2 The Generalized Burgers’Equation

Travelling Wave Solutions Of Burgers’Equation For Gee-Lyon Fluid Flows

Dongming Wei

, Ken Holladay

Received 26 August 2011

Abstract

In this work we present some analytic and semi-analytic traveling wave so- lutions of a generalized Burger’ equation for isothermal unidirectional ‡ow of viscous non-Newtonian ‡uids obeying the Gee-Lyon nonlinear rheological equa- tion. 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

@t +u@u

@x = @

@x

1 0

@u

@x (1)

where (t) = (1 +ct²)t, 0< c <1. The solution can be written as the following:

0

= 2

(u2 u1)ln u₂ u u u1

4

(u2 u1)extra b;2u (u₂+u₁) u2 u1

(2) where

extra(b; ) = Re 0

@arctan(p

1 isinh(2b)) p 1 isinh(2b)

1

A (3)

in which the constant b is de…ned by sinh(2b) = p ⁸

c (u2 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 Sub ject Classi…cations: 35L67, 76A10, 35Q53.

yDepartment of Mathematics, University of New Orleans, New Orleans, USA

zDepartment of Computer Science, University of New Orleans, New Orleans, USA

129

satisfying the upstream and downstream boundary conditions

!lim+1u( ) =u1; lim

! 1u( ) =u2; lim

j j!+1

du d ( ) = 0 with =x t; =^u1+u₂ 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 layer between upstream and downstream can be given by = ^{8 0}

(u2 u1)f^{1+c[8(u2 u1)]2}g which forc= 0 gives the corresponding classical estimate = _(u⁸₂ ⁰_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

Dt = div( ) 5p+g (4)

where u= (u1; u2; u3)is the ‡uid velocity,

= 0

@ ¹¹21 ¹²22 ¹³23

31 32 33

1

A; andDu= 0

@d11 d12 d13

d21 d22 d23

d₃₁ d₃₂ d₃₃ 1 A

are the stress tensor and the strain tensor, is the density,gthe external force,pthe scalar pressure, andd_ij =¹₂(^@u_@xⁱ

j+^@u_@x^j

i),1 i; j 3. For unidirectional ‡ows,we assume that u= (u1;0;0), ij = 0fori6= 1orj 6= 1,g= (g1;0;0), and5p= (_@x^@p

1;0;0). The Navier-Stokes equation (4), in this case, takes the following simple scalar form

Du₁

Dt = d ₁₁ dx1

@p

@x1

+g₁ (5)

where ^Du_Dt¹ = ^@u_@t¹+u1@u1

@x1. Rheological relationships between andDuare frequently used to determine the type of ‡uids. Polyethylene and polystyrene melts can be de- scribed 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

0dij = ( ij+cj ^{kl lk}jⁿ²)tij,1 i; j 3 (6)

where ₀, n, and c are constants, ij = 1, ifi=j

0, ifi6=j , see [2], [10] or [12]. The temperature dependence of the viscosity is expressed by ₀=Ae^RTE . In this work, we refer to ‡uid ‡ow satisfying the rheological equation (6) as Gee-Lyon ‡ows.

Ifc= 0, then the ‡uid is said to be a Newtonian ‡uid; it is non-Newtonian ifc6= 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) reduces to ₀d11= (1 +cj ¹¹j^{n 11}. Letu1, x1,g1 be denoted byu,x, g respectively.

Then from (5), let ^@p_@x+g= 0, we have the generalized Burgers’equation

@u

@t +u@u

@x = @

@x

1 0

@u

@x (7)

where (t) = (1 +cjtjⁿ)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

@t +u@u

@x = @²u

@x². (8)

It is well known that if we impose lim

!+1u( ) =u₁, lim

! 1u( ) =u₂, lim

j j!+1 du d ( ) = 0, andu₁< u₂, (8) has the celebrated traveling wave solution

0 = _(u²

2 u1)ln^u_{u u}^{2 u}

1, which is equivalent to

u( ) =u₁+u₂exp[ ₂ (u₂ u₁)]

1 + exp[ ₂ (u₂ u₁)] (9)

where =x t,u₁ andu₂are the downstream and upstream ‡uid velocities.

It can be shown that there exists a thin transition layer of thickness of order

8

u2 u1 for (9). This thickness can be referred to as the shock thickness, which tends 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 semi- analytic solutions to (7) for c6= 0, andn= 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) forc6= 0, andn= 2.

