Applied Mathematics E-Notes, 12(2012), 129-135 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Travelling Wave Solutions Of Burgers’Equation For
Gee-Lyon Fluid Flows
Dongming Weiy, Ken Holladayz
Received 26 August 2011
Abstract
In this work we present some analytic and semi-analytic traveling wave solutions of a generalized Burger’ equation for isothermal unidirectional ‡ow of
viscous non-Newtonian ‡uids obeying the Gee-Lyon nonlinear rheological equation. The solutions include the corresponding well-known traveling wave solution
of the Burgers’ equation for Newtonian ‡ow as a special case. We also derive
estimates of shock thickness for the non-Newtonian ‡ows.
1
Introduction
In this work we derive a traveling wave solution to the following generalized Burgers’
equation
@u
@u
@
@u
1
+u
=
(1)
0
@t
@x
@x
@x
where
(t) = (1 + ct2 )t, 0 < c < 1. The solution can be written as the following:
=
0
where
2
(u2
u1 )
ln
u2 u
u u1
4
(u2
u1 )
extra b;
2u
(u2 + u1 )
u2 u1
1
)
arctan( p
1 i sinh(2b)
A
extra(b; ) = Re @ p
1 i sinh(2b)
0
(2)
(3)
in which the constant b is de…ned by sinh(2b) = pc (u82 u1 ) . It is well-known that for
c = 0, equation (1) is the classical Burgers’equation for Newtonian ‡uid ‡ows and the
traveling wave solution is
=
0
2
(u2
u1 )
ln(
u2 u
)
u u1
Mathematics Subject Classi…cations: 35L67, 76A10, 35Q53.
of Mathematics, University of New Orleans, New Orleans, USA
z Department of Computer Science, University of New Orleans, New Orleans, USA
y Department
129
130
Traveling Solutions of Burgers’Equation
satisfying the upstream and downstream boundary conditions
lim u( ) = u1 ;
lim u( ) = u2 ;
!+1
! 1
lim
j j!+1
du
( )=0
d
2
.
with = x
t;
= u1 +u
2
It is interesting to note the if the second term in our solution (2) is dropped, the
…rst term coincides with the classical solution. So the solution to the Non-Newtonian
‡ow equals the solution to the Newtonian ‡ow plus an extra term “extra(b; )”. We
also show that using the …rst order approximation, the thickness of the transition
8 0
layer between upstream and downstream can be given by = (u u ) 1+c[
2
2
1 f
8 (u2 u1 )] g
8 0
which for c = 0 gives the corresponding classical estimate = (u2 u1 ) for Newtonian
‡uid ‡ows. Similar results for power-law ‡ows have been established in [13]. Although
the pro…les of the transition layer for both power-law ‡ows and Gee-Lyon ‡ows look
similar, the mathematical solutions describing these pro…les are quite di¤erent.
2
The Generalized Burgers’Equation
The general Navier-Stokes equation for incompressible viscous ‡ows is given by
Du
= div( )
Dt
5p + g
(4)
where u = (u1 ; u2 ; u3 ) is the ‡uid velocity,
0
=@
11
12
21
22
31
32
13
1
23 A ;
33
0
d11
and Du = @d21
d31
d12
d22
d32
1
d13
d23 A
d33
are the stress tensor and the strain tensor, is the density, g the external force, p the
@u
@ui
+ @xji ), 1 i; j 3. For unidirectional ‡ows,we assume
scalar pressure, and dij = 21 ( @x
j
@p
that u = (u1 ; 0; 0), ij = 0 for i 6= 1 or j 6= 1, g = (g1 ; 0; 0), and 5p = ( @x
; 0; 0). The
1
Navier-Stokes equation (4), in this case, takes the following simple scalar form
Du1
d 11
=
Dt
dx1
@p
+ g1
@x1
(5)
@u1
@u1
1
where Du
and Du are frequently
Dt = @t + u1 @x1 . Rheological relationships between
used to determine the type of ‡uids. Polyethylene and polystyrene melts can be described approximately by a rheological equation proposed by Rabinowitch and later
generalized by Gee and Lyon [12], taking into account that the viscosity of these ‡uids
depends highly on the temperature and the high stress levels. The rheological equation
proposed by Gee and Lyon is given by
0 dij
=(
ij
+ cj
kl lk j
n
2
)tij , 1
i; j
3
(6)
D. Wei and K. Holladay
131
1, if i = j
, see [2], [10] or [12]. The
0, if i 6= j
E
temperature dependence of the viscosity is expressed by 0 = Ae RT . In this work, we
refer to ‡uid ‡ow satisfying the rheological equation (6) as Gee-Lyon ‡ows.
where
0,
n, and c are constants,
ij
=
If c = 0, then the ‡uid is said to be a Newtonian ‡uid; it is non-Newtonian if c 6= 0.
For many important industrial polymer ‡uids, the values of A, E, R, c and n have
been experimentally determined. For unidirectional ‡ows, the rheological equation (6)
n
reduces to 0 d11 = (1 + c j 11 j 11 . Let u1 , x1 , g1 be denoted by u, x, g respectively.
