Acta Geophysica vol. 61, no. 3, Jun. 2013, pp. 649-667

DOI: 10.2478/s11600-013-0108-2

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© 2013 Institute of Geophysics, Polish Academy of Sciences

Boundary Conditions Effect on Linearized Mud-Flow Shallow Model

Cristiana DI CRISTO1, Michele IERVOLINO2,

and Andrea VACCA2

1Dipartimento di Ingegneria Civile e Meccanica,

Università degli Studi di Cassino e del Lazio Meridionale, Cassino, Italy

e-mail: dicristo@unicas.it (corresponding author) 2Dipartimento di Ingegneria Civile, Design, Edilizia e Ambiente,

Seconda Università di Napoli, Aversa, Italy

e-mails: michele.iervolino@unina2.it, vacca@unina.it

A b s t r a c t

The occurrence of roll-waves in mud-flows is investigated based on

the formulation of the marginal stability threshold of a linearized one-

dimensional viscoplastic (shear-thinning) flow model. Since for this kind

of non-Newtonian rheological models this threshold may occur in

a hypocritical flow, the downstream boundary condition may have a non-

negligible effect on the spatial growth/decay of the perturbation. The pa-

per presents the solution of the 1D linearized flow of a Herschel and

Bulkley fluid in a channel of finite length, in the neighbourhood of

a hypocritical base uniform flow. Both linearly stable and unstable condi-

tions are considered. The analytical solution is found applying the

Laplace transform method and obtaining the first-order analytical expres-

sions of the upstream and downstream channel response functions in the

time domain. The effects of both the yield stress and the rheological law

exponent are discussed, recovering as particular cases both power-law

and Bingham fluids. The theoretical achievements may be used to extend

semi-empirical criteria commonly employed for predicting roll waves

occurrence in clear water even to mud-flows.

Key words: boundary conditions, Herschel and Bulkley fluid, linear sta-

bility, roll-waves, shallow flow model.

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1. INTRODUCTION

Mud-flows involve massive movement of solid particles carried by the fluid

as a mixture with great destructive potential. These frequent natural hazards

may be triggered by torrential rains, mountain slides or volcanic eruptions.

Eyewitnesses often describe them as a sequence of surges impacting on de-

fense structures and destroying houses and fields (Takahashi 1991).

In order to analyze this kind of complex flows, two main different strate-

gies are commonly adopted. The first one, the so-called “two-phase” or

“two-fluid” model, is based on mixture theory and deduces the mathematical

model through an average of mass and momentum balance laws for fluid and

solid constituents (Anderson and Jackson 1967, Drew 1983, Anderson et al.

1995, Jackson 2000, Iverson 1997, Pitman and Le 2005, Greco et al. 2012).

The terms representing interaction between the phases are modeled either

through experimental correlations or considering simplified conditions of

flow (Moreno and Bombardelli 2012, Bialik et al. 2012).

A different schematization of the mud-flow, without analyzing the inter-

action among its single components, considers the mixture as an ideal con-

tinuum. The presence of the solid material is accounted for through a non-

Newtonian behavior of the mixture, which can be described by a linear

(Bingham) or non-linear (Herschel and Bulkley) viscoplastic shear-thinning

model (Johnson 1970, O’Brien and Julien 1988, Liu and Mei 1989, Huang

and Garcia 1997, 1998, Balmforth and Liu 2004, Ancey et al. 2012). In the

present paper the latter strategy is followed and the mud-flow is considered

as a homogeneous medium with a non-Newtonian rheology. Moreover, lam-

inar conditions are assumed, as those often encountered in real situations

(Ng and Mei 1994).

Despite the non-Newtonian behavior of the flowing material, the devel-

opment of multiple surges in mud-flows shows strong similarities with the

roll-waves trains observed in clear-water flow (Takahashi 1991). As far as

the water is concerned, it is now widely accepted that the roll-waves genera-

tion can be ascribed to linear instability of the uniform base flow. In the one-

dimensional shallow-water framework (Bose and Dey 2013), the temporal

normal mode analysis applied to linearized Saint–Venant equations allowed

to detect the limiting values of dimensionless governing parameter, i.e.,

Froude (F) and Reynolds (Re) numbers, above which unstable conditions are

expected (Brock 1969, Berlamont 1976, Ponce and Simons 1977, Berlamont

and Vanderstappen 1981, Venutelli 2011, Di Cristo et al. 2012a). In turbu-

lent conditions of the base flow and independently of the Reynolds number,

it has been found that roll waves can be expected only in hypercritical condi-

tions, i.e., F > 1. Owing to the convective nature of the instability (Di Cristo

and Vacca 2005), the marginal stability condition can be equivalently de-

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

651

duced through a spatial linear analysis which considers the evolution either

of a pointwise instantaneous (Vedernikov 1946, Stoker 1957, Liggett 1975,

Ridolfi et al. 2006, Di Cristo et al. 2012b) or of a periodically time-varying

(Supino 1960, Di Cristo and Vacca 2005) perturbation. From these analyses,

according to the hyperbolic character of the mathematical model and the sign

of the slopes of the characteristics in unstable flow (both positive), it follows

