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Solution of the Time-Fractional Navier-Stokes Equation
V.B.L. Chaurasia1 and Devendra Kumar2
1Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, India E-mail: [email protected]
2Department of Mathematics, Jagan Nath Gupta Institute of Engineering and Technology, Jaipur, Rajasthan, India
E-mail: [email protected] (Received:12-4-11 /Accepted:25-6-11)
Abstract
In this paper we obtain the solution of a time-fractional Navier-Stokes equa- tion. 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 func- tion, 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 de- scribed 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
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 diffu- sion and wave equations. In recent work Debnath and Bhatta [6] solve some linear inhomogeneous fractional partial differential equations in fluid mechan- ics. 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:
∂u
∂ t+ (u.∇) u = −1
ρ∇p + ν∇2u, (1)
∇.u = 0, (2)
whereρ 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 time- fractional Navier-Stokes equations.
In this paper, we derive an analytical solution for the time fractional Navier- Stokes equation in a circular cylinder, where the first time derivative in the clas- sical 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
a D−αt f(t) = 1 Γ(α)
Z t
a
(t−τ)α−1f(τ) dτ, (t > a) (3) where R(α) >0.
The right-sided Riemann-Liouville fractional derivative of order α is defined as
RL
a Dαt f(t) = d dt
!n
(In−αa f(t)) (Re(α) > 0, n = [Re(α)] + 1), (4) where [α] represents the integral part of the number α.
The following fractional derivative of orderα > 0 is introduced by Caputo [3] in the form (if m −1 < α ≤m, Re (α) > 0, m ∈N):
c
0Dαt f(t) = 1 Γ(m−α)
Z t
0
f(m)(τ) dτ (t−τ)α+1−m
= dmf(t)
dtm , if α = m (5)
where dmdtf(t)m is the m-th derivative of order m of the function f(t) with respect to t. The Laplace transform of this derivative given in [21] in the form
L{c0Dαt f(t) ; s} = sα¯f(s) −
m−1
X
r=0
sα−r−1f(r)( 0 + ),(m−1 < α ≤ m). (6) A generalization of the Riemann-Liouville fractional derivative operator (4) and Caputo fractional derivative operator (5) is given by Hilfer [11], by intro- ducing a right-sided fractional derivative operator of two parameters of order 0< α < 1 and 0 ≤ β ≤ 1 in the form
0Dα,βa+ f(t) = Iβ(1−α)a+ d dt
I(1−β) (1−α)
a+ f(t)
!
, (7)
It is interesting to observe that forβ= 0, (7) reduces to the classical Riemann- Liouville 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]
L{Dα,β0+ f(t) ; s} = sα¯f(s) − sβ(α−1)I(1−β) (1−α)
0+ f(0 + ), (0 < α < 1), (8) where the initial value term I(1−β) (1−α)
0+ 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]
H0(f(r)) = f∗(ξn) =
Z R
0
r f(r) J0(rξn) dr, (9) where ξn are the roots of the equation J0(r) = 0, then at each point of the interval at which f(r) is continuous
f(r) = 2 R2
∞
X
n=1
f∗(ξn) J0(ξnr)
J21(ξnR), (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 d2f dr2 + 1
r df dr
!
= −ξn2f∗(r) + Rξnf(R) J1(ξnR). (11) The Bessel function of the first kind of order p is defined as [1]:
Jp(x) =
∞
X
n=0
(−1)n n ! Γ(n + p + 1)
x 2
2n+p
, |x| < ∞. (12)
3 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 coor- dinates (r,θ,z) where z-axis coincides with the axis of the cylinder. Assuming thatur =uθ = 0 anduz = u(r,t), the momentum equation in cylindrical coor- dinates reduces to:
ρ ∂u
∂ t = −∂p
∂z + µ ∂2u
∂r2 + 1 r
∂u
∂r
!
, (13)
whereρ is fluid density, µis fluid kinematic viscosity , u(r,t) is the velocity in the axial direction and ∂p∂z is a constant.
The initial and boundary conditions are:
u(r,0) = f(r),
u(r, t) = 0, at r=R,
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
ρ0Dα,βt u = P + µ ∂2u
∂ r2 + 1r∂u∂r,(15)where P = − ∂p∂z.
The initial and boundary conditions are
I(1−β)(1−α)0+ 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
ρ0Dα,βt u∗(ξn,t) = P
ξnR J1(ξnR) − µ ξn2u∗(ξn,t). (17) If we apply the Laplace transform to equation (17) with respect to the variable t and use the initial conditions (16), it yields
ρsαu¯∗(ξn,s) − ρsβ(α−1)f∗(ξn) = P
ξnsR J1(ξnR)− µ ξn2u¯∗(ξn,s). (18) Solving for ¯u∗(ξn,s),it gives
¯
u∗(ξn,s) = P R J1(ξnR) ρ ξn
s−1
sα+ µ ξρ2n + f∗(ξn)sβ(α−1)
sα+ µξρ2n . (19) On taking the inverse Laplace transform of (19) and applying the formula
L−1
( sβ−1 sα+a
)
=tα−βEα,α−β+1(−atα), it is seen that
u∗(ξn,t) = PR J1(ξnR)
ρ ξn tαEα,α+1 −µξn2 ρ tα
!
