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International Journal of Non–Linear Mechanics

journal homepage: www.elsevier.com/locate/nlm

Optimal systems of Lie subalgebras for a two-phase mass flow

Sayonita GhoshHajraa,⁎, Santosh Kandelb, Shiva P. Pudasainic

a Department of Mathematics, Hamline University, 1536 Hewitt Avenue, MS-B1807, Saint Paul, MN 55104, USAb Institute für Mathematik, University of Zürich, Winterthurerstrasse 190, CH-8057, Zürichc Department of Geophysics, Steinmann Institute, University of Bonn, Meckenheimer Allee 176, D-53115, Bonn, Germany

A R T I C L E I N F O

Keywords:Lie symmetryLie algebraOptimal systemTwo-phase mass flowDebris flow

A B S T R A C T

We apply the Lie symmetry method to a two-phase mass flow model (Pudasaini, 2012 [18]) and construct one-,two- and three-dimensional optimal systems of Lie subalgebras corresponding to the non-linear PDEs. As anoptimal system contains structurally important information about different types of invariant solutions, itprovides precise insights into all possible invariant solutions emerging from infinitesimal symmetries. We usethe optimal system of one-dimensional Lie subalgebras to reduce the two-phase mass flow model to othersystems of PDEs. Using the fact that the Lie bracket contains information about further reduction, we furtherreduce to systems of ODEs and PDEs. We solve a system numerically and present results for different physicaland Lie parameters. Simulations reveal fluid and solid dynamics are distinctly sensitive to different Lieparameters, whereas both phases are influenced by the solid and the fluid pressure parameters. Higher pressuregradients result in higher flow velocities and lower flow heights. Fluid velocities dominate solid velocities, butthe solid heights are higher than the fluid heights. Results provide an overall picture of the physical process, andthe coupled dynamics of the solid and fluid phase velocities and the flow heights. These are physicallymeaningful results in sheared inclined channel flow of coupled two-phase mixture. This confirms theconsistency of the obtained similarity solutions and potential applicability of the models and the constructedoptimal systems.

1. Introduction

Rapid gravity mass flows and gravity currents such as debris flows,debris floods, and water waves are common natural phenomena. Dueto their complex nature these events pose greater challenges toenvironmental, geophysical and engineering communities [4–7,10,16,17,21,22,31]. In this paper, we are mainly concerned withdebris flows, which are effectively two-phase gravity-driven mass flowsconsisting of a mixture of solid particles and viscous fluid. These flowsare mainly described by the relative motion between the solid and thefluid phases. Their evolution primarily depends on the solid–fluidmixture composition, mixture interactions, and mixture dynamicsmodelled by the driving forces. The debris flows are extremelydestructive natural hazards. Hence, we need a reliable way to predictthe dynamics of the flow. There has been extensive (field, experimental,theoretical, and numerical) studies investigating the dynamics, andconsequences of these flows including their industrial applications[9,21,25,26,28]. In [18] Pudasaini presented a new theory, whichdescribes the different interactions between the solid and the fluid intwo-phase debris flows including several fundamentally important and

dominant physical aspects which constitutes the most generalized two-phase flow model, as a set of partial differential equations in theconservative form [18,19,21] available to date. This theory supports thestrong interactions between the solid and the fluid phases.

In this paper, we apply the Lie symmetry method to the two-phasemass flow model [18] and construct optimal systems of subalgebrascorresponding to this system of non-linear PDEs. Lie symmetrymethod is an important tool for examining differential equations andconstructing their solutions [1,8]. The heart of this method lies in theconstruction of the invariant solutions, the solutions which areinvariant under the Lie subgroups of the Lie group of symmetries,and the construction of new solutions from known solutions by usingsymmetries. Invariant solutions are constructed by reducing the givensystem of PDEs into another system of PDEs or ODEs with the aid ofLie subgroups and solving the reduced system. Since almost every Liegroup of symmetries has infinitely many Lie subgroups, it seems thequest for invariant solutions is a very challenging task. As a symmetrytransformation maps one solution to another, practically, it is sufficientto find the invariant solutions which are not related by any member ofthe Lie group of symmetries. This suggests to look for ways to

http://dx.doi.org/10.1016/j.ijnonlinmec.2016.10.005Received 21 August 2016; Received in revised form 10 October 2016; Accepted 10 October 2016

⁎ Corresponding author.E-mail address: sghoshhajra01@hamline.edu (S. GhoshHajra).

International Journal of Non–Linear Mechanics 88 (2017) 109–121

0020-7462/ © 2016 Elsevier Ltd. All rights reserved.Available online 13 October 2016

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determine Lie subgroups which generates fundamentally distinctinvariant solutions. It is of great importance from the mathematicalpoint of view as well as constraining the physical and engineeringapplications, as it provides precise insights into all possible invariantsolutions of the system. This problem can be solved by classifying theLie subgroups of the Lie group of symmetries, often called the optimalsystem of s-parameter subgroups [11,12]. It possesses an infinitesimalcounterpart called the optimal system of s-parameter subalgebras.

In the literature, mostly, single-phase, inviscid and pressure-drivenshallow water flows in horizontal channels with no friction models areinvestigated. For example, in [15] similarity forms for two-layershallow water equations are derived and discussed where the couplingis via the reduced gravity parameter. In [1], GhoshHajra et al. discuss atwo-phase mass flow model whose driving forces consist of gravity,buoyancy, and hydraulic pressure gradients for the solid and fluidphases respectively. The two-phase mass flow model utilized there isderived from a general two-phase mass flow model [18] as a mixture ofsediment particles and viscous fluid with strong non-linear phaseinteractions. These interactions pose great challenges in constructingexact solutions as compared to the effectively single-phase gravity massflows. Thus, the problem, which is the study of the phase interactionbetween the solid and the fluid in a two-phase mass flow, is highly non-trivial and novel.

Using an elementary Lie transformation of the model [18],GhoshHajra et al. [1] constructed several similarity forms and simi-larity variables, which generalizes several similarity variables andsimilarity forms obtained in the literature by using Lie symmetrygroup transformations [2,3,13–15,23,24,27,30]. They also constructedseveral reduced homogeneous and non-homogeneous systems of ODEsand provided some analytical and numerical solutions. Here, weconsider the simplified model from [1] and advance further byconstructing their optimal systems of Lie subalgebras and providingsimulations and applications. For this, we first briefly discuss the two-phase mass flow model and the symmetry Lie algebra of the givensystem of PDEs from [1]. Then, we construct the optimal system of Liesubalgebras of dimensions one, two, and three. Further, we analyze indetail the reduced system of PDEs obtained using the one-dimensionalLie subalgebras. We construct several reduced systems of ODEs andPDEs. We also present simulation results for some systems in order tohave an in-depth and quantitative analyses of these systems. Thisallows for the analysis of impact of the model parameters in thephysical–mathematical model [18] and the Lie parameters in theoptimal system under consideration. As all the reduced ODE systemsobtained here are non-homogeneous, these systems are more complexboth from the mathematical structure and the underlying physics thanthe systems presented and discussed in [1]. Consequently, the newsystems potentially cover wider spectrum of applications.

2. The two-phase mass flow model and the symmetry Liealgebra

We consider the general two-phase debris flow model [18] that wasreduced to one-dimensional inclined channel flow [21]. For complete-ness, we briefly recall the basic features of the model equations [1].1

The depth-averaged mass and momentum conservation equations forthe solid and fluid phases are as follows [18]:

ht

QX

ht

QX

∂∂

+ ∂∂

= 0,∂∂

+∂∂

= 0,s s f f

(1)

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Qt X

Q hX

βh h h h S

Qt X

Q hX

βh h h h S

∂∂

+ ∂∂

( ) + ∂∂ 2

( + ) = ,

∂∂

+ ∂∂

( ) + ∂∂ 2

( + ) = ,

ss s

ss s f s s

ff f

ff f s f f

2 −1

2 −1

(2)

where S S,s f are the solid and fluid net driving forces, given byS ζ δ γ ζ S ζ= sin − tan (1 − )cos , = sin .s f The dynamical variables andparameters are

h α h h α h Q h u α hu Q h u α hu β εKp

β εp p ζ p γ p α α γρρ

= , = ; = = , = = ; = ,

= , = cos , = (1 − ) , = 1 − , = .

s s f f s s s s s f f f f f s b

f b b b b f sf

s

s

f f s f

Here, t is the time, X and Z are coordinates along and normal to theslope with inclination ζ. The solid particles and fluid constituents aredenoted by the suffices s and f respectively. The mixture flow depth is h,and the solid and fluid velocities are us and uf respectively. Thedensities and volume fractions are ρs, ρf, and αs, αf respectively, L andH denote the typical length and depth of the flow with the aspect ratioε H L= / . Both K, the earth pressure coefficient, and δtan , where δ is thefriction angle, include frictional behavior of the solid-phase. Here pbfand pbs are associated with the effective basal fluid and solid pressures,β β,s f are the hydraulic pressure parameters associated with the solid-and the fluid-phases respectively, and γ is the density ratio, γ(1 − )indicates the buoyancy reduced solid normal load. The solid and fluidfractions in the mixture are h h,s f , and the solid and fluid fluxes areQ Q,s f .