that the perturbation propagates only in the downstream direction (with an

exponential growth) and without any influence of the downstream boundary

conditions. As a matter of fact, as far as clear-water flows are concerned the

spatial domain has been assumed unbounded or semi-unbounded.

The exponential growth in the downstream direction of the both instan-

taneous (Montuori 1963) and oscillating (Di Cristo et al. 2008, 2010) dis-

turbance predicted by the linearized model has been moreover used to

support design principles, the so called minimum channel length criteria, in

order to predict the occurrence of the roll-waves in channels. The compari-

son between theoretical estimates and experimental data (laboratory and

field) shows the validity of minimum-channel length criteria for clear-water

and even for frictional granular flows (Di Cristo et al. 2009).

Similarly to the water, the temporal stability analysis of a viscoplastic

flow model has been widely applied to individuate the occurrence of unsta-

ble conditions in mud-flows. Ng and Mei (1994) considered a power-law

fluid in absence of yield stress, then their results were generalized by Hwang

et al. (1994) accounting for the effect of surface tension. Pascal (2006) and

Di Cristo et al. (2013b) considered a power-law fluid accounting for the

seepage through the porous bottom channel whereas the influence of a super-

ficial shear stress on the stability of a power-law fluid was studied by Pascal

and D’Alessio (2007). In presence of yield stress, the marginal stability con-

dition has been deduced, as far as a Bingham fluid is concerned, by Trow-

bridge (1987), and by Coussot (1994) for a Herschel and Bulkley fluid. Even

if the marginal stability condition is different depending on the specific con-

sidered model, it is worth of note that many viscoplastic shear-thinning flow

models show instability in hypocritical conditions. More recently, with ref-

erence to a Herschel and Bulkley fluid, the spatial evolution of a pointwise

instantaneous disturbance in an unbounded domain has been studied in Di

Cristo et al. (2013a), showing the convective nature of the instability. More-

over, independently of the governing dimensionless numbers and the rheo-

logical parameters, in unstable condition of flow, similarly to the clear-water

case, an exponential spatial growth of the disturbance has been found.

From the above literature review and supported by some experimental

evidences (Coussot 1994, Zanuttigh and Lamberti 2007), it may be conjec-

tured that the roll-waves occurrence in a mud-flow could be predicted

through a minimum channel length criterion. However, while in hypercritical

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condition of flow, similarly to the clear-water case, the wave growth can be

estimated considering an unbounded domain, in hypocritical conditions, ow-

ing to the negative definiteness of one of the slopes of the two characteris-

tics, the effects of downstream boundary condition has to be properly

accounted for.

In the present paper, the generalization of the solution deduced for stable

hypocritical turbulent clear-water flow for a viscoplastic flow model is pre-

sented. In particular, a non-linear viscoplastic (or Herschel–Bulkley) model

is considered, which allows to better reproduce the shear-thinning behavior

of concentrated mud (Huang and Garcia 1998). Starting from an initial uni-

form flow condition, the solution of the linearized hypocritical flow of

a two-equations Herschel and Bulkley fluid in a channel of finite length is

derived. The analytical solution is found applying the Laplace transform

method and obtaining first-order analytical spatio-temporal expressions of

upstream and downstream channel response functions. Moreover, both stable

and unstable conditions are investigated in unified way.

The paper is structured as follows. Section 2 introduces the considered

mathematical model. Analytical solution of the problem is deduced in Sec-

tion 3. Analysis and discussion of the results are presented in Section 4.

Finally, conclusions are summarized in Section 5.

2. GOVERNING EQUATIONS

Let us consider a homogeneous and uncompressible fluid flowing over

a fixed bed without lateral inflow or outflow. The density (ρ), the consis-

tency (µ) of the flowing material and the bed slope tan(θ) are assumed to be

constant, along with the yield stress ( cτ ) and the exponent of the power law

(n), the latter smaller than unity (Bird et al. 1983). Let us suppose that:

spatial variations occur over scales larger than the flow depth,

the flow resistance by the sidewalls is negligible with respect to that by

bottom,

the momentum correction factor is equal to unity.