+ f∗(ξn)tα−β(α−1)−1Eα,α−β(α−1)
−µξn2 ρ tα
!
. (20)
Inverting the finite Hankel, using (10) leads to the exact solution u(r,t) = 2P tρRα P∞n=1 ξJ0(ξnr)
nJ1(ξnR)Eα,α+1−µξρn2tα +2tα−β(α−1)−1
R2
∞
X
n=1
f∗(ξn) J0(ξr)
J21(ξnR)Eα,α−β(α−1) −µξn2 ρ tα
!
. (21)
After some computations, the solution of the time-fractional Navier-Stokes equation (21) gets simplified into
u(r,t) = 2P tα ρR
∞
X
n=0
(−1)n
22n(n !)2 gn(t) r2n+ 2tα−β(α−1)−1 R2
∞
X
n=0
(−1)n
22n(n !)2 hn(t) r2n, (22) where
gn(t) =
∞
X
k=1
(ξk)2n−1 Eα,α+1
−µξ2 n
ρ tα
J1(ξkR) , (23)
and
hn(t) =
∞
X
k=1
(ξk)2nf∗(ξk)
Eα,α−β(α−1)
−µξ2k ρ tα
J21(ξkR) . (24) In case of α = 1, β = 1 and using the relations
E1,1(z) = ez, (25)
E1,2(z) = ez−1
z , (26)
the solution of the Navier-Stokes equation (13) assumes the form u(r,t) = 2P
µR
∞
X
n=1
J0(ξnr) ξn3J1(ξnR)
"
1− e−
µξ2 n ρ t #
+ 2 R2
∞
X
n=1
f∗(ξn) J0(ξnr) J21(ξnR)e−
µξ2 n ρ t
. (27)
In particular, if we set β = 1, then we arrive at the result recently obtained by Odibat and Momani [19].
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
0Dα,βt u = P +µ ∂2u
∂r2 + 1 r
∂u
∂r
!
, (28)
subject to the initial and boundary conditions I(1−β)(1−α)0+ 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
n=1
J0(ξnr)
ξnJ1(ξn)Eα,α+1
−µξ2ntα. (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
0Dα,βt u = 1 +µ ∂2u
∂r2 + 1 r
∂u
∂r
!
(31) subject to the initial and boundary conditions
I(1−β)(1−α)0+ 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 R
0
r (R2−r2) J0(ξnr) dr,
=
∞
X
m=0
(−1)m (m !)2
ξn2m 22m
Z R
0
(R2r2m+1− r2m+3)dr,
= 2R2 ξn2
∞
X
m=0
(−1)m m ! Γ(m + 3)
ξnR 2
!2m+2
,
= 2R2
ξn2 J2(ξnR). (33)
In view of (22), the exact solution of equation (31) is given by
u(r,t) = 2tα R
∞
X
n=0
(−1)n
22n(n !)2 gn(t)r2n+ 2tα−β(α−1)−1
∞
X
n=0
(−1)n
22n(n !)2 hn(t) r2n, (34) where
gn(t) =
∞
X
k=1
(ξk)2n−1 Eα,α+1(−µξk2tα)
J1(ξkR) , (35)
hn(t) =
∞
X
k=1
(ξk)2n J2(ξk) ξk2
Eα,α−β(α−1)(−µξk2tα)
J21(ξkR) . (36) Example 3. Consider the following time-fractional Navier-Stokes equation
0Dα,βt u = µ ∂2u
∂r2 + 1 r
∂u
∂r
!
, (37)
subject to the initial and boundary conditions
I(1−β)(1−α)0+ u (r,0 + ) = J0(ξ1r),
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(ξ1r) assumes the form
f∗(ξn) =
Z 1
0
r J0(ξ1r) J0(ξnr) dr,
=
( 1
2J21(ξ1r), 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
n=0
(−1)n
22n(n !)2 hn(t) r2n, (40) where
hn(t) = (ξ1)2nEα,α−β(α−1)
−µξ12tα. (41) Therefore, the exact solution in a closed form is
u(r,t) = tα−β(α−1)−1J0(ξ1r) Eα,α−β(α−1)−µξ12tα. (42)
5 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 Navier- Stokes 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 Victo- ria, Canada for his kind help and valuable suggestions in the preparation of this paper.
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