Gen. Math. Notes, Vol. 4, No. 2, June 2011, pp.49-59
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2011
www.i-csrs.org
Available free online at http://www.geman.in
Solution of the Time-Fractional Navier-Stokes
Equation
V.B.L. Chaurasia1 and Devendra Kumar2
1
Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, India
E-mail: vblchaurasia@gmail.com
2
Department of Mathematics, Jagan Nath Gupta Institute of Engineering
and Technology, Jaipur, Rajasthan, India
E-mail: devendra.maths@gmail.com
(Received:12-4-11 /Accepted:25-6-11)
Abstract
In this paper we obtain the solution of a time-fractional Navier-Stokes equation. The solution is derived by the application of Laplace and finite Hankel
transforms. The results are obtained in compact and elegant forms in terms
of the Mittag-Leffler and Bessel functions, which are suitable for numerical
computation. The results derived here are quite general in nature and capable
of yielding a very large number of known (may be new also) results, hitherto
scattered in the literature.
Keywords: Time-fractional Navier-Stokes equation, Mittag-Leffler function, Caputo fractional derivative, Laplace transform, finite Hankel transform.
1
Introduction
Fractional calculus is a field of applied mathematics that deals with derivatives
and integrals of arbitrary orders. In recent years, it has turned out that many
phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional
calculus. For example, fractional derivatives have been used successfully to
model frequency dependent damping behavior of many viscoelastic materials
and the nonlinear oscillation of earthquake. They are also used in modeling of
many chemical processes, mathematical biology and many other problems in
50
V.B.L. Chaurasia et al.
physics and engineering. Consequently, considerable attention has been given
to the solutions of fractional differential equations of physical interest. Many
authors including Podlubny [21], Hilfer [11], Kilbas et al. [14], Kiryakova [15],
Beyer and Kempfle [2], Schneider and Wyss [24], Mainardi [17], Huang and Liu
[12, 13] Chaurasia and Kumar [4] and Chaurasia and Singh [5] discussed some
examples of homogeneous fractional differential equations, homogeneous diffusion and wave equations. In recent work Debnath and Bhatta [6] solve some
linear inhomogeneous fractional partial differential equations in fluid mechanics. The solutions are obtained with the help of the joint Laplace and Fourier
transform combined with Mittag-Leffler function. The Navier-Stokes equation
is the primary equation of computational fluid dynamics, relating pressure
and external forces acting on a fluid to the response of the fluid flow. The
Navier-Stokes and continuity equations are given by:
1
∂u
+ (u. ∇) u = − ∇p + ν ∇2 u,
∂t
ρ
(1)
∇. u = 0,
(2)
where ρ is the density, p is the pressure, ν is the kinematics viscosity, u is the
velocity and t is the time.
In recent papers El-Shahed and Salem [8] and Odibat and Momani [19] have
generalized the classical Navier-Stokes equations by replacing the first time
derivative by a fractional derivative of order α, 0 < α < 1. They use Laplace,
Fourier sine and finite Hankel transforms to obtain exact solution for the timefractional Navier-Stokes equations.
In this paper, we derive an analytical solution for the time fractional NavierStokes equation in a circular cylinder, where the first time derivative in the classical Navier-Stokes equation is replaced by the generalized Riemann-Liouville
fractional derivative of order 0 < α < 1 and type 0 ≤ β ≤ 1. The solutions are
obtained by the application of Laplace and finite Hankel transforms in terms
of Bessel and Mittag-Leffler functions.
2
Mathematics Prerequisites
The right-sided Riemann-Liouville fractional integral of order α is defined by
Miller and Ross [18, p.45], Samko et al. [22]:
RL −α
a Dt
where R(α) > 0.
1 Zt
(t − τ )α−1 f(τ ) dτ, (t > a)
f(t) =
Γ(α) a
(3)
51
Solution of the Time-Fractional...
The right-sided Riemann-Liouville fractional derivative of order α is defined
as
RL α
a Dt
d
dt
f(t) =
!n
f(t))
(In−α
a
(Re(α) > 0, n = [Re(α)] + 1),
(4)
where [α] represents the integral part of the number α.