Moreover, the curvature effects are neglected (Di Nucci et al. 2007, Di

Nucci and Russo Spena 2011). Assuming as reference length and velocity

scales the uniform flow values of depth (h0) and velocity (u0) and denoting

with 0b bτ τ τ= the dimensionless tangential stress at the bottom (Fig. 1),

with 0τ the corresponding value in uniform condition, the dimensionless

depth-averaged momentum and mass conservation equations can be written

as (Ancey et al. 2012, Di Cristo et al. 2013a):

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

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Fig. 1. A sketch of the viscoplastic fluid flowing over the fixed inclined bed in the

initial uniform flow condition.

2

12( , ) 1 0 ,bhu hu h h h

+ + f n =t x x Re hF

τχ∂ ∂ ∂ ⎛ ⎞+ −⎜ ⎟∂ ∂ ∂ ⎝ ⎠ (1)

0 ,h h u

+ u + h =t x x

∂ ∂ ∂∂ ∂ ∂ (2)

1/ 1/ 1 1/2 2 1

1 1tan 1

n n n

b c c

c c b b

F n u n

Re n h n

τ τ ττ θ τ τ τ

+⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ = − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3)

with

( ) ( )( )( ) ( )1 1

2 1 1 1, .

1 1

n

n

n nf n

n n nχ χ χ +

⎡ ⎤+ += ⎢ ⎥+ + −⎢ ⎥⎣ ⎦ (4)

In Eqs. (1)-(3), 0/cχ τ τ= denotes the yield parameter (always less then

unity), 0 0 cosF u gh θ= and 2

0 0

n nRe u hρ μ−= are the Froude and the

Reynolds numbers, respectively. The power-law rheological model follows

from Eqs. (3)-(4) for χ = 0, whereas Bingham fluid rheology may be ob-

tained assuming n = 1.

For sake of simplicity, in what follows we assume uniform flow as base

state, and we superpose on it a small perturbation (up, hp). The linearization

of Eqs. (1) and (2) leads to following hyperbolic system of first-order differ-

ential equations:

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1

2

0 0

( , )10 ,

p p p b b

p p

u u h f n+ + u h

t x x Re u h h hF

τ τχ ⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + =⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ (5)

0p p ph h u

+ + =t x x

∂ ∂ ∂∂ ∂ ∂ (6)

with

( )( )

2 2

0

1 1,

1 2 2

bn n

nu h n n n

χ χτχ χ

− + +∂ ⎛ ⎞ =⎜ ⎟∂ + + +⎝ ⎠ (7)

( )( )

2 2

0

1 11 .

1 2 2

bn n

nh h n n n

χ χτχ χ

− + +⎡ ⎤∂ ⎛ ⎞ = − +⎢ ⎥⎜ ⎟∂ + + +⎝ ⎠ ⎣ ⎦ (8)

Expressions (7) and (8) have been deduced differentiating Eq. (3). It is

worth of note that depending on the value of the Froude number, hypocritical

(F < 1), critical (F = 1) or hypercritical (F > 1) conditions of the base uni-

form flow may occur. In hypercritical conditions both characteristic lines

have positive slope, while for hypocritical flows the signs are opposite. In

the critical condition the first characteristic is positive, while the second one

vanishes.

The first order system Eqs. (5)-(6) can be rewritten in terms of hp only as

follows:

2 2 2

32

2 2 2

( , )( , )12 1 0 ,

p p p p ph h h h hf nf n

+ + + =x t Re t Re xt F x

χχ∂ ∂ ∂ ∂ ∂⎛ ⎞− +⎜ ⎟∂ ∂ ∂ ∂∂ ∂⎝ ⎠ (9)

where the expressions of the positive functions f2(n, χ) and f3(n, χ) – with

f3(n, χ) > f2(n, χ) – are:

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

1 1 2 1 1 1, ,

11 2 2 1

n

n

n n n nf n n

n n nn n n

χ χχ χχ χ χ +⎡ ⎤− + + + += ⎢ ⎥+ ++ + + −⎢ ⎥⎣ ⎦ (10)

( ) ( )( )( ) ( )( )( )

3 1 2 2

2 1 1 1 11, 1 2 .

