λp is pressure drift factor, and λu , λw are velocity drift factors. αs , αf (= 1 − αs ) denote

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the volume fractions for the solid and the fluid components in the mixture, denoted by the suffices

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s and f respectively. f = (fx , fz ), where fx and fz are the components of the gravitational

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acceleration, Ληu = νse αs + λu νf αf , Ληw = νse αs + λw νf αf are the mixture viscosities, where νse

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for mixture and my is the exponential factor introduced for smooth transition between yielded

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and unyielded regions (Papanastasiou, 1987; Domnik and Pudasaini, 2012; Domnik et al., 2013).

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The generalized bulk mixture viscosity for the mixture mass flow is constructed (Pokhrel et al.,

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2018) as

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is the effective kinematic viscosity for the solid, and νf is the kinematic viscosity for the fluid. τy s νse = νs + (1 − e−my ||Dm || ), where νs is kinematic viscosity for solid, τy s = τc + τp pm /Λp is a ||Dm || pressure dependent yield stress, where Λp = αs + λp αf , τc is cohesion, Dm is the strain-rate tensor

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1 1 Ληm = νse αs (Λu + Λw ) + νf αf (λu Λu + λw Λw ), 2 2

(11)

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where Λu = 1/(αs + λu αf ), Λw = 1/(αs + λw αf ). If the velocity and pressure drifts are close to

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unity, i.e., λp , λu , λw ≈ 1, then (8)-(10) reduce to 6

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∂um ∂wm + = 0, ∂x ∂z

(12)

 ∂um ∂wm i + , (13) ∂z ∂x   2 ) ∂wm ∂pm ∂ h  ∂um ∂wm i ∂ ∂wm ∂(um wm ) ∂(wm Λm , (14) + + = fz − + Λm + +2 ∂t ∂x ∂z ∂z ∂x ∂z ∂x ∂z ∂z



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∂um ∂(u2m ) ∂(um wm ) ∂pm ∂  ∂um  ∂ h + + = fx − +2 Λm + Λm ∂t ∂x ∂z ∂x ∂x ∂x ∂z

where Λm = νse αs + νf αf . As discussed in Pokhrel et al. (2018), and Khattri and Pudasaini

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(2018), even the simplest set of equations (12)-(14) is important because this includes the first

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

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3.2

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Now, we write (12)-(14) in terms of stream function. Partially differentiating (13) with respect to

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z, and (14) with respect to x, and assuming constant fx , fz that is valid for inclined channel, we

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get

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Stream function - vorticity formulation

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∂  ∂um  ∂ 2 (u2m ) ∂ 2 (um wm ) ∂ 2 pm ∂2  ∂um  ∂ 2 h  ∂um ∂wm i + + = − +2 Λ + 2 Λm + , (15) m ∂z ∂t ∂z∂x ∂z 2 ∂z∂x ∂z∂x ∂x ∂z ∂z ∂x 2 ) ∂  ∂wm  ∂ 2 (um wm ) ∂ 2 (wm ∂ 2 pm ∂ 2 h  ∂um ∂wm i ∂2  ∂wm  + + = − + Λ + +2 Λ . (16) m m ∂x ∂t ∂x2 ∂x∂z ∂x∂z ∂x2 ∂z ∂x ∂x∂z ∂z

Subtracting (15) from (16) yields       ∂ ∂wm ∂um ∂ ∂wm ∂um ∂ ∂wm ∂um − + um − + wm − ∂t ∂x ∂z ∂x ∂x ∂z ∂z ∂x ∂z       ∂um ∂wm ∂wm ∂um ∂ ∂ ∂um ∂wm pppppp + 2 + − + wm − um + ∂x ∂z ∂x ∂z ∂x ∂z ∂x ∂z

     ∂wm ∂um ∂ 2 ∂wm ∂um ∂Λm ∂ ∂wm ∂um − + Λm 2 − +2 − ∂x ∂z ∂z ∂x ∂z ∂x ∂x ∂x ∂z ∂Λm ∂  ∂wm ∂um  ∂Λm ∂  ∂um ∂wm  ∂Λm ∂  ∂um ∂wm  pppppp + 2 − +2 + −2 + ∂z ∂z ∂x ∂z ∂x ∂z ∂x ∂z ∂z ∂x ∂x ∂z  2      ∂ Λm ∂ 2 Λm ∂um ∂wm ∂ 2 Λm ∂wm ∂um pppppp + − + + 2 − . (17) ∂x2 ∂z 2 ∂z ∂x ∂z∂x ∂z ∂x



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∂2 pppp = Λm 2 ∂x

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It is important to note that due to continuity, the pressure term disappears from (17). Using the

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continuity equation (12), in fifth and sixth terms on right hand side of (17), and the substitution

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of vorticity defined by (6) in it, we obtain a vorticity-transport equation ∂ω ∂ω ∂ω + um + wm ∂t ∂x ∂z  ∂ 2 ω ∂ 2 ω   ∂Λ ∂ω ∂Λ ∂ω   ∂ 2 Λ ∂ 2 Λm  ∂ 2 ψ ∂ 2 ψ  ∂ 2 Λm ∂ 2 ψ m m m pp = Λm + +2 + + − − 2 −4 . 2 2 2 2 2 ∂x ∂z ∂x ∂x ∂z ∂z ∂x ∂z ∂z ∂x ∂z∂x ∂z∂x

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(18)

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Thus, the system of partial differential equations (12)-(14) in (um , wm , pm ) has now been converted into the simplified stream function - vorticity (ψ, ω) formulation in the form ∂2ψ ∂2ψ + = −ω, ∂x2 ∂z 2

(19)

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 ∂ 2 ω ∂ 2 ω   ∂Λ ∂ω ∂Λ ∂ω  ∂ω ∂ψ ∂ω ∂ψ ∂ω m m + − = Λm + 2 +2 + 2 ∂t ∂z ∂x ∂x ∂z ∂x ∂z ∂x ∂x ∂z ∂z  2   ∂2Λ  2 2 ∂ ψ ∂ ψ ∂ Λm ∂ 2 Λm ∂ 2 ψ m pppppppppppppppppppppp− − − − 4 . ∂x2 ∂z 2 ∂x2 ∂z 2 ∂z∂x ∂z∂x

Equations (19) and (20) form a close system for ψ and ω. There is no pressure term in the model

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(19) and (20). Thus, Pokhrel et al. (2018) model equations have been replaced by a set of just

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two partial differential equations, in place of the three for the velocity components and pressure.

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The two equations are coupled through the vorticity equation, and the vorticity ω is acting as

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the source term in the vorticity Poisson equation (19). The system of equations (19) and (20) is

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advantageous over Pokhrel et al. (2018), or (12)-(14), because it has only two unknown variables

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ψ and ω to be determined.

