Fractional calculus applied in solving instability phenomenon in fluid dynamics

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),
ISSN 0976 – 6316(Online), Volume 6, Issue 5, May (2015), pp. 34-44 © IAEME
34
FRACTIONAL CALCULUS APPLIED IN SOLVING
INSTABILITY PHENOMENON IN FLUID DYNAMICS
Ravi Singh Sengar1
, Manoj Sharma2
, Ashutosh Trivedi3
1,
M.Tech Scholar, Civil Engineering (Construction Technology and Management),
IPS-CTM Gwalior
2,3
Dept. of Civil Engineering, IPS-CTM Gwalior
ABSTRACT
The purpose of present paper is to find applications of Fractional Calculus approach in Fluid
Mechanics. In this paper by generalizing the instability phenomenon in fluid flow through porous
media with mean capillary pressure with transforming the problem into Fractional partial differential
equation and solving it by applying Fractional Calculus and special functions.
Keywords: Fluid flow through porous media, Laplace transform, Fourier sine transform, Mittag -
Leffler function, Fox-Wright function, Fractional time derivative.
1. INTRODUCTION AND MATHEMATICAL PREREQUISITES
Using ideas of ordinary calculus, we can differentiate a function, say, f(x)=x to the Ist
or IInd
order. We can also establish a meaning or some potential applications of the results. However, can
we differentiate the same function to, say, the 1/2 order? Better still, can we establish a meaning or
some potential applications of the results? We may not achieve that through ordinary calculus, but
we may through fractional calculus—a more generalized form of calculus. Fractional Calculus is the
branch of calculus that generalizes the derivative of a function to non-integer order. In other words
the fractional calculus operators deal with integrals and derivatives of arbitrary (i.e. real or complex)
order. The name "fractional calculus" is actually a misnomer; the designation, "integration and
differentiation of arbitrary order" is more appropriate. The subject calculus independently discovered
in 17th
century by Isaac Newton and Gottfried Wilhelm Leibnitz, the question raised by Leibnitz for
a fractional derivative was an ongoing topic for more than three hundred years. For a long time,
fractional calculus has been regarded as a pure mathematical realm without real applications. But, in
recent decades, such a state of affairs has been changed. It has been found that fractional calculus can
be useful and even powerful, and an outline of the simple history about fractional calculus.. The
various researcher investigated their investigations dealing with the theory and applications of
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ISSN 0976 – 6308 (Print)
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Volume 6, Issue 5, May (2015), pp. 34-44
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35
fractional calculus Free shear flows are inhomogeneous flows with mean velocity gradients that
develop in the absence of cigarette, and the buoyant jet issuing from an erupting volcano - all
illustrate both the omnipresence of free turbulent shear flows and the range of scales of such flows in
the natural environment. Examples of the multitude of engineering free shear flows are the wakes
behind moving bodies and the exhausts from jet engines. Most combustion processes and many
mixing processes involve turbulent free shear flows. Free shear flows in the real world are most often
turbulent. The tendency of free shear flows to become and remain turbulent can be greatly modified
by the presence of density gradients in the flow, especially if gravitational effects are also important.
Free share flows deals with incompressible constant-density flows away from walls, which include
shear layers, jets and wakes behind bodies. Hydrodynamic stability is of fundamental importance in
fluid dynamics and is a well-established subject of scientific investigation that continues to draw
great curiosity of the fluid mechanics community. Hydrodynamic instabilities of prototypical
character are, for example, the Rayleigh-Bènard, the Taylor-Couette, the Bènard-Marangoni, the
Rayleigh-Taylor, and the Kelvin-Helmholtz instabilities. Modeling of various instability mechanisms
in biological and biomedical systems is currently a very active and rapidly developing area of
research with important biotechnological and medical applications (biofilm engineering, wound
healing, etc.). The understanding of breaking symmetry in hemodynamics could have important
consequences for vascular biology and diseases and its implication for vascular interventions
(grafting, stenting, etc.). When in a porous medium filled with one fluid and another fluid is injected
which is immiscible in nature in ordinary condition, then instability occurs in the flow depending
upon viscosity difference in two flowing phases. When a fluid flow through porous medium
displaced by another fluid of lesser viscosity then instead of regular displacement of whole front
protuberance take place which shoot through the porous medium at a relatively high speed. This
phenomenon is called fingering phenomenon (or instability phenomenon). Many researchers have
studied this phenomenon with different point of view. Fractional calculus is now considered as a
practical technique in many branches of science including physics (Oldham and Spainier [13]). A
growing number of works in science and engineering deal with dynamical system described by
fractional order equations that involve derivatives and integrals of non-integer order (Bensonet al.
[2], Metzler and Klafter [9], Zaslavsky [24]). These new models are more adequate than the
previously used Integer order models, because fractional order derivatives and integrals describe the
memory and hereditary properties of different substances (Poddulony [14]). This is the most
significant advantage of the fractional order models in comparison with integer order models, in
which such effects are neglected. In the context of flow in porous media, fractional space derivatives
model large motions through highly conductive layers or fractures, while fractional time derivatives
describe particles that remain motionless for extended period of time (Meerscheart et al. [8]).
The phenomenon of instability in polyphasic flow is playing very important role in the study
of fluid flow through porous media in two ways viz. with capillary pressure and without capillary
pressure. The statistical view point was studied by Scheidegger and Johnson [21], Bhathawala and
Shama Parveen [3] considering instability phenomenon in porous media without mean Capillary
pressure. Verma [23] has also studied the behavior of instability in a displacement process through
heterogeneous porous media and existence and uniqueness of solution of the problem was discussed
by Atkinson and Peletier [1]. El-Shahed and Salem [11, 12] have used the fractional calculus
approach in fluid dynamics, which has been described by fractional partial differential equation and
the exact solution of these equations, have been obtained by using the Laplace transform, Fourier
transform. Flow in a porous medium is described by Darcy’s Law (El-Shahed and Salem [11]) which
relates the movement of fluid to the pressure gradients acting on a parcel of fluid. Darcy’s Law is
based on a series of experiments by Henry Darcy in the mid-19th century showing that the flow
through a porous medium is linearly proportional to the applied pressure gradient and inversely
proportional to the viscosity of the fluid. In one dimension, q represents “mass flow rate by unit area”

