DOI : 10.17577/IJERTV3IS10406
- Open Access
- Total Downloads : 563
- Authors : Vatsala Mathur, Kavita Khandelwal
- Paper ID : IJERTV3IS10406
- Volume & Issue : Volume 03, Issue 01 (January 2014)
- Published (First Online): 16-01-2014
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Exact Solution for the Flow of Oldroyd-B Fluid Between Coaxial Cylinders
Vatsala Mathur, Kavita Khandelwal
Department of Mathematics
Malaviya National Institute of Technology, Jaipur-302017
Abstract
This paper deals with unsteady unidirectional transient
Newtons law of viscosity ( dv
dy
) , are described
flow of Oldroyd-B fluid between two infinitely long co- axial circular cylinders. At time t=0+, the motion is produced by a constant pressure gradient & the inner cylinder start moving along its axis of symmetry with the constant velocity. The velocity field of the flow of fluid with fractional derivative is obtained by using Hankel and Laplace transforms. The obtained result is presented in terms of the generalized G-functions. Key difference this paper brings from previous work is that in this case inner cylinder is moving with constant velocity. The influence of different values of
as non- Newtonian fluids. Examples of such fluids are blood, saliva, semen, lava, gums, slurries, emulsions, synovial fluid, butter, cheese, jam, ketchup, soup, mayonnaise etc. These fluids have complex molecular structure with non-linear viscoelastic behavior. To study non- Newtonian fluids, many models have been used. Out of these, differential type [1] and rate type [2] have received most of the attention.
Tong [3] used constitutive relation for the flow of non-Newtonian fluid with fractional derivative in an annular pipe as follows
parameters, constants and fractional coefficients on the
(1 ) (1 ) v(r,t),
(1)
velocity field is also analyzed using graphical illustration.
t r t r
where is tangential tension, is the viscosity, v is
Keywords – Coaxial cylinders; Fractional calculus;
the velocity, and r
times respectively.
are relaxation and retardation
Hankel transform; Laplace transform; Oldroyd-B fluids; Velocity field.
1. Introduction
t r t r
Fluids are generally classified based on their
The starting point of the fractional derivative model of non-Newtonian fluid is usually a classical differential equation being modified by replacing the time derivative of an integer order by the so-called RiemannLiouville fractional calculus operator [4]. Using fractional approach, the constitutive relation of the generalized Oldroyd-B fluid can be written as
rheological properties. The simplest classification is Newtonian fluid. These fluids are represented using Navier-Stokes theory. The fluids which do not obey
(1 D ) (1 D ) v(r, t),
(2)
t
t
where D and D
are fractional operators and are
v 1 p
defined as [5]
1 d t f ( )
t r r z ,
(4)
(1 ) dt (t ) d ,
0 1;
where is the constant density of the fluid.
t
D f (t) 0
(3)
d
dt
f (t),
1,
Putting eq. (2) in eq. (4), we get
v
t
2 1
where (.) is the Gamma function. When 1,
1 Dt t A
A (1 )
1 r Dt r
r v(r, t),
r
(5)
eq. (2) simplified as eq. (1).
For non-Newtonian fluids, the first exact
where
is the kinematical viscosity and
solution corresponding to motions of Oldroyd-B fluids in cylindrical domains seem to be those of Waters &
A p
z
is the constant pressure gradient that acts
King [6]. Fetecau [7] worked on unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. He used the theorem of Steklov to obtain exact solutions for flows satisfying no-slip boundary conditions. The unsteady rotational flow of a generalized second grade fluid through a circular cylinder has been considered by Kamran [8]. Exact solutions for the velocity field and the shear stress corresponding to the unsteady flows of a generalized Oldroyd-B fluid in an infinite circular cylinder subject to a longitudinal time dependent shear stress have been obtained by Rubbab [9]. M. Kamran [10] concluded unsteady linearly accelerating flow of a fractional second grade fluid through a circular cylinder. Various other studies have been done recently on non- Newtonian fluids [11-17].
on the liquid in the z-direction.
3. Flow through the annular region
Let us consider an incompressible Oldroyd-B fluid in infinite coaxial circular cylinders. At time t 0, fluid
is assumed to be stationary. At time t 0 , a constant pressure gradient applied and the inner cylinder moves with constant velocity and the outer cylinder held fixed. Consider that the radius of inner and outer cylinders are R1 and R2 ( R1 ) respectively.
The initial and boundary conditions are
The aim of this paper is to study the flow of Oldroyd-B fluid with fractional derivative between two
v(r,0) 0, t v(r,0) 0,
R1 r R2 ,
(6)
coaxial cylinders. The solution is obtained by using
v(R1, t) f ,
v(R2 , t) 0, t 0,
(7)
Hankel and Laplace transform. At time t 0 a constant pressure gradient applied and the inner cylinder moves with constant velocity & the outer cylinder held fixed. The obtained result is presented in terms of the generalized G functions.