The following fractional derivative of order α > 0 is introduced by Caputo
[3] in the form (if m − 1 < α ≤ m, Re (α) > 0, m ∈ N):
c α
0 Dt
Z t
f (m) (τ ) dτ
1
Γ(m − α) 0 (t − τ )α+1−m
f(t) =
=
dm f(t)
, if α = m
dtm
(5)
m
where ddtf(t)
is the m-th derivative of order m of the function f(t) with respect
m
to t. The Laplace transform of this derivative given in [21] in the form
L {c0 Dαt f(t) ; s} = sα f̄(s) −
m−1
X
sα−r−1 f (r) ( 0 + ), (m − 1 < α ≤ m) . (6)
r=0
A generalization of the Riemann-Liouville fractional derivative operator (4)
and Caputo fractional derivative operator (5) is given by Hilfer [11], by introducing a right-sided fractional derivative operator of two parameters of order
0 < α < 1 and 0 ≤ β ≤ 1 in the form
α,β
0 Da+
f(t) =
β(1−α)
Ia+
!
d (1−β) (1−α)
Ia+
f(t) ,
dt
(7)
It is interesting to observe that for β = 0, (7) reduces to the classical RiemannLiouville fractional derivative operator (4). On the other hand, for β = 1 it
yields the Caputo fractional derivative operator defined by (5). The Laplace
transform formula for this operator is given by Hilfer [11]
(1−β) (1−α)
α
β(α−1)
L {Dα,β
I0+
0+ f(t) ; s} = s f̄(s) − s
(1−β) (1−α)
f(0 + ), (0 < α < 1), (8)
where the initial value term I0+
f(0 + ), involves the Riemann-Liouville
fractional integral operator of order (1−β) (1−α) evaluated in the limit as t
→ 0+.
If f(r) satisfies Dirchlet conditions in the interval 0 < r < R and its finite
Hankel transform is defined by [1]
52
V.B.L. Chaurasia et al.
∗
H0 (f(r)) = f (ξn ) =
Z
R
r f(r) J0 (r ξn ) dr,
(9)
0
where ξn are the roots of the equation J0 (r) = 0, then at each point of the
interval at which f(r) is continuous
∞
2 X
J0 (ξn r)
f(r) = 2
,
f ∗ (ξn ) 2
R n=1
J1 (ξn R)
(10)
where the sum is taken over all positive roots of J0 (r) = 0, J0 and J1 are
Bessel functions of first kind. In application of the finite Hankel transforms to
ordinary or partial differential equations, it is useful to have the formula [1]:
H0
d2 f
1 df
2 +
r dr
dr
!
2
= − ξn f ∗ (r) + R ξn f(R) J1 (ξn R).
(11)
The Bessel function of the first kind of order p is defined as [1]:
Jp (x) =
∞
X
n=0
3
(−1)n
n ! Γ(n + p + 1)
2n+p
x
2
, | x | < ∞.
(12)
Analytical Solution for the Time-Fractional
Navier-Stokes Equation
Let us consider the unsteady flow of a viscous fluid past a circular cylinder of
radius R and length L. We refer all motion to a set of cylindrical polar coordinates (r,θ,z) where z-axis coincides with the axis of the cylinder. Assuming
that ur =uθ = 0 and uz = u(r,t), the momentum equation in cylindrical coordinates reduces to:
∂u
∂p
ρ
= −
+µ
∂t
∂z
1 ∂u
∂ 2u
+
2
∂r
r ∂r
!
,
(13)
where ρ is fluid density, µ is fluid kinematic viscosity , u(r,t) is the velocity in
the axial direction and ∂p
is a constant.
∂z
The initial and boundary conditions are:
u (r, 0) = f (r),
u (r, t) = 0, at r = R,
53
Solution of the Time-Fractional...
u (0, t) is f inite, at r = 0.
(14)
If the generalized Riemann-Liouville fractional derivative model is used to
present the time derivative term, the equation (13) assumes the form
α,β
ρ0 Dt u = P + µ
∂ 2u
∂ r2
where P = − ∂p
.
