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Applied Mathematical Sciences, Vol. 7, 2013, no. 33, 1623 - 1641

HIKARI Ltd, www.m-hikari.com

Gravity-Driven Flow of a Shear-Thinning Power–

Law Fluid over a Permeable Plane

Cristiana Di Cristo

Dipartimento di Ingegneria Civile e Meccanica

Università degli Studi di Cassino e del Lazio Meridionale

Via Di Biasio 43, 03043 Cassino (FR), Italy

dicristo@unicas.it

Michele Iervolino

Dipartimento di Ingegneria Civile, Design, Edilizia ed Ambiente

Seconda Università di Napoli, Via Roma 29, 81031 Aversa (CE), Italy

michele.iervolino@unina2.it

Andrea Vacca

Dipartimento di Ingegneria Civile, Design, Edilizia ed Ambiente

Seconda Università di Napoli, Via Roma 29, 81031 Aversa (CE), Italy

vacca@unina.it

Copyright © 2013 Cristiana Di Cristo et al. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The flow of a thin layer of power-law fluid on a porous inclined plane is

considered. The unsteady equations of motion are depth-integrated according to

the von Karman momentum integral method. The variation of the velocity

distribution with the depth is accounted for, and it is furthermore assumed that the

flow through the porous medium is governed by the modified Darcy’s law. The

stability condition is deduced considering the hierarchy of kinematic and gravity

waves. The response of the linearized model to a Dirac-delta disturbance in

unbounded domain is analytically deduced, in both stable and unstable conditions

1624 C. Di Cristo, M. Iervolino and A. Vacca

of flow. The influence of the effect of power-law exponent and bottom

permeability on the disturbance propagation is finally analyzed, suggesting

indications about the choice of the bottom permeability in order to improve the

performance of industrial processes.

Keywords: Power-law fluid, Darcy’s law, shallow flow models, linear analysis.

1 Introduction

Steady flow of a uniform layer either of Newtonian or of non-Newtonian fluid in

an inclined open channel may become unstable under certain conditions. The

instability degenerates in progressive bores connected by sections of gradually

varying flow, known as roll-waves. The prediction of long-wave instability, for

both Newtonian and non-Newtonian fluid, is of utmost importance in many

environmental and industrial processes. For instance, in debris-flows the presence

of such waves increases their destructive power, while in coating applications the

interfacial instability can cause an uneven distribution of material. In contrast, in

other industrial sectors large amplitude wave structures on the surface can

optimize the process, see for example mass and heat exchangers. Due to the

relevant impact of the instability presence, it is important to determine the critical

conditions for its occurrence and to predict the evolution the waves’ dynamics.

Benjamin [3] and Yih [39] firstly investigated the long-wave instability of a

falling Newtonian laminar film on an impermeable inclined plan, showing that it

may be unstable to infinitesimal periodic perturbations. In particular, Yih [39]

obtained critical conditions of the Reynolds number for the onset of instability

from the solution of the Orr-Sommerfeld equation. The stability conditions of a

Newtonian turbulent film on an inclined impermeable channel, considering

linearized depth-integrated equations, were analyzed both through temporal [4, 5]

and spatial analyses [23, 9, 10, 12].

For non-Newtonian fluids modeled as power-law, the stability conditions on an

impermeable inclined plane have been found by Ng and Mei [26] and by Hwang

et al. [21]. The latter Authors, still using a linear analysis, showed that, for a fixed

value of the power-law exponent (n), the stability characteristics in terms of

generalized Reynolds and Weber numbers are the same as those of Newtonian

fluids, i.e. an increase of the Reynolds number, or a decrease of the Weber

number, destabilizes the film flow. Furthermore, the decreasing of the only n

magnitude causes more unstable film flow and gives rise to faster waves. The

wave dynamics for a power-law flow has been investigated by Danadapat and

Mukhopadhyay [8], who showed the presence of both kinematic and dynamic

wave processes which may either act together or individually to dominate the flow

field depending on the order of magnitude of different parameters. Perazzo and

Gravity-driven flow of a shear-thinning power-law fluid 1625

Gratton [32] analyzed, within the lubrication approximation, the family of

traveling wave solutions. They derived general formulas for the traveling waves,

founding seventeen different kinds of solutions. Miladinova et al. [25] analyzed in

a periodic domain the nonlinear evolution of waves in power-law films.