2. Governing equations
where f is constant.
Making the change of unknown function
v(r,t) V (r) u(r,t),
where
(8)
A 2 2
A (R2 R2 )
We consider the unsteady flow of an incompressible Oldroyd-B fluid in coaxial cylinders. Further, following assumptions are considered during this mathematical
V (r)
4 (R2 r
) 2 1 ln(r / R2 ).
4 ln(R2 / R1 )
(9)
study. The fluid velocity at the direction of the pipe radius is assumed to be zero. The flows are assumed to be axisymmetric. The axial velocity is assumed to be only relevant to the cylinder radius.
The equation of axial flow motion is
Putting eq. (8) in eq. (5), we obtain
u(r,t)
2 1
t
2A J
(R s )
(1 Dt ) (1 r Dt )r r u(r,t) A
u(s ,0) 0 1 n 1 , u(s ,0) 0. (16)
t r
(1 )
n s 4 J (R s ) t n
t
n 0 2 n
r A (1 ) .
Putting eq. (8) in eqs. (6) & (7), we obtain
(10)
Applying Laplace transform of eq. (15) and using eq. (16), we obtain
2A J (R s ) (1 s s2 s 1)
u(sn , s)
0 1 n 1 n r
s2 J (R s ) s2 (s s 1 s2 s2 s )
u(r,0) V (r),
tu(r,0) 0,
(11)
n 0 2 n n
2f 1
n n r
s(s s 1 s2 s2 s )
u(R1,t) f ,
u(R2 ,t) 0.
(12)
2 f
n n r
1
-
r
s1 (s s 1 s2 s2 s )
The Hankel Transform method with respect to r is used and is defined as follows
-
2Ag (sn )
s2
n n r
1
s1 (s s 1 s2 s2 s )
n
-
2r Ag(sn )
n n r
1
. (17)
R s2 s1 (s s 1 s2 s2 s )
2 n
n n r
u ru(r, s)1 (sn , r)dr.
R1
(13)
Applying Inverse-Laplace transform of eq. (17) and taking into account the following result [18]
The inverse Hankel Transform is
1 qb
(qa d )c
2 2 d j(c j)
t (c j )ab1
2 s J
(R s )u(s , s) (s , r)
;
u(r, s)
n 0 2 n n 1 n , (14)
j0 (c)( j 1) [(c j)a b]
2 n1
J 2 (R s ) J 2 (R s )
where
0 1 n
0 2 n
Re(ac b) 0,
d 1. qa
(18)
1 (sn , r) Y0 (R1sn )J0 (sn r) J0 (R1sn )Y0 (sn r), sn
We obtain
is the positive root of 1 (sn , R2 ) 0.
2A J (R s )
1
u(sn ,t)
0 1 n
1
s2 J (R s )
s2
Applying the Hankel transform in eq. (10), we obtain
n 0 2 n
n
1 s2 m 1 m m
(1)m n
k G
(1,t)
u(sn , t) 2
2
s2
k r
, k m 2,m 1
(1 Dt ) t
sn (1 r Dt )u(sn , t) f
n m 0
k 0
2 1 s2 m 1 m m
2r f 1
f (1)m n
k G
(1,t)
-
t s2
k r
, k m 2,m 1
(1 )
n m 0
k 0
2 1 s2 m 1 m m
2Ag (s )t
2 Ag (s ) t
f (1)m n
k G
(1,t)
n r n ,
(15)
r s2
k r
, k m
2,m 1
s 2 (1 ) s 2 (1 )
n m 0
k 0
n n 2Ag (s ) 1 s2 m 1 m m
n (1)m n
k G
(1,t)
where
s2 s2
k r
, k m
2,m 1
n n m 0
k 0
2 Ag (s ) 1 s2 m 1 m m
R s
r n (1)m n
k G
(1,t) . (19)
g(s ) 2 n {Y (R s )J (R s ) J (R s )Y (R s )} 1 .
s2 s2
k r
, k m
2,m 1
n 2 0 1 n 1 2 n 0 1 n 1 2 n
n n m 0 k 0
Applying the Hankel transform of eq. (11), we obtain
The expression of the velocity field can be written as
J 2(R s ) (s , r) be clearly seen that the velocity decreases, when
v(r,t) V (r) 2 0
2 n 1 n
2
kinematic viscosity increases.