∂z
(15)
The initial and boundary conditions are
+ 1
r
∂u
,
∂r
(1−β)(1−α)
I0+
u(r, 0 + ) = f(r),
u (r, t) = 0, at r = R,
u (0, t) is f inite, at r = 0.
(16)
Applying the finite Hankel transform to equation (15) with respect to the
variable r, using (11) and boundary conditions (16), we get
α,β
ρ 0 Dt u∗ (ξn , t) =
P
R J1 (ξn R) − µ ξn2 u∗ (ξn , t).
ξn
(17)
If we apply the Laplace transform to equation (17) with respect to the variable
t and use the initial conditions (16), it yields
ρ sα ū∗ (ξn , s) − ρ sβ(α−1) f ∗ (ξn ) =
P
R J1 (ξn R) − µ ξn2 ū∗ (ξn , s).
ξn s
(18)
Solving for ū∗ (ξn , s),it gives
ū∗ (ξn , s) =
P R J1 (ξn R)
s−1
f ∗ (ξn )sβ(α−1)
+ .
2
2
ρ ξn
sα + µρξn
sα + µξρn
(19)
On taking the inverse Laplace transform of (19) and applying the formula
(
−1
L
sβ−1
sα + a
)
= tα−β Eα,α−β+1 (−atα ),
it is seen that
PR J1 (ξn R) α
−µξn2 α
u (ξn , t) =
t Eα,α+1
t
ρ ξn
ρ
∗
!
54
V.B.L. Chaurasia et al.
−µξn2 α
t
ρ
+ f ∗ (ξn )tα−β(α−1)−1 Eα,α−β(α−1)
!
.
(20)
Inverting the finite Hankel, using (10) leads
to the
exact solution
2
J0 (ξn r)
−µξn
2P tα P∞
α
u(r, t) = ρR
t
n=1 ξn J1 (ξn R) Eα,α+1
ρ
∞
2tα−β(α−1)−1 X
J0 (ξ r)
Eα,α−β(α−1)
+
f ∗ (ξn ) 2
2
R
J1 (ξn R)
n=1
−µξn2 α
t .
ρ
!
(21)
After some computations, the solution of the time-fractional Navier-Stokes
equation (21) gets simplified into
u(r, t) =
∞
∞
2P tα X
2tα−β(α−1)−1 X
(−1)n
(−1)n
2n
2n
g
(t)
r
+
n
2 hn (t) r ,
2n
ρR n=0 22n (n !)2
R2
n=0 2 (n !)
(22)
where
∞
X
gn (t) =
(ξk )2n−1
k=1
Eα,α+1
2
−µξn
tα
ρ
J1 (ξk R)
,
(23)
and
hn (t) =
∞
X
(ξk )2n f ∗ (ξk )
Eα,α−β(α−1)
k=1
−µξk2 α
t
ρ
J21 (ξk R)
.
(24)
In case of α = 1, β = 1 and using the relations
E1,1 (z) = ez ,
(25)
ez − 1
,
z
the solution of the Navier-Stokes equation (13) assumes the form
E1,2 (z) =
∞
−
2P X
J0 (ξn r)
u(r, t) =
1− e
3
µR n=1 ξn J1 (ξn R)
"
∞
2 X
J0 (ξn r) −
+ 2
f ∗ (ξn ) 2
e
R n=1
J1 (ξn R)
2
µξn
t
ρ
(26)
2
µξn
t
ρ
#
.
(27)
In particular, if we set β = 1, then we arrive at the result recently obtained
by Odibat and Momani [19].
55
Solution of the Time-Fractional...
4
Applications
In this section, we present three special cases to demonstrate the behavior of
the solution of the time-fractional Navier-Stokes equation. The solutions are
obtained in terms of Bessel and Mittag-Leffler functions.
Example 1. Consider the following time fractional Navier-Stokes equation
α,β
0 Dt
u = P+µ
∂ 2u
1 ∂u
+
2
∂r
r ∂r
!