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A novel pressure Poisson equation in terms of stream function and mixture viscosity

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4.1

Generalized pressure Poisson equation

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Pressure Poisson equation plays important role in the dynamic simulation of mass flows (Domnik

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et al., 2013; Khattri and Pudasaini, 2018). So, next, we derive a pressure Poisson equation in

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terms of stream function ψ, and mixture viscosity Λm . Differentiating (13) with respect to x, and

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(14) with respect to z, we get

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∂ 2 pm ∂2  ∂um  ∂ 2 h  ∂um ∂wm i ∂  ∂um  ∂ 2 (u2m ) ∂ 2 (um wm ) + + = − +2 Λ + Λm + , (21) m ∂x ∂t ∂x2 ∂x∂z ∂x2 ∂x2 ∂x ∂x∂z ∂z ∂x 2 ) ∂  ∂wm  ∂ 2 (um wm ) ∂ 2 (wm ∂ 2 pm ∂ 2 h  ∂um ∂wm i ∂2  ∂wm  + + = − + Λ + +2 Λ . (22) m m ∂z ∂t ∂z∂x ∂z 2 ∂z 2 ∂z∂x ∂z ∂x ∂z 2 ∂z Adding (21) and (22) yields 2 ) ∂  ∂um ∂wm  ∂ 2 (u2m ) ∂ 2 (wm ∂ 2 (um wm ) + + + + 2 ∂t ∂x ∂z ∂x2 ∂z 2 ∂x∂z  ∂2p ∂ 2 pm  ∂2  ∂um  ∂2  ∂wm  ∂ 2 h  ∂um ∂wm i m pp = − + +2 Λ +2 Λ +2 Λm + . (23) m m ∂x2 ∂z 2 ∂x2 ∂x ∂z 2 ∂z ∂x∂z ∂z ∂x

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Using um = ∂ψ/∂z, wm = −∂ψ/∂x, and the continuity equation (12), the left hand side of (23)

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reduces to 8

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The terms on right hand side of (23), other than the pressure term, can be written as         ∂um ∂2 ∂wm ∂2 ∂um ∂2 ∂wm ∂2 + 2 2 Λm +2 Λm +2 Λm 2 2 Λm ∂x ∂x ∂z ∂z ∂x∂z ∂z ∂x∂z ∂x     ∂ 2 um ∂Λm ∂um ∂ ∂ 2 wm ∂Λm ∂wm ∂ Λm + +2 Λm + ppppppppppppp = 2 ∂x ∂x2 ∂x ∂x ∂z ∂z 2 ∂z ∂z     ∂ ∂ ∂ 2 um ∂Λm ∂um ∂ 2 wm ∂Λm ∂wm ppppppppppppppp + 2 Λm + +2 Λm + ∂z ∂x∂z ∂x ∂z ∂x ∂z∂x ∂z ∂x     ∂ ∂um ∂wm ∂Λm ∂um ∂Λm ∂wm ∂ Λm + + + ppppppppppppp = 2 ∂x ∂x ∂x ∂z ∂x ∂x ∂z ∂x     ∂ ∂ ∂um ∂wm ∂Λm ∂um ∂Λm ∂wm ppppppppppppppp + 2 Λm + + + . ∂z ∂z ∂x ∂z ∂x ∂z ∂z ∂z

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2 ) ∂ 2 (um wm ) ∂ 2 (u2m ) ∂ 2 (wm ∂  ∂um  ∂  ∂wm  ∂  ∂wm ∂um  + +2 = 2 u +2 w +2 u +w m m m m ∂x2 ∂z 2 ∂x∂z ∂x ∂x ∂z ∂z ∂x ∂z ∂z        ∂um ∂wm ∂ ∂wm ∂ ∂um ∂ um +2 wm +2 wm ppppp = 2 + ∂x ∂x ∂z ∂z ∂z ∂x ∂z     ∂ ∂um ∂wm ∂wm 2 ∂um ∂wm ppppp = 2wm +2 +2 + ∂z ∂x ∂z ∂z ∂z ∂x  2 2 ∂ ψ ∂2ψ ∂2ψ ppppp = 2 −2 2 . (24) ∂z∂x ∂x ∂z 2

Again, using um = ∂ψ/∂z, wm = −∂ψ/∂x, and the continuity equation (12), this reduces to     ∂ ∂Λm ∂um ∂Λm ∂wm ∂ ∂Λm ∂um ∂Λm ∂wm 2 + +2 + ∂x ∂x ∂x ∂z ∂x ∂z ∂x ∂z ∂z ∂z

 ∂2ψ ∂2ψ  ∂2Λ ∂Λm ∂ω ∂Λm ∂  ∂ 2 ψ ∂ 2 ψ   ∂ 2 Λm ∂ 2 Λm  ∂ 2 ψ m −2 + +2 − −2 − ∂x ∂z ∂z ∂x ∂x2 ∂z 2 ∂x2 ∂z 2 ∂z∂x ∂x2 ∂z 2 ∂x∂z  ∂Λ ∂ω ∂Λ ∂ω   ∂2Λ  ∂2ψ ∂2ψ  ∂2Λ ∂ 2 Λm  ∂ 2 ψ m m m m − +2 − − 2 − . (25) pp = 2 ∂z ∂x ∂x ∂z ∂x2 ∂z 2 ∂z∂x ∂x2 ∂z 2 ∂x∂z pp = 2

So, with (24) and (25), the equation (23) can be written as:

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191

 2 2   ∂ 2 pm ∂ 2 pm ∂2ψ ∂2ψ ∂ ψ ∂Λm ∂ω ∂Λm ∂ω + = − +2 − ∂x2 ∂z 2 ∂x2 ∂z 2 ∂z∂x ∂z ∂x ∂x ∂z   2   2 ∂ Λm ∂ 2 Λm ∂ 2 ψ ∂ ψ ∂ 2 ψ ∂ 2 Λm +2 − −2 − . ∂x2 ∂z 2 ∂z∂x ∂x2 ∂z 2 ∂x∂z

192

(26)

Or, ∂ 2 pm ∂ 2 pm + = f (ψ, ω, Λm ), ∂x2 ∂z 2

(27) 9

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where f (ψ, ω, Λm ) is given by right hand side of (26). This is a generalized pressure Poisson

194

equation. Thus, the pressure is a function of the stream function, vorticity and the mixture

195

viscosity.

196

From (19) and (20), we obtain ψ and ω. Then, for a given mixture viscosity Λm , (27) provides

197

the mixture pressure.

198

4.2

199

Equations (19), (20) and (27) constitute a full system of equations in stream function, vorticity,

200

and pressure (ψ, ω, pm ) describing the dynamics of the mixture motion. However, unlike the

201

original system (12)-(14), in (19), (20) and (27) the pressure is decoupled and can be computed

202

separately with the knowledge of ψ and ω. This is advantageous over the original system (in

203

Pokhrel et al. (2018)).

204

5

205

Reduction of the full and complex system is required for the construction of exact and analyti-

206

cal solutions of the model equations. So, next, we construct several reduced systems and their

207

associated exact and analytical solutions.

208

5.1

209

If ω is constant, say, ω = ω0 , then (19) and (20) reduce to

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Reduced systems

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The full system of equations

Constant vorticity

∂2ψ ∂2ψ + = −ω0 , ∂x2 ∂z 2  2   2  ∂ Λm ∂ 2 Λm ∂ ψ ∂2ψ ∂ 2 Λm ∂ 2 ψ − · − = −4 . ∂x2 ∂z 2 ∂x2 ∂z 2 ∂z∂x ∂z∂x

(28) (29)

Below, we will exclusively analyze this system of equations.

211

5.2

212

Furthermore, if there is only fluid, the mixture viscosity (Λm ) is a constant because Λm = νf ,

213

then (29) vanishes, and what remains is (28). If there is only solid; Λm = νse . But, now there are

214

two possibilities. (i) If solid viscosity is a constant, i.e., Λm = νse = constant, then (29) vanishes,

215

and only (28) remains. (ii) However, if νse is a variable, then both (28) and (29) remain. Then,

216

the situation becomes much more complex.