1 1 2 21

n

n

n n n nf n n

n n n n n n

χ χχ χ χ χχ +⎡ ⎤+ + − + +⎛ ⎞= +⎜ ⎟⎢ ⎥+ + + + +−⎢ ⎥ ⎝ ⎠⎣ ⎦ (11)

Coussot (1994) has shown that, for a Herschel and Bulkley model, un-

stable conditions of flow occur whenever the Froude number exceeds the

value:

( )( )*

2 2 2

1 1

2 1

n nF n

n n n n

χ χχ χ

− + += + + + + (12)

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

655

or equivalently

( )( ) ( )2*

3 2

,.

, ,

f nF

f n f n

χχ χ= − (13)

Therefore the base uniform flow is stable (resp. unstable) when F < F*

(resp. F > F*). For shear-thinning material (n < 1) and independently of the

yield stress it is easy to verify that the marginal condition (F = F*) occurs

only in hypocritical condition of flow (F < 1). In what follows only hypocrit-

ical (stable and unstable) conditions will be considered.

3. SOLUTION OF THE LINEARIZED FLOW MODEL IN A FINITE

CHANNEL LENGTH

Denoting with L the dimensionless channel length, we look for the solution

of the following differential problem:

( )2 2 2

2 20 , 0, ,

p p p p p

tt xt xx t x

h h h h ha + a a + a a = x L

x t t xt x

∂ ∂ ∂ ∂ ∂+ + ∀ ∈∂ ∂ ∂ ∂∂ ∂ (14)

where

32

2

( , )( , )11, 2, 1 , , ,tt xt xx t x

f nf na a a a a

Re ReF

χχ= = = − = = (15)

with homogeneous initial conditions and inhomogeneous Dirichlet boundary

ones:

( ) ( ) ( )t 0

,,0 0 , 0 , 0, ,

p

p

h x th x x L

t =

∂= = ∀ ∈∂ (16)

( ) ( ) ( ) ( )0, , , , 0 ,p u p dh t h t h L t h t t= = ∀ > (17)

Denoting with { }( , ) ( , )f x s f x t= L the Laplace transform of a function

f (x, t), with s > 0 the Laplace variable (Dyke 1999), and applying the

Laplace transform to the Eq. (14) we are led to the following ordinary differ-

ential equation

( ) ( ) ( )2

2

20 , 0, ,

p p

xx xt x tt t p

d h dha + a s a a s a s h = x L

dxdx+ + + ∀ ∈ (18)

with the boundary conditions:

( ) ( ) ( ) ( )0, , , , 0 .p u p dh s h s h L s h s s= = ∀ > (19)

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The solution of the ordinary second order differential Eq. (18) can be

expressed as (Treves 1966):

( , ) ( ) ( ) ,x x

ph x s c s e c s eλ λ+ −+ −= + (20)

in which

= ( ) ( ) ,p s q sλ± ± (21)

where p(s) and q(s) are given by

( ) ( ) ( ) ( )2

1 2 3 4 5, .p s s q s s sα α α α α= + = + + (22)

The coefficients αi (i = 1, ..., 5) in Eq. (22) are

2 2

1 2 3 4 52 2 2

4 2 4, , , , ,

2 2 4 4 4

xt x xt xx tt xt x xx t x

xx xx xx xx xx

a a a a a a a a a a

a a a a aα α α α α− −= − = − = = = (23)

or, equivalently, accounting for Eq. (15):

( ) ( )( ) ( ) ( ) ( )

2 2 23

1 2 32 2 22

22 2 42 3

4 52 2 22 2

,, , ,

21 1 1

, ,1 , ,

* 41 1

f n χF F F

ReF F F

f n f nF F F

Re F ReF F

α α αχ χα α

= = =− − −⎛ ⎞= + =⎜ ⎟⎝ ⎠− −

(24)

Inspection of Eq. (24) leads to recognize that in hypocritical conditions

of flow (F < 1) all αi (i = 1, ..., 5) coefficients are positive, and therefore the

q function is positive definite. The expressions of c+(s) and c–(s) in

Eq. (20) can be found by imposing the boundary conditions (19) and read:

( ) ( ) ( ) ( ) ( ) ( ), ,

L L

u d d u

L L L L

h s e h s h s h s ec s c s

e e e e

λ λλ λ λ λ

− +

− + − ++ −− −= =− − (25)

Accounting for Eq. (20) the solution of problem (14)-(17) in the trans-

formed space can be therefore rewritten as follows:

( ) ( ) ( ) ( ) ( ), , , ,p u u d dh x s h s g x s h s g x s= + (26)

in which the functions ( )ug x,s and ( )dg x,s

( ) ,x L x L

u L L

e eg x,s

e e

λ λ λ λλ λ

− + + −

+ −

+ +−= − (27)