36
and is defined as,
q = −
where K is permeability, a parameter intrinsic to the porous network. The unit of permeability K is
. δ is the kinematics viscosity has dimension , e.g., and is the non-hydrostatic
part of pressure gradient has dimension M e.g., g . thus the mass flow rate by
unite area(q) has dimension = Here, we considered homogeneous dimensions
but in fractional calculus dimensions are inhomogeneous.
Mathematical Prerequisites
The Laplace Transform (Sneddon [20]) is defined as,
=
!
"
# $# %
> 0 1.1
Fourier transform of Weyl fractional derivative ! 8
9
. # is given by (Metzler and
Klafter,2000,p 59A12,2004)
: ! 8
9
, # }= |=|9
> =, #
The Fourier sine transform (Debnath [5]) is given as,
? @, #
=
2
√C
? , #
!
"
sin @ $ 1.2
The Error Function ([16]) of x is given as
E
=
2
C
exp −#
"
$# 1.3
And the complimentary error function of x is given as
E =
I
J exp −#
!
$# 1.4
In 1903, the Swedish mathematician Gosta Mittag-Leffler [10] introduced the function Eα (z) which
is given as ----
Eα z = N
OP
Γ R@ + 1
!
PT"
, 1.5
Where z is a complex variable and Γ(α) is a gamma function of α. The Mittag–Leffler function is
direct generalization of the exponential function to which it reduces for α = 1. For 0 < α < 1, Eα z
interpolates between the pure exponential and a hyper geometric function
V
. Its significance is