n1 J0 (R1sn ) J0 (R2sn )
A J0 (R1sn )
s2 J (R s ) 1
8
=0.003
n 0 2 n
=0.005
s2 m1 m m
7 =0.007
6
1 (1)m n
kG
(1,t)
k r
, k m 2,m 1
m0 k 0 5
v1(r)
-.-.
v2(r)
—-
v3(r)
—
s2 m1 m m 4
f (1)m n
k G
(1,t)
k r
, k m 2,m 1 3
m0
k 0
s2 m1 m m 2
f (1)m n
kG
(1,t)
1
r k r , k m
2,m 1
m0 k 0
Ag (sn )
m s2
n
m1 m
0
m k 1 r
s2
(1)
k r G ,k m 2,m1( ,t)
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
n m0
Ag(s )
k 0
2 m1
m s m m k
Fig. 2: Profiles of the velocity v(r,t) given by
1 eq. (20) for R =0.3, R =0.5, f=-3, t=6s, =9, =4,
r n (1) n r G ,k m 2,m1( ,t) . (20) 1 2 r
n
s2
4. Results
m0
k 0 k
=0.3, =0.3, A=4 and different values of .
Figure 3 is showing the dependency of the relaxation time on the fluid motion. It can also be clearly seen that the velocity decreases, when
As shown in below diagrams, the velocity
v(r,t)
increases.
given by eq. (20) has been drawn against r for different values of the time t , f and some other relevant parameters.
Figure 1 is showing the time dependency on the fluid motion. It can also be clearly seen that the velocity increases, when time increases.
60
=16
=18
=20
50
40
v1(r)
-.-.
v2(r)
—-
v3(r)
—
30
20
90
t=5s
t=7s
80 t=9s 10
70
0
60 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
v1(r)
-.-.
v2(r)
—-
v3(r)
—
50
40
30
20
10
0
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
Fig. 3: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3, t=5s,=0.04,
r=7,=0.3, =0.3, A=4 and different values of
.
Figure 4 is showing the dependency of the
retardation time r on the fluid motion. It can also be
Fig. 1: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3, =0.035, =12,
r=2.2, =0.9, =0.6, A=4 and different values of t.
clearly seen that the velocity increases, when r
increases.
Figure 2 is showing the dependency of the kinematic viscosity on the fluid motion. It can also
35
r=1
=2
450
=0.3
=0.4
r
r
30 =3
400 =0.5
350
25
300
v1(r)
-.-.
v2(r)
—-
v3(r)
—
v1(r)
-.-.
v2(r)
—-
v3(r)
—
20 250
15 200
150
10
100
5
50
0
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
0
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
Fig. 4: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3,t=5s, =0.04,
=8, =0.3, =0.9, A=4 and different values of
r.
Figure 5 is showing the dependency of the fractional parameter on the fluid motion. It can also be clearly seen that the velocity increases, when increases.
Fig. 6: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3, t=6s,=0.04,
=8, r=5.5, =1, A=4 and different values of .
Figure 7 is showing the dependency of f on the fluid motion. It can also be clearly seen that the velocity decreases, when f increases.
35
60
=0.2
=0.3
=0.4
50
f = -9
f = -7
f = -5
30
25
20
v1(r)
-.-.
v2(r)
—-
v3(r)
—
40
15
v1(r)
-.-.
v2(r)
—-
v3(r)
—
30 10
20 5
0
10 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
0
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
Fig. 5: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3, t=6s, =0.045,
=25, r=8, =0.5, A=4 and different values of
.
Figure 6 is showing the dependency of the fractional parameter on the fluid motion. It can also be clearly seen that the velocity decreases, when
increases.
Fig. 7: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, t=5s, =0.045, =14,
r=2.8, =0.8, =0.5, A=4 and different values of f.
Figure 8 is showing the dependency of A on the fluid motion. It can also be clearly seen that the velocity increases, when A increases.
40
A=2
A=4
35 A=6
30
25
v1(r)
-.-.
v2(r)
—-
v3(r)
—
20
15
10
5
0
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5
r
Fig. 8: Profiles of the velocity v(r,t) given by eq. (20) for R1=0.3, R2=0.5, f=-3, t=5s,=0.04,
=11,r=2.5,=0.9, =0.6 and different values of
A.
In all of above, the root sn has been approximated
1
(2n 1)
by 2(R R ) .
5. Conclusions
The main objective of this paper is to provide exact solution for the velocity field for Oldroyd-B fluid between two coaxial circular cylinders where inner cylinder is moving with constant velocity & the outer cylinder is fixed. This solution is obtained by using Hankel transform and Laplace transform methods and the result is presented in terms of generalized G functions. This solution satisfies the governing equation and all imposed initial and boundary conditions. The velocity field is also analyzed using graphical illustration for various parameters, constants and fractional coefficients.
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