Ind. Eng. Chem. Res. 2004, 43, 8407-8410
8407
Power-Law Flow Past a Cylinder at Large Distances S. J. D. D’Alessio* and L. A. Finlay Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Derived in this paper is the behavior of the two-dimensional steady flow of a power-law fluid past a circular cylinder at large distances. An approximate far-field solution for a weakly nonNewtonian fluid has been constructed. Some numerical calculations to illustrate the impact of the proposed approximate solution are also included. 1. Introduction This work is concerned with the steady two-dimensional flow of a viscoinelastic incompressible power-law fluid past a nonrotating circular cylinder. The Newtonian version of this problem is well documented, where numerous authors have contributed experimental, numerical, and analytical investigations (see the recent texts by Zdravkovich1,2 for a comprehensive listing of previous studies). The purpose of this paper is to propose an approximate analytical solution valid at large distances from the body. The methodology used to achieve this involves linearizing the equations of motion using the known asymptotic solution for the Newtonian case, which is justified if the power-law index, n, is close to unity. Appearing in the linearized equations is the parameter ) 1 - n, which is assumed to be small. Treating as a small parameter allows further simplications, and the resulting equations admit an analytical solution. 2. Governing Equations We consider the steady flow of a non-Newtonian power-law viscoinelastic fluid past a circular cylinder of radius a. The oncoming uniform flow of speed U is in the x direction and is perpendicular to the axis of the cylinder. Because the flow is assumed to be twodimensional and symmetrical, it is worthwhile to formulate the governing equations in terms of a stream function, ψ, and vorticity, ζ. Further, it is beneficial to employ modified polar coordinates where x ) eξ cos θ and y ) eξ sin θ in dimensionless form. This maps the cylinder surface to ξ ) 0 and the infinite exterior region to the semi-infinite rectangular strip ξ > 0, -π e θ e π. For symmetrical flow, only the region 0 e θ e π needs to be considered. Here, θ is the usual polar angle, while r ) eξ. These coordinates are better suited for analytical work because the coordinate ξ scales the flow field in such a way that it does not affect distances close to the surface while contracting distances far from the cylinder. In terms of the coordinates (ξ, θ), the equations of motion in dimensionless form can then be written as (see ref 3) * To whom correspondence should be addressed. Tel.: (519) 888-4567 ext. 5014. Fax: (519) 746-0274. E-mail: sdalessio@ uwaterloo.ca.
∂2ψ ∂2ψ + 2 + e2ξζ ) 0 2 ∂ξ ∂θ I22
(
)
(1)
(
)
n - 1 ∂I2 ∂ζ ∂I2 ∂ζ ∂2ζ ∂2ζ + 2 + I + 2 2 2 ∂ξ ∂ξ ∂θ ∂θ ∂ξ ∂θ (n - 1)(n - 3) H - (n - 1)I2G ) 2
(
)
R (5-n)/2 ∂ψ ∂ζ ∂ψ ∂ζ I2 (2) ∂θ ∂ξ ∂ξ ∂θ 2n where
(
∂I2 ∂I2 +B ∂ξ ∂θ
(
)
G ) eξ sin θ A)
(
)
H ) e-2ξ A
(
)
∂G1 ∂G2 ∂G1 ∂G2 + + eξ cos θ ∂ξ ∂θ ∂θ ∂ξ
) ( ) (
)
∂I2 ∂ψ ∂I2 ∂ψ ∂I2 ∂2ψ 1 ∂I2 ∂2ψ ∂2ψ + + 2 ∂ξ ∂ξ2 ∂ξ ∂ξ ∂θ ∂θ ∂θ ∂ξ ∂θ ∂θ2
B)-
(
)
∂I2 ∂ψ ∂I2 ∂ψ 1 ∂I2 ∂2ψ ∂2ψ - 2 + + 2 2 ∂θ ∂ξ ∂θ ∂ξ ∂ξ ∂θ ∂θ ∂I2 ∂2ψ ∂ξ ∂ξ ∂θ G1 ) e-3ξ(A sin θ + B cos θ) G2 ) e-3ξ(B sin θ - A cos θ)
and
I2 ) e-4ξ
[( ) ( ) ∂2 ψ ∂ξ2