Numerical calculations indicated the occurrence of a saturation of non-linear

interactions with a final formation of finite-amplitude permanent waves. The free-

surface evolution shares strong similarities with that of a Newtonian liquid.

However, the shape and the amplitude of permanent wave are strongly influenced

by the power-law exponent.

Since the critical conditions for the onset of instability in inclined flows and the

development of the unstable waves are influenced by the structure of the solid

surface at the bottom of the fluid layer, several theoretical investigations

incorporating the bottom porosity have been carried out.

With reference to a Newtonian fluid, Pascal [29], considering depth-integrated

equations, analyzed the effect of a permeable bottom on the linear stability of a

laminar flow, assuming that the flow through the porous medium is governed by

Darcy's law. The results revealed that increasing the permeability of the inclined

plane promotes the instability of the fluid layer flowing above. Sadiq and Usha

[35], confirming that the substrate porosity generally destabilizes the flowing film,

gave a physical explanation in terms of a friction reduction due to the presence of

the permeable bottom. Moreover, through a weakly non-linear stability analysis,

the Authors predicted the existence of both supercritical stable and subcritical

unstable regions. Numerical simulations confirmed the results found on the basis

of linear and weakly non-linear stability analyses. The stability and the evolution

of a thin liquid free surface film flowing down an inclined heated solid porous

wall have been analyzed by Thiele et al. [37]. It has been shown that the substrate

porosity has still a destabilizing influence on the liquid film in presence either of

isothermal or heated substrate. Liu and Liu [24] studied the linear stability of

three-dimensional disturbances on a porous inclined plane using the non-modal

stability theory. Results showed that the critical conditions of both the surface-

mode and the shear-mode instabilities are dependent on permeability for

streamwise disturbances, while the spanwise disturbances have no unstable

eigenvalues. Pascal and D’Alessio [31], generalizing the findings of D’Alessio et

al. [7], analyzed the stability problem in presence of a porous surface exhibiting

periodic undulations, through Floquet–Bloch theory. Several non-linear

simulations of the evolution of perturbed steady flows were conducted, which

confirmed the onset of instability as predicted by the linear analysis. Ogden et al.

[27], using the method of weighted residuals, analyzed the stability of a thin film

flowing down an inclined plane considering the effects of bottom waviness,

heating and permeability.

The effect of the porous bottom on the stability of a power-law fluid layer flowing

down an inclined plane was studied by Pascal [30], considering a one-dimensional

model and accounting for the variation of the velocity distribution with the depth.

1626 C. Di Cristo, M. Iervolino and A. Vacca

The boundary condition at the bottom of the fluid layer is deduced starting from

the assumption that the flow through the porous medium is governed by the

modified Darcy’s law and assuming that the characteristic length-scale of the pore

space is much smaller than the depth of the fluid layer above. It has been shown

that the effect of the permeability of the bed and the shear-thinning nature of the

fluid is to destabilize the flow. Moreover, numerical solution of the non-linear

depth-integrated model indicated that infinitesimal disturbances are amplified in

time and evolve into roll-waves flows. The two-dimensional analysis of Sadiq and

Usha [36] confirmed that the substrate porosity in general destabilizes the film

flow system and the shear-thinning rheology enhances this destabilizing effect.

The influence of an externally applied electric field on the stability of a thin fluid

film over an inclined porous plane is analyzed in Zakaria et al. [41]. Both linear

and non-linear stability analyses in the long-wave limit have been carried out. It

has been found that the permeability parameter, as well as the inclination of the

plane, plays a destabilizing role, while a damping was observed as a consequence

of increasing the electrical conductivity in both linear and non-linear behavior.

From the above literature review it follows that the onset of instability in

Newtonian and non-Newtonian power-law fluid flows over a permeable bottom

has been thoroughly investigated. However, the complete solution of the

corresponding linearized flow model has not been deduced. As far as the

impermeable bed case is concerned, the solution of the linearized flow model

allowed to enlighten the essential flow features of the waves for both Newtonian

[17, 18, 38, 34, 13, 14] and non-Newtonian [11, 15, 16] fluids.

In the present paper the wave dynamics of a shear-thinning power-law fluid over a

permeable surface is analyzed. In facts, the study of the waves evolution of a

power law fluid flowing on a porous bottom may improve the modeling of

important environmental flows and indicate design solutions able to increase the

industrial processes performances. The presented analysis is carried out

considering depth-averaged equations and assuming that the fluid layer depth is

much larger than the characteristic length scale of the pore space below it.