3 The Integral Equation for the Traveling Waves and the Solution

Let u(x; t) =u( ), with =x t. Then ^@u_@t = ^du_d ^d_dt = ^du_d and ^@u_@x = ^du_d ^d_dx = ^du_d . Substituting ^@u_@t = ^du_d and ^@u_@x =^du_d into equation (7), we get

du

d +u( )du d = 1 d

d

1 0

du

d . (10)

Therefore

d d

1

2u² u 1 ₁

0

du

d = 0;

which gives

1

2u² u 1 ₁

0

du

d =A (11)

where A is an arbitrary integration constant. Applying the downstream and upstream boundary conditions: lim

!+1u( ) = u1, lim

! 1u( ) = u2, and lim

j j!+1 du

d ( ) = 0 to equation (11), we get

1 0

du

d =

2(u² 2 u 2A) =

2(u u1)(u u1) (12) where = ¹₂(u₁+u₂)and A= ¹₂u₁u₂,u₁ andu₂ are the given constants. We have

0du

d = (₂(u u1)(u u1)), which gives

0

=

Z du

(₂(u u1)(u u1)). (13) Without loss of generality, in the following, we assume that u1 < u < u2. For c = 0, (13) gives

1

2 =

Z du

(u u1)(u u1) = 1 u1 u2

ln u u₁ u2 u where = ⁰, which gives the classical traveling wave solution

u( ) = u₁+u₂exp[ ₂ (u₂ u₁)]

1 + exp[ ₂ (u2 u1)]

to Burgers’equation for Newtonian ‡ows.

In the following, we are interested in …nding solutions to (13) forc6= 0andn= 2.

Letu=^u2₂^u1 +^u2+u₂ ¹. Then

2(u u₁)(u u₂) = (u₂ u₁)²

8 1 + c ²(u₂ u₁)⁴

2⁶ [( + 1)( 1)]² ( +1)( 1)

and (13) becomes

0

= 8

(u2 u1)²

Z d

n1 + ^{c 2(u}₂²6 ^u1)4[( + 1)( 1)]²o

( + 1)( 1) .

Let the constant b be de…ned by sinh 2b = p ⁸

c (u2 u1)², and de…ne (t; b) = (1 +

t²

sinh²(2b))t. We have the decomposition 1

(( + 1)( 1); b) = 1 ( + 1)( 1)

1

( cosh(b) isinh(b))( cosh(b) +isinh(b))

2 1

( + cosh(b) isinh(b))( + cosh(b) +isinh(b)): By using Mathematica, we …nd that

Z d

(( + 1)( 1); b) = ln( 1 + 1) +

1 2

0 BB

@

arctan p

1 isinh(2b)

p 1 isinh(2b) +

arctan p

1+isinh(2b)

p 1 +isinh(2b) 1 CC A

Let

extra(b; ) = Re 0

@arctan(p

1 isinh(2b)) p 1 isinh(2b)

1 A.

Then we have

0

= 8

(u₂ u₁)² 1

2ln( 1

+ 1) ₀extra(b; ) . Therefore

0 = _(u ⁸

2 u1)²

h1

2ln(^u_{u u}^{2 u}

1) extra(b;^2u_u^(u2+u1)

2 u1 )i

and the traveling wave solution of (1) is implicitly de…ned by

0

= 4

(u2 u1)²ln(u₂ u u u1

) 8

(u2 u1)²extra(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, and0:25respectively and the red curve represents the classical solution corresponding to b= 0:0.

1 . 5 1 . 0 0 . 5 0 . 5 1 . 0 1 . 5

0 . 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 order derivative ^du_d

=0. From ^du_d = ¹

0 2(u u₁)(u u₂) and u(0) = ^u1+u₂ ², we get ^du_d

=0 = ¹

0 2(u2 u1)² . Let denote the thickness of the transition layer, using the Taylor expansion, we have

u₂ u₁=u(

2) u(

2) = du

d ₌₀+O( ²).

Therefore we have

= u₂ u₁

du

d (0) = ⁰(u₂ u₁)

(₈(u2 u1)²)= 8 ₀ (u2 u1)n

1 +c ₈(u2 u1) ²o,

Which is the …rst order approximation of the thickness of the transition layer for power- law ‡ows. This estimate, forc= 0, gives the well-known estimate = _(u^{8 0}

2 u1) 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.

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 vis- coelastic 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 power- law ‡uid, International Journal of Nonlinear Science, 6(3)(2008), 255–264.

[8] E. Hopf, The partial di¤erential equationu_t+uu_x= u_xx, Comm. Pure and Appl.

Math., 3(1950), 201–230.

[9] M. U. Tyn and L. Debnath, Partial Di¤erential Equations for Scientists and En- gineers, 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.