2
+
∂2ψ ∂θ2
2
-2
(
∂2ψ ∂2ψ ∂ψ ∂2ψ -4 2 2 ∂ξ ∂ξ2 ∂ξ ∂θ
) [( ) ( ) ( ) ]
∂2ψ ∂ψ 2 ∂ψ 2 ∂2ψ + 4 + + ∂ξ ∂θ ∂ξ ∂θ ∂θ2
2
-8
]
∂ψ ∂2ψ (3) ∂θ ∂ξ ∂θ
In the above equations, I2 is a strictly positive invariant and R is the Reynolds number defined by
R)
FU2-n(2a)n H
with H denoting the consistency and F the density. The
10.1021/ie040122q CCC: $27.50 © 2004 American Chemical Society Published on Web 11/17/2004
8408
Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004
dimensionless stream function and vorticity, ψ and ζ, are related to their dimensional counterparts, ψ′ and ζ′, through ψ′ ) aUψ and ζ′ ) Uζ/a. The dimensionless velocity components u and v in the directions of ξ and θ, respectively, are then
u ) e-ξ
CD )
[
2n+1 R
n
2 (n - 1) R
CD )
and the vorticity is given by
ζ)e
∂v ∂u v+ ∂ξ ∂θ
(
)
Note that when n ) 1, eqs 1 and 2 collapse to the familiar Navier-Stokes equations.
To investigate the far-field behavior, it is instructive to first consider the case of a Newtonian fluid. The first approximation to the asymptotic solution was obtained by Oseen4 by linearizing the Navier-Stokes equations and making use of the fact that
∂ζ - ζ) sin θ dθ ∫0π(∂ξ 0
∂2ψ ∂2ψ + 2 + e2ξζ ) 0 2 ∂ξ ∂θ
∂2ψ ∂2ψ + 2 + e2ξζ ) 0 2 ∂ξ ∂θ
(
(
)
]
)
( )
R CD 2n π
Cn )
1-n
Setting ζ(ξ,θ) ) eM(ξ,θ)χ(ξ,θ) with
M(ξ,θ) ) (5)
Cn (2n-1)ξ cos θ e 2
the vorticity equation (9) then becomes
and are valid for any Reynolds number at sufficiently large distances. Here, the equations are decoupled and can be solved analytically. Imai5 generated an asymptotic expansion for ψ and ζ. The first term in this expansion, which is equivalent to Oseen’s solution, is given by
CD θ ψ ∼ eξ sin θ + - erf(Q) 2 π
∂ζ ∂ζ - sin θ (9) ∂ξ ∂θ
where
(4)
∂ζ ∂2ζ ∂2ζ R ξ ∂ζ + 2 ) e cos θ - sin θ 2 2 ∂ξ ∂θ ∂ξ ∂θ
(8)
∂ζ ∂2ζ ∂2ζ + 2 - 2(n - 1) ) 2 ∂ξ ∂ξ ∂θ
in accordance with the flow approaching a uniform velocity. The equations then take the form
CDR -ξ -Q2 e Qe 4xπ
]
4 R
Cne(2n-1)ξ cos θ
∂ψ ∂ψ ∼ eξ sin θ, ∼ eξ cos θ as ξ f ∞ ∂ξ ∂θ
ζ∼-
∂I
An approximate solution at large distances for a nonNewtonian power-law fluid can be constructed in a similar manner. If the power-law index, n, is close to unity, we can linearize eqs 1 and 2 using the Newtonian asymptotic solution given by expressions (6) and (7). Doing this and noting that to leading order I2 ∼ CD2e-4ξ/ π2 as ξ f ∞, we obtain the linearized set of equations
3. Far-Field Behavior
[
[
∫0π I2(n-3)/2 ∂ξ2ζ 0 sin θ dθ
When n ) 1, this yields the Newtonian result
∂ψ ∂ψ , v ) -e-ξ ∂θ ∂ξ
-ξ
]
∂ζ - ζ) sin θ dθ + ∫0π I2(n-1)/2 (∂ξ 0
(6)
2
∂2χ ∂2χ Cn 2(2n-1)ξ + χ) e 4 ∂ξ2 ∂θ2 2(2M - 1)
∂χ + 2M(1 - 2M)χ (10) ∂ξ
For n close to unity, ) 1 - n will be a small parameter and so we can approximate χ by χ0, where χ0 satisfies
∂2χ0
(7)
∂ξ2
+
∂2χ0 ∂θ2
-
Cn2 2(2n-1)ξ e χ0 ) 0 4
(11)
Equation 11 can then be easily solved by separation of variables; the solution is
where
Q)
xR2 e
ξ/2
sin(θ/2)
CD denotes the drag coefficient, and erf(Q) is the error function defined as
erf(Q) )
2 xπ
∫0Qe-x
2
dx
The drag coefficient for a non-Newtonian power-law fluid is given by
χ0(z,θ) ) A0K0(z) + ∞
∑ (Am cos mθ + Bm sin mθ)Kν (z) m
m)1
(12)
where
z)
Cn
e(2n-1)ξ, νm )
2(2n - 1)
m 2n - 1
and Kνm(z) refers to the modified Bessel functions of order νm of the second kind.
Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 8409
For symmetrical flow with n close to unity, the above equation can be expressed alternatively as ∞
χ0(z,θ) )
∑ Bm sin[(m + 1/2)θ]Kν +1/2(z) m
m)0
(13)
Retaining the first term and using the asymptotic form of Kνm+1/2(z) for large z (see ref 6)
Kνm+1/2(z) ∼
x2zπ e
-z
and again exploiting the smallness of , we can express the final result for the vorticity as
(2n - 1)xπB0 -2(n-1/2)ξ 2 ζ0(ξ,θ) ∼ e Q/e-Q/ Cn
Figure 1. Scaled stream function for n ) 0.9, 1, and 1.1 with R ) 5 and ξ ) 4.
(14)
where
Q/(ξ,θ) )
x
Cn (n-1/2)ξ sin(θ/2) e 2n - 1
To arrive at this result, we have used the approximation
1 - (2n - 1) cos θ ) 1 - cos θ + 2 cos θ ≈ 1 - cos θ for small . Note that the form of eq 14 bears a close resemblance to that of eq 7. The next step is to solve for the stream function. Setting ψ ) eξ sin θ + ψ* and substituting ζ0 for ζ in eq 8, we find that ψ*, the perturbed stream function, satisfies
(2n - 1)xπB0 2ξ -2(n-1/2)ξ 2 ∂2ψ* ∂2ψ* + )e e Q/e-Q/ 2 2 C ∂ξ ∂θ n (15)
Figure 2. Scaled vorticity for n ) 0.9, 1, and 1.1 with R ) 5 and ξ ) 4.
containing the cylinder. The result is
B0 ) -
A straightforward calculation reveals that
[
]
∂ψ* ∂2ψ* n Q/ + Q/2 ∂Q/ ∂Q 2 /
For small , we can ignore the last term on the righthand side and replace ψ* by ψ0* to denote this. Equation 15 then transforms to
)2
∂Q/
4(2n - 1)2xπB0 2
Cn
2(2n - 1)2π
and hence
Cn ∂2ψ* ∂2ψ* ∂2ψ* + ) e2(n-1/2)ξ 2 2 4(2n - 1) ∂ξ ∂θ ∂Q/2
∂2ψ/0
Cn2CD
Q/e-Q/
2
(16)
where we have utilized the approximation e4ξ ≈ 1 for small . Equation 16 can then be easily integrated twice. This will yield an expression that is also similar to eq 6, given by
CD θ - erf(Q/) 2 π
[
ψ ∼ eξ sin θ + ζ∼-
CnCD
]
(18)
e-2(n-1/2)ξQ/e-Q/
2
2(2n - 1)xπ
(19)
as ξ f ∞. Note that when n ) 1, then Q/ ) Q, Cn ) R/2, and eqs 18 and 19 reduce to the Newtonian results (6) and (7), as expected. To investigate the impact of these formulas, we first present plots of scaled distributions of ψ and ζ for R ) 5 along the boundary ξ ) 4 for n ) 0.9, n ) 1, and n ) 1.1. The scaled stream function is defined by
θ 2 (ψ - eξ sin θ) ∼ - erf(Q/) CD π while the scaled vorticity is defined as
ψ/0(ξ,θ)
∼
(2n - 1)2πB0 Cn2
[
θ erf(Q/) π
]
(17)
All that remains to be determined is the constant B0. This constant can be related to the drag coefficient, CD, by integrating the stresses around a large contour
( )
CD 4xπeξζ ∼ CDR π
1-ne-2(n-1)ξQ
/e
-Q/2
2n-1(2n - 1)
Plotted in Figures 1 and 2 are the right-hand sides of the above expressions. We see that the departure from
8410
Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004
the Newtonian result (n ) 1) is much more noticeable in the vorticity than in the stream function. We next present some calculated results for CD using the numerical procedure outlined in ref 3 for the cases R ) 5 and 20. In these calculations, CD was computed using asymptotic far-field conditions. A computationally efficient method of implementing eqs 18 and 19 is in the form of gradient conditions. If ξ∞ denotes the outer boundary of the computational domain and h the uniform grid spacing in the ξ direction, then these gradient conditions can be expressed as
ψ(ξ∞,θ) ) ehψ(ξ∞-h,θ) ζ(ξ∞,θ) ) e-(n-1/2)he-s(θ)(1-e
-(2n-1)h)
ζ(ξ∞-h,θ)
where
s(θ) )
Cn sin2(θ/2)e(2n-1)ξ∞ 2n - 1