Moreover, the domain is assumed unbounded and the linearized equations are

perturbed with an impulsive point-wise disturbance (Green’s function), in both

stable and unstable conditions. The paper is structured as follows. The section 2

introduces the adopted mathematical model. In section 3 the linearized equations

are deduced along with the analytical expression of the Green’s function. Analysis

and discussion of the results are presented in section 4. Finally, conclusions are

summarized in section 5.

2 Governing Equations

We consider the two-dimensional laminar flow of a thin layer of power-law fluid

over a porous fixed bed inclined at angle θ with respect to the horizontal plane.

Gravity-driven flow of a shear-thinning power-law fluid 1627

Let us denote with x the axis along the bottom and with z the axis normal to it.

The characteristic depth, H, is assumed to be much smaller than the characteristic

longitudinal length of the flow, L, i.e. 1

<<

=

LH

ε

. Neglecting the terms of order

ε

2, the continuity and the x-momentum conservation equations for an

incompressible power-law fluid reads [30]:

0=

∂

∂

∂

∂

z

w

+

x

u (1)

( )

∂

∂

∂

∂

∂

∂

+

∂

∂

−=

∂

∂

∂

∂

∂

∂−

z

u

z

u

zx

h

gg

z

uw

+

x

u

+

t

un1

2

)cos()sin(

ρ

µ

θθ

(2)

in which t is the time, h the flow depth, u and w are the x and z velocity

components, respectively. In (1) and (2), g denotes the gravity,

ρ

, n and µ are the

density, the index and the consistency of the power-law fluid, respectively. In

what follows only shear-thinning fluid will be considered, i.e. 1

≤

n. Neglecting

the effect of surface tension, the boundary conditions at z =h (x,t) for the

streamwise and normal components are:

0=

∂

∂

=hz

z

u (3)

( ) ( )

x

h

h,xu

t

h

h,xw ∂

∂

+

∂

∂

= (4)

At the fluid layer-porous medium interface (z = 0), the following expression

provided by Rao and Mishra [33] for a non-Newtonian fluid, which extended the

formula proposed by Beavers and Joseph [2] for Newtonian fluids, is considered

( )

( ) ( )

,00, ,0, 11

0

==

∂

∂

+

=

xwxu

k

z

u

n

M

z

χ

(5)

in which kM is the modified permeability of the porous medium with a dimension

of length to power n+1,

and

χ

is a dimensionless parameter depending on the

structure of the porous medium. The modified permeability kM depends on the

particle diameter (d) and the porosity (

Φ

) of the porous medium along with the

fluid rheology [6]:

( )

1

131325

6+

Φ−

Φ

+

Φ

=

n

n

M

d

n

n

k (6)

Under uniform flow condition, denoted with a subscript 0, being the variations

along the x-axis null, the corresponding velocity profile can be easily deduced

from Eq.(2) and reads:

1628 C. Di Cristo, M. Iervolino and A. Vacca

( ) ( )

( ) ( )

[ ]

+

+

−−

+++

+

=+

γ

γ

n

n

h

z

nnn

unn

un

n

1

11

121

12

z 1

0

0

0 (7)

where the overbar denotes the dept averaged value. Accounting for Eqs. (6) and

(7), the dimensionless permeability parameter

γ

is given by:

( )

( )

d

~

n

n

h

knn

M

)n/(

)n/(

Φ

ΦΦ

χ

γ

−

+

==

+

+

+

131325

61

0

11

11

(8)

with

(

)

0

hdd

~

χ

= the modified dimensionless diameter of the bottom particles.

Depth-integrating the continuity equation (1), applying the Leibnitz integral rule,

and accounting for the kinematic condition (4), the following equation is deduced:

0 =

∂

∂

+

∂

∂

x

hu

t

h (9)

Similarly, integrating the x-momentum equation and assuming that the

relationship (8) between the velocity and the fluid depth holds also in unsteady

flow, when these quantities are time-dependent and non-uniform, the following

depth-integrated equation is deduced:

( )

( )

( )

0

h

u

12

12

cossin

n

n

2=

++

+

+

∂

∂

−−

∂

∂

+

∂

∂

nn

n

x

h

ghuh

xt

uh

γρ

µ

θθβ

(10)

beingthe momentum correction factor:

( ) ( )( )