37
realized during the last two 100 years due to its involvement in the problems of physics, chemistry,
biology, engineering and applied sciences. Mittag–Leffler function naturally occurs as the solution of
fractional order differential equation or fractional order integral equations. The generalization of
Eα z was studied by Wiman [23] in 1905 and defined the function as
Eα,β z = N
OP
Γ R@ + W
!
PT"
, α, β ∈ C; Re α > 0, Re β
> 0 1.6
Which is known as Wiman’s function or generalized Mittag–Leffler function as _R, (z )=Eα (z ).
The Laplace transform of (1.6) takes in the following form (Shukla and Prajapati [20])
!
"
#àbc
_`,c
a
#`
$#
=
d ! ` c
` − ab
1.7
Where _`,c
a
O =
g
Vg _`,c h .
The Fox-Wright function (Craven and Csordas [4]) is given as under
pΨq ( )
= ∑
∏ k lm n b om
p
gq
∏ k rm n b st
p
gq
u
v !
, 1.8!
vT"
Where Γ(x) denotes the Gamma function of x and p and q are nonnegative integers. If we set bj = 1 (j
= 1, 2, 3, ..., p)
and dj = 1 (j = 1, 2, 3, ..., q ) then (1.8) reduces to the familiar generalized hyper geometric function
(Craven and Csordas [4]).
pFq x ,…,xz;{ ,…{| ; = ∑
} u…. }p u
~ u…. ~p u
u
v !
1.9!
vT"
For the study of generalized Navier - Stokes equations, El - Shahed and Salem [12] used the very
special case of (1.1), given as
€ R, W; h = N
ha
d ! Γ α• + β
1.10
!
aT"
The Laplace Transforms of (1.10) is given by
!
"
€ R, W; # $#
=
1
_`,c ‚
1
ƒ 1.11
The relationship between the Wright function and the Complementary Error function is given as,
€ ‚−
1
2
, 1; hƒ = E h 2⁄ 1.12

38
Riemann-Liouville fractional integrals of order µ (Khan and Abukhammash [7]) is given as
Let f (x) ∈ L(a, b), µ ∈ C (Re(µ) > 0) then
x…
†
= … +}
†
=
1
Γ μ
#
− # †
}
$# > x 1.13
is called R-L left-sided fractional integral of order µ.
Let f ( ) ∈ L (a, b), µ ∈ C (Re(µ) > 0) then
…~
†
=
1
Γ μ
#
# − †
~
$#
< { 1.14
is called R-L right-sided fractional integral of order µ.
The Laplace Transform of the fractional derivative (El - Shahed and Salem [12]) is defined by
!
"
8`
# $# = `
− N ` a
8`
0 @ − 1 < R < @
P
aT"
1.15
Theorem (Asymptotic expansion of Wiman function _`,c(z )): Let 0 < α < 1 and β be an arbitrary
complex number
then
_`,c h =
1
2RC‰
exp ‚∈`ƒ ∈
c
`
∈ −h
$ ∈ 1.16
We also use following integral (El - Shahed and Salem [12]) in terms of Wright function as,
@ sin @
!
"
_`,`b −@ Š#`
$@ =
C
2Š#`
‹ ‚
−R
2
, 1 ;
−
√Š#`
ƒ 1.17
2. STATEMENT OF THE PROBLEM
If water is injected into oil saturated porous medium, then as a result perturbation (instability)
occurs and develops the finger flow (Scheidegger and Johnson [19]). In this paper, our aim is to
study one dimensional flow, x-indicating the direction of fluid flow with the origin at the surface, due
to presence of large quantity of water at x = 0. We assume that water saturation at x = 0 is almost
equal to one i.e.1 and water saturation remain constant during the displacement process. Our picky
interest in this paper is to look at the possibilities of transforming the problem in form of fractional
partial differential equation with proper initial and boundary conditions.
3. FORMATION OF THE PROBLEM
The seepage velocity of water (Vw) and oil (δw) are given by Darcy’s law [17] as

