Journal of Applied Mathematics and Physics (ZAMP) Vol. 23, 1972

Birkh~iuser Verlag Basel

The Axisymmetric Free Laminar Jet of a Power-Law Fluid By Robert W. Serth, Dept. of Chemical Engineering, University of Puerto Rico, Mayaguez, R R., USA

Nomenclature d D f F J K n Re u

orifice diameter characteristic length preliminary dependent variable final dependent variable m o m e n t u m in x-direction coefficient in power law model exponen~ in power law model generalized Reynolds n u m b e r x-component of velocity; u o = characteristic velocity; up = initial velocity; ~ = u/u o; u m= velocity on centerline of jet.

v x

y-component of velocity; ~ = Re "+1 v/u o coordinate in axial direction; Y~=x/D; ;?=distance from orifice in experimental jet

y

coordinate in radial direction; y = Re n+l y/D function of fmwtion o f ~ parameter in final similarity transformation preliminary independent variable final independent variable fluid density shear stress stream function

1

1

fl y q p r

Introduction

An analysis of the two-dimensional free laminar jet of a power-law fluid has been presented by Lemieux and Unny [1], who showed that the velocity along the centerline of the jet decays as the ( - 1/3 n) power of the distance downstream, and that the jet spreads at a rate proportional to the distance downstream to the 2/3 n power. Here n is the exponent in the power-law model. These results are a direct generalization of the classical solution for a Newtonian fluid [2]. Although ~he results were derived in Ref. [1] only for pseudoplastic fluids, they can easily be shown to be valid for dilatant fluids as well. The purpose of this paper is to investigate the extension of the above results to the case of the axisymmetric jet. One potential application of the present results is in the investigation of the degree of validity, in boundary layer type flows, of theological equations of state

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Robert W.Serth

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which have been determined in standard viscometric flow systems. In the same vein, when dealing with viscoelastic fluids it is of some interest to determine under what conditions the elastic properties of the fluid can be disregarded. This has recently been a point of contention in the literature, with different theoretical analyses predicting contradictory results (see, for example, [3], [4]). Of course, no experimental results pertaining to this question are available. However, the free jet is considerably more amenable to direct measurement than is the laminar boundary layer flow over an external surface, which has been employed for the above purpose by Shah, et al. [5]. The total absence of velocity measurements in non-Newtonian fluids in the latter class of flows (Shah, et al., measured heat-transfer rates) testifies to the experimental difficulties involved with such flows.

A Similarity Transformation The boundary layer equations for the axisymmetric laminar jet of a power-law fluid may be written in dimensionless form as [2], [6]

a~ ~ + T

=~

(1)

[' 0~" l

0fi ~ ~ -~-+ - ~ a~ -

1 9 r ~ a~ LY \

a~! J'

(2)

where ~2= x / D = u/u o 1

y = R e "+1 y / D

(3)

1 = Re n+l v/u o 2-n

Re = e u o D K

n

Here x is distance downstream measured along the centerline of the jet, y is radial distance measured outward from the centerline, u and v are velocity components in the x- and y-directions, respectively, and u o and D are a characteristic velocity and length. K and n are parameters in the power-law fluid model, which in the present situation takes the form

z = K I OY I

OY'

(4)

Vol. 23, 1972

The Axisymmetric Free Laminar Jet of a Power-law Fluid

133

where z is the shear stress. In addition, the momentum integral equation for the jet reduces to [2] 2 oe

J=2neu~DZRe "+1S fiZpd~=const

(5)

0

where J is the total momentum in the x-direction. The boundary conditions are ~>0,

~=0,

~=~-=0

f>0,

~oo,

fi~0.

oy

(6)

If a stream function, 0, is introduced in the usual manner and a similarity transformation is made of the form ~ = ~(s

~]=~/fi(ff).

(7)

Then one obtains 1

/~(x)-x 2"-1,

n, 89

f i ( 2 ) ~ e (. . . . t)N

n= 89

(8)

Equation (8) immediately yields the form of the velocity decay in the jet, viz., ~

1 2n-1,

n:~= 89

fi~e-( .... t)~,

n=89

(9)

For n<89 however, equation (9) predicts that the velocity increases with distance downstream, which is physically unreasonable. Furthermore, it can be shown that for n < 89 the similar solutions cannot satisfy the boundary conditions (6). Hence, similar solutions of the form (7), are possible only for n>89 Since the integral condition (5) appears to effectively proscribe any further generalization of the transformation (7), similar solutions cannot be obtained for n <89 and more laborious techniques must be employed to obtain solutions (if they exist) for these cases. The remainder of this article will be restricted to the cases n >89

Similar Solutions

Upon carrying out the transformation (8), the problem is reduced to the form 1_

F'

r" = ~ - -

{FF'F ~ ~,~ I

(io)

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Robert W. Serth

ZAMP

with boundary conditions F(O) = F'(O) = O,

F'(O) = 1

F'

lim - - - - 0. Here, F is a dimensionless stream function and ~ a dimensionless independent variable related to the original variables by 2-n •3(n--1) y

n>-a2, n@l

( 2 n _ l ) a ( . - l ) ~2n-t ~=7Y/2,

n=l

(11)

.=89

Y = 7} e ~ '

where 7 is a constant.

u Um Q.8

Q.6 0.4

~ 1.5

Z.0 O.Z 0

1

Z

S

4

5

Figure 1 Similar velocityprofiles in axisymmetricfree laminar jet.