@p
+ g = 0, we have the generalized Burgers’equation
Then from (5), let @x
@u
@u
+u
@t
@x
=
@
@x
1
0
@u
@x
(7)
n
where (t) = (1 + c jtj )t, 0 < c < 1. Equation (7) is referred to as the generalized
Burgers’ equation for Gee-Lyon ‡ows. For c = 0, = 0 , (7) reduces to Burgers’
equation for Newtonian ‡ows
@u
@2u
@u
+u
=
.
@t
@x
@x2
(8)
It is well known that if we impose lim u( ) = u1 , lim u( ) = u2 ,
!+1
! 1
0, and u1 < u2 , (8) has the celebrated traveling wave solution
which is equivalent to
u( ) =
where
=x
u1 + u2 exp[
1 + exp[
2
2
(u2
(u2
0
=
lim du ( ) =
j j!+1 d
u2 u
2
(u2 u1 ) ln u u1 ,
u1 )]
u1 )]
(9)
t, u1 and u2 are the downstream and upstream ‡uid velocities.
It can be shown that there exists a thin transition layer of thickness of order
can be referred to as the shock thickness, which tends
u2 u1 for (9). This thickness
to zero as ! 0, and for …xed , ! 1, as (u2 u1 ) ! 0. See, for example, [7] or
[10] for a derivation of (9) and analysis of (8). In this work, we …nd analytic and semianalytic solutions to (7) for c 6= 0, and n = 2, and we derive the corresponding order
of thickness for the transition layers in non-isothermal ‡ow of viscous non-Newtonian
‡uids. Applications of these types of ‡ows are abundant in studying ‡ows in drilling
‡uids, food, oil, polymers, etc; see e.g. [1], [2], and [11]. There are numerous papers
devoted to the study of equation (5) in the literature on shock formation and traveling
waves in Newtonian ‡ows dating back to the original papers of Burgers, Cole, and
Hopf, see [3], [5] and [8]. A generalized Burgers’ equation for non-Newtonian ‡ows
based on the Maxwell model has recently been studied in [4]. We have not found any
paper which deals with Burgers’equation (7) for c 6= 0, and n = 2.
8
132
3
Traveling Solutions of Burgers’Equation
The Integral Equation for the Traveling Waves and
the Solution
du d
du
Let u(x; t) = u( ), with = x
t. Then @u
@t = d dt =
d and
du
@u
du
@u
and
=
into
equation
(7),
we
get
Substituting @t =
d
@x
d
du
du
1 d
+ u( )
=
d
d
d
1
0
du
d
@u
@x
=
.
du d
d dx
=
du
d .
(10)
Therefore
d
d
1 2
u
2
1
u
1
0
du
d
= 0;
which gives
1 2
u
2
u
1
1
0
du
d
=A
(11)
where A is an arbitrary integration constant. Applying the downstream and upstream
boundary conditions: lim u( ) = u1 , lim u( ) = u2 , and lim du
d ( ) = 0 to
!+1
equation (11), we get
1
0
du
d
! 1
=
2
(u2
2 u
j j!+1
2A) =
2
(u
u1 )(u
u1 )
(12)
where = 21 (u1 + u2 ) and A = 12 u1 u2 , u1 and u2 are the given constants. We have
du
u1 )(u u1 )), which gives
0 d = ( 2 (u
=
0
Z
( 2 (u
du
u1 )(u
u1 ))
.
(13)
Without loss of generality, in the following, we assume that u1 < u < u2 . For c = 0,
(13) gives
Z
1
u u1
du
1
=
=
ln
2
(u u1 )(u u1 )
u1 u2
u2 u
where
=
0
, which gives the classical traveling wave solution
u( ) =
u1 + u2 exp[
1 + exp[
2
2
(u2
(u2
u1 )]
u1 )]
to Burgers’equation for Newtonian ‡ows.
In the following, we are interested in …nding solutions to (13) for c 6= 0 and n = 2.
1
Let u = u2 2 u1 + u2 +u
. Then
2
2
(u
u1 )(u
u2 ) =
u1 )2
(u2
8
1+
c 2 (u2 u1 )4
[( + 1)(
26
2
1)]
( +1)(
1)
D. Wei and K. Holladay
133
and (13) becomes
=
0
8
(u2
u1
)2
Z
n
1+
d
c
2 (u
2
u1 ) 4
26
Let the constant b be de…ned by sinh 2b =
2
t
)t.
sinh2 (2b)
2
[( + 1)(
p
1)]
8
,
c (u2 u1 )2
o
.