39
Œ•
= −
Ž•
••
Ž
•‘•
•
3.1
Œ’
= −
Ž’
•’
Ž
•‘’
•
3.2
And equation of continuity
∅
•”•
•#
+
•Œ•
•
= 0 3.3
∅
•”’
•#
+
•Œ’
•
= 0 3.4
Where K is the permeability of the homogeneous medium, Kw and Ko are the relative
permeability of the water and oil, Sw and So are saturation of water and oil respectively, Pw and Po
are the pressure in water and oil, phases δw and δo are the kinematics viscosities of water and oil
respectively and φ is the porosity of medium.
For inhomogeneous dimensions, considering the Time-fractional partial differential equations
of the two phases as under:
∅
•`
”•
•#`
+
•Œ•
•
= 0, 0 < R < 1 3.5
∅
•`
”"
•#`
+
•Œ’
•
= 0, 0 < R < 1 3.6
For α = 1, equations (3.5) and (3.6) reduce to equations of continuity (3.3) and (3.4) respectively and
from the definition of phase saturation [17], we have
”• + ”" = 1 3.7
The capillary pressure PC is defined as pressure discontinuity between the flowing phases across
their common interface and assumes the function of the phase saturation is a continuous linear
functional relation as
‘ = W ”• 3.8
‘ = ‘’ − ‘•, 3.9
where β is constant.
Relationship between phase saturation and relative permeability [18] is given by
Ž• = ”•
Ž’ = 1 − ”• 3.10
= ”’
4. FORMATION OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATION
Now, we put the values of Œ• and Œ’ (from (3.1) and (3.2)) in (3.5) and (3.6) respectively, we get

40
∅
•`
”•
•#`
=
•
•
•
Ž•
••
Ž
•‘•
•
– 0 < R < 1 4.1
∅
•`
”’
•#`
=
•
•
•
Ž’
•’
Ž
•‘’
•
– 0 < R < 1 4.2
Eliminating
— ˜
—
from (4.1) and (3.9),
∅
•`
”•
•#`
=
•
•
•Ž
Ž•
••
‚
•‘’
•
−
•‘
•
ƒ– 4.3
From (4.2), (4.3) and (3.7) we obtained
•
•
™Ž •
Ž•
••
+
Ž’
•’
–
•‘’
•
−
Ž•
••
Ž
•‘
•
š = 0 4.4
Integrating (4.4) with respect to x, we get
Ž •
Ž•
••
+
Ž’
•’
–
•‘’
•
−
Ž•
••
Ž
•‘
•
= −› 4.5
Where B is the constant of integration, whose value can be determined.
Equation (4.5) can be written as
•‘’
•
=
−›
Ž
Ž•
••
•1 +
Ž’
Ž•
••
•’
–
+
•‘
•
1 +
Ž’
Ž•
••
•’
4.6
Substituting the value of
— œ
—
from (4.6) in (4.3), we ar
∅
•`
”•
•#`
+
•
•
•
Ž
Ž’
•’
•‘
•
1 +
Ž’
Ž•
••
•’
+
›
1 + •
Ž’
Ž•
••
•’
–
ž = 0 4.7
Pressure of oil (‘’) can be written as,
‘’ =
1
2
‘’+‘• +
1
2
‘’−‘•
= ‘ +
1
2
‘ 4.8
where P is the mean pressure, which is constant.
From (4.5) and (4.8) we get,
› =
Ž
2
•
Ž•
••
−
Ž’
•’
–
•‘
•
4.9
Substituting (4.9) in (4.7), we get
∅
•`
”•
•#`
+
1
2
•
•
•Ž
Ž•
••
•‘
•”•
•”•
•
–
= 0 0 < R < 1 4.10