1

2312

2

12

2

12

12 2

2

2

>

++

+

+

+

++

+

=nn

n

n

n

nn

n

γ

γ

γ

β

(11)

Introducing the following dimensionless quantities:

,,,,

000

0

0u

u

u

h

h

h

h

u

tt

h

x

x=

′

=

′

=

′

=

′ (12)

the equations (9) and (10) in dimensionless form read:

0 =

′

∂

′

′

∂

+

′

∂

′

∂

x

hu

t

h (13)

( )

( )

0

12

121tan

2

1

n

n

2

2

2

2=

′

′

++

+

+

′

−

′

+

′′

′

∂

∂

+

′

∂

′′

∂

h

u

nn

n

Re

h

F

h

F

uh

xt

uh

γ

θ

β

(14)

in which

µ

ρ

θ

nnhu

gh

u

F0

2

0

0

0Re ,

)cos(

−

== (15)

denote the Froude and Reynolds numbers of the base flow, respectively.

Gravity-driven flow of a shear-thinning power-law fluid 1629

2 Impulsive response of the linearized flow model

Superposing to the base flow a disturbance in terms of both average velocity (p

u

′

)

and flow depth (p

h′), with both 1 and <<

′′ pp hu , the linearization of (13) and (14)

leads to the following three-wave equation in p

h′ only:

( )

12

12

Re''

'

*

'

n

21

∂

∂

+

∂

∂

++

+

−=

′

∂

∂

+

′

∂

∂

∂

∂

+

′

∂

∂

x

h

c

t

h

nn

nn

h

x

c

tx

c

t

pp

p

γ

(16)

where 2,1

c are the speeds of the gravity waves

( )

2

2,1

1

1F

c+−±=

βββ

(17)

and c* denotes the speed of the kinematic wave

n

n

c12

*

+

= (18)

While the speeds of both the kinematic (c*) and the faster gravity wave (

β

>

1

c)

are always positive, the speed of the slower gravity one

β

<

2

c is negative

whenever F is smaller than the critical value

β

1=

c

F. Accounting for Eq.(11),

the critical Froude number, may be expressed as

( ) ( )( )

2312

2

12

2

12

121 2

2

++

+

+

+

++

+

=nn

n

n

n

nn

n

Fc

γ

γ

γ

(19)

Figure (1) depicts the dependence of Fc on the power-law exponent for different

values of the porosity

Φ,

for d

~

= 1. In all the analysis presented here and in the

next section, the modified dimensionless particle diameter is assumed equal to

one. Independently on the porosity value, the critical Froude number is always

smaller than one. For a fixed value of the porosity, Fc decreases with the power-

law exponent, while for a fixed value of n it increases with

Φ.

For Newtonian

fluids (n=1) on impermeable bottom (

Φ

=0) it results Fc=0.91 [22].

1630 C. Di Cristo, M. Iervolino and A. Vacca

Figure 1. Critical Froude number as a function of the power-law exponent and the

bottom porosity (d

~

= 1).

The wave hierarchy in (16) allows to easily deduce the unstable conditions of the

system (13)-(14), individuated as the cases for which the gravity wave speed

overwhelms the faster kinematic one, i.e. 1

cc

*

>

[40, 19]. With reference to (17)

and (18), the inequality 1

cc

*

>

furnishes limiting stability Froude number ( *

F),

which represents the lower bound above which instabilities appear:

( ) ( )

1

12

*

*

−−−=

βββ

c

F (20)

With the aid of expressions (8), (11) and (18), it is easily verified that F* is real

provided that

γ

> 0. For sake of clarity, Figure 2 represents the celerity of the two

gravity waves along with the one of the kinematic wave for the case n = 1 and

Φ

= 0 (Newtonian fluid on impermeable bottom), in which *

F=0.577.

Gravity-driven flow of a shear-thinning power-law fluid 1631

Figure 2. Wave celerities as a function of the Froude number (

Φ

= 0, n = 1).

It follows that, similarly to the critical Froude number Fc , also F* is a function of

the pair (n,

Φ

)

for a fixed value of

d

~

.

Figure (3) shows F* as function of the

porosity

Φ

and of the power law exponent n, for d

~

= 1. It can be observed that an

increase of the bottom porosity or a decrease of the power-law index destabilizes

the flow, accordingly with the normal mode analysis results of Pascal [30].