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Reduced mixture viscosity and an isotropic stream function

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217

The effective solid viscosity can be estimated by assuming the shear-rate ||Dm || ≈ um /h, where h

218

is the local flow depth, and neglecting the axial (normal) strain: νse = νs +

containing the mixture viscosity Λm can be simplified (estimated) as:       τy ∂2 ∂2 ∂z ∂2 h ∂ 2 Λm = νs + ≈ τy ≈ τy , ∂x2 ∂x2 ∂um /∂z ∂x2 ∂um ∂x2 um       τy ∂2 ∂z ∂2 h ∂ 2 Λm ∂2 ≈ 2 τy ≈ τy 2 , = 2 νs + ∂z 2 ∂z ∂um /∂z ∂z ∂um ∂z um    ∂ ∂ 2 Λm ∂ h = τy , ∂z ∂x ∂z ∂x um    ∂ 2 Λm ∂ ∂ h = τy . ∂x∂z ∂x ∂z um

222

223

(31) (32) (33)

Λm with x and z, i.e., ∂ 2 Λm /(∂z ∂x) = ∂ 2 Λm /(∂x ∂z), gives      1 ∂h ∂ ∂ 1 ∂ = h . ∂z um ∂x ∂x ∂z um Integration yields

1 ∂h ∂ =h um ∂x ∂z

224

(30)

So, assuming a uniform flow, since h is independent of z, and the continuous differentiability of

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220

So, assuming locally negligible variations of νs and τy , for νs and νf as parameters, the terms

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219

τy τy τy h . ≈ νs + ≈ νs + ||Ds || ∂um /∂z um

Or,



1 um



=h


h ∂um ∂ u−1 . m =− 2 ∂z um ∂z

1 ∂h 1 ∂um =− . h ∂x um ∂z

225

Integrating it again, to get

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h um = c,

(34)

226

where c is constant. The equation (34) is a typical flux. This satisfies mass flux (always for

227

incompressible mixture). So, since h is independent of z, the first term on left hand side of (29)

228

with (32), (33) and (34), becomes       h 1 ∂2h ∂ 2 Λm ∂ 2 Λm ∂ h∂ h i ∂ h∂ h i ∂2 1 i − τy ≈ τy − = τy −h 2 ∂x2 ∂z 2 ∂x ∂x um ∂z ∂z um um ∂x2 ∂z um  2  τy h ∂ 2 h h ∂ h h ∂2h ppppppppppppp = τy − = . (35) c ∂x2 c ∂z 2 c ∂x2 11

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230

Similarly, the first term on the right hand side of (29) becomes       τy h ∂ ∂h ∂ ∂ h ∂h ∂ 2 Λm 1 ∂h = τy = τy = = 0. ∂z ∂x ∂z um ∂x ∂z c ∂x c ∂z ∂x So, (29) reduces to  2   2  ∂ Λm ∂ 2 Λm ∂ ψ ∂2ψ · = 0. − − ∂x2 ∂z 2 ∂x2 ∂z 2

(36)

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229

This implies that either the first or, second term on the left hand side must be equal to zero.

232

But, (35) shows that, in general, the first term does not vanish. So, the second term must vanish,

233

which gives

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231

∂2ψ ∂2ψ = . ∂x2 ∂z 2 234

Using (37) in (28), we get

235

re-

∂2ψ ω0 =− . 2 ∂x 2

(37)

(38)

On integration, we obtain a quadratic stream function as: ψ=−

ω0 2 x + c1 x + c2 , 4

(39)

where c1 and c2 are constants. The stream function given by (39) are shown in Fig. 2 for c1 = 2,

237

c2 = 3; and for different values of ω0 , ω0 = 1.5 and ω0 = −2.5. As the flow velocity field is tangent

238

to the stream function, the solutions are realistic: Both panels may correspond to the flow release

239

from silo gate and down slope motion on an inclined channel but with different velocity profiles.

240

It is important to note that, as (39) indicates, for a known stream function ψ, the vorticity ω0 is

241

known.

(b)

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Figure 2: The stream function ψ given by (39) for different vorticities; showing concave downward or concave upward stream functions.

12

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242

5.3

243

A unique stream function can be constructed for the reduced system discussed above. In general,

244

(37) suggests the following theorem:

247

of

246

∂2ψ ∂2ψ Theorem 5.1. The only stream function ψ that satisfies − = 0, and the mass balance ∂x2 ∂z 2 ∂um ∂wm equation + = 0 is ψ = a x z. ∂x ∂z Proof. Since, ψ = a x z, so ∂2ψ = 0; ∂z 2

248

∂2ψ = 0, ∂x2

and thus,

Furthermore, by the definition um = ∂ψ/∂z, wm = −∂ψ/∂x, we have

which implies

∂wm ∂2ψ =− = −a, ∂z ∂z∂x

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∂2ψ ∂um = = a; ∂x ∂x∂z 250

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∂2ψ ∂2ψ − = 0. ∂x2 ∂z 2 249

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A general stream function for the reduced system

∂um ∂wm + = 0. ∂x ∂z

251

This proves the theorem.

252

Furthermore, from (38), ψ = a xz implies ω0 = 0. So, the vorticity for the reduced system is

253

determined.

254

255

∂2ψ ∂2ψ Corollary 5.1. ψ = a xz + bx + cz + d also satisfies − = 0, and the mass balance ∂x2 ∂z 2 ∂um ∂wm equation + = 0. ∂x ∂z

5.4

Velocity field for the reduced system

257

From the dynamical point of view, velocity and pressure fields are the most important quantities

258

that describe the flow. So, we construct the velocity field. The velocity field v = (um , wm ) =

259

(ax, −az) = a(x, −z) associated with ψ = a xz is shown in Fig. 3. The stream lines are given by

260

z = c/(ax) = λ(1/x), where c is a parameter.

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Figure 3: The stream lines of the function ψ = a x z, and the velocity vector field v = (ax, −az) with a = 1. The colors in arrows indicate the velocity magnitudes. .

Fig. 3 displays realistic velocity field for debris and granular flows down a channel after the flow

262

release (Pudasaini et al., 2007; Domnik and Pudasaini, 2012; Domnik et al., 2013). It shows

263

the rapid increase in the flow velocities as mass slides down-slope, and shears both in x and z

264

directions. The flow depth is determined by the relation ψ = a x z with h = z at free surface.

265

6

266

6.1

267

The general pressure Poisson equation given by (27) is rather complex. In technical applications,

268

this can be simplified for a constant vorticity and mixture viscosity. With a constant vorticity,

269

ω = ω0 , (27) can be written as

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Reduction of the pressure Poisson equation Pressure Poisson equation as a function of stream function

 2 2  2   2  ∂2ψ ∂2ψ ∂ ψ ∂ Λm ∂ 2 Λm ∂ 2 ψ ∂ ψ ∂ 2 ψ ∂ 2 Λm ∂ 2 pm ∂ 2 pm + = − +2 − −2 − ∂x2 ∂z 2 ∂x2 ∂z 2 ∂z∂x ∂x2 ∂z 2 ∂z∂x ∂x2 ∂z 2 ∂x∂z pppppppppppp = f (ψ, Λm ),

(40)

which is only a function of the stream function ψ and the mixture viscosity. In simple situation,

271

the mixture can be assumed homogeneous with constant viscosity. So, assuming Λm as constant,

272

(29) further reduces to

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∂ 2 pm ∂ 2 pm ∂2ψ ∂2ψ + = − ∂x2 ∂z 2 ∂x2 ∂z 2



∂2ψ ∂z∂x

2

,

(41)

273

which is a pressure Poisson equation in ψ. For a known stream function ψ, the advantage of this

274

reduced equation is, the pressure pm can be computed. The pressure Poisson equation (41) can 14

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275

further be reduced. One possibility is that the mixture flow may be irrotational. Then, with

276

vorticity w0 = 0, (28) reduces to ∂2ψ ∂2ψ + = 0. ∂x2 ∂z 2

(42)

Or, even for a given value of ω0 , an analytical solution of ψ can be obtained from (28). Once

278

ψ is known, pm can be obtained from the pressure Poisson equation (41). So, since the stream

279

function ψ retains the flow properties of the mixture, the velocity field v = (um , wm ), and the

280

pressure Poisson equation (41) capture the basic flow dynamics.