The above initial value problem was solved numerically by a Runge-Kutta procedure. Values of ~t/G,=F'(~.)/~, where ~,,=fi(~,0) are presented in Figure1. For n = l , a closed form solution is available [-2], viz., d2 F({) = 2 + (4/2) 2.

(12)

(Note that the variable ~ defined here differs from that in [2] by a factor of lf2.) The values obtained by numerical integration agree to six decimal places with those computed from (12).

Vol. 23, 1972

The AxisymmetricFree Laminar Jet of a Power-lawFluid

135

During the course of the numerical calculations, it became apparent that for n > 1, the boundary condition at infinity could not be met. Instead, it must be replaced by F'(~)/~ = 0 for ~ = ~,, where ~, depends on n. This phenomenon of a finite "boundary layer" in dilatant fluids has also been noted in other types of boundary-layer flow [6], [7], [8], [91. Although solutions can be obtained for n > 2, it has been shown [5], [8] that for n > 2, the boundary-layer approximations are incompatible with the fluid model (4). Hence, such solutions are probably not relevant to the jet flow. The solutions for the velocity components fi and ~ read -2

n+l

--1

u = ( 2 n - - 1 ) 3("-1) 7 3(,-a) ~2n-1 2-3n

F'(~)/~

2n-1

~ ]

v = ( 2 n - - 1 ) 3(n-1) 7 3(n-1) (x) -1 [ F ' ( ~ ) -

n> 89 n # l ,

(13)

n=l,

(14)

n -----89

(15)

~l= ~2 (2)-1 F'(~)/~ = 7 (X) -~ [ F ' ( ~ ) - F(~)/~-] = 7- ~ e - r~ F'(~)/~. = r [F'(r

F(~)/Q

The parameter 7 is related to the momentum, J, as follows: 2(2n-1)

J-

2neuZ DZy 3(n-1) O~F,2(y ~ 2 2 So2 ~ d~'

n>89 n * l

Re "z-I (2 n - 1) 3(,= t)

o~ F,2(~ j=2zceuZ DZRe-172 S ~ i d~,

n=l

(16)

oo

j=27reuZ D2Re-~7--~ ~o F'2(~) ,~ It will be noted that for n - x2, ~ does not decay, in the sense that its maximum value is independent of ~, and is merely shifted to iarger values of y for increasing 2. Such behavior, if it occurs, must lead to a breakdown of the assumptions underlying equation (2), since for sufficiently large 2, Ov/Oy will be of the same magnitude as Ou/Oy. Of course, the boundary-layer approximations must break down for sufficiently large axial distances in any case, but the above results suggest that for n close to 89 this may occur at much smaller distances than in the Newtonian case. Finally, it should be noted that as n approaches 89 the convergence of the solution to the boundary condition at infinity becomes very slow, so that the determination of y from equation (16) is difficult. F r o m an experimental point of view, however, this is not a serious drawback, since the value of 7, as well as the apparent origin of the jet, -

136

Robert W. Serth

ZAMP

can be determined from the velocity profile along the jet axis. In fact, only axial and radial velocity profiles are required for a comparison of experiment with theory.

Experimental Results The only relevant experimental data known to the author are those obtained in Ref, [10]. These data, which are presented in Figure 2, consist of a single axial velocity profile in a 0.12 per cent aqueous solution of Separan AP 30, a high-molecularweight polyacrylamide manufactured by Dow Chemical Co. The jet was formed by forcing the fluid through a ~ inch diameter orifice into a body of quiescent fluid of the same composition. The velocity was measured with a hot-film anemometer. Details of the experimenta! apparatus have been given elsewhere [10], [11]. 1.(1 0.8 u~ 0.6

Urn

04 02 Figure 2 Axial velocitydistribution in 0.12 per cent Separan AP30 jet.

i

i

4

8

F

1Z

i

lfi

r

i

Z0 A Z4 x_ d

i

Z8

i

3Z

~

38

i

40

Although the above fluid exhibited significant viscoelastic behavior, its shear stress-shear rate relationship in simple shear flow was well described by the fluid model (4). Values of n =0.57 and K = 1.25 dyne sec cm -2 were obtained for shear rates between 1 and 800 (sec) -1 by means of the Cone-and-plate attachement for the Weissenberg Rheogoniometer. The validity of the power-law relationship for Separan solutions over much wider ranges of shear rate has been demonstrated by Gupta et al. [12], and Seyer and Metzner [13]. In Figure 2, up denotes the initial velocity at which the jet issues from the orifice (about 4 ft/sec in this case), ~ is the axial distance measured from the orifice, and d is the orifice diameter. The initial sharp drop in velocity is due to the expansion of the jet as it issues from the orifice, the so-called "die swell" or "Barus Phenomenon." From Figure 2, it can be seen that the jet achieves its affine structure after about 10 diameters downstream. This is the region to which the theoretical results apply. For a value of n =0.57, the power-law theory predicts that the velocity decay should be proportional to (~)-7.1~. In fact, however, the decay is exponential, as can be seen by reference to Figure 3, which is a semi-logarithmic plot of the experimental

Vol, 23, 1972

The Axisymmetric Free Laminar Jet of a Power-lawFluid 0.5

137

"

0.4 03

~

02't

Figure 3 Exponential decay of the center line velocity in a 0.12 per cent Separan AP30 jet.

0.1

5

10 15 ~--10

20

2b

30

velocity profile. If the power-law theory for n=89 applied with Uo=Up and D=d, the value of 7 in equation (15) is found from the slope of the least-squares line (shown in Figure 3) to be 7 =0.051. Setting ~ and ~ equal to zero in (15) gives ~(0, 0 ) = 7 - ~ = 52.5. Extrapolating the least-squares line to ~m=52.5 yields a value of 2/d=-1010, which is the apparent origin of the jet. Thus, 2 and 2 are related by ~-+1010. Unfortunately, no radial velocity profiles were obtained for this jet, so that no further comparison with the theoretical results is possible. Clearly, much more extensive experimental work is needed before the significance of these results can be fully assessed. However, the present results serve to illustrate the application of the theory and to demonstrate the existence of the predicted exponential velocity decay. Conclusions It has been shown that similar solutions to the boundary layer equations for the axisymmetric laminar jet of a power-law fluid are possible only for n_>_89 For n> 89 the value of the axial component of the velocity is proportional to (2) -1/2"-1 and the rate of spread of the jet is proportional to (~)1/2,-t. For n -~-2, the velocity decay and the rate of spread of the jet are exponential. The existence of the exponential velocity decay in real fluids has been demonstrated.

138

Robert W. Serth

ZAMP

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

P.F. LEMIEUX and T.F. UNNY, J. appl. Mech. 35, 810 (1968). H. SCHLICHTING, Boundary Layer Theory, 4th edn. (McGraw-Hill, 1960). K.R. F~TER, Z. angew. Math. Phys. 20, 712 (1969), K.WALTERS, Z. angew. Math. Phys. 21, 276 (1970). M.J. SHAH, E.E. PETERSEN, and A. AcRwos, A. I. Ch.E. J. 8, 542 (1962). A. AcRrvos, M. J. S~AH, and E. E. PETERSEN,A. I. Ch.E. J. 6, 312 (1960). R.W. SERTH and K.M. K~SER, Chem. Eng. Sci. 22, 945 (1967). R.A. SPrNELLI, Z~ angew. Math. Phys. 20, 479 (1969). V.G. Fox, L.E. ERICKSON, and L.T. FAN, A. I. Ch. E. J. 15, 327 (1969). R.W. SERTH, Ph. D. Thesis, State University of New York at Buffalo 1968. R.W. SERTI~ and K.M. KISER, A. I. Ch. E. J. 16, 163 (1970). M.K. GUPTA, A.B. METZNER, and J. P. HARTNETT, Int. J. Heat Mass Transf I0, 1211 (1967). F.A. SEVER and A.B. METZNER, Canad. J. Chem. Eng. 45, 121 (1967).

Summary The solution of the boundary-layer equations for an axisymmetric free laminar jet of a power-law fluid is investigated. It is shown that similar solutions are possible only for n__>89where n is the exponent in the power-law model. For n>89 the axial velocity decay is proportional to the ( - 1 / 2 n - 1 ) power of the distance downstream and the jet spreads at a rate proportional to the (1/2 n - 1) power of the distance. For n = 89 the velocity decay and the rate of spread are exponential. Experimental data are presented which demonstrate that the predicted exponential decay occurs in real fluids.

Zusammenfassung Ahnliche L6sungen der Grenzschichtgleichungen fiir einen achsensymmetrischen laminaren Strahl von Potenzgesetz-Fltissigkeiten werden gegeben. Diese L6sungen werden mit vorhandenen experimentellen Daten verglichen. (Received: June 17, 1971)