Figures (1) and (3) collectively lead to recognize that the marginal stability

condition occurs in hypocritical condition of flow, i.e. c* FF <.

Figure 3. Limiting stability Froude number as a function of the power-law

exponent and the bottom porosity ( d

~

= 1).

1632 C. Di Cristo, M. Iervolino and A. Vacca

In order to analyze the wave dynamics in a power-law fluid flowing on a

permeable bottom, we deduce the analytical expression of the impulsive response,

G(x’,t’), of the flow model (16) to an instantaneous and point-wise disturbance at

x=0 and t=0. With reference to an unbounded domain with homogeneous initial

conditions, rewriting Eq.(16) in terms of the characteristic variables:

'' ''

2

1

xtcxtc

−

=

−

=

η

ξ

(21)

we are led to solve the following problem

( ) ( ) ( )

)()(

2

1

~

1

1

Re412

12

~

1

2

*

2

2

2

1

2n

2

ηδξδ

β

β

γηξ

−

−

−

++

+

+

∂∂

∂

c

=G

F

F

cF

n

nn

nG (22)

in which

( ) ( )

( ) ( )

( )

−

−

+−

−

−

++

+

=

'

*

2

1

'

2

1

*

n

12

12

Re2

','','

~t

c

c

x

c

c

nn

nn

etxGtxG

β

β

β

β

β

γ

(23)

and

(

)

⋅

δ

denotes the Dirac function.

Equation (22) may be easily solved through bilateral Laplace transform [28].

Moreover, imposing the causality constraint, i.e. G(x′, t′) = 0 for

ℜ

∈

′

∀

≤

′

xt 0 and

accounting for the sign of the coefficients in (22), the following expressions of the

impulsive response are finally achieved:

- for

*

FF

<

( )

(

)

(

)

( )

( ) ( )

( )

( ) ( ) ( )( )

−−

−

−

++

+

−

−

−

=

−

−

+−

−

−

++

+

''''1

1

Re212

12

2

''''

','

21

2

*

2

2

1

n

0

'

*

2

1

'

2

1

*

n

12

12

Re2

1

21

tcxxtc

F

F

cF

n

nn

n

I

e

c

tcxHxtcH

txG

t

c

c

x

c

c

nn

nn

β

γ

β

β

β

β

β

β

γ

(24)

- for *

FF

≥

( )

(

)

(

)

( )

( ) ( )

( )

( ) ( ) ( )( )

−−

−

−

++

+

−

−

−

=

−

−

+−

−

−

++

+

''''1

1

Re212

12

2

''''

','

21

2

*

2

2

1

n

0

'

*

2

1

'

2

1

*

n

12

12

Re2

1

21

tcxxtc

F

F

cF

n

nn

n

J

e

c

tcxHxtcH

txG

t

c

c

x

c

c

nn

nn

β

γ

β

β

β

β

β

β

γ

(25)

In (24) and (25)

(

)

⋅

Hdenotes the Heaviside function,

(

)

⋅

0

J the zero-order Bessel

functions of the first kind and

(

)

⋅

0

I the modified one [1]. The analytical

expression of the Green function (24)-(25) shows that the slower (resp. faster)

Gravity-driven flow of a shear-thinning power-law fluid 1633

gravity wave represents the upstream (resp. downstream) front of the perturbed

portion of the phase plane. Moreover, being

β

>

*

c, it follows that the exponential

wave causes a temporal growth of the disturbance on all rays of the (x′, t′) plane

such that:

(

)

βc

βc

β

t'

x'

*−

−

+>

2

1 (26)

From (24) and (26) it follows that for flow conditions with *

cc

>

1, i.e. *

FF

<

, the

disturbance will always decay in the time on any ray of the (x′, t′) plane (stable

conditions of flow), even accounting for the maximum asymptotic growth of the

modified Bessel function I0. When *

cc

=

1, i.e. *

FF

=

, on the upstream front

1

ct'x'

=

the disturbance will propagate with constant amplitude (marginal

stability condition), while whenever the kinematic wave exceeds the fastest

gravity one, i.e. *

FF

>

, the disturbance will grow in time on all t'x' rays

satisfying the following inequalities

(

)

1

2

1c

t'

x'

βc

βc

β

*

<<

−

−

+ (27)

The existence of at least one x′/t′ ray on which the disturbance grows in time

allows reinterpreting the stability condition deduced through the waves hierarchy

in terms of asymptotic behavior of the Green function [20].