281

6.2

282

Using the method of separation of variables (Senger and Fogg, 1990), a solution for (42) can be

283

constructed. Let ψ = X(x) Z(z) be the general solution of (42). As the variables are separated,

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we obtain

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Analytical solution for ψ

Z 00 X 00 =− = k (constant). X Z

286

Then, the stream function can be written as 
 c1 e−px + c2 epx (c3 cos pz + c4 sin pz) , if k = p2 ,       ψ(x, z) = (c5 cos px + c6 sin px) (c7 e−pz + c8 epz ) , if k = −p2 ,        (c x + c ) (c z + c ), if k = 0, 9 10 11 12

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(43)

where c1 , · · · , c12 are constants. The stream functions given by (43) that involve the trigonometric

287

terms are not physically meaningful. So, the only feasible solution of the stream function in (43)

288

is the third algebraic solution. This has been constructed in Theorem 5.1 and Corollary 5.1.

289

7

290

As discussed earlier, ψ = a x z is a legitimate choice for the stream function. Then, even for a

291

variable Λm with (35), (40) reduces to

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A new pressure Poisson equation for shear mixture flow

  ∂ 2 pm ∂ 2 pm ∂2h + = a λh 2 − a , ∂x2 ∂z 2 ∂x

(44)

292

where λ = (2τy /c). This is a two-dimensional pressure diffusion with yield stress associated source

293

term. The importance of the new pressure Poisson equation (44) induced by the flow field intensity

294

a, the yield strength τy , and free surface geometry contribution h ∂ 2 h/∂x2 will be discussed later.

295

The flow depth h and the diffusion of the free surface ∂ 2 h/∂x2 with the non-linear diffusion

296

coefficient (2aτy h)/c =: D plays crucial role in characterizing the pressure Poisson equation (44). 15

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7.1

Analysis of the pressure Poisson equation

298

Appearance of the right hand side in the pressure Poisson equation (44) is important for several

299

regions.

300

A. Mixture with negligible yield strength: First, for frictionless or strengthless material

301

(τy = 0), (44) simplifies to a constant source pressure Poisson equation (with non-zero right hand

302

side):

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∂ 2 pm ∂ 2 pm + = −a2 . ∂x2 ∂z 2

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(45)

B. Exact knowledge of the flow boundary: Second, due to the presence of the flow depth h,

304

we know exactly the region in the channel occupied by the flowing mass. This is very crucial to

305

define the boundary for the pressure Poisson equation, because along the free surface h = h(x),

306

pm = 0, because, due to the tractionless free surface boundary condition (Pudasaini, 2012), the

307

pressure vanishes at the free surface. This is important in applications (Domnik and Pudasaini,

308

2012; Domnik et al., 2013; Khattri and Pudasaini, 2019).

311

312 313 314

315 316

317 318 319 320

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C. Source or sink: Third, the sign and magnitude of ∂ 2 h/∂x2 is crucial. Because, depending ∂2h ∂2h on this, λ 2 − a ≷ 0; which will result in completely different scenarios. λ 2 − a > 0 implies ∂x ∂x ∂2h positive, but λ 2 − a < 0 implies negative source for the pressure Poisson equation. This ∂x ∂2h represents a source and sink in (44). For example, if h ∝ 1/x, ∂ 2 h/∂x2 ∝ 1/x3 . Then, λ 2 − a ∂x can be > 0. This is the scenario of flow release and motion down the channel. If for some choices ∂2h of h, λ and a, λ 2 − a < 0, then, the result changes completely. ∂x Consider an example h = α/x for α > 0 (or, h = α e−x ), and x is sufficiently larger than 0. Then, ∂ 2 h/∂x2 = 2α/x3 is well defined. With this, (44) reduces to   ∂ 2 pm ∂ 2 pm 2λα2 + =a − a = f (x). ∂x2 ∂z 2 x4

(46)

There are two possibilities. If x is small, then 2λα2 /x4 dominates a, but if x is large a dominates 2λα2 /x4 (assuming a > 0). So, in these two regimes, pressure behaves completely differently, with 2λα2 smooth transition at ≈ a. Furthermore, if a ∼ 0, then for x relatively greater than 0, z is x4 large. This means that for thick flow, the pressure Poisson equation reduces to Laplace equation

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321

4pm ≈ 0, that can be solved easily.

322

As we will see later with the construction of several analytical solutions of (44) offers a major

323

contribution that can be applied in solving technical problem related to the diffusion of pressure in

324

the sheared particle fluid mixture flow with yield strength down a channel (slope with the explicit

325

knowledge of the free surface). 16

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8

Construction of analytical solutions for pressure

327

In the following sections, we extensively analyze and develop different analytical and exact solu-

328

tions for the shear flow induced pressure Poisson equation (44). Next, we construct some analytical

329

solutions to the special pressure Poisson equation (44). Pressure can be diffused much faster in

330

x or z direction (Pudasaini et al., 2005). So, we construct analytical/exact solutions for such

331

situations.

332

8.1

333

In this situation,

334

Integrating twice, we get

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Negligible pressure diffusion in flow depth direction ∂ 2 pm ∂ 2 pm  . Then (44) can be written as ∂z 2 ∂x2   ∂ 2 pm ∂2h = a λh 2 − a . ∂x2 ∂x

!

re-

pm (x) = aλ

Z

∂h h dx − aλ ∂x

Z

Z 

∂h ∂x

2

dx

dx −

a2 x + c1 x + c2 , 2

(47)

where c1 , c2 are constants of integration. This implies that for a given hydraulic pressure gradient

336

∂h/∂x, the solution for pressure is known explicitly and exactly. The explicit form of the pressure

337

in (47) is important. Because, in practical applications, it is much harder to obtain or measure

338

real pressure pm of the flow than its hydraulic pressure gradient ∂h/∂x. The later can much

339

easily be measured by mapping the free surface of the flow which is a geometrical property, than

340

to internally (or at base) measure the pressure by means of complex, but perhaps less accurate

341

pressure measurements. So, the solution (47) offers a great technical advancement. In particular,

342

if h = α/x, then from (47) the exact solution is obtained:

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pm (x) =

343

aλα2 a2 2 − x0 + c1 x0 + c2 . 2 3x20

(49)

Applying the condition pm = pr > 0 at x = xr > 0 (right boundary of the flow domain), we get

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(48)

Applying the condition pm = p0 > 0 at x = x0 > 0 (left boundary of the flow domain), we get p0 =

344

aλα2 a2 2 − x + c1 x + c2 . 3x2 2

pr =

aλα2 a2 2 − xr + c1 xr + c2 . 3x2r 2

Solving (49) and (50), we obtain c1 , c2 : 1 x0 + xr a2 pr − p0 aλα2 2 2 + (x0 + xr ) − , 3 2 xr − x0 x0 xr  2  2 1 a2 pr x0 − 2p0 x0 + p0 xr 2 x0 + xr + x0 xr c2 = − aλα − xr x0 + . 2 2 3 2 xr − x0 x0 xr c1 =

17

(50)

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Figure 4: Distribution of the mixture pressure along the slope given by (51).

This implies the fully determined analytical solution from (48): pr −p0 1  a2 aλα2 x0 +xr aλα2  1 − (x−x0 ).(51) pm (x) = p0 − − (x−x )(x−x )+ (x−x0 )+ 0 r xr −x0 3 x2 x20 2 3 x20 x2r

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re-

.

347

The pressure distribution (per unit mixture density) along the slope given by (51) is shown in

348

Fig. 4 for parameters p0 = 1.08, pr = 1.10, intermediate values x0 = 0.2, xr = 1.4, a = 0.16,

349

λ = 335, α = 0.02. It reveals an interesting evolution of the mixture pressure along the channel.

350

When the flow is released from a silo gate, the mixture pressure increases slowly as the flow moves

351

downslope up to x = 5 m. As the flow further moves downslope, the mixture pressure decreases

352

slowly in the down-slope direction. Such a solution is meaningful and in line with previous studies

353

(Domnik and Pudasaini, 2012; Domnik et al., 2013; Khattri and Pudasaini, 2019).