41
Taking Ž ˜
˜
— Ÿ
— ˜
= −λ then (4.10) reduces in the form,
• ”•
•
=
1 •`
”•
•#`
, 4.11
Where C =
¢
Equation (4.11) is the desired fractional partial differential equation of motion for water
saturation, which governed by the ﬂow of two immiscible phases in a homogenous porous medium
and appropriate initial and boundary conditions are associated with the description as to
”• , 0 = 0,
”• 0, = •£
< 1, 4.12
lim
→∞
”• , = 0; 0 < < ∞
5. FORMULATION AND SOLUTION OF PROBLEM
Here we are formulating the generalized fractional partial differential equation by
generalizing of Prajapati’s et. al.[16] model equation which is given as it is in equation (4.11)
From (4.11), we have
•`
”•
•#`
= Š
• ”•
•
Now, we generalizing the above equation by using fractional calculus operators in following form
—¦
˜ ,
— ¦ + x
—§
˜ ,
— § =
Š
—¨
˜ ,
— ¨ + > , # 5.1
Or
" 8`
”• , # + x " 8
c
”• , #
= Š ! 8
9
”• , # + > , # 5.2
Where " 8`
is a Riemann-Liouville Fractional Derivative , ! 8
9
is a Weyl Fractional
derivative> , # is a constant which describes the nonlinearity in the system.
If we put x = 0, © = 2 x@$ > , # = it converts into Prajapati’s et. al.[16] paper
•`
”• , #
•#`
+ x
•c
”• , #
•#c
= Š
•9
”• , #
• 9
+ > , # 5.3
Solution-: Applying Laplace transformation both sides in equation (5.2), we get

42
`
”• , − N ` ª
8
P
ªT"
”• , « + x c”• , − N x c ª
8
P
ªT"
”• , «
= Š ! 8
9
”• , + ,¬¬¬¬¬¬¬¬¬ 5.4
Apply boundary conditions from (4.12)
`
”• , + x c ”• , = Š !8
9
”• , + ,¬¬¬¬¬¬¬¬¬ 5.5
As is usual, It is convenient to employ the symbol – to indicate the Laplace transform with respect to
the variable t. Also ,¬¬¬¬¬¬¬¬¬ is Laplace transform of > , # [17].
Now we apply the Fourier transform with respect to the space variable x to the above equation both
sides
`
”•
∗
Ž, + x c
”•
∗
Ž,
= −Š|Ž|9
”•
∗
Ž, + =,
∗
5.6
”•
∗
Ž, =
® v,
∗
¦b } §b~
where b= Š|Ž|9
5.7
”•
∗
Ž,
=
=,
∗
` + x c + {
5.8
Applying Inverse Laplace transform both sides
•
∗
Ž, # = ∑ −x ª
J >∗
=, # − ¯"
¯`b ` c ª
_`, ` c ªb`
ªb!
ªT" −{¯`
$¯
Where b= Š|Ž|9
5.9
Now, Applying Inverse Fourier transforms both sides, we get the desired result,
°± ², ³ = N
−´ µ
¶·
!
µT¸
¹ºb º » µ ¼
³
¸
½ ¾¿²
∞
!
>∗
=, #
− ¯ _`,`b ` c ª ~À¦
ªb
Á¿Á¹
5.9
Which is a new and generalized result of Prajapati’s et. al.[16] results.
Now if we put a=0, © = 2 and > = 0
It converts into Prajapati’s et. al.[16] results.
This is easy to write in the form of wright functions
”• , # = •œ
‹ Â−
R
2
,1; −
#`
Ã 5.10
On setting α = 1 and using (1.12), (5.7) reduces to,
”• , # = •œ
E /2√C t) (5.11)
Which is same as Prajapati’s et. al.[16] results.

43
CONCLUSION
The constitutive relationship model of generalization of the instability phenomenon in fluid
flow through porous media with mean capillary pressure by applying fractional calculus is obtained.
The exact solution of the generalized fractional partial differential equation in terms of Wright
function by means of Laplace transform, Fourier transform with proper initial and boundary
conditions has been found. If we set α = 1 then equation (5.10) reduces to (5.11), this method
certainly useful than conventional method as the conventional method derived only for α = 1
(equations (3.3) and (3.4)) whose solution given by equation (5.11). While this fractional calculus
together with Fourier and Laplace transforms method presented in this paper also applicable for 0 <
α < 1 whose solution given by equation (5.10).
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Fractional calculus applied in solving instability phenomenon in fluid dynamics

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