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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