354

8.2

355

In this situation,

∂ 2 pm ∂ 2 pm  . Then, (44) can be written as ∂x2 ∂z 2   ∂ 2 pm ∂2h = a λh 2 − a , ∂z 2 ∂x

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Negligible pressure diffusion in the down-slope

which, when integrated twice, results in (since ∂h/∂x is independent of z)   ∂2h a λ h 2 − a z 2 + c1 z + c2 . pm = 2 ∂x

(52)

357

Applying the boundary condition as ∂pm /∂z = k at z = 0, we get c1 = k, and at the free-surface

358

z = h, p = 0, we get a c2 = − 2

  ∂2h λ h 2 − a h2 − kh. ∂x 18

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359

It is important to note that, in application the pressure gradient at the lower boundary can be

365

condition. So, we have constructed an exact analytical solution for the non-hydrostatic pressure

366

distribution in mixture flow. This is important and realistic. Furthermore, as λ = 2τy /c, the pres-

362

363

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361

of

364

determined relatively easily to obtain k. So, (52) becomes   ∂2h a λ h 2 − a (z 2 − h2 ) + k(z − h). (53) pm (z) = 2 ∂x   a ∂2h In general, depending on whether λ h 2 − a = 0 or not, we obtain a linear or quadratic 2 ∂x pressure through depth.   ∂2h It is crucial to note that if λ h 2 − a 6= 0, then pm changes quadratically with flow depth. ∂x Otherwise, pm changes linearly with depth which is the most often used hydrostatic (or lithostatic)

360

370

for the first time. The analytical solution (53) can be applied to study the non-linear variation of

371

the mixture pressure through the flow depth.

372

8.3

374 375

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Reconstruction of the full pressure field

From (47) and (53), we can reconstruct the full pressure field pm (x, z) = pm (x) pm (z) as:  Z Z   ∂2h a2 2 pm (x, z) = aλ h 2 dx dx − x + c1 x + c2 ∂x 2     a ∂2h 2 2 · λ h 2 − a (z − h ) + k(z − h) . 2 ∂x

(54)

Once h is known, then h ∂ 2 h/∂x2 is known. For example, if h = α/x, then h ∂ 2 h/∂x2 is known. For such a situation (54) can be calculated explicitly. So,    aλα2 1 1 pr − p0 a2 pm (x, z) = p0 − (x − x0 ) + − − (x − x0 )(x − xr ) 2 xr − x0 3 x2 x0 2      aλα2 x0 + xr aλα2 a2  2 α α + (x − x0 ) · z − 2 +k z− − . (55) 3 x4 2 x x x20 x2r

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369

sure is high for high yield strength flow. Moreover, (53) impliesthat the pressure field is quadratic  2 ∂ h a λ h 2 − a that varies with the with flow depth, but with a complex non-linear coefficient 2 ∂x flow geometry or the dynamics of free surface. Such a special pressure profile is contributed here

367

376

Using the same parameters that have been used to plot Fig. 4 (i.e., p0 = 1.08, pr = 1.10, x0 = 0.2,

377

xr = 1.4, a = 0.16, λ = 335, α = 0.02), and a new parameter k = 0.50; Fig. 5 is obtained which

378

shows the mixture pressure (per unit density) distribution in the channel employing (55). As the

379

mixture is released from a silo gate, the pressure increases from the free surface to the channel

380

bottom. The pressure forms a layered structure. As the mixture mass flows downslope, it shears 19

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Figure 5: The mixture pressure pm given by (55). The pressure forms a layered structure that decreases non-linearly from basal to free surface.

.

so that the pressure decreases from upstream to downstream. Figure 5 is the extension of Fig. 4.

382

Both figures are in line with previously presented results (Domnik and Pudasaini, 2012; Khattri

383

and Pudasaini, 2019), but here, we have presented such a pressure field explicitly and analytically.

384

9

385

In the far down-slope, the hydraulic pressure gradient can be negligible (Pudasaini et al., 2007;

386

Domnik and Pudasaini, 2012; Khattri and Pudasaini, 2019). So, an alternative solution can be

387

constructed for the pressure field (44) with the condition in Section 8.1 but for negligible hydraulic

389

Alternative solution of the pressure field

pressure gradients. So, if (∂ 2 pm /∂z 2 ) ∼ 0, then (44) reduces to "  2     2 # ∂ 2 pm ∂ h ∂ ∂h ∂h = a λ 2 − a = aλ h − − a2 . ∂x2 ∂x ∂x ∂x ∂x

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Integrating,

∂pm = aλ ∂x

Z "

∂ ∂x

   2 # ∂h ∂h dx − a2 x + k, h − ∂x ∂x

390

where k is constant of integration. Assuming slowly varying free surface of flow, i.e., ∂h/∂x small

391

implies (∂h/∂x)2 ∼ 0, so, we get

  ∂pm ∂h = aλ h − a2 x + k. ∂x ∂x 20

(56)

Journal Pre-proof

Here, h(∂h/∂x) is load due to hydraulic pressure (Pudasaini and Hutter, 2007), and aλ is pressure

393

coefficient. Thus, the dynamic pressure gradient ∂pm /∂x is proportional to the hydraulic pressure

394

gradient, which is consistent. The pressure gradient is increasing or decreasing depending on

395

whether ∂h/∂x is increasing or decreasing. However, if we have some more information on ∂h/∂x,

396

we can obtain even more general model than (56).

397

Equation (56) can be integrated explicitly if h is known. Homogeneous pressure is often an applied

398

condition in open channel flow. So, if (∂pm /∂x) ∼ 0, then

399

ax ∂h = − k0 . ∂x λ

On integration

pro

h

of

392

p h(x) = ± cx2 − k 0 x + k1 ,

(57)

where c = a/λ, and k 0 , c1 are constants of integration. For k 0 = 0, k1 = 2, the solution of (57)

401

are plotted in Fig. 6 with c = −1 in panel (a), which represents to the mass release from the silo

402

gate as the flow depth h decreases along the down-slope distance (Domnik and Pudasaini, 2012),

403

and c = 1 in panel (b) represents the flow depth increasing in deposition regimes (Pudasaini et

404

al., 2007; Domnik et al., 2013). Nevertheless, note that for the homogeneous pressure along x,

405

(44) could be directly solved. Such a solution is also described below in (60).

406

10

407

Now, we analyze (44) in detail with respect to the associated source term.

urn al P

re-

400

Dynamics of the source term in pressure Poisson equation

(b)

Jo

(a)

Figure 6: The flow depths given by (57).

21

Journal Pre-proof

408

Case I: For a = 0, (44) becomes a simple pressure equation, or a two-dimensional diffusion

409

equation for which solution can be constructed with standard methods.

410

Case II: For

412

reduces to

of

411

2τy h ∂ 2 h − a = 0, again (44) becomes a pressure equation, but now it is interesting c ∂x2 to analyze the flow dynamics, particularly, the associated free surface structure. Equation (44)

∂2h λ = , 2 ∂x h

415

Putting

∂h ∂2h ∂v ∂h ∂v = v, = =v , in (58), yields 2 ∂x ∂x ∂h ∂x ∂h v

416

pro

414

ac . This is a second order non-linear autonomous ordinary differential equation. This 2τy can be solved exactly to obtain the analytical solution (λ 6= 0):

where λ =

Integrating it, we obtain

λ ∂v = . ∂h h

(59)

re-

413

v2 = log(c1 h), which, on substitution v = ∂v/∂x, takes the form 2λ

urn al P

1 ∂h p √ = log(c1 h). 2λ ∂x

417

Integrating again, we get

1 √ 2λ

Z

∂h p = log(c1 h) 2

419

420

421

Z

∂x + c2 = x + c2 .