( + 1)(
and de…ne
1)
(t; b) = (1 +
We have the decomposition
1
(( + 1)(
1
( + 1)(
=
1); b)
(
1)
1
i sinh(b))(
cosh(b)
cosh(b) + i sinh(b))
1
:
i sinh(b))( + cosh(b) + i sinh(b))
2
( + cosh(b)
By using Mathematica, we …nd that
Z
d
(( + 1)(
1); b)
Let
=
ln(
0
1
)+
+1
1
p
p
arctan
arctan
1 i sinh(2b)
1+i sinh(2b) C
1B
B p
C
p
+
@
2
1 i sinh(2b)
1 + i sinh(2b) A
0
1
)
arctan( p
1 i sinh(2b)
A.
extra(b; ) = Re @ p
1 i sinh(2b)
Then we have
=
0
8
(u2
h
u1
)2
Therefore
= (u2 8 u1 )2 21 ln( uu2 uu1 )
0
solution of (1) is implicitly de…ned by
=
0
4
(u2
u1
)2
ln(
u2 u
)
u u1
1
1
ln(
)
2
+1
0 extra(b;
) .
i
2 +u1 )
extra(b; 2u u(u
)
and the traveling wave
2 u1
8
(u2
u1
)2
extra(b;
2u
(u2 + u1 )
).
u2 u1
We have omitted the integration constants in the above solutions. For simplicity, we
plot the pro…le of the transition layer of u = u( ) and provide the following graphic
representation of the pro…les of the transition layers. The blue curves correspond to
b = 0:5, 0:35, and 0:25 respectively and the red curve represents the classical solution
corresponding to b = 0:0.
134
Traveling Solutions of Burgers’Equation
0 .5
1 .5
1 .0
0 .5
0 .5
1 .0
1 .5
0 .5
4
The Order of Thickness of the Transition Layers
The transition layer thickness or the shock thickness can be estimated by using the …rst
1
2
order derivative du
. From du
u1 )(u u2 ) and u(0) = u1 +u
, we
d
d =
2 (u
2
0
=0
get
du
d
=0
1
=
2 (u2
0
2
u1 ) . Let
denote the thickness of the transition layer,
using the Taylor expansion, we have
u2
u1 = u(
2
)
u( ) =
2
du
d
+ O( 2 ).
=0
Therefore we have
=
u2
u1
du
d (0)
=
0 (u2
( 8 (u2
u1 )
=
u1 )2 )
(u2
8 0
n
u1 ) 1 + c 8 (u2
u1 )
2
o,
Which is the …rst order approximation of the thickness of the transition layer for powerlaw ‡ows. This estimate, for c = 0, gives the well-known estimate = (u82 0u1 ) for the
thickness of the transition layer of Newtonian ‡ows.
5
Conclusion
In this work, we consider a generalized Burgers’equation for Gee-Lyon ‡uid ‡ows, and
derive a new general traveling wave solution of this equation. As special cases of this
solution, we show several analytic solutions and pro…les of the thickness of the transition
layer of the solution. We de…ned a …rst order approximation of the thickness of the
transition layer or the thickness of the shock which generalized the known estimate for
the shock thickness of the corresponding Burgers’solution for Newtonian ‡ows.
D. Wei and K. Holladay
135
References
[1] A. Ishak and N. Bachok, Power-law ‡uid ‡ow on a moving wall, European Journal
of Scienti…c Research, ISSN 1450-216X, 34(1)(2009), 55–60.
[2] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids,
Vol. 1-2, Wiley, New York, 2nd ed. 1987.
[3] J. M. Burgers, A mathematical model illustrating the theory of turbulence. edited
by Richard von Mises and Theodore von Kármám, Advances in Applied Mechanics,
pp. 171–199. Academic Press, Inc., New York, N. Y., 1948.
[4] V. Camacho, R. D. Guy and J. Jacobsen, Traveling waves and shocks in a viscoelastic generalization of Burgers’equation, SIAM J. Appl. Math., 68(5)(2008),
1316–1332.
[5] J. D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quart.
Appl. Math., 9(3)(1951), 225–236.
[6] L. Debnath, Nonlinear Partial Di¤erential Equations for Scientists and Engineers,
2nd ed., Birkhausser, Boston, 2005.
[7] M. Guedda and Z. Hommouch, Similarity ‡ow solutions of a non-Newtonian powerlaw ‡uid, International Journal of Nonlinear Science, 6(3)(2008), 255–264.
[8] E. Hopf, The partial di¤erential equation ut +uux = uxx , Comm. Pure and Appl.
Math., 3(1950), 201–230.
[9] M. U. Tyn and L. Debnath, Partial Di¤erential Equations for Scientists and Engineers, 3rd ed., P T R Prentice-Hall, Upper Saddle River, New Jersey, 1987.
[10] W. L. Wilkinson, Non-Newtonian Fluids, Pergamon Press, New York, 1960.
[11] M. V. Ochoa, Analysis of Drilling Fluid Rheology and Tool Joint E¤ect to Reduce
Errors in Hydraulics Calculations, Ph.D. Dissertation, Texas A and M University,
August 2006.
[12] R. E. Gee and J. B. Lyon, Nonisothermal ‡ow of viscous non-Newtonian ‡uids,
Industrial and Engineering Chemistry, 49(6)(1957), 956–960.
[13] D. Wei and H. Borden, Traveling wave solutions of Burgers’equation for power-law
non-Newtonian ‡ows, AMEN, 11(2011), 133-138.