( ) , x x

d L L

e eg x,s

e e

λ λλ λ+ −

+ −−= − (28)

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

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may be interpreted as the Laplace transforms of the channel response to

Dirac at the upstream and downstream locations, respectively (Tsai and Yen

2001). Applying the inverse Laplace transform to Eq. (26), the solution in

the time domain is finally obtained:

( ) ( ) ( ) ( ) ( )0 0

, , , .

t t

p u u d dh x t h g x t d h g x t dτ τ τ τ τ τ= − + −∫ ∫ (29)

In order to determine the analytical expression of gu(x, t) and gd (x, t),

let us preliminarily express the denominator of Eqs. (27)-(28) in convergent

series (Gradshteyn and Ryzhik 1963):

-2 q

0

1 1e .

m L

L L Lme e e

λ λ λ+ − +

+∞

==− ∑ (30)

Moreover, denoting with

( ) 2

24

1

3 *

1 ,2 2

f n, χα Fβ α Re F

⎛ ⎞= = +⎜ ⎟⎝ ⎠ (31)

( )( )

2

2 22 2

4 3 5

2 2 2 2

3 *

141 ,

4 4

f n, χ Fα α α Fβ α Re F

− ⎛ ⎞−= = −⎜ ⎟⎝ ⎠ (32)

and accounting for the following identity, obtained considering Eqs. (22),

(31), and (32):

( ) ( ) 121 2 2

3 1 2 ,/q s α s β β⎡ ⎤= + −⎣ ⎦ (33)

the Laplace transforms of the channel response to Dirac at the upstream and

downstream locations can be rewritten as follows:

( ) ( ) ( ) ( ) ( ),1 ,2 ,3 ,4 ,u u u u ug x,s g x,s g x,s g x,s g x,s= − + − (34)

( ) ( ) ( ) ( ) ( )1 2 3 4,

d d, d, d, d,g x,s g x,s g x,s g x,s g x,s= − + − (35)

where

( ) ( ) ( ) ( )1 21 2 3 12

1

0

,/s x mL x s

u,

m

g x,s e eα α α β+∞+ − + +

== ∑ (36)

( ) ( ) ( ) ( )1 23 11 2

2 1

2

0

,/m L x ss x

u,

m

g x,s e eα βα α +∞ − + − +⎡ ⎤+ ⎣ ⎦

== ∑ (37)

( ) ( ) ( ) ( ) ( ) ( )21 21 2

3 1 21 2 3 1

2 21 2

3

0

,/

/mL x ss x mL x s/

u,

m

g x,s e e eα β βα α α β+∞ ⎡ ⎤− + + −+ − + +⎢ ⎥⎣ ⎦

== −∑ (38)

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( ) ( ) ( ) ( ) ( ) ( )21 2 1 23 1 2 3 11 2

2 1 2 11 2

4

0

,/ /m L x s m L x ss x /

u,

m

g x,s e e eα β β α βα α +∞ ⎡ ⎤− + − + −⎡ ⎤ − + − +⎡ ⎤⎣ ⎦+ ⎢ ⎥⎣ ⎦ ⎣ ⎦

== −∑ (39)

and

( ) ( )( ) ( ) ( )1 23 11 2

2 1

1

0

,/m L x L ss x L

d,

m

g x,s e eα βα α +∞ − + − − +⎡ ⎤+ − ⎣ ⎦

== ∑ (40)

( ) ( )( ) ( ) ( )1 21 2 3 12

2

0

,/s x L mL x L α s β

d,

m

g x,s e eα α +∞+ − − + + +

== ∑ (41)

( ) ( )( ) ( ) ( ) ( ) ( )21 21 2

3 1 21 2 3 1

2 21 2

3

0

,/

/mL x L ss x L mL x L s/

d,

m

g x,s e e eα β βα α α β+∞ ⎡ ⎤− − + + −+ − − − + +⎢ ⎥⎣ ⎦

== −∑ (42)

( ) ( )( ) ( ) ( ) ( ) ( )21 21 2

3 1 21 2 3 1

2 21 2

4

0

./

/mL x L ss x L mL x L s/

d,

m

g x,s e e eα β βα α α β+∞ ⎡ ⎤− + + + −+ − − + + +⎢ ⎥⎣ ⎦

== −∑ (43)

It is worth of note that, in hypocritical conditions, while the coefficients αi (i = 1, ..., 5) and β1 are always positive definite, the sign of the β2 coeffi-

cient (see Eq. (32)) depends on the stable or unstable character of the base

flow (i.e., β2 > 0 for F < F*; β2 < 0 for F > F*).