Setting n ) 1 in these conditions gives the Newtonian condition. Computations for n * 1 were carried out using both the Newtonian condition and the nonNewtonian condition. In these computations, the values ξ∞ ) 4 and h ) 0.05 were used. For R ) 5, we obtained
n ) 0.8, CD ) 4.02 (4.00), n ) 1.2, CD ) 4.17 (4.16)
relations represent an extension of eqs 6 and 7 to include the dependence on the power-law index when n is close to unity, which corresponds to a weakly nonNewtonian fluid. One advantage offered by these relationships is that when computations are carried out for this flow arrangement, these approximate solutions can be implemented as appropriate far-field conditions. Using these expressions has the effect of bringing infinity closer and in doing so helps reduce the otherwise unbounded domain to a more manageable one for numerical purposes. Also presented were some representative calculations and plots to illustrate the impact of the proposed solution as well as a method of implementing the proposed solution in a numerical procedure. The results obtained confirm that the Newtonian condition can safely be used for situations involving a weakly non-Newtonian fluid. However, we wish to point out that if accurate solutions for the far-field flow are required, for either numerical or analytical purposes, then the approximate solutions given by eqs 18 and 19 should be used and not the Newtonian solution. Acknowledgment Financial support for this research was provided by the Natural Sciences and Engineering Research Council of Canada. We also acknowledge Prof. R. P. Chhabra for informing us of an error in the formula for the drag coefficient in a previous paper (ref 3).
while for R ) 20, the following results were obtained:
n ) 0.9, CD ) 2.03 (2.02), n ) 1.1, CD ) 2.13 (2.13) The quantities in parentheses represent the values computed using the Newtonian condition. These results clearly indicate that using the Newtonian condition for a weakly non-Newtonian fluid introduces a very small change. Thus, despite the significant departures from the Newtonian case shown in Figures 1 and 2 along the outer boundary, they have a negligible influence near the surface where CD is computed. Hence, the flow near the surface is not affected significantly by the nonNewtonian far-field conditions. 4. Conclusions An approximate solution for the flow of a power-law fluid past a circular cylinder at large distances has been derived and is given by expressions (18) and (19). These
Literature Cited (1) Zdravkovich, M. M. Flow Around Circular Cylinders. Vol. 1: Fundamentals; Oxford University Press: Oxford, U.K., 1997. (2) Zdravkovich, M. M. Flow Around Circular Cylinders. Vol. 2: Applications; Oxford University Press: Oxford, U.K., 2003. (3) D’Alessio, S. J. D.; Pascal, J. P. Steady flow of a power-law fluid past a cylinder. Acta Mech. 1996, 117, 87-100. (4) Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: Cambridge, U.K., 1932. (5) Imai, I. On the asymptotic behaviour of viscous fluid flow at a great distance from a cylindrical body, with special reference to Filon’s paradox. Proc. R. Soc. London 1951, A208, 487-516. (6) Abramowitz, M., Stegun, I. A., Eds. Handbook of Mathematical Functions: Applied Mathematics Series-55; NBS: Washington, DC, 1964.
Received for review April 22, 2004 Revised manuscript received October 14, 2004 Accepted November 3, 2004 IE040122Q