For convenience, GhoshHajra et al. [1] introduced the transforma-tion

x X S t y X S t Q Q h S t Q Q h S t= − /2, = − /2, = − , = − .s f s s s s f f f f2 2

(3)

Here, x and y are the moving spatial coordinates for the solid and fluidrespectively. Since Ss and Sf are independent, x and y are considered asindependent variables [1]. Using (3), Eqs. (1)–(2) become a homo-geneous system of partial differential equation:

ht

Qx

ht

Qy

∂∂

+ ∂∂

= 0,∂∂

+∂∂

= 0,s s f f

(4)

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

Qt x

Q hx

βh h h

Qt y

Q hy

βh h h

∂∂

+ ∂∂

( ) + ∂∂ 2

( + ) = 0,

∂∂

+ ∂∂

( ) + ∂∂ 2

( + ) = 0.

ss s

ss s f

ff f

ff f s

2 −1

2 −1

(5)

Replacing Q u h Q u h= , =s s s f f f , introducing the suffices s f1≔ , 2≔ forthe variables and parameters associated with the solid and fluidcomponents respectively, and dropping the hats, (4)–(5) changes to

ht

u hx

∂∂

+ ∂( )∂

= 0,1 1 1(6)

ht

u hy

∂∂

+ ∂( )∂

= 0,2 2 2

ut

u ux

β hx

β hx

β hh

hx

ut

u uy

β hy

β hy

β hh

hy

∂∂

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0,

∂∂

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0.

11

11

1 1 2 1 2

1

1

22

22

2 2 1 2 1

2

2

(7)

As discussed in [1], the mass flow model (6)–(7) which is deduced fromthe mixture model [18] is different from the other existing models. Thefourth and fifth terms associated with β /2 in (7) that emerge from thepressure-gradients, include buoyancy (through γ1 − ), friction (throughK and δtan ), net driving forces (S S,s f ) and the coordinate transforma-tions (3) that incorporate gravity, friction, and buoyancy. These forceshave mechanical significance in explaining the physics of the two-phasegravity mass flows that are not discussed in previous models, and in1 This discussion follows [1, pp. 326–327].

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

110

[1], they are considered for the mixture flows.GhoshHajra et al. [1] initiated the use of Lie symmetry method to

study the system (6)–(7). They computed the most general symmetryLie algebra g of these systems of PDEs, and applied the symmetries toreduce the system into systems of ODEs. The Lie algebra g is fivedimensional with a basis V V V V V{ , , , , }1 2 3 4 5 , where

Vx

Vy

Vt

V xx

yy

hh

hh

uu

uu

V tt

hh

hh

uu

uu

= ∂∂

, = ∂∂

, = ∂∂

,

= ∂∂

+ ∂∂

+ 2 ∂∂

+ 2 ∂∂

+ ∂∂

+ ∂∂

,

= ∂∂

− 2 ∂∂

− 2 ∂∂

− ∂∂

− ∂∂

.

1 2 3

4 11

22

11

22

5 11

22

11

22

The Lie bracket operation on the basis V V V V V{ , , , , }1 2 3 4 5 is shown inTable 1.

3. Optimal system of Lie subalgebras

Generally, it is difficult to gain insight on all the possible invariantsolutions as there can be infinitely many Lie subgroups of the Lie groupof symmetries G of a given system. Hence, it is desirable to partition allthe possible invariant solutions into disjoint sets so that two solutionsbelonging to the same set are similar (one solution can be transformedinto the other by an element of the symmetry Lie group) and belongingto the different sets are not similar (these solutions are not related byany member of the symmetry Lie group). This classification problemhas a solution, namely the construction of an optimal system of s-parameter subgroups [11,12]. Often, it is easier to work with thesymmetry Lie algebra, i.e., construction of optimal systems of s-parameter Lie subalgebras of the symmetry Lie algebra, and in nicesituations, an optimal system of s-dimensional Lie subalgebras issufficient to construct an optimal system of s-parameter subgroups.

An optimal system of s-dimensional Lie subalgebras (s-parametersubgroups) contains structurally important information about differenttypes of invariant solutions [11,12]. Let g be the Lie algebra of G. Anoptimal system of s-parameter Lie subalgebras is a list of s-dimensionalLie subalgebras of g such that every s-dimensional Lie subalgebra of gis equivalent to a unique member of the list and no two Lie subalgebrasin the list are equivalent to each other. Two s-dimensional Liesubalgebras h and h′ are equivalent if there is an element g of G suchthat h h′ = Ad ( )g where Adg is the adjoint action of g on g. We can usethe exponential map Exp from g to G [11,12] to define the adjointaction of a generic g G∈ .

Assume that g V= Exp( ) for some gV ∈ . Note that any gV ∈ definesa linear operator g gVad( ): → given by

V W V Wad( )( ) = [ , ]

where [ , ] is the Lie bracket. Now, represent Vad( ) by a matrix bychoosing a basis of g. Then (see [11]),

∑e Vn

Ad = = ad( )!

.gV

n

nad( )

=0

∞

(8)

Let g be the symmetry Lie algebra with basis V V V V V{ , , , , }1 2 3 4 5 ofSection 2 and identify this with 5 as a vector space using the mapV e↦i iwhere e e{ ,…, }1 5 is the standard basis of 5. Then, from Table 1, we get

the following matrix representations of Vad( )i :

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

V V

V V

V

ad( ) =

0 0 0 1 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

, ad( ) =

0 0 0 0 00 0 0 1 00 0 0 0 00 0 0 0 00 0 0 0 0

,

ad( ) =

0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

, ad( ) =

− 1 0 0 0 00 − 1 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

,

ad( ) =

0 0 0 0 00 0 0 0 00 0 − 1 0 00 0 0 0 00 0 0 0 0

.

1 2

3 4

5

(9)

Let ε i, = 1, 2,…,5i , be real constants and g ε V= Exp( )i i i . Then, bycomputing the exponential of matrices ε Vad( )i i , we get

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

εε

ee

e

Ad =

1 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

, Ad =

1 0 0 0 00 1 0 00 0 1 0 00 0 0 1 00 0 0 0 1

,

Ad =

1 0 0 0 00 1 0 0 00 0 1 0 ϵ0 0 0 1 00 0 0 0 1

, Ad =

0 0 0 00 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

,

Ad =

1 0 0 0 00 1 0 0 00 0 0 00 0 0 1 00 0 0 0 1

.

g g

g g

ε

ε

g ε

12

3

−−

−

1 2

3 4

4

4

55

(10)

The adjoint action Adgi of gi on the basis V V{ ,…, }1 5 of g is summarizedin Table 2.

Construction of optimal systems of s-parameter subalgebras of a Liealgebra g is a difficult task. To our knowledge, there is no systematicway to compute an optimal system when s > 3. However, when s ≤ 3,we can compute optimal systems of s-parameter subalgebras. The Liesubalgebras can be computed by solving some algebraic equations, andthe equivalent Lie subalgebras can be identified by applying the adjointaction Adg on the set of these Lie subalgebras [12]. In connection to thecoupled system of two-phase mass flows (6)–(7), this is one of thecontributions of this paper. Here, we construct one-, two-, and three-dimensional optimal systems of Lie subalgebras for the physical system(6)–(7).

3.1. Optimal system of one-dimensional Lie subalgebras

Since any nonzero element of g generates one dimensional Liesubalgebra, it is possible to identify whether two one-dimensional Liesubalgebras are equivalent by studying whether their generators arerelated by the adjoint action Adg for some g of the symmetry Lie group.In other words, it is sufficient to concentrate on how a general Adg

transforms a general element of g and use that information to identify

Table 1Lie brackets for the generators V V V V V{ , , , , }1 2 3 4 5 of the Lie algebra g.

[ , ] V1 V2 V3 V4 V5

V1 0 0 0 V1 0V2 0 0 0 V2 0V3 0 0 0 0 V3

V4 −V1 −V2 0 0 0V5 0 0 −V3 0 0

Table 2The adjoint action iAd , = 1, 2,…,5gi , on the basis V V V V V{ , , , , }.1 2 3 4 5

Ad V1 V2 V3 V4 V5

g1 V1 V2 V3 V ε V+4 1 1 V5

g2 V1 V2 V3 V ε V+4 2 2 V5

g3 V1 V2 V3 V4 V ε V+5 3 3g4 e Vε− 4 1 e Vε− 4 2 V3 V4 V5

g5 V1 V2 e Vε− 5 3 V4 V5

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

111

equivalent Lie subalgebras.Let V a V a V a V a V a V= + + + +1 1 2 2 3 3 4 4 5 5 bea general element of g where a i, = 1, 2,…,5i , are real constants. Wecan think of V as a column vector with entries a a,…,1 5. LetA ε ε ε ε ε( , , , , ) = Ad ∘Ad ∘Ad ∘Ad ∘Adg g g g g1 2 3 4 5 5 4 3 2 1, then multiplying thematrices in (10) we get

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥A ε ε ε ε ε

e ε ee ε e

e ε e( , , , , ) =

0 0 00 0 00 0 00 0 0 1 00 0 0 0 1

.

ε ε

ε ε

ε ε1 2 3 4 5

−1

−

−2

−

−3

−

4 4

4 4

5 5

(11)

Moreover, A ε ε ε ε ε( , , , , )1 2 3 4 5 transforms V as follows:

A ε ε ε ε ε V a ε a e V a ε a e V

a ε a e V a V a V

( , , , , ) = ( + ) + ( + )

+ ( + ) + + .

ε ε

ε

1 2 3 4 5 1 1 4−

1 2 2 4−

2

3 3 5−

3 4 4 5 5

4 4

5 (12)

By definition V and A ε ε ε ε ε V( , , , , )1 2 3 4 5 generate equivalent one-dimensional Lie subalgebras for any ε ε,…, .1 5 This gives the freedomof choosing various values of εi so that a representative of theequivalence class of V might be much simpler than V . The followinglemmas will be useful for this purpose.