Denoting with δ(·) and H(·) the Dirac and the Heaviside function, apply-

ing the complex-shift and time-shift theorems and accounting for the follow-

ing standardized formulae (Dyke 1999):

{ } ( ) 1 with 0 ,ase t a aδ− − = − >L (44)

{ } ( ) ( )21 2 2

12 2with 0 and 0 , a s b as b

e e H t a I b t a a bt a

− − − − ⎡ ⎤− = − − > >⎢ ⎥⎣ ⎦−L (45)

{ } ( ) ( )21 2 2

12 2with 0 and 0 , a s b as b

e e H t a J b t a a bt a

− − − − − ⎡ ⎤− = − − − > <⎢ ⎥⎣ ⎦−L (46)

with J1(·)and I1(·) the first-order Bessel functions of the first kind and the

modified one, respectively, the analytical expression of the Laplace anti-

transform of Eqs. (36)-(37) and (40)-(41) can be found and reads

( ) ( ) ( )( )1 22 1 32 1 2

,1 1 3

0

2 ,/x mL x /

u

m

g x,t e δ t x mL xα β α α α+∞ − +

== + − +∑ (47)

( ) ( ) ( )( )1 22 1 32 1 1 2

2 1 3

0

2 1 ,/x m L x /

u,

m

g x,t e δ t x m L xα β α α α+∞ − + −⎡ ⎤⎣ ⎦

== + − + −⎡ ⎤⎣ ⎦∑ (48)

( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 22 1 32 1 1 2

1 1 3

0

2 1 ,/x L L m x L /

d,

m

g x,t e δ t x L m L x Lα β α α α+∞ − − + − +⎡ ⎤⎣ ⎦

== + − − + − +⎡ ⎤⎣ ⎦∑ (49)

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

659

( ) ( ) ( ) ( ) ( )( )1 22 1 32 1 2

2 1 3

0

2 ./

x L mL x L /

d,

m

g x,t e δ t x L mL x Lα β α α α+∞ − − + +

== + − − + +∑ (50)

As far as the Laplace inverse transform of gu,i(x, t) and gd,i(x, t) for

i = 3 and 4, owing to the sign dependence of the β2 coefficient on the Froude

number, the further following distinction is needed:

For stable condition of flow (i.e., F < F*)

( ) ( ) ( ) ( ) ( ) ( ) ( )2 1 1 1222

1 2 1 1220

1

,x t m m

u,i u,i u,im

mu,i

g x,t e I t x k H t x kt x k

α α β β β β α αα+∞− −=

⎛ ⎞⎡ ⎤= + − + −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠+ −∑ (51)

( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )

2 1 1 1

2

22

1

0 22

1 2 1 1

mx L t d,i

d,i

mm m

d,i d,i

β

t x L - kg x,t e

I t x L k H t x L k

α α β β αβ α α

+∞− − −=

×+ −⎡ ⎤⎣ ⎦=⎛ ⎞⎡ ⎤× + − − + − −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∑

, (52)

For unstable condition of flow (i.e., F > F*)

( ) ( ) ( ) ( )( ) ( ) ( )

2 1 1 1

2

22

1

0 22

1 2 1 1

m

x t u,i

u,i

mm m

u,i u,i

t x k

g x,t e

J t x k H t x k

α α β β

βαβ α α

+∞− −=

− ×+ −=⎛ ⎞⎡ ⎤× − + − + −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

∑, (53)

( ) ( )( ) ( ) ( )( ) ( ) ( )

2 1 1 1

2

22

1

0 22

1 2 1 1( ) ( )

mx L t d,i

d,i

mm m

d,i d,i

β

t x L kg x,t e

J t x L k H t x L k

α α β β αβ α α

+∞− − −=

− ×+ − −⎡ ⎤⎣ ⎦= ⎛ ⎞⎡ ⎤× − + − − + − −⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠∑

, (54)

where

[ ]1/2 1/2

,3 3 ,4 3(2 ) ; 2( 1) ,m m

u uk mL x k m L xα α= + = + − (55)

[ ] [ ]1/ 2 1/ 2

,3 3 ,4 32( 1) ; 2 .m m

d dk m L x L k mL x Lα α= + − − = + + (56)

Expressions (47)-(54) represent the extension of the solution of the line-

arized Saint–Venant equations in stable conditions (Dooge and

Napiórkowski 1984, Dooge et al. 1987, Napiórkowski and Dooge 1988, Tsai

and Yen 2001) to a viscoplastic fluid in both stable and unstable conditions.