4 Results and discussion

The spatio-temporal behavior of the Green function G(x′, t′) is exemplified in

Figure 4 for fixed values of Reynolds number (Re = 10), bed porosity and

modified dimensionless particle diameter (

Φ

= 0.2, d

~

= 1) and the power-law

exponent (n = 0.6). For the presented case the limiting stability Froude number is

∗ = 0.397, while the critical Froude number is Fc = 0.943. In Figure 4 two

different conditions are considered: F = 0.1 (Fig. 4a), which corresponds to a

stable flow and F = 1 (Fig. 4b), which corresponds to a hypercritical unstable

flow. For sake of completeness, the ray expressed by r.h.s. of (26) is also

represented. It is observed that the Green’s function behavior is completely

different under stable and unstable conditions. As far as stable base flow is

considered (Fig.4a), the instantaneous maximum perturbation height progressively

moves from the downstream ray x′/t′=c1 towards the one corresponding to the

kinematic wave celerity x′/t′ = c*. Similarly to [15], it may be shown that the

Green function tends to assume asymptotically in time a Gaussian shape with the

centroid moving with c*. Consistently, the Green function tends to vanish at the

two boundaries of the characteristic fan.

On the other hand, the behavior of the function in the considered hypercritical

unstable conditions (Fig.4b) of flow is dominated by the monotonic growth along

1634 C. Di Cristo, M. Iervolino and A. Vacca

the downstream end of the characteristic fan x′/t′ = c*. The progressive decay of

the perturbation is observed instead on the upstream end of the fan x′/t′ = c2. This

figure gives also a visual representation of the convective nature of the instability,

since the t′ axis simply remains outside from the perturbed region for t′ > 0. In

fact, since the r.h.s. of (26) denotes a positive quantity, the x′/t′ = 0 ray always

falls in the region where the exponential wave produces a dampening effect.

Figure 4. Space-time Green’s function behavior for n = 0.6,

Φ

= 0.2, Re = 10,

d

~

= 1. F = 0.1(a) and F = 1(b).

In order to analyze the influence of both the power-law exponent n and the bed

porosity

Φ

on wave propagation, we look at the temporal evolution of the

disturbance on the upstream (x′/t′=c2) and downstream (x′/t′=c1) fronts of the

perturbed region. Indeed, on these rays, since the arguments of Bessel functions I0

(in stable conditions of flow) and J0 (in unstable conditions of flow) vanish, the

disturbance propagates as the exponential waves with a temporal growth rate

given by:

Gravity-driven flow of a shear-thinning power-law fluid 1635

• Downstream front

x′/t′ = c1

( )

−

−

−

++

+

=1

12

12

Re2 1

*

n

1

β

β

γ

η

c

c

nn

nn (28)

• Upstream front

x′/t′ = c2

( )

( )( )

( )

−

−

−−

++

+

=1

12

12

Re2 2

1

*2

n

2

β

ββ

γ

η

c

cc

nn

nn (29)

While on the upstream front, owing to the c2 definition (17), the perturbation

always decays, i.e.

η

2 < 0, either negative (i.e.

η

1<0, stable conditions) or positive

(i.e.

η

1>0, unstable conditions) temporal growth may occur on the downstream

one, dependently on value of the kinematic and the fastest gravity wave celerities,

or, equivalently, of the Froude number. In what follows, the influence of n and

Φ

on both

η

1 and

η

2 coefficients is analyzed, for three different Froude number

values (Figures 5,6 and 7), assuming a unitary value of both Reynolds number and

modified dimensionless particle diameter.

Figure 5. Growth/decay factors for d

~

= 1, Re = 1, F = 0.1.

In particular, Figures 5 represents the disturbance growth rate on both the

upstream and downstream fronts as a function of the bottom porosity

Φ

, for F=0.1

and considering three different values of for power- law index n, namely 0.2, 0.6

and 1.0. Independently on the bottom porosity and the power-law index, all the

conditions represented in Figure 5 are characterized by negative values of both

η

1

and

η

2, i.e. the base uniform flow is stable. On both fronts, the decay rate

monotonically increases with the power law index, while it monotonically reduces

with the porosity.

1636 C. Di Cristo, M. Iervolino and A. Vacca

Figure 6. Growth/decay factors for d

~

= 1, Re = 1, F = 0.55.