Lemma 3.1. When a ≠ 04 , a V a V a V a V a V+ + + +1 1 2 2 3 3 4 4 5 5 anda V a V a V+ +3 3 4 4 5 5 generate equivalent Lie subalgebras.

Proof. Choosing ε = − ,aa1

14

ε = − aa2

24

and ε = 03 , we see thata V a V a V a V a V+ + + +1 1 2 2 3 3 4 4 5 5 and V a e V a V+ +ε

4 3−

3 5 55 generateequivalent one-dimensional Lie subalgebras.□Lemma 3.1 says theequivalence class of one-dimensional Lie subalgebras containing theLie subalgebra a V a V a V V a V⟨ + + + + ⟩1 1 2 2 3 3 4 5 5 is independent of a1 anda .2 The key reasons behind this lemma are the Lie bracket relationsV V V[ , ] = −4 1 1 and V V V[ , ] = −4 2 2, which is reflected in Ad , Adg g1 2 and

Ad .g4

Lemma 3.2. The Lie subalgebra a V V a V⟨ + + ⟩3 3 4 5 5 is equivalent toexactly one of the Lie subalgebras V V a V V V a V⟨ ⟩, ⟨ + ⟩, ⟨ + + ⟩4 4 5 5 3 4 5 5and V V a V⟨ − + + ⟩3 4 5 5 .

Proof.When a a= 0 =3 5, the Lie subalgebra is V⟨ ⟩4 .When a = 03 , the Liesubalgebra is V a V⟨ + ⟩4 5 5 .

When a ≠ 03 , taking ε ε ε= = = 01 2 3 , we observe

A ε ε ε ε ε a V V a V a ε V V a V( , , , , )( + + ) = + + .ε1 2 3 4 5 3 3 4 5 5 3

−3 4 5 55 (13)

Choosing ε5 so that either a e = 1ε3

− 5 or a e = −1ε3

− 5 , it follows that theLie subalgebra a V V a V⟨ + + ⟩3 3 4 5 5 is equivalent to V V a V⟨ + + ⟩3 4 5 5 or

V V a V⟨ − + + ⟩3 4 5 5 . This proves the lemma.□Lemmas 3.1 and 3.2 imply the Lie subalgebra a V a V⟨ + +1 1 2 2

a V a V a V+ + ⟩3 3 4 4 5 5 , a ≠ 04 , is equivalent to one of the following typesof one-dimensional Lie subalgebras

V V k V V V k V V V k V⟨ ⟩, ⟨ + ⟩, ⟨ + + ⟩, ⟨ − + + ⟩4 4 1 5 3 4 2 5 3 4 3 5

where k ≠ 01 , k2 and k3 are constants.Similar analysis can be carried out for a ≠ 05 , it can be shown

a V a V a V a V a V⟨ + + + + ⟩1 1 2 2 3 3 4 4 5 5 and a V a V a V a V⟨ + + + ⟩1 1 2 2 4 4 5 5 gen-erate equivalent one-dimensional Lie subalgebras. Similarly,a V a V a V a V⟨ + + + ⟩1 1 2 2 3 3 5 5 is equivalent to one of the following:

V k V V V k V V k V V V k V V V⟨ + + ⟩, ⟨ − + + ⟩, ⟨ + + ⟩, ⟨ − + ⟩1 4 2 5 1 5 2 5 6 1 2 5 7 1 2 5

where k k k, ,4 5 6 and k7 are constants. Analyzing the rest of the casesin the similar way we get:

Proposition 3.3. An optimal system of one-dimensional Liesubalgebras contains Lie subalgebras

V V V V V V V⟨ ⟩, ⟨ ⟩, ⟨ ⟩, ⟨ + ⟩, ⟨ − + ⟩,1 3 5 3 4 3 4

and the following families of subalgebras

k V V k V k V kV k V V k V k V V kV k V V k V k V V kk V V V k k V V V kk V V V k k V V V k

{⟨ + ⟩: ∈ }, {⟨ + ⟩: ∈ },{⟨ + + ⟩: ∈ }, {⟨ − + + ⟩: ∈ },{⟨ + + ⟩: ∈ }, {⟨ + − ⟩: ∈ },{⟨ + + ⟩: ∈ ⧹{1, −1}}, {⟨ − + ⟩: ∈ ⧹{1, −1}},{⟨ + + ⟩: ∈ ⧹{1, −1}}, {⟨ + − ⟩: ∈ ⧹{1, −1}}.

1 1 2 1 4 2 5 2

1 3 2 5 3 1 4 2 5 4

1 5 2 3 5 1 6 2 3 6

7 1 2 5 7 8 1 2 5 8

9 1 2 3 9 10 1 2 3 10

3.2. Optimal system of two-dimensional Lie subalgebras

Next, we compute an optimal system of two-dimensional Liesubalgebras. The key idea is to analyze only those two-dimensionalLie subalgebras which are extensions of the one-dimensional Liesubalgebras from the list in Proposition 3.3 rather than all the two-dimensional Lie subalgebras [12]. This suggests, we first compute allthe two-dimensional Lie subalgebras which are extensions of the one-dimensional Lie subalgebras from the list in Proposition 3.3, then,remove the redundancies by studying action of a general Adg on theseLie subalgebras.

The two-dimensional Lie subalgebras which are extension of one-dimensional Lie subalgebras from Proposition 3.3 can be computed asfollows. Let W⟨ ⟩ be a one-dimensional Lie subalgebra from Proposition3.3 and consider the problem of finding all two-dimensional Liesubalgebras containing W⟨ ⟩. To solve this problem, we must find allpossible gV ∈ such that V W⟨ , ⟩ is a two-dimensional Lie subalgebra. Aswe want V W⟨ , ⟩ to be a two-dimensional Lie subalgebra, there must beconstants λ and μ such that

W V λW V[ , ] = + μ . (14)

In summary, we must find V such that V W⟨ , ⟩ is a two dimensionalvector space, and λ and μ satisfy (14). This will lead to a set of algebraicequations whose solutions will give the two-dimensional Lie subalge-bras containing W⟨ ⟩.

To illustrate, we compute all two-dimensional Lie subalgebraswhich contain the one-dimensional Lie subalgebra V⟨ ⟩.4 LetV a V a V a V a V= + + +1 1 2 2 3 3 5 5 be a second basis element of a desiredtwo-dimensional Lie subalgebra. Table 1 gives V V a V a V[ , ] = − −4 1 1 2 2,which provides the left hand side of Eq. (14). The problem, here, is tofind a a a a, , ,1 2 3 5 not simultaneously zero and λ and μ such that

μa a μa a μa= − , = − , = 0,1 1 2 2 3

λ μa= 0, = 0.5

Case 1: When μ = 0 then, a a= = 01 2 . Any a3 and a5, not simulta-neously zero, can be chosen arbitrarily. This results in the two-dimensional Lie subalgebras a V a V V a a⟨ + , ⟩, ( , ) ∈ ⧹{(0, 0)}3 3 5 5 4 3 4

2 .Case 2: When μ ≠ 0 then, a a= = 03 5 . As V ≠ 0, μ = −1 is the only

choice and any a1 and a2, not simultaneously zero can be chosen as freeparameters. This results in another two-dimensional Lie subalgebras

a V a V V a a⟨ + , ⟩, ( , ) ∈ ⧹{(0, 0)}1 1 2 2 4 1 22 .

This process, applied to all the one-dimensional Lie subalgebrasfrom Proposition 3.3, computes the two-dimensional Lie subalgebraswhich contain one-dimensional Lie subalgebras from Proposition 3.3.Next, we must eliminate redundancies by studying equivalence classesof these Lie subalgebras to compute an optimal system. We achieve thisby analyzing how these Lie subalgebras transform when a general Adg

applied to them.To demonstrate this, we compute the equivalence class of the

subalgebra a V a V V⟨ + , ⟩3 3 5 5 4 for fixed a a( , ) ∈ ⧹{(0, 0)}3 52 and find a

simplified representative of the equivalence class.Let A A ε ε ε ε ε= ( , , , , )1 2 3 4 5 be as in (12), then

A a V a V e a a ε V a V( + ) = ( + ) + andε3 3 5 5

−3 5 3 3 5 55

A V ε e V ε e V V( ) = + + .ε ε4 1

−1 2

−2 44 4

This means, by definition, a V a V V⟨ + , ⟩3 3 5 5 4 ande a a ε V a V ε e V ε e V V⟨ ( + ) + , + + ⟩ε ε ε−

3 5 3 3 5 5 1−

1 2−

2 45 4 4 are equivalent Liesubalgebras. Following this process, we obtain all two-dimensional

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

112

Lie subalgebras from the two cases (Case 1, Case 2) above which areequivalent to a V a V V⟨ + , ⟩3 3 5 5 4 by choosing various values of εi's. Thefollowing lemma is useful to determine the equivalence classes of two-dimensional subalgebras.

Lemma 3.4. For all a ≠ 05 , a V a V V⟨ + , ⟩3 3 5 5 4 is equivalent to V V⟨ , ⟩5 4 .

Proof. Take ε ε ε= = = 01 2 5 and ε3 such that a a ε+ = 03 5 3 . Then,A a V a V a V( + ) =3 3 5 5 5 5. Hence, for this particular A, we haveA a V a V V V a V V V(⟨ + , ⟩) = ⟨ , ⟩ = ⟨ , ⟩3 3 5 5 4 4 5 5 4 5 . □

Lemma 3.4 says V V⟨ , ⟩4 5 can be taken as a simple representative ofthe equivalence class of the Lie subalgebras a V a V V⟨ + , ⟩3 3 5 5 4 whena ≠ 05 .

