R s
= = = =
S
=
Ri Re
u w
= = z,y,z =
radius of the biggest sphere in a packing radius of internal sphere in Happel’s model number Reynolds local pore cross section local cross section of the porous medium velocity dimensionless velocity defined in eq 10 coordinates in Cartesian space
GREEKLETTERS = relaxation factor = ratio between the radius of the internal sphere to the y external sphere in Happel’s model e = porosity 1.1 = viscosity p = density
p
literature Cited
Happel, J., A.I.Ch.E. J. 4 (2), 197 (1958). Brenner, H., “Low Reynolds Number HydrodyHappe!, namics, pp 358-430, Prentice-Hall, Englewood Cliffs, N. J.,
Jl;,
146.5
J. J., McCabe, W. L., Monrad, C. C., Chem. Eng. Progr. (2), 91 (1951);, Scheidegger, A. E., The Physics of Flow through Porous Media,” pp 68-74, 112-155, University of Toronto Press, Toronto, 1960. Susskind, H., Becker, W., A.1.Ch.E. J. 13 (6), 1115 (1967). Ma;;]
47
RECEIVED for review July 23, 1971 ACCEPTED June 20, 1972 This work is based on an M.Sc. thesis submitted to the TechnionIsrael Institute of Technology by A. Marmur.
On the Asymptotic Solution to the Falling Film Stability Problem Byron E. Anrhur Department of Chemical Engineering, University of Delaware, Newark, Del. I971 1
The asymptotic forms of the solution to the falling film stability problem for large and small Reynolds number fluctuations (Re) and surface tension parameter ({) are found and discussed. It i s shown that when Re + are confined to a region near the surface of thickness 0 (Re-1’3) and that as { + 0) the region becomes of thickness 0 (Re-’” {l’”). The corresponding forms of the solution then show that the wavelength and the dimensional growth rate of the maximum instability approach constants as Re + 03. The line of wave inception i s proportional to As { + 03 when Re = 0 ( l ) , the fluctuations are not confined to a boundary layer, and the unstable wave numbers are confined to a region between a = 0 and a = CYN,where CXN =
0(
p z ) .
T h e problem of linear stability of the falling liquid film was first properly formulated by Benjamin (1957). Benjamin showed that time-independent flow of the vertical falling film is unstable a t all flow rates. A landmark paper on the subject was that of Yih (1963), who obtained a solution to the QrrSommerfeld equation and its boundary conditions for the falling film quite simply in the form of an expansion about 0 wave number. Yih (and earlier, Benjamin) found that the critical Reynolds number of the falling films is “6 cot p, where p is the angle the plane of the film makes with the horizontal. Thus, the vertical ( p = ~ / 2 falling ) film is unstable a t all Reynolds numbers. The discrepancy between this result and results of experimental measurements which do indicate a nonzero critical Reynolds number (Binnie, 1957; Tailby and Portalski, 1962) probably lies in the effect of surface-active agents (Benjamin, 1964; Whitaker, 1964). Anshus and Acrivos (1967) showed that surface-active material present in even minute amounts could change the stability characteristics considerably. A quite complete review of the literature on falling film stability was recently published by Krantz and Goren (1971), and we shall not attempt to duplicate that effort here. The perturbation analysis of Yih (1963) , while simple in concept and use, is inherently limited in its application to the region of small wave number (CY)or small Reynolds number 502
Ind. Eng. Chem. Fundam., Vol. 1 1 , No.
4, 1972
(Re). Using Yih’s technique one is able to determine the neutral stability curve on the a-Re plane and the wave number, growth rate, and wave velocity of the fastest growing disturbance when CY and the product a R e are small. The numerical results of Whitaker (1964) and of ,4nshus and Goren (1966), while they do pertain to regions where the parameters cy and Re are finite, have the usual deficiencies of numerical results. It is the purpose of this paper to present several asymptotic descriptions (corresponding to large and small values of parameters) of the neutral-stability and maximum-instability waves. Basic Equations
The Orr-Sommerfeld equation, together with the boundary conditions for a film with a free surface, are (see, e.g., Whitaker, 1964, and Yih, 1963) ,$,I,, - 2&” + cu’4 = iaRe[(a
-
c)(+”
icrRe[-(c‘ ,$ =
-
a24) - a”+]
3
+ a/~yz)(+“- CY%)+ 3$1
4 ’ = Oaty
=
1
(1) (li and ii)
4”’
+ (iaRec’ - 3a2)+’ -t ia
- (3 cot p C’
6ttt
+ a2Re-’’/r)~ = 0 a t y = 0
(liv)
Here +, a , and c are, as in standard works (e.g., Lin, 1955), the nondimensional stream function, wave number, and comples wave velocity, respectively. The quantity G(y) represents the nondimensional undisturbed velocity profile, which for the falling film is equal to ”2 (1 - y2). c’ E c - 3/2 represents the difference between wave velocity and surface fluid velocity. The term involving { in boundary condition liv expresses the effect of surface tension, whereas that involving {* in (liii) espresses the effect of insoluble surface-active agent. All quantities in eq 1 are nondimensional. The mode of stability to be examined here is the “surface mode’’ described by Yih (1963), and differs essentially from the “shear mode” which apparently governs the stability characteristics of boundary layers and of bounded shear flows.
+
+
- 3a02~e2n-2k16’
(iaoco~ei+n-m-2k
iao (3 cot pRen+n-3k +
a02{Re3n+m-3k-
‘I3) 6 =
co
o a t y*
=
o
(2iv) Evaluation of the three exponents involves only the insistence that the equation and boundary conditions retain their essential character in their asymptotic form. To this end the following three criteria are used: I. the equation must be fourth order; 11. the third boundary condition must keep the (3/c’)+ term; and 111. surface tension (represented by {) must remain in the problem. The necessity of requirements I and I11 is obvious. Requirement I1 is not so obvious and therefore deserves some discussion. It has been