2

Substituting log(c1 h) = −t2 , c1 h = e−t , c1 ∂h = e−t (−2t ∂t), in the integral, we get  √  Z √ 1 π 2 −t2 √ − e ∂t = 2λ(x + c2 ). c1 2 π With the definition of error function erf , this can be written as r 2λ erf (t) = −2c1 (x + c2 ). π

Jo

418

(58)

Moreover, with the definition of t, we obtain: p − log(c1 h) = erf −1

−2c1

r

−2c1

r

! 2λ (x + c2 ) . π

Hence, the solution for h is given by h(x) =



1 exp − erf −1 c1 22

!!2 

2λ (x + c2 ) π

.

pro

of

Journal Pre-proof

422

Or, equivalently "

1 h(x) = exp 2λ

(

re-

Figure 7: The flow depth h given by (60).

−c1 − 2λ erf

−1



r

±i

 2p λ exp(c1 λ) (c2 + x)2 π

)#

,

(60)

where c1 , c2 are constants of integration. So, we have constructed a complex, and exact/analytical

424

solution for free surface of the flow.

425

Equation (58) indicates that for λ = 0, h = c2 x + c1 , which is linear in x. Equation (60) models

426

the flow depth along the slope with λ 6= 0. For c1 = 1.0, c2 = −1.5, λ = 0.7, (60) is plotted in

427

428

urn al P

423

Fig. 7 which implies release and shear flow of the viscous mixture material.   2τy h ∂ 2 h Case III: −a ≶0 c ∂x2

429

This requires a full solution of (44) as discussed earlier.

430

Case IV: Negative source: The source term becomes negative either for very small h, or τy ∼ 0, or (∂ 2 h/∂x2 ) ∼ 0. Then, we obtain the pressure Poisson equation with simple constant

Jo

431

∂ 2 pm ∂ 2 pm + = −a2 . ∂x2 ∂z 2

432

However, the analytical solution is not that easy to obtain, if it exists at all.

433

Case V:

2τy h ∂ 2 h  a: Then, the pressure Poisson equation reduces to c ∂x2   2a τy ∂ 2 pm ∂ 2 pm ∂2h + = h , ∂x2 ∂z 2 c ∂x2

(61)

(62)

434

with a complex source term. The flow depth h can be relatively easily determined in experiments

435

just from the measurement of the free surface (Pudasaini et al., 2007; Pudasaini and Kroner, 23

Journal Pre-proof

436

2008), then the complex pressure distribution of the flow domain can be obtained from (62).

437

Case VI:

2τy h ∂ 2 h  a: Then, again, a negative source of pressure Poisson equation is obtained c ∂x2 ∂ 2 pm ∂ 2 pm + = −a2 . ∂x2 ∂z 2

Case VII:

of

438

(63)

2τy h ∂ 2 h − a = ξ, a constant: Then c ∂x2

pro

∂2h (ξ + a)c = . ∂x2 2τy h

(64)

439

This is similar to Case II.

440

11

441

As we have seen several important aspects of the newly constructed pressure Poisson equation

442

(44), further analysis of its nature is desirable. Dimensional analysis helps to understand the flow

443

regimes, intrinsic flow dynamics and the relative importance of some terms in the model equation

444

as compared to the other terms (Pudasaini and Hutter, 2007). To further analyze the pressure

446 447

re-

urn al P

445

Dimensional analysis of pressure Poisson equation

Poisson equation (44), we introduce a non-dimensional analysis and variables with hats (Pudasaini ˆ and Hutter, 2007): x = x ˆL, z = zˆH, pm = pˆm p0 , h = hH, τy = τˆy τy , where L, H are typical flow 0

length and depth, and p0 , τy 0 are typical pressure and yield strength scales. Then, (44) becomes 2ˆ p0 ∂ 2 pˆm 2aH 2 2 p0 ∂ 2 pˆm ˆ ∂ h − p0 a + = L ( τ ˆ τ h) ˆ2 . y y 0 L2 ∂ x ˆ2 H 2 ∂ zˆ2 c ∂x ˆ2 H 2

(65)

448

With the definition of the aspect ratio, ε = H/L, the yield strength to pressure ratio, τy 0 /p0 = τpy ,

449

and the hydrostatic pressure, because pressure is already normalized with density in (8) - (10),

450

g H = ph , we obtain:

Jo

ε

2 pˆ m 2 ∂x ˆ

2∂

ˆ  ph 2 ∂ 2 pˆm a y
ˆ ∂2h 3 2ˆ a ˆ2 . + =ε τ τˆy h 2 − ∂ zˆ2 cˆ p ∂x ˆ p0

451

By further defining ph /p0 = php as hydrostatic to full dynamic pressure ratio, the pressure Poisson

452

equation (65) in dimensionless form yields ε

453

2∂

2p

m ∂x2

∂ 2 pm + = ε3 ∂z 2



2a τy c



τpy h

∂ 2 h  h 2 2 − pp a , ∂x2

where, for simplicity, the hats and suffix have been dropped. 24

(66)

Journal Pre-proof

454

I. Reduction with flow thickness

455

By considering some special situations, the pressure Poisson equation (66) can be reduced to

456

simpler equations that could be solved analytically. One of such possibilities is associated with

457

flow thickness.

459

of

A. Thin flow: Importantly, if ε  1, then (66) represents a thin flow, and thus  2 ∂ 2 pm = − php a2 . ∂z 2 Integrating, we obtain

pro

458

 2 z 2 pm (z) = − php a2 + c1 z + c2 . 2

(67)

For a 6= 0, php 6= 0, this represents a quadratic pressure profile through depth, and thus, is an

461

important extension of classical hydrostatic pressure distribution which is linear in z, which can

462

be obtained from (67) for php = 0. This justifies the physical relevance of the new model (44) and

463

the solution (67). It has an important implication: For the dynamic pressure dominated flow, the

464

total pressure distribution must be quadratic.

B. Thick flow: For ε  1, the terms without ε are negligible. So, (66) reduces to    2  ∂ 2 pm 2a ∂ h y =ε τy τp h 2 . 2 ∂x c ∂x

urn al P

465

re-

460

466

This means, for a given flow depth, the pressure along the channel is known (or, can be obtained).

467

For example, linear h implies pm (x) = c1 x+c2 , or, ∂pm /∂x = c1 , the constant pressure gradient in

468

the downstream, which is often used in free surface channel flow approximation. For a non-linear,

469

or a general distribution of the flow depth profile, the dynamic pressure gradient is non-linear.

470

II. Reduction with mechanics

471

For ε ≈ 1, (66) reduces to

 ∂2h
2 ∂ 2 pm ∂ 2 pm  2a y + = τ τ − a php . y p h 2 2 2 ∂x ∂z c ∂x

(68)

Since, τy /p = τpy , for high yield strength, the term with τpy remains, but for lower yield strength

473

(water or dilute flow), τpy = 0. Furthermore, php = ph /p for high hydrostatic pressure, the term

474

with php remains, but for low hydrostatic pressure php = 0. Next, we analyze in detail on how

475

pressure and yield strength control the flow dynamics in (68).

476

A. Pressure dominated flow: If p  τy and p  ph , then (68) reduces to Laplace equation for

477

thin, low yield strength fluid

Jo

472

∂ 2 pm ∂ 2 pm + = 0. ∂x2 ∂z 2 25

Journal Pre-proof

478

B. Thick, low yield strength flow: If p  τy , or p . ph , then (68) reduces to  2 ∂ 2 pm ∂ 2 pm h , + = − ap p ∂x2 ∂z 2

480

which is the pressure Poisson equation for thick, low yield strength mixture. C. Thin, high yield strength flow: If p . τy , or p  ph , then (68) reduces to   ∂ 2 pm ∂ 2 pm 2a ∂2h y + = τ τ h , y p ∂x2 ∂z 2 c ∂x2

of

479

that describes the pressure Poisson equation for thin, high yield strength mixture flow.

482

D. General situation: If p ≈ τy , or p ≈ ph , then no further reduction is possible and (68)

483

represents the pressure Poisson equation for general situation which models the thick, high yield

484

strength mixture flow with substantial pressure. Three distinctions with geometrical and mechan-

485

ical considerations can be associated with incipient flow (B), main flow (A, C), and depositional

486

regimes (D).