Corollary 3.5. We can take V V⟨ , ⟩3 4 and V V⟨ , ⟩4 5 as the representativeof equivalence classes of Lie subalgebras a V a V V⟨ + , ⟩3 3 5 5 4 for any

a a( , ) ∈ ⧹{(0, 0)}3 52 .We consider only one representative from each

equivalence class to get an optimal system. The simplest choices areV V⟨ , ⟩3 4 and V V⟨ , ⟩4 5 for the Lie subalgebras a V a V V⟨ + , ⟩3 3 5 5 4 for

a a( , ) ∈ ⧹{(0, 0)}3 52 .

A similar analysis to all the other two-dimensional Lie subalgebrascontaining one-dimensional Lie subalgebras from Proposition 3.3 leadsto the following conclusion.

Proposition 3.6. An optimal system of the two-dimensional Liesubalgebras can be represented by the following Lie subalgebras:

V V V V V V V V V V⟨ , ⟩, ⟨ , ⟩, ⟨ , ⟩, ⟨ , ⟩, ⟨ , ⟩1 2 1 3 1 5 3 4 4 5

and the families of Lie subalgebras

k V V a V V k a

k V V a V V k a

k V V a V V k a

V V V a V V a

V a V V a

V k V a V a V V a a

k V V k V k

V k V a V V k a

V k V V V a V k a

V k V V V a V k a

V k V V V a V k a

V k V V a V V k a

k V V V a V V k a

k V V V a V V k a

k V V V a V V k a

V a V a V V a a

k V V V a V V a k

{⟨ + , + ⟩: ( , ) ∈ },

{⟨ + , + ⟩: ( , ) ∈ },

{⟨ + , + ⟩: ( , ) ∈ },

{⟨ − − + , + ⟩: ∈ },

{⟨ , + ⟩: ∈ },

{⟨ + , + + ⟩: ( , ) ∈ ,

∈ ⧹{0}}, {⟨ , + ⟩: ∈ ⧹{0}},

{⟨ + , + ⟩: ∈ ⧹{0}, ∈ },

{⟨ + + , + ⟩: ( , ) ∈ },

{⟨ − + + , + ⟩: ( , ) ∈ },

{⟨ + + , + ⟩: ( , ) ∈ },

{⟨ + − , + ⟩: ( , ) ∈ },

{⟨ + + , + ⟩: ( , ) ∈ },

{⟨ − + , + ⟩: ( , ) ∈ },

{⟨ + + , + ⟩: ( , ) ∈ },

{⟨ , + + ⟩: ( , ) ∈ },

{⟨ + − , + ⟩: ≠ −1, ≠ 0}.

1 1 2 1 1 3 1 12

1 1 2 2 1 5 1 22

1 1 2 3 3 4 1 32

1 2 3 4 1 2 4

1 5 3 4 5

4 3 5 6 1 7 2 3 6 72

3 1 4 3 5 3

4 3 5 8 1 2 3 8

1 4 2 5 1 9 2 4 92

1 5 2 5 1 10 2 5 102

1 6 2 3 1 11 2 6 112

1 7 2 3 12 2 3 7 122

8 1 2 5 13 1 2 8 132

9 1 2 5 14 1 2 9 142

10 1 2 3 15 1 3 10 152

3 17 1 18 2 5 17 182

11 1 2 3 16 2 3 16 11

3.3. Optimal system of three-dimensional Lie subalgebras

The construction of an optimal system of three-dimensional Liesubalgebras is exactly similar to construction of an optimal system oftwo-dimensional Lie subalgebras in the previous subsection: First,extend the two-dimensional Lie subalgebras from the proposition(Proposition 3.6) to three-dimensional Lie subalgebras in all possibleways, and then, remove all the redundancies by applying a general Adg

[12]. The only difference is, here, there are more algebraic equations.The result is summarized in:

Proposition 3.7. The following subalgebras form an optimal systemof three-dimensional Lie subalgebras:

V V CV V V V CV V V V V

V V CV V V V V V V V CV

V V V k V V V V CV k

k V V V CV V k

V V V a V V V V a

V a V V V a V k V a V a V V CV V

k a a V k V a V V V k

V V a V V CV V a V V k V V k

V k V V V a V V k a

V k V V V a V V k a

V k V V V a V V V k a

V V k V CV V k

V k V V a V V V V k a

k V V V a V V V k a

V a V a V V CV V

a a k V V V a V V V k a

k V V V a V V V V a k

k V V V a V V V V a k

V V a V V V a

⟨ , , + ⟩, ⟨ , , + ⟩, ⟨ , , ⟩,

⟨ , , + ⟩, ⟨ , , ⟩, ⟨ , , + ⟩,

⟨ , , ⟩, {⟨ + , , + ⟩, ∈ },

{⟨ + , , + ⟩, ∈ },

{⟨ − − + , + , + ⟩, ≠ 1},

{⟨ , + , ⟩, ∈ }, {⟨ + , + + , + ⟩,

≠ 0, ( , ) ∈ }, {⟨ + , + , ⟩, ≠ 0},

{⟨ + , + , + ⟩, ∈ }, {⟨ , + , ⟩, ∈ },

{⟨ + + , + , ⟩, ( , ) ∈ },

{⟨ − + + , + , ⟩, ( , ) ∈ },

{⟨ + + , + , + ⟩, ≠ },

{⟨ , + , + ⟩, ∈ },

{⟨ + − , + , + ⟩, ∈ , ≠ 0},

{⟨ + + , + , ⟩, ( , ) ∈ },

{⟨ , + + , + ⟩,

( , ) ∈ ⧹(0, 0)}{⟨ − + , + , ⟩, ( , ) ∈ },

{⟨ + + , + , + ⟩, ≠ 0, ∈ },

{⟨ + − , + , + ⟩, ≠ 0, −1; ≠ 0},

{⟨ − , + , ⟩, ≠ −1}.

1 2 4 5 1 2 3 4 1 2 3

3 4 1 2 3 4 5 1 3 4 5

2 4 5 1 1 2 3 4 5 1

1 1 2 3 1 5 1

1 2 3 4 1 2 4 5 4

1 5 3 4 5 5 4 3 5 6 1 7 2 3 1 2

3 6 72

4 3 5 7 2 3 1 3

4 5 8 1 2 1 3 8 5 1 4 2 4 4

1 4 2 5 1 9 2 3 4 92

1 5 2 5 1 10 2 3 5 102

1 6 2 3 1 11 2 4 5 6 11

3 1 6 2 2 5 6

1 7 2 3 12 2 3 4 5 7 12

8 1 2 5 13 1 2 3 8 132

3 17 1 18 2 5 1 2

17 182

9 1 2 5 14 1 2 3 9 142

10 1 2 3 15 1 3 4 5 15 10

11 1 2 3 16 2 3 4 5 16 11

2 3 16 2 3 1 16

4. Reduction of PDEs

In this section, we reduce the two-phase mass flow system of PDEsinto another systems of PDEs. The reduction procedure works for anyinfinitesimal symmetry and it goes as follows. The procedure assigns toan infinitesimal symmetry a characteristics equation, which is a systemof ODEs. The solutions of the characteristics equation lead to newindependent variables (similarity variables) and dependent variables(similarity forms). Substitution of these similarity variables andsimilarity forms in the given system of PDEs results in a reducedsystem.

The crucial fact discussed in Section 3 is that many infinitesimalsymmetries will lead to similar solutions. For fundamentally differentsolutions emerging from infinitesimal symmetries, it is enough to usean optimal system of Lie subalgebras as it contains information aboutdifferent types of solutions. In what follows, we restrict our analysis tothe optimal system of one dimensional Lie subalgebras computed inSection 3.1 to reduce the system of PDEs (6)–(7). We present thereduced systems (although each of these systems can be computedeasily with some tedious computation) because anyone who is furtherinterested in studying these systems can directly solve and analyze oneor more of these systems that lead to fundamentally different quanti-tative solutions. We note that it is also possible to use the optimalsystems of two- and three-dimensional Lie subalgebras for the reduc-tion but we do not discuss that in this paper.

(I) Reduced PDEs associated to V kV V+ +1 2 5: The characteristicsequation for this case is

dx dyk

dtt

dhh

dhh

duu

duu1

= = =−2

=−2

=−

=−

.1

1

2

2

1

1

2

2

The similarity variables (w t, ′) are given by w y kx tt e= − , ′ = ,x and thesimilarity forms (u h,i i) satisfy u u e h h e= , =i i

xi i

x− −2 . This reduces thePDEs (6)–(7) to:

t

h

ht

tu h

u ht

ku h

u hw u h

t

h

ht u h

u hw

′ ∂∂ ′

− ′( )

∂( )∂ ′

+( )

∂( )∂

= 3 ,

′ ∂∂ ′

− 1( )

∂( )∂

= 0,

2

12

1

1 12

1 1

1 12

1 1

1 1

2

22

2

2 22

2 2

(15)

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

113

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

tu

tu

ut

ku

uw

t β

h

t βh h

ht

kβ

h

kβh h

hw

t β

h

ht

kβ

h

hw u

βh

βh

tu

ut u

uw

β

h

hw

β

h

hw

βh h

hw

′− ′ ∂

∂ ′+ ∂

∂−

′+

′2

∂∂ ′

+ +2

∂∂

−′

2

∂∂ ′

+2

∂∂

= 1 +2

+2

,

′ ∂∂ ′

− 1 ∂∂

− ∂∂

−2

∂∂

−2

∂∂

= 0.