4. CHANNEL RESPONSE FUNCTIONS

The solution of the linearized flow model is represented by the sum of two

convolution integrals of the upstream and downstream boundary conditions,

with the two response functions gu(x, t) and gd (x, t) as kernels (see Eq. (29)).

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660

As stated for the clear-water case by Tsai and Yen (2001), these functions

express the impulse response of the channel to a prescribed boundary forcing

and embody the main mechanisms of wave dynamics: propagation, distor-

tion, attenuation or amplification and reflection. The wave structure of the

considered flow model is characterized by the presence of two families of

gravity waves, which travel with the following celerities:

1 21 2 1 2

3 1 3 1

1 1 1 11 and 1 ,

/ /

c cF Fα α α α

= = + = − = −− + (57)

which represent the slopes of the characteristics of the differential Eq. (14).

For hypocritical flow herein considered, the two characteristics’ slopes are

opposite in sign, with c1 > 0 and c2 < 0, so the first gravity wave moves

downstream and the second one upstream.

The role of these waves in the channel response functions may be easily

detected by looking at Eqs. (47)-(50) and (51)-(54): indeed, they produce an

infinite series of Dirac-delta and exponential-Bessel functions that propagate

and are reflected successively at each boundary with a round-trip time of

0

1 2

.L L

tc c

= − (58)

Consequently, any impulse disturbance at one of the channel ends pro-

duces two forms of response waves: the first one expresses the direct trans-

mission of the impulse disturbance, the second one arises from the reflection

at the opposite boundary of the previously generated waves.

Concerning the evolution of the wave body, the interaction between ex-

ponential and Bessel functions is responsible for the attenuation/amplifica-

tion (occurring respectively for F < F* or F > F*) in time and in space of

the wave peak. The rate of attenuation/amplification can be recognized from

Eqs. (47)-(50) and (51)-(54) to further depend on the rheology of the flow, as

indicated by the expression of αi coefficients in Eq. (24). Due to the depend-

ence of the amplitude/attenuation factor from each individual wave, the

resulting wave dynamics is characterized by distortion in space and time.

To illustrate in detail the features of gu(x, t) and gd (x, t) and their depend-

ence on the rheology, the following two flow conditions are considered in

a channel with length L = 50: a stable one characterized by Re = 10 and

F = 0.2 (correspondingly, t0 = 10.1) and an unstable one with Re = 10 and

F = 0.4 (t0 = 47.6). Moreover, three different rheologies are compared: the

first one corresponds to a Herschel and Bulkley fluid with n = 0.5 and

χ = 0.25, the second one to a power-law fluid with n = 0.5, the third one to

a Bingham fluid with χ = 0.25. The above values of the rheological parame-

ters lead to critical Froude numbers F* = 0.25, 0.33, and 0.39, respectively.

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

661

The comparison between the first and the second rheological model aims to

enlighten the effect of the yield stress on the flow response for a constant

value of the exponent n, whereas the comparison between the first and the

third configuration reflects the influence of the shear-thinning character of

the fluid for a constant value of the yield parameter.

Interesting observations can be drawn from Figs. 2 and 3 which report

the upstream gu(x, t) and the downstream gd (x, t) response functions, re-

spectively, evaluated at x = L/2 for the considered fluids and the stable flow

condition. The gu(x, t) is null for t < x1/c1 (Fig. 2), i.e., before the fastest

gravity wave arrives at the considered location. The response is then charac-

terized by an abrupt peak which successively decays in time. The occurrence

of this peak may be explained by considering the first term (since no reflec-

tion has occurred) of the infinite sum in Eq. (51) which, for sufficiently

small values of x, is a monotonically decreasing function of t. In the investi-

gated conditions, before the decay is completed, an abrupt discontinuity is

observed, due to the reflected gravity wave traveling upstream from the

channel end with celerity c2.

Fig. 2. Time evolution of the upstream response function at x = L/2 – stable hypo-

critical flow (Re = 10, F = 0.20).

Fig. 3. Time evolution of the downstream response function at x = L/2 – stable hypo-

critical flow (Re = 10, F = 0.20).

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As far as the effect of the rheology is concerned, the channel response

function shows similar qualitative behavior for the different fluids, but sig-

nificant differences in magnitude are observed. In particular, we observe that

the Bingham fluid is characterized by the highest value of the response func-

tion at the arrival of the first gravity wave and its decay is also the fastest.