If the Froude number is increased up to F = 0.55 (Figure 6), the two considered

shear-thinning cases, independently on the porosity value, exhibit an unstable

behavior, i.e.

η

1 > 0. For the Newtonian fluid (n = 1) such a flow condition can be

either stable or unstable, depending on the bottom porosity value, enlightening the

destabilizing effect of the porosity. In unstable conditions (

Φ

> 0.55), the growth

rate increases with the bed porosity. Conversely, for both the non-Newtonian

fluids, the growth rate on the downstream is a monotone decreasing function of

Φ

with the maximum at

Φ

= 0 (impermeable bottom). As it will be successively

clarified, the observed different dependence of the

η

1 coefficient on

Φ

between

the shear-thinning and the Newtonian cases is essentially due to the chosen

Froude number value and not to the rheology itself. Finally, Figure 6 shows that

on the upstream front, the decay rate

η

2 reduces with the porosity and increases

with power-law index, similarly to the cases described in Figure 5.

Gravity-driven flow of a shear-thinning power-law fluid 1637

Figure 7. Growth/decay factors for d

~

= 1, Re = 1, F = 1.0.

For all the considered values of the-power law index and independently of the

bottom porosity, an unitary value of the Froude number determines unstable

conditions of flow (Figure 7). The decay rate on the upstream front behaves

similarly the cases shown in Figures 5 and 6. On the downstream front, while it is

confirmed the reduction of the growth factor with the porosity even for the

Newtonian fluid, an increment of power-law index leads to a larger growth rate.

Figures (6) and (7) collectively indicate that for Froude numbers slightly above

the instability threshold (Figure 6), for n > 0.2 the growth rate of the downstream

wave monotonically decreases with the power-law index. The opposite behavior

is encountered as the Froude number is increased (Figure 7).

On the other hand, the growth rate of the downstream wave exhibits a non-

monotone dependence on the bottom porosity. In order to specifically analyze this

aspect, in Figures 8 and 9 the plot of

η

1 versus

Φ

is reported, for different values

of the Froude number, with reference to either a Newtonian (Figure 8) and to a

shear-thinning (n = 0.6, Figure 9) fluid.

1638 C. Di Cristo, M. Iervolino and A. Vacca

Figure 8. Growth rate at the downstream front as a function of bottom porosity.

Newtonian fluid (n = 1, d

~

= 1, Re = 1). Symbols denote function maxima.

Figure 9. Growth rate at the downstream front as a function of bottom porosity.

Shear-thinning fluid (n = 0.6, d

~

= 1, Re = 1). Symbols denote function maxima.

From Figure 8 it follows that for Froude numbers slightly above the limiting

stability value, the maximum growth rate occurs at relatively high values of

Φ

. The increase of the Froude number induces a reduction of the porosity value at

which the maximum occurs, until for sufficiently high values of F the growth rate

becomes a monotonic decreasing function of the porosity with the maximum

Gravity-driven flow of a shear-thinning power-law fluid 1639

value localized at

Φ

= 0. Further increase of the Froude number does not modify

the maximum location. The comparison of Figures 8 and 9 suggests that the

dynamics discussed above is observed even in non-Newtonian fluid, although in a

smaller range of Froude numbers. The knowledge of the porosity at which the

growth factor in unstable conditions attains its maximum, for a given Froude

number, may be of utmost importance in many industrial processes in order to

control of long-wave instability.

5 Conclusions

The flow of a thin power-law fluid layer on a porous inclined plane has been

considered. The boundary condition at the bottom is fixed assuming that the flow

through the porous medium is governed by the modified Darcy’s law and that the

characteristic length scale of the pore space is much smaller than the depth of the

fluid layer above. The considered depth-integrated unsteady equations account for

the variation of the velocity distribution with the depth in the momentum

correction coefficient expression. The analytical solution of the linearised flow

model of an initial uniform condition in an unbounded domain perturbed by a

pointwise instantaneous disturbance has been derived. The discussion of the main

features of the analytical solution allowed to clarify the role of the bottom

porosity on the value of the growth/decay rate governing the perturbation

evolution on the boundaries of the propagation domain. It has been found that on

the upstream propagation front the decay rate monotonically increases with the

power-law index, while it monotonically reduces with the porosity. On the

downstream propagation front, in unstable conditions of flow, the growth rate

may be, for a fixed rheological index value, a non-monotone function of the

bottom porosity dependently of the Froude number value.

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Received: January, 2013