487

III. A further solution to the general pressure Poisson equation

re-

Suppose λ = (2a/c)τy τpy and ζ = a php . Then, (68) becomes    2 ∂2 λ 2 ∂ 2 pm ∂h pm − h + = −λ − ζ 2. 2 2 ∂x 2 ∂z ∂x

urn al P

488

pro

481

(69)

489

This implies that for the pressure pm in the vicinity of λ1 h2 /2, i.e., pm ≈ λ1 h2 /2, further reduction

490

of (68) is possible:

∂ 2 h2 ∂ 2 h2 = −2λ (λ1 − λ) 2 + λ1 ∂x ∂z 2

491

∂h ∂x

2

− 2ζ 2 .

(70)

Since h is independent of z, (70) simplifies to:

∂2h (λ1 − λ)h 2 = −λ1 ∂x



∂h ∂x

2

− ζ 2.

(71)

We write (71) as

Jo

492



∂2h h 2 = −a ∂x



∂h ∂x

2

− e,

(72)

493

where a = λ1 /(λ1 − λ), e = ζ 2 /(λ1 − λ). Suppose ∂h/∂x = v. So, ∂ 2 h/∂x2 = v (∂v/∂h). Then,

494

from (72), we get

495

hv

 ∂v e = −a v 2 + . ∂h a

Integrating it, we obtain the solution for v in terms of h: v2 =

c e − , 2a h a 26

(73)

Journal Pre-proof

(b)

pro

of

(a)

re-

Figure 8: The non-dimensional debris flow depth given by (75).

(b)

urn al P

(a)

Figure 9: The non-dimensional debris flow depth given by (75) for different parameters than in Fig. 8.

where, c is constant of integration. Since ∂h/∂x = v, (73) allows us to construct an analytical

497

solution for flow depth:

498

Jo

496

∂h = ∂x

√

c − bh2a , ha

(74)

where b = e/a. Integrating (74), we obtain k+x=

ha+1 √ 2 F1 (a + 1) c



 1 b 1 a+1 3 , ; + ; h2a , 2 2a 2 2a c

(75)

499

which provides an analytical solution for the flow depth h, where, 2 F1 is the hyper-geometric

500

function. Figure 8 plots the flow depth along the downslope distance for a debris flow with the 27

Journal Pre-proof

chosen parameters, k = −5.0, a = 2.5, c = 1.0 with b = 1.0, and b = 0. Panel (a) shows the

502

change in flow depth as the flow moves downslope after flow release from a silo gate, where as

503

panel (b) displays the flow depth during front bore propagation showing the physically reasonable

504

solutions (Pudasaini et al., 2007; Pudasaini, 2011).

505

Similarly, Fig. 9 plots the flow depth along the downslope distance for a debris flow with another

506

set of chosen parameters, k = −5.0, c = 1.0 with a = 2.0, b = −20.5, and a = −0.95, b = 2.0. Panel

507

(a) shows the propagation of the front in an undisturbed flow, where the flow height decreases as

508

it moves downslope due to shearing. Panel (b) shows the deposition of the debris material, e.g., in

509

front of the obstacle placed at the downslope distance of x = 2 m. There is a higher deposition at

510

the front of the obstacle and the deposition decreases upstream. These are observable phenomena

511

in debris mixture flow (Pudasaini, 2011; Kattel et al., 2016, 2018; Khattri and Pudasaini, 2019).

512

12

513

Here, we considered a generalized quasi two-phase, full two-dimensional bulk mixture model for

514

mass flow down a channel (Pokhrel et al., 2018), which is a set of highly non-linear partial differ-

515

ential equations for three variables: mixture pressure and mixture velocities. The model includes

516

the mixture flow rheology containing mixture velocities, and mixture pressures, which are writ-

517

ten in conservative form to describe the complex motion of mixture of viscous fluid and granular

518

particles. By employing the model, we constructed the vorticity-transport equation and pressure

519

Poisson equation for stream function, and these two equations become a close system for two

520

variables, namely, the stream function and the vorticity. The stream function vorticity-transport

521

equation is a non-linear partial differential equation which exclusively includes the newly con-

522

structed mixture viscosity in Pokhrel et al. (2018). We constructed a novel generalized pressure

523

Poisson equation which is a function of the stream function, vorticity and the mixture viscosity

524

that can compute the mixture pressure for given mixture viscosity. We also formed a full system of

525

equations in three variables, namely the stream function, vorticity and pressure so as to describe

526

the dynamics of the mixture motion. Next, the pressure is decoupled so that as an advantage,

527

it could be computed separately with the knowledge of stream function and vorticity. We also

528

developed the model equations for a constant vorticity, and for more complex situation with a

529

variable mixture viscosity. With constant viscosity and constant vorticity, we further reduced the

530

pressure Poisson equation that includes stream function only. For a given stream function, mix-

531

ture pressure can also be obtained from the pressure Poisson equation. Since the stream function

532

retains the flow properties of the mixture, the velocity field and the pressure Poisson equation

re-

pro

of

501

Jo

urn al P

Discussion and summary

28

Journal Pre-proof

captures the basic flow dynamics. We obtained the solution of this model equation by considering

534

a variable solid viscosity, and obtained an isotropic stream function which is a function of only

535

one variable, and then analyzed the flow behavior with velocity field. The velocity field for the

536

reduced systems shows the rapid increase and shearing both in downslope and normal directions.

537

In the case of constant vorticity, we reduced the generalized pressure Poisson equation in terms of

538

stream function and mixture viscosity. Furthermore, with a constant viscosity, i.e., with the situ-

539

ation of homogeneous mixture mass flow, the pressure Poisson model could be reduced in terms

540

of stream function only. The advantage of the reduced model is that the mixture pressure can be

541

calculated for a given stream function. Analytical solution of the stream function was constructed

542

in the case of irrotational flow.

543

We developed a new pressure Poisson model equation for shear mixture flow with a legitimate

544

choice of the stream function which includes free surface geometry contributions, the flow field

545

intensity and the yield strength. We analyzed the distinctive features of the model with respect

546

to the source terms. We further analyzed the pressure Poisson equation with negligible pressure

547

diffusion in flow depth direction, and constructed its exact solution in explicit form for a given

548

hydraulic pressure gradient, which is important and offers a great technical advancement. We also

549

modeled the pressure Poisson equation with negligible pressure diffusion in down-slope direction,

550

and constructed its exact/analytical solution for the non-hydrostatic pressure distribution. Fur-

551

thermore, we justified that the pressure is high for high yield strength flow. Moreover, the pressure

552

field is quadratic with flow depth, but with a complex non-linear coefficient that varies with the

553

flow geometry or the dynamics of free surface. Such a special pressure profile is a novel contribu-

554

tion. Different solutions for the pressure Poisson equations were obtained and their behaviors were

555

studied. We analyzed the new pressure Poisson equation for shear mixture flow by converting it

556

into dimensionless form. In the case of thin flow, a quadratic pressure profile through the depth

557

were obtained. This is an important extension of classical hydrostatic pressure distribution which

558

is linear in flow depth justifying the physical relevance of the new model. We also analyzed in

559

detail on how pressure and yield strength control the flow dynamics.

560

Acknowledgments: We gratefully acknowledge the financial support provided by the German

561

Research Foundation (DFG), Germany, through the research project PU 386/5-1: “A novel and

562

unified solution to multi-phase mass flows”: UMultiSol . Puskar R. Pokhrel acknowledges University

563

Grant Commission (UGC), Nepal for the financial support provided as a PhD fellowship (PhD -

564

2071/072 - Sci. & Tech. - 01).