,
(28)
subject to the initial and boundary conditions
(1−β)(1−α)
I0+
u(r, 0+) = 0,
u (r, t) = 0, at r = 1,
u (0, t) is f inite, at r = 0.
(29)
In view of (21), the exact solution of equation (28) is given by
u(r, t) = 2P tα
∞
X
J0 (ξn r)
Eα,α+1 −µξn2 tα .
n=1 ξn J1 (ξn )
(30)
In particular, if β = 1, then we arrive at the result given by El-Shahed and
Salem [8].
Example 2. Consider the following time fractional Navier-Stokes equation
α,β
0 Dt
u = 1+µ
∂ 2u
1 ∂u
+
∂r2
r ∂r
!
(31)
subject to the initial and boundary conditions
(1−β)(1−α)
I0+
u (r, 0 + ) = R2 − r2 ,
u (r, t) = 0, at r = R,
u (0, t) is f inite, at r = 0.
(32)
According to (10), the finite Hankel transform of the function f(r) = R2 − r2
assumes the form
f ∗ (ξn ) =
Z
0
R
r (R2 − r2 ) J0 (ξn r) dr,
56
V.B.L. Chaurasia et al.
∞
X
(−1)m ξn2m Z R 2 2m+1
=
(R r
− r2m+3 )dr,
2 2m
0
m=0 (m !) 2
∞
2R2 X
(−1)m
= 2
ξn m=0 m ! Γ(m + 3)
ξn R
2
!2m+2
,
2R2
J2 (ξn R).
ξn2
=
(33)
In view of (22), the exact solution of equation (31) is given by
u(r, t) =
∞
∞
X
(−1)n
2tα X
(−1)n
2n
2n
α−β(α−1)−1
g
(t)r
+
2
t
n
2 hn (t) r ,
2n
R n=0 22n (n !)2
n=0 2 (n !)
(34)
where
∞
X
gn (t) =
(ξk )2n−1
k=1
hn (t) =
∞
X
(ξk )2n
k=1
Eα,α+1 (−µξk2 tα )
,
J1 (ξk R)
J2 (ξk ) Eα,α−β(α−1) (−µξk2 tα )
.
ξk2
J21 (ξk R)
(35)
(36)
Example 3. Consider the following time-fractional Navier-Stokes equation
α,β
0 Dt
∂ 2u
1 ∂u
+
,
2
∂r
r ∂r
!
u = µ
(37)
subject to the initial and boundary conditions
(1−β)(1−α)
I0+
u (r, 0 + ) = J0 (ξ1 r),
u (r, t) = 0, at r = 1,
u (0, t) is f inite, at r = 0.
(38)
According to (10), the finite Hankel transform of the function f(r) = J0 (ξ1 r)
assumes the form
f ∗ (ξn ) =
Z
0
1
r J0 (ξ1 r) J0 (ξn r) dr,
57
Solution of the Time-Fractional...
(
=
1
2
J21 (ξ1 r), n = 1
0
n > 1,
(39)
from the orthogonality property of Bessel functions. In view of (22), the exact
solution of equation (37)is given by
u(r, t) = tα−β(α−1)−1
∞
X
(−1)n
2n
2 hn (t) r ,
2n
n=0 2 (n !)
(40)
where
hn (t) = (ξ1 )2n Eα,α−β(α−1) −µξ12 tα .
(41)
Therefore, the exact solution in a closed form is
u(r, t) = tα−β(α−1)−1 J0 (ξ1 r) Eα,α−β(α−1) −µξ12 tα .
5
(42)
Conclusion
In this paper, we have presented a solution of a time-fractional Navier-Stokes
equation. The solution has been developed in terms of the Mittag-Leffler
and Bessel functions in a compact and elegant form with the help of Laplace
transform and finite finite Hankel transform. Most of the results obtained
are in a form suitable for numerical computation. The time-fractional NavierStokes equation discussed in this article, contains a number of known (may
be new also) time-fractional Navier-Stokes equations. The result obtained in
the present paper provides an extension of the results given by El-Shahed and
Salem [8] and Odibat and Momani [19].
Acknowledgements
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valuable suggestions in the preparation of
this paper.
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