The comparison of the computed response functions for the Herschel and

Bulkley and power-law fluids shows that the presence of a finite yield stress

increases the magnitude of the upstream response function. On the other

hand, the comparison of the computed response function of Herschel and

Bulkley and Bingham fluids shows that the increase of n produces an higher

peak in the response associated to the first gravity wave, but also a faster

decrease in time of the magnitude of the response function as the repeated

reflections goes on.

These results are consistent with the time-asymptotic stable hypocritical

impulse response obtained by Di Cristo et al. (2013a) for unbounded chan-

nel. In fact, the maximum wave height is shown to monotonically decrease

with n, and in the range herein concerned, also with χ. It may be therefore

argued that this analysis is representative of conditions in which the down-

stream reflection does not affect significantly the main features of the up-

stream channel response.

From Fig. 3 it can be noted that, similarly to the previous case, gd (x, t) is

null for t < (x – L)/c2. Indeed, the Bingham fluid appears to be the most sen-

sitive to disturbances at the downstream end of the channel, since the corre-

sponding response function attains greater magnitude than the two other flu-

ids. On the other hand, presence of a finite yield stress produces a significant

reduction in the downstream channel response if compared with that of pow-

er-law fluid, which is also characterized by the slowest temporal decay of the

response function.

Figures 4 and 5 present the upstream gu(x, t) and the downstream gd (x, t)

response functions, respectively, evaluated at x = L/2 for the same fluids,

but for the unstable flow condition.

As far as the effect of the fluid rheology is concerned, it can be observed

that the magnitude of the upstream channel response for the Herschel and

Bulkley fluid overwhelms the other two, with the Bingham fluid being the

least sensitive to the disturbance. Moreover, the responses function exhibits

the effect of reflection from the downstream channel end, which can be seen

in the jump discontinuity occurring at t ≈ 31. However, the peak of the re-

sponse function is always associated to the arrival of the first traveling wave.

Moreover, the dependence of the upstream channel response on the rhe-

ology is consistent with the spatial and temporal growth factors obtained by

Di Cristo et al. (2013a) for an unbounded channel. Indeed, in this analysis

the growth factor is a monotonically increasing function of χ, whereas for the

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BOUNDARY CONDITION EFFECT ON SHALLOW MUD FLOWS

663

Fig. 4. Time evolution of the upstream response function at x = L/2 – unstable hypo-

critical flow (Re = 10, F = 0.40).

Fig. 5. Time evolution of the downstream response function at x = L/2 – unstable

hypocritical flow (Re = 10, F = 0.40).

considered Froude number and for χ = 0.25, it is easily verified that it

decreases with the rheological exponent in the range n ∈ (0.5, 1).

About the downstream response function in the unstable case (Fig. 5),

we firstly note that its magnitude is much smaller than the upstream one.

Comparing the results for the three considered rheologies, the decay of the

peak with respect to the distance from the outlet section appears to be more

pronounced for the Herschel and Bulkley fluid than for the power-law,

whereas the Bingham fluid is characterized by the least sensitive response to

the downstream forcing.

It is worth of note that a detailed inspection of Eqs. (47)-(50) and (51)-

(54) reveals that under stable flow condition, the magnitude of the upstream

and downstream flow response reduces moving sufficiently away from the

boundary x = 0 or x = L, respectively. On the other hand, the unstable na-

ture flow with F > F* is characterized by the increasing magnitude of up-

stream response function along the channel, witnessing the amplification of

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C. DI CRISTO et al.

664

upstream disturbances in the flow direction. Finally, as far as the down-

stream response function in unstable flows is considered, it further reduces

as we move upstream into the channel, witnessing that the most pronounced

effect of the downstream boundary condition has to be expected close to the

downstream channel end.

5. CONCLUSIONS

In the present paper the analytical solution of the linearized equations

describing 1D hypocritical flow of a Herschel and Bulkley fluid in a finite

channel length has been deduced. Both linearly stable and unstable condi-

tions of the base uniform flow are treated in a unified way. The study has

been carried out applying the Laplace transform method. The influence of

the rheological parameters on the spatio-temporal expressions of upstream

and downstream channel response functions has been analyzed in detail,

comparing the Herschel and Bulkley rheology with both power-law and

Bingham fluid models. The theoretical findings of the present study may

represent an important tool to properly account for the effect of downstream

boundary condition in engineering criteria for the roll waves occurrence in

mud-flows. Moreover, in stable condition the presented analysis may be use-

ful also for comparing different approximations of the wave model.

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Received 12 November 2012

Accepted 10 January 2013

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