Jo

urn al P

re-

pro

of

533

29

Journal Pre-proof

565

References [1] Alvarez M, Gatica GN and Ruiz-Baier R (2016): Analysis of a vorticity-based fully-mixed

567

formulation for the 3D Brinkman-Darcy problem. Comput. Methods Appl. Mech. Eng., 307,

568

68-95.

of

566

[2] Anaya V, Gatica GN, Mora D and Ruiz-Baier R (2015): An augmented velocity-vorticity-

570

pressure formulation for the Brinkman equations. Int. J. Numer. Meth. Fluids, 79, 109-137.

571

[3] Anaya V, Mora D, Reales C and Ruiz-Baier R (2016): Mixed Methods for a Stream-

572

Function-Vorticity Formulation of the Axisymmetric Brinkman Equations. J. Sci. Comput.,

573

71, 348-364.

576 577

Inc., SanDiego, California, 381p.

re-

575

[4] Anderson MP and Woessner WM (1992): Applied Groundwater Modeling. Acadeic Press

[5] Chen Y and Xie X (2016): Vorticity vector-potential method for 3D viscous incompressible flows in time-dependent curvilinear coordinates. J. Comput. Phys., 312, 50-81.

urn al P

574

pro

569

578

[6] Cockburn B and Cui J (2012): An analysis of HDG methods for the vorticity-velocity-

579

pressure formulation of the Stokes problem in three dimensions. Math. Comp., 81, 1355-

580

1368.

581

[7] Domnik B and Pudasaini SP (2012): Full two-dimensional rapid chute flows of simple vis-

582

coplastic granular materials with a pressure-dependent dynamic slip-velocity and their nu-

583

merical simulations. J. Non-Newtonian Fluid Mech., 173-174, 72-86. [8] Domnik B, Pudasaini SP, Katzenbach R and Miller SA (2013): Coupling of full two-

585

dimensional and depth-averaged models for granular flows. J. Non-Newtonian Fluid Me-

586

chanics, 201, 56-68.

587 588

589 590

591 592

Jo

584

[9] Ern A, Guermond JL and Quartapelle L (1999): Vorticity-velocity formulations of the Stokes problem in 3D. Math. Methods Appl. Sci., 22, 531 -546. [10] Evans DG and Raffensperger JP(1992): On the Stream Function for Variable-Density Groundwater Flow. Water Resour. Res., 29(8), 2141-2145. [11] Fogg GA and Senger RK (1985): Automatic generation of flow nets with conventional ground-water modeling algorithms. Ground Water, 23(3), 336-344. 30

Journal Pre-proof

593 594

[12] Hutter K, Svendsen B and Rickenmann R (1996): Debris flow modelling: A review. Continuum Mech, Thermodyn., 8, 1-35. [13] Iverson RM (1997): The physics of debris flows. Rev. Geo-phys., 35(3), 245-296.

596

[14] Iverson RM and Denlinger RP (2001): Flow of variably fluidized granular masses across

597

three-dimensional terrain: 1. Coulomb mixture theory. J. Geophys. Res., 106(B1), 537-552.

598

[15] Kattel P, Khattri KB, Pokhrel PR, Kafle J, Tuladhar BM and Pudasaini SP (2016): Simu-

599

lating glacial lake outburst floods with a two-phase mass flow model. Annals of Glaciology,

600

57(71), 349-358.

603 604

pro

602

[16] Kattel P, Kafle J, Fischer J-T, Mergili M, Tuladhar BM and Pudasaini SP (2018): Interaction of two-phase debris flow with obstacles. Engineering Geology, 242, 197-242.

re-

601

of

595

[17] Khattri KB and Pudasaini SP (2018): An extended quasi-two-phase mass flow model. Int. J. Non-Linear Mech., 106, 205-222.

[18] Khattri KB and Pudasaini SP (2019): Channel flow simulation of a mixture with a full-

606

dimensional generalized quasi two-phase model. Mathematics and Computers in Simulation,

607

doi: org/10.1016/j.matcom.2019.03.014.

urn al P

605

608

[19] Mergili M, Fischer J-T, Krenn J and Pudasaini SP (2017): r.avaflow v1, an advanced open-

609

source computational framework for the propagation and interaction of two-phase mass

610

flows. Geosci. Model Dev., 10, 553-569.

[20] Mergili M, Emmer A, Juicov A, Cochachin A, Fischer J.-T., Huggel C and Pudasaini

612

SP(2018 a): How well can we simulate complex hydro-geomorphic process chains? The

613

2012 multi-lake outburst flood in the Santa Cruz Valley (Cordillera Blanca, Per). Earth

614

Surface Processes and Landforms, 43(7), 1373-1389.

Jo

611

615

[21] Mergili M, Frank B, Fischer J.-T., Huggel C and Pudasaini SP(2018 b): Computational ex-

616

periments on the 1962 and 1970 landslide events at Huascarn (Peru) with r.avaflow: Lessons

617

learned for predictive mass flow simulations. Geomorphology, 322, 15-28.

618

[22] Papanastasiou TC (1987): Flows of materials with yield, J. Rheol., 31, 385.

619

[23] Pitman EB and Le L (2005): A two-fluid model for avalanche and debris flows. Phil. Trans.

620

R. Soc. A, 363(3), 1573-1601. 31

Journal Pre-proof

623 624

625 626

627 628

629 630

two-phase bulk mixture model for mass flow. Int. J. of Non-Linear Mech., 99, 229-239. [25] Pudasaini SP (2012): A general two-phase debris flow model. Journal of Geophysical Research, 117, F03010.

of

622

[24] Pokhrel PR, Khattri KB, Tuladhar BM and Pudasaini SP (2018): A generalized quasi

[26] Pudasaini SP (2011): Some exact solutions for debris and avalanche flows. Phys. Fluids, 23(4), 043301.

[27] Pudasaini SP and Kroner C (2008): Shock waves in rapid flows of dense granular materials:

pro

621

Theoretical predictions and experimental results. Phys. Rev. E., 78(4), 041308. [28] Pudasaini SP and Hutter K (2007): Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches, 602 pp., Springer, New York.

[29] Pudasaini SP, Hutter K, Hsiau S-S, Tai S-C, Wang Y and Katzenbach R (2007): Rapid

632

flow of dry granular materials down inclined chutes impinging on rigid walls. Phys. Fluids,

633

19(5), 053302.

635

636 637

[30] Pudasaini SP, Wang Y and Hutter K (2005): Modelling debris flows down general channels.

urn al P

634

re-

631

Nat. Hazards Earth Syst. Sci., 5, 799-819.

[31] Savage SB and Hutter K (1989): The motion of a finite mass of granular material down a rough incline. J. Fluid Mech., 199, 177-215.

638

[32] Senger RK and Fogg GE (1990): Stream Functions and Equivalent Freshwater Heads for

639

Modeling Regional Flow of Variable-Density Groundwater 1. Review of Theory and Verifi-

640

cation. Water Resour. Res., 26, 2089-2096.

642

643 644

[33] Slichter CS (1897): Theoretical investigations of motions of ground waters flow, 19th Annual Report, 1897-1898, part II, pp. 329-378, U.S. Geol. Surv., Washington, D.C. [34] Takahashi T (1991): Debris Flow. IAHR-AIRH Monograph Series A, Balkema, Rotterdam,

Jo

641

Netherlands.

32

Journal Pre-proof We have formulated a stream function - vorticity and vorticity-transport equation for rapid flow of mixture of viscous fluid and solid particles.

-

A novel pressure Poisson equation is derived in terms of stream function, vorticity and ratedependent mixture viscosity for shear flows including yield strength.

-

The pressure Poisson equation is characterized by the non-linear diffusion of the free surface.

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Mixture pressures are derived analytically for pressure dominated flow; thick, low yield strength flow; and thin, high yield strength flows.

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Exact/analytical solutions are constructed for pressure and flow depths for incipient, shearing and free surface flows, propagating bore and deposition.

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