2

12

13

1

13

1 1

12

1

1 2

1 1

12

1

1 2

1

1

22

2 1

22

2

12

1

1

1

2

2

22

2

23

2 2

22

2 2

12

1 2

1 2

2

(16)

There are some particularly interesting aspects associated with thesystem (15)–(16). First, although this system has only one spatialvariable w that combines (together with k) the two variables x and yfrom the system (6)–(7), the new system turns non-homogeneous forboth the mass and momentum balance equations. Also, the coefficientsin the reduced PDEs are highly non-linear both in similarity variablesand similarity forms. These aspects made the reduced set a verychallenging system of equations. This discussion also holds for severalother reduced systems that follows.

(II) Reduced PDEs associated to V kV V− + +1 2 5:Characteristics equation:

dx dyk

dtt

dhh

dhh

duu

duu−1

= = =−2

=−2

=−

=−

.1

1

2

2

1

1

2

2

Similarity variables:

w t w y kx tt e, ′ with = + , ′ = .x−

Similarity forms:

u h u u e h h e, with = , = .i i i ix

i ix2

Reduced PDEs:

t

h

ht

tu h

u ht

ku h

u hw u h

t

h

ht u h

u hw

′ ∂∂ ′

+ ′( )

∂( )∂ ′

−( )

∂( )∂

= −3 ,

′ ∂∂ ′

− 1( )

∂( )∂

= 0,

2

12

1

1 12

1 1

1 12

1 1

1 1

2

22

2

2 22

2 2

(17)

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

tu

tu

ut

ku

uw

t β

h

t βh h

ht

kβ

h

kβh h

hw

t β

h

ht

kβ

h

hw u

βh

βh

tu

ut u

uw

β

h

hw

β

h

hw

βh h

hw

′+ ′ ∂

∂ ′− ∂

∂+

′+

′2

∂∂ ′

− +2

∂∂

+′

2

∂∂ ′

−2

∂∂

= − 1 −2

−2

,

′ ∂∂ ′

− 1 ∂∂

− ∂∂

−2

∂∂

−2

∂∂

= 0.

2

12

13

1

13

1 1

12

1

1 2

1 1

12

1

1 2

1

1

22

2 1

22

2

12

1

1

1

2

2

22

2

23

2 2

22

2 2

12

1 2

1 2

2

(18)

(III) Reduced PDEs associated to V V+2 5:Characteristics equation:

dy dtt

dhh

dhh

duu

duu1

= =−2

=−2

=−

=−

.1

1

2

2

1

1

2

2

Similarity variables:

x t tt e, ′ with ′ = .y

Similarity forms:

u h u u e h h e, with = , = .i i i iy

i iy− −2

Reduced PDEs:

t

h

ht u h

u hx

t

h

ht

tu h

u ht u h

′ ∂∂ ′

− 1( )

∂( )∂

= 0,′ ∂

∂ ′− ′

( )∂( )

∂ ′= 3 ,

2

12

1

1 12

1 12

22

2

2 22

2 2

2 2 (19)

⎛⎝⎜

⎞⎠⎟

tu

ut u

ux

β

h

hx

β

h

hx

βh h

hx

tu

tu

ut

β t

h

ht

β t

h

ht

β th h

ht u

βh

βh

′ ∂∂ ′

− 1 ∂∂

− ∂∂

−2

∂∂

−2

∂∂

= 0,

′− ′ ∂

∂ ′−

′ ∂∂ ′

−′

2

∂∂ ′

−′

2∂∂ ′

= 1 +2

+2

.

2

12

1

13

1 1

12

1 1

22

2 1

1 2

1

2

22

23

2 2

22

2 2

12

1 2

1 2

2

22

2

2

2

1

(20)

(IV) Reduced PDEs associated to V V− +2 5:Characteristics equation:

dy dtt

dhh

dhh

duu

duu−1

= =−2

=−2

=−

=−

.1

1

2

2

1

1

2

2

Similarity variables:

x t tt e, ′ with ′ = .y−

Similarity forms:

u h u u e h h e, with = , = .i i i iy

i iy2

Reduced PDEs:

t

h

ht u h

u hx

t

h

ht

tu h

u ht u h

′ ∂∂ ′

− 1( )

∂( )∂

= 0,′ ∂

∂ ′+ ′

( )∂( )

∂ ′= −3 ,

2

12

1

1 12

1 12

22

2

2 22

2 2

2 2 (21)

⎛⎝⎜

⎞⎠⎟

tu

ut u

ux

β

h

hx

β

h

hx

βh h

hx

tu

tu

ut

β t

h

ht

β t

h

ht

β th h

ht

uβh

βh

′ ∂∂ ′

− 1 ∂∂

− ∂∂

−2

∂∂

−2

∂∂

= 0,

′+ ′ ∂

∂ ′+

′ ∂∂ ′

+′

2

∂∂ ′

+′

2∂∂ ′

= − 1 −2

−2

.

2

12

1

13

1 1

12

1 1

22

2 1

1 2

1

2

22

23

2 2

22

2 2

12

1 2

1 2

2

22

2

2

2

1 (22)

(V) Reduced PDEs associated to V5:Characteristics equation:

dtt

dhh

dhh

duu

duu

= −2

= −2

= − = − .1

1

2

2

1

1

2

2

Similarity forms:

u h u tu h t h, with = , = .i i i i i i2

Reduced PDEs:

t ht

u hx

h t ht

u hx

h∂∂

+ ∂( )∂

= 2 , ∂∂

+ ∂( )∂

= 2 ,1 1 11

2 2 22 (23)

t ut

u ux

β hx

β hx

β hh

hx

u

t ut

u uy

β hy

β hy

β hh

hy

u

∂∂

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= ,

∂∂

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= .

11

11

1 1 2 1 2

1

11

22

22

2 2 1 2 1

2

22

(24)

(VI) Reduced PDEs with the symmetry V V+3 4:Characteristics equation:

dxx

dyy

dt dhh

dhh

duu

duu

= =1

=2

=2

= = .1

1

2

2

1

1

2

2

Similarity variables:

w t y wx t x e, ′ with = , ′ = .t

Similarity forms:

u h xu u x h h, with = , = .i i i i i i2

Reduced PDEs:

t ht

w u hw

t u ht

u h t ht

u hw

′ ∂∂ ′

− ∂( )∂

− ′ ∂( )∂ ′

= −3 , ′ ∂∂ ′

+ ∂( )∂

= 0,1 1 1 1 11 1

2 2 2(25)

t ut

u w uw

β w hw

βw h

wβ

w hh

hw

t u ut

β t ht

βt h

tβ

t hh

ht

u β h β h

t ut

u uw

β hw

β hw

β hh

hw

′ ∂∂ ′

− ∂∂

− ∂∂

−2

∂∂

−2

∂∂

− ′ ∂∂ ′

− ′ ∂∂ ′

−2

′ ∂∂ ′

−2

′ ∂∂ ′

= − − 2 − 2 ,

′ ∂∂ ′

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0.

11

11

1 1 2 1 2

1

11

1

11 1 2 1 2

1

112

1 1 1 2

22

22

2 2 1 2 1

2

2

(26)

Similarly, we can deal with V V− +3 4.(VII) Reduced PDEs with the symmetry V kV+4 5:Characteristics equation:

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

114

dxx

dyy

dtkt

dhh k

dhh k

duu k

duu k

= = =2 (1 − )

=2 (1 − )

=(1 − )

=(1 − )

.1

1

2

2

1

1

2

2

Similarity variables:

w t y wx t x t, ′ with = , ′ = .k

Similarity forms:

u h x u u x h h, with = , = .i ik

i ik

i i(1− ) 2(1− )

Reduced PDEs:

ht

w u hw

kt u ht

k u h ht

u hw

∂∂ ′

− ∂( )∂

− ′ ∂( )∂ ′

= −3(1 − ) , ∂∂ ′

+ ∂( )∂

= 0,1 1 1 1 11 1

2 2 2

(27)

ut

u w uw

β w hw

βw h

wβ

w hh

hw

u kt ut

β kt ht

βkt h

tβ

kt hh

ht

k u β h β h

ut

u uw

β hw

β hw

β hh

hw

∂∂ ′

− ∂∂

− ∂∂

−2

∂∂

−2

∂∂

− ′ ∂∂ ′

− ′ ∂∂ ′

−2

′ ∂∂ ′

−2

′ ∂∂ ′

= (1 − ){− − 2 − 2 },

∂∂ ′

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0.

11

11

1 1 2 1 2

1

11

11

1

1 2 1 2

1

112

1 1 1 2

22

22

2 2 1 2 1

2

2

(28)

We observe that this system reduces to a homogeneous system for k=1,it is expected as V V x y t+ = + +

x y t4 5∂∂

∂∂

∂∂ is a scaling transformation in

the independent variables.(VIII) Reduced PDEs with the symmetry kV V V+ +1 2 3:Characteristics equation:

dxk

dy dt=1

=1

.

Similarity variables:

w t w y xk

t t xk

, ′ with = − , ′ = − .

Similarity forms:

u h u u h h, with = , = .i i i i i i

Reduced PDEs:

ht k

u hw k

u ht

ht

u hw

∂∂ ′

− 1 ∂( )∂

− 1 ∂( )∂ ′

= 0, ∂∂ ′

+ ∂( )∂

= 0,1 1 1 1 1 2 2 2(29)

ut k

u uw k

β hw

βk

hw

βk

hh

hw k

u ut k

β ht

βk

ht

βk

hh

ht

ut

u uw

β hw

β hw

β hh

hw

∂∂ ′

− 1 ∂∂

− 1 ∂∂

−2

∂∂

−2

∂∂

− 1 ∂∂ ′

− 1 ∂∂ ′

−2

∂∂ ′

−2

∂∂ ′

= 0,

∂∂ ′

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0.

11

11

1 1 2 1 2

1

11

11

1

1 2 1 2

1

1

22

22

2 2 1 2 1

2

2

(30)

As kV V V+ +1 2 3 is a linear combination of translations in the indepen-dent variables, the reduced system is a homogeneous system asexpected. Similarly, we can deal with kV V V+ −1 2 3, kV V V− +1 2 3,kV V V− −1 2 3.

(IX) Reduced PDEs with the symmetry V V+1 3:Characteristics equation:

dx dt1

=1

.

Similarity variables:

y t t t x, ′ with ′ = − .

Similarity forms:

u h u u h h, with = , = .i i i i i i

Reduced PDEs:

ht

u ht

ht

u ht

∂∂ ′

− ∂( )∂ ′

= 0, ∂∂ ′

− ∂( )∂ ′

= 0,1 1 1 2 2 2(31)

ut

u ut

β ht

β ht

β hh

ht

ut

u uy

β hy

β hy

β hh

hy

∂∂ ′

− ∂∂ ′

− ∂∂ ′

−2

∂∂ ′

−2

∂∂ ′

= 0,

∂∂ ′

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= 0.

11

11

1 1 2 1 2

1

1

22

22

2 2 1 2 1

2

2

(32)

Similarly, we can compute the reduced PDEs for the symmetriesV V−1 3, V V+2 3, V V−2 3 and kV V+1 2 and each of these symmetriesyields a homogeneous system. These cases are similar to the onediscussed in [1].

5. Further reduction

Here, we discuss the further reduction of some of the reducedsystem of PDEs from Section 4. There are at least two ways to deal withthis problem. One way, the challenging one is to compute all thepossible symmetries of each of the reduced systems. Instead, we followa different approach, where we compute some symmetries from theknowledge of the Lie brackets and use them for further reduction.Some of the reduced systems are ODEs and some are still PDEs.

The Lie bracket, which is an abstract mathematical object, surpris-ingly contains information about whether a second reduction ispossible. More precisely, if V and W are two infinitesimal symmetriessuch that V W λV[ , ] = for some constant λ, then, W induces asymmetry of the reduced system of PDEs obtained by using V[11,29]. Following this fact, we compute symmetries of some reducedsystems of PDEs from Section 4 and use them for reduction.

Recall from Section 2

V kV V V V kV V V[ + + , ] = 0 and [ + + , ] = 0.1 2 5 2 1 2 5 5

Thus, V2 and V5 induce infinitesimal symmetries of the reduced system(15)–(16). The infinitesimal symmetries obtained in this way are

wt

th

hh

hu

uu

u∂

∂and ′ ∂

∂ ′− 2 ∂

∂− 2 ∂

∂− ∂

∂− ∂

∂.1

12

21

12

2

We can also check directly that these are symmetries of (15)–(16).(I) ODEs associated with (15)–(16): The discussion in the previous

paragraph implies (15)–(16) has a two dimensional Lie algebra ofinfinitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b

wb t

th

hh

hu

uu

u∂

∂+ ′ ∂

∂ ′− 2 ∂

∂− 2 ∂

∂− ∂

∂− ∂

∂1 2 11

22

11

22 (33)

where b1 and b2 are real constants.Characteristics equation:

dwb

dtb t

dhb h

dhb h

dub u

dub u

= ′′

=−2

=−2

=−

=−

.1 2

1

2 1

2

2 2

1

2 1

2

2 2

In what follows, we fix l b b= /2 1. For the infinitesimal symmetry (33),we can compute

Similarity variable:

v e vtwith = ′,lw

Similarity forms:

u h h h e u u e i, with = , = for = 1, 2.∼ ∼∼ ∼i i i i

lwi i

lw−2 −

Substitution of these similarity variables and similarity forms in (15)–(16) results in the following system of ODEs:

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

115

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

kl vu kl vhlvu lvh

β v kl

h h

β v kl v kl u

lβ v β lv h h lvu

dh dvdh dvdu dvdu dv

kl u hlu h

kl u kl β h kl β h

l u β h β h

1 − ( + 1) 0 − ( + 1) 00 1 + 0

− (1 + )(1

+ 0.5 / )

− 0.5 (1 + ) 1 − (1 + ) 0

0.5 (1 + 0.5 / ) 0 (1 + )

////

=

3(1 + )− 3

(1 + ) + 2(1 + ) + 2(1 + )

− ( + 2 + 2 )

.

∼∼

∼

∼

∼∼

∼∼

∼

∼

∼∼

∼ ∼∼ ∼

∼∼

∼∼

∼ ∼

∼ ∼

1 1

2 2

1

2 1

1 1

2 2 1 2 2

1

2

1

2

1 1

2 2

12

1 1 1 2

22

2 1 2 2 (34)

Similar analysis can be done for the other systems of PDEs in Section 4.We will only write the outcomes below.

(II) ODEs associated with (17)–(18):Lie algebra of infinitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b

wb t

th

hh

hu

uu

u∂

∂+ ′ ∂

∂ ′− 2 ∂

∂− 2 ∂

∂− ∂

∂− ∂

∂1 2 11

22

11

22

Characteristics equation:

dwb

dtb t

dhb h

dhb h

dub u

dub u

= ′′

=−2

=−2

=−

=−

.1 2

1

2 1

2

2 2

1

2 1

2

2 2

Similarity variable:

v e vtwhere = ′.lw

Similarity forms:

u h h h e u u e i, with = , = for = 1, 2.∼ ∼∼ ∼i i i i

lwi i

lw−2 −

Reduced system of ODEs:

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

kl vu kl vhlvu lvh

β v kl h

h

β v kl v kl u

lβ v β lv h h lvu

dh dvdh dvdu dvdu dv

kl u hlu h

kl u kl β h kl β h

l u β h β h

1 + ( + 1) 0 ( + 1) 00 1 + 0

(1 + )(1 + 0.5 /

)

0.5 (1 + ) 1 + (1 + ) 0

0.5 (1 + 0.5 / ) 0 (1 + )

////

=

− 3(1 + )− 3

− (1 + ) − 2(1 + ) − 2(1 + )

− ( + 2 + 2 )

.

∼∼

∼

∼

∼∼

∼∼

∼

∼

∼∼

∼

∼∼ ∼

∼∼

∼∼

∼ ∼

∼ ∼

1 1

2 2

1 2

1

1 1

2 2 1 2 2

1

2

1

2

1 1

2 2

12

1 1 1 2

22

2 1 2 2 (35)

(III) ODEs associated with (19)–(20):Lie algebra of infinitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b

xb t

th

hh

hu

uu

u∂∂

+ ′ ∂∂ ′

− 2 ∂∂

− 2 ∂∂

− ∂∂

− ∂∂1 2 1

12

21

12

2

Characteristics equation:

dxb

dtb t

dhb h

dhb h

dub u

dub u

= ′′

=−2

=−2

=−

=−

.1 2

1

2 1

2

2 2

1

2 1

2

2 2

Similarity variable:

v e vtwhere = ′.lx

Similarity forms:

u h h h e u u e i, with = , = for = 1, 2.∼ ∼∼ ∼i i i i

lxi i

lx−2 −

Reduced system of ODEs:

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

lvu lvhvu vh

β lv β lvh h β lv lvu

β v β v β vh h vu

dh dvdh dvdu dvdu dv

lu hu h

l u β h β h

u β h β h

1 + 0 00 1 − 0 −

+ 0.5 / 0.5 1 + 0

− 0.5 − − 0.5 / 0 1 −

////

=

− 33

− ( + 2 + 2 )

+ 2 + 2

.

∼∼

∼

∼∼∼

∼∼

∼

∼

∼∼

∼ ∼

∼ ∼

∼∼

∼∼∼ ∼

∼ ∼

1 1

2 2

1 1 2 1 1 1

2 2 2 1 2 2

1

2

1

2

1 1

2 2

12

1 1 1 2

22

2 1 2 2 (36)

(IV) ODEs associated with (21)–(22):A Lie algebra of infinitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b

xb t

th

hh

hu

uu

u∂∂

+ ′ ∂∂ ′

− 2 ∂∂

− 2 ∂∂

− ∂∂

− ∂∂1 2 1

12

21

12

2

Characteristics equation:

dxb

dtb t

dhb h

dhb h

dub u

dub u

= ′′

=−2

=−2

=−

=−

.1 2

1

2 1

2

2 2

1

2 1

2

2 2

Similarity variable:

v e vtwhere = ′.lx

Similarity forms:

u h h h e u u e i, with = , = for = 1, 2.∼ ∼∼ ∼i i i i

lxi i

lx−2 −

Reduced system of ODEs:

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

lvu lvhvu vh

β lv β lvh h β lv lvu

β v β v β vh h vu

dh dvdh dvdu dvdu dv

lu hu h

l u β h β h

u β h β h

1 + 0 00 1 + 0

+ 0.5 / 0.5 1 + 0

0.5 + 0.5 / 0 1 +

////

=

− 3− 3

− ( + 2 + 2 )

− ( + 2 + 2 )

.

∼∼

∼

∼∼∼

∼∼

∼

∼

∼∼

∼ ∼

∼ ∼

∼∼

∼∼

∼ ∼

∼ ∼

1 1

2 2

1 1 2 1 1 1

2 2 2 1 2 2

1

2

1

2

1 1

2 2

12

1 1 1 2

22

2 1 2 2 (37)

(V) PDEs associated with (23)–(24):A Lie algebra of infitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b

xb

yb x

xy

yh

hh

hu

uu

u∂∂

+ ∂∂

+ ∂∂

+ ∂∂

+ 2 ∂∂

+ 2 ∂∂

+ ∂∂

+ ∂∂1 2 3 1

12

21

12

2

Characteristics equation:

dxb b x

dyb b y

dhb h

dhb h

dub u

dub u+

=+

=2

=2

= = .1 3 2 3

1

3 1

2

3 2

1

3 1

2

3 2

Similarity variable:

w t w b b x b b y, with ( + ) = ( + ).1 3 2 3

Similarity forms:

u h h b b x h u b b x u i, with = ( + ) , = ( + ) for = 1, 2.∼ ∼∼ ∼i i i i i i1 3

21 3

Reduced system of PDEs:

t ht

wb u hw

h b u h∂∂

− ∂( )∂

= 2 − 3 ,∼ ∼

∼ ∼ ∼ ∼13

1 11 3 1 1 (38)

t ht

b u hw

h∂∂

+ ∂( )∂

= 2 ,∼∼ ∼ ∼2

32 2

2

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

116

t ut

b wu uw

β b w hw

β wb h

wβ w

b hh

hw

b u β b h β b h u

t ut

u b uw

β b hw

βb h

wβ h

hb h

wu

∂∂

− ∂∂

− ∂∂

−2

∂∂

−2

∂∂

= − − 2 − 2 + ,

∂∂

+ ∂∂

+ ∂∂

+2

∂∂

+2

∂∂

= .

∼ ∼ ∼

∼ ∼

∼ ∼ ∼ ∼

∼ ∼ ∼∼

∼

∼ ∼

∼ ∼ ∼∼

∼

13 1

11 3

1 13

2 13

2

1

1

3 12

1 3 1 1 3 2 1

22 3

22 3

2 23

1 2 1

23

22

(39)

(VI) PDEs associated with (25)–(26):A Lie algebra of infitesimal symmetries:

⎛⎝⎜

⎞⎠⎟b t

tb h

hh

hu

uu

u′ ∂∂ ′

+ 2 ∂∂

+ 2 ∂∂

+ ∂∂

+ ∂∂1 2 1

12

21

12

2

Characteristics equation:

dtb t

dhb h

dhb h

dub u

dub u

′′

=2

=2

= = .1

1

2 1

2

2 2

1

2 1

2

2 2

Similarity forms:

u h h t h u t u i, with = ( ′) , = ( ′) for = 1, 2.∼ ∼∼ ∼i i i

li i

li

2

Reduced system of PDEs:

t ht

wt u hw

t u ht

t u h l lh

t ht

t u hw

lh

′ ∂∂ ′

− ′ ∂( )∂

− ′ ∂( )∂ ′

= −3 ′ (1 − ) − 2 ,

′ ∂∂ ′

− ′ ∂( )∂

= −2 ,

∼ ∼ ∼

∼

∼ ∼ ∼ ∼ ∼

∼ ∼ ∼

l l l

l

1 1 1 ( +1) 1 11 1 1

2 2 22 (40)

t ut

t wu uw

β t w hw

β wt h

wβ w

t hh

hw

t u ut

β t ht

β t ht

βt h

hht

l t u β t h β t h lu

t ut

u t uw

β t hw

βt h

wβ h

ht h

wlu

′ ∂∂ ′

− ′ ∂∂

− ′ ∂∂

−2

′ ∂∂

−2

′ ∂∂

− ′ ∂∂ ′

− ′ ∂∂ ′

−′2

∂∂ ′

−2

′ ∂∂ ′

= ( − 1)( ′ + 2 ′ + 2 ′ ) − ,

′ ∂∂ ′

+ ′ ∂∂

+ ′ ∂∂

+2

′ ∂∂

+2

′ ∂∂

= − .

∼ ∼ ∼ ∼ ∼

∼ ∼

∼ ∼ ∼ ∼

∼ ∼ ∼∼

∼

∼ ∼ ∼∼

∼

∼ ∼

∼ ∼ ∼∼

∼

l l l l l

ll

l

l l l

l l l l

11

11

1 1 2 1 2

1

1 +11

1

1+1 1 1

+12 1 +1 2

1

1

12

1 1 1 2 1

22

22

2 2 1 2 1

2

22

(41)

(VII) PDEs associated with (27)–(28):A Lie algebra of infitesimal symmetries:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

b hh

hh

uu

uu

b tt

hh

hh

uu

uu

2 ∂∂

+ 2 ∂∂

+ ∂∂

+ ∂∂

+ ′ ∂∂ ′

− 2 ∂∂

− 2 ∂∂

− ∂∂

− ∂∂

1 11

22

11

22

2 11

22

11

22

Characteristics equation:

dtb t

dhb b h

dhb b h

dub b u

dub b u

′′

=2( − )

=2( − )

=( − )

=( − )

.2

1

1 2 1

2

1 2 2

1

1 2 1

2

1 2 2

Similarity forms:

u h h t h u t u i, with = ′ , = ′ for = 1, 2.∼ ∼∼ ∼i i i

li i

li

(2 −2) ( −1)

Reduced system of PDEs:

t ht

wt u hw

kt u ht

l h kl t u h

t ht

t u hw

l h

′ ∂∂ ′

− ′ ∂( )∂

− ′ ∂( )∂ ′

= −2( − 1) + 3( − 1) ′ ,

′ ∂∂ ′

− ′ ∂( )∂

= −(2 − 2) ,

∼ ∼ ∼

∼

∼ ∼ ∼ ∼ ∼

∼ ∼ ∼

l l l

l

1 1 1 +1 1 11 1 1

2 2 22 (42)

t ut

t wu uw

β t w hw

β wt h

wβ w

t hh

hw

t ku ut

β kt ht

β kt h

tβ k

t hh

ht

kl t u β h β h l u

t ut

u t uw

β t hw

βt h

wβ h

ht h

wl u

′ ∂∂ ′

− ′ ∂∂

− ′ ∂∂

−2

′ ∂∂

−2

′ ∂∂

− ′ ∂∂ ′

− ′ ∂∂ ′

−2

′ ∂∂ ′

−2

′ ∂∂ ′

= ( − 1) ′ ( + 2 + 2 ) − ( − 1) ,

′ ∂∂ ′

+ ′ ∂∂

+ ′ ∂∂

+2

′ ∂∂

+2

′ ∂∂

= −( − 1) .

∼ ∼ ∼ ∼ ∼

∼ ∼

∼ ∼ ∼ ∼

∼ ∼ ∼∼

∼

∼ ∼ ∼∼

∼

∼ ∼

∼ ∼ ∼∼

∼

l l l l l

l l l

l

l l l l

11

11

1 1 2 1 2

1

1 +11

1

1+1 1 1 +1 2 1 +1 2

1

1

12

1 1 1 2 1

22

22

2 2 1 2 1

2

22

(43)

We do not discuss further reduction of the reduced homogeneoussystems associated with kV V V+ +1 2 3 and V V+1 3 as they are similar tothe ones in [1].

6. Simulation results and discussions

Some conditions can be imposed in the above systems to furtherreduce them and seek some (very) special and restricted analyticalsolutions. For example, we can choose the parameter values for k l β, , 1,and β2 so that some terms drop out. Nevertheless, we continue with thesystem as it is and solve it numerically to access the performance of thereduced system of ODEs. This will provide an overall picture of thephysical process, mainly, the coupled dynamics of the solid and fluidphase velocities and the flow heights. Since the different optimalsystems are associated with fundamentally and quantitatively differentphysical aspects, their qualitative behaviors can be similar as theyrepresent the solution of the same physical system of equations (6)–(7)governing the down-slope mass flows as a mixture of sediment particlesand viscous fluid. So, it suffices to discuss the numerical simulations ofa representative optimal system. Without loss of generality, we choosethe system (34). The results are presented for different solid and fluidpressure parameters β1 and β2. Further results are presented for theoptimal Lie parameters k and b b( , )1 2 . The physical parameters β β( , )1 2characterize the dynamical system (6)–(7) and the parametersk b b( , , )1 2 are associated to the Lie symmetry and its optimal structure.Since l b b= /2 1 always appears in combination, there are essentially twophysical pressure parameters β β( , )1 2 and two Lie parameters k l( , ) thatgovern the system (34). The flow is released from the top-left with theinitial flow heights and velocities as indicated in the simulation Figs. 1–3. This corresponds to the mass release from a silo as in Pudasaini et al.[20] and in Dominik and Pudasaini [32]. In what follows (u u h h, , ,s f s f )

represent the similarity forms (u u h h, , ,∼ ∼ ∼ ∼s f s f ) and x represents the

similarity variable v in (34).Fig. 1 shows the dynamics and interactions of the solid- and fluid-

phase heights (left) and velocities (right) as given by the solution of thesystem (34). Increasing fluid pressure parameter β2 results in decreaseof the respective flow heights and increase of the flow velocities. Ingeneral, these results are consistent with the physics of coupled two-phase mass flow, because for the downslope shear flow of granularmaterial, as flow thins, the velocity generally increases. The phase flowdepths and flow velocities evolve non-linearly. After the release of thedebris mass, the flow depth first decreases rapidly then slowly in thefarther downstream. Similarly, the velocities first increase quickly justbelow the silo gate and then slowly in the farther downstream. As thefluid pressure parameter increases from top to the bottom panels, thedifference between the solid and fluid phase heights increases, so doesthe difference between the solid and fluid phase velocities. In general,fluid heights are smaller than the solid heights, and the associated fluidvelocities are higher than the solid velocities. These are observablephenomena in two-phase debris motion, because higher fluid pressureparameter makes fluid weaker than solid. As the homogeneous PDEs(6)–(7) are explicitly driven by the hydraulic pressure gradient, thehigher pressure gradients (either for solid, or fluid, or both) result inlarger driving forces leading to the higher flow velocities for thecorresponding phases, and thus, the decreased flow depths as themixture mass moves down slope. These physical aspects are clearlycaptured by the solutions presented in Fig. 1.

Next, we discuss the effect of the optimal Lie parameters on the flowdynamics. Fig. 2 indicates as k increases the solid flow heights increase.However, the fluid heights remain almost unchanged. Consequently,due to significantly reduced shearing, the solid could not thin furtherresulting in an increased flow height. Interestingly, within the range ofthe chosen flow domain and parameters, the dynamics are much lesssensitive to the fluid-phase than for the solid-phase. Small change in kresults in large change in solid dynamics (flow heights and velocities).

S. GhoshHajra et al. International Journal of Non–Linear Mechanics 88 (2017) 109–121

117

For larger values of k, the solid dynamics tends to cease while the fluiddynamics continues without much change. This is the situation for adebris release in a gentle slope for which the solid mass moves anddeforms very slowly due to (basal) friction. However, as the fluid isweak it deforms and moves quickly along the slope. These are realisticresults. For small k, both the solid and fluid after traveling a certaindistance along the slope the flow heights do not change so much andtend to maintain constant heights. But due to the inclined channel theflow accelerates and velocities increase continuously along the channelslope. These are observable phenomena in geophysical mass flows[20,17]. Nevertheless, for larger k (=2), as solid moves very slowly theflow depth remains virtually unchanged. This behavior is similar to thecreeping as often observed in rock glaciers and frictional granular flowsin gentle slope. So, k is somehow associated with the frictional strengthof the solid phase. As in Fig. 1, fluid velocities are larger than the solidvelocities resulting in larger heights for the solid and smaller heightsfor fluid. Depending on the value of k, both the flow heights and

velocities evolve strongly or weakly non-linearly. Further increase of kresults in constant values for the solid heights and velocities.Interestingly, independent of the value of k, by setting the solidpressure parameter to zero (β = 01 ) the solid height and velocityremain constant (the initial values), not shown here. The same holdstrue for the fluid, i.e., by setting the fluid pressure parameter to zero(β = 02 ) we obtain the constant initial solutions for fluid height andvelocity. This conforms the consistency of the obtained similaritysolution, because as the flows are driven by the hydraulic pressureparameters, the motion must cease as these parameters become zero.

Further, we present the results associated with the effect of theoptimal Lie parameter b2 on the flow dynamics. Fig. 3 shows thedecreasing value of the parameter b2 results in the complete change inthe flow dynamics (both the flow heights and velocities) more for thefluid and less for the solid. Here, as the value of b2 decreases the flowheights decrease first rapidly in the vicinity of the flow release (silogate), then slowly in the farther downslope leading to the more

Fig. 1. Effect of the fluid pressure parameters β2 on the flow dynamics when β = 0.251 and the Lie parameters b b= 3, = −1.21 2 , and k=1: the panels show the dynamics and interactions

of the solid- and fluid-phase heights (left) and velocities (right).

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constant flow heights. Similarly, the flow velocities first increasequickly close to the silo gate, then relatively slowly in the far down-stream. The differences between the solid and fluid heights, and thesolid and fluid velocities increase quickly after the flow release. Thisresulted in the rapid drop of the flow heights in the upslope thatdecreased gently in the farther downslope along the channel. Anotherinteresting aspect is that as the value of b2 decreases the fluid flowdynamics turns quickly from highly non-linear (in the vicinity of flowrelease) to weakly non-linear (farther downslope) for the flow depths,whereas the fluid velocities show increasingly non-linear state. So, theflow dynamics are fundamentally different in the above three figures.Simulations reveal that the fluid dynamics is more sensitive to theparameter b2, whereas the solid dynamics is more sensitive to theparameter k. However, both the solid and fluid phase dynamics areequally influenced by the solid and fluid pressure parameters (β β,1 2).In all the simulations, the fluid velocities dominate the respective solidvelocities as soon as the flow is released. The difference may reduce or

increase depending on the parameter selection. For all the flowsituations, solid flow heights are dominating the fluid flow heightswhich is expected. Because, for the sheared inclined channel flow, thehigher flow velocities correspond to the lower flow heights, it isconsistent with the physics of coupled two-phase mass flows. Largerfluid velocities are expected in the two-phase mass flows down theslope because the fluid is mechanically relatively weak as compared tothe solid-phase in the mixture. In Figs. 1 and 2, the phase velocities(mostly) tend to saturate in the farther down-slope, whereas in Fig. 3the solid and fluid phases accelerate as these phases rapidly thin in thefar downstream. Both of these situations are possible in the mass flows.So, from the physical point of view the results presented in Figs. 1–3are meaningful.

Finally, we would like to point out that usually the solutionsobtained by group analysis could be inferred by looking at theequations if the model equations have been sufficiently analyzed and,most, if not all, solutions have already been constructed. This can be

Fig. 2. Effect of the optimal Lie parameter k, the Lie parameters b b( , ) = (3, −1.2)1 2 , and the solid and fluid pressure parameters β β( , ) = (0.25, 0.64)1 2 on the flow dynamics: The panels

show the dynamics and interactions of the solid- and fluid-phase heights (left) and velocities (right).

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the case for the single-phase fluid dynamical equations for whichseveral analytical and exact solutions have already been developed.This includes the pressure driven Euler, viscous Navier–Stokes, andshallow water equations for a fluid. However, our system of PDEs dealswith model equations for the flow of a strongly coupled mixturematerial consisting of the non-Newtonian viscous fluid and frictionalgranular material. Furthermore, in our model, there are stronginteractions between the solid and fluid phases, e.g., due to buoyancy,hydraulic pressure gradients, and friction.

7. Summary

In this paper, we applied the Lie symmetry method to the two-phase mass flow model [18] and constructed some optimal systems ofsubalgebras corresponding to this system of non-linear PDEs. Anoptimal system provides precise insights into all possible invariantsolutions, hence it is of great importance from the mathematical point

of view as well as constraining the system for the physical andengineering applications. For different solutions emerging from in-finitesimal symmetries, it is enough to use an optimal system of Liesubalgebras as it contains information about different types of invar-iant solutions. In particular, this allows to partition all the possibleinvariant solutions into disjoint sets.

We constructed one-, two-, and three-dimensional optimal systemsof Lie subalgebras of the physical system. To construct optimal systemsof Lie subalgebras, we first computed one-dimensional ones and followan inductive procedure. More precisely, we constructed the otheroptimal systems by computing all one dimensional higher Lie sub-algebras and removing the redundancies by the action of a generaladjoint operator. This involved an analysis of equivalence classes ofthese Lie subalgebras, and resulted in an optimal system by consider-ing only one representative from each equivalence class. We reducedthe two-phase mass flow system of PDEs into another systems of PDEs.Using the fact that the Lie bracket contains information about further

Fig. 3. Effect of the Lie parameter b2, the Lie parameters b k( , ) = (3, 1)1 , and the pressure parameters β β( , ) = (0.25, 0.64)1 2 on the flow dynamics: the panels show the dynamics and

interactions of the solid- and fluid-phase heights (left) and velocities (right).

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reduction, we further reduce to systems of ODEs and PDEs.We solved a system numerically and analyzed in detail with its

implications, in particular presented simulation results for flows thatcorrespond to the mass release from a silo. This provided an overallpicture of the physical process, the coupled dynamics of the solid andfluid phase velocities and the flow heights. The results are presented fordifferent solid and fluid pressure parameters and Lie parameters. Thephysical parameters characterize the dynamical system, whereas theother parameters are associated to the optimal structure of the Liesymmetry. Simulations reveal that the fluid dynamics is more sensitiveto the Lie parameter b2, whereas the solid dynamics is more sensitive toanother Lie parameter k. However, both solid and fluid phase dynamicsare equally influenced by the solid and fluid pressure parameters(β β,1 2). Increasing fluid pressure parameter results in decrease of therespective flow heights and increase of the flow velocities. Higherpressure gradients result in higher flow velocities, and hence, itdecreases flow depths as the mixture mass moves down slope. Zeropressure parameters result in the constant initial solutions for bothsolid and fluid heights and velocities. These physical aspects are clearlycaptured by the simulations of the optimal system. This confirms theconsistency of the obtained similarity solutions.

In all the simulations, the fluid velocities dominate the respectivesolid velocities as soon as the flow is released. The difference mayreduce or increase depending on the parameter selection. For all flowsituations in consideration, solid flow heights are dominating fluid flowheights which is expected. Because, for the sheared inclined channelflow, higher flow velocities correspond to lower flow heights, consistentwith the physics of coupled two-phase mass flows. Larger fluidvelocities are expected in the two-phase mass flows down the slope,because the fluid is mechanically relatively weak as compared to thesolid-phase in the mixture. The phase heights (mostly) tend to saturatein the farther down-slope, whereas the phases mostly accelerate as theflow rapidly thins in the far downstream. These situations are possiblein mass flows. So, from the physical point of view the results presentedhere are meaningful.

Acknowledgment

Sayonita GhoshHajra gratefully acknowledges University of Utahfor the support where a part of this project was done. Santosh Kandel isgrateful to the Max Planck Institute for Mathematics, Bonn, where apart of this work was carried out. Shiva P. Pudasaini acknowledges thefinancial support provided by the German Research Foundation (DFG)through the research project, PU 386/3-1: “Development of a GIS-based Open Source Simulation Tool for Modeling General Avalancheand Debris Flows over Natural Topography” within a transnationalresearch project, D-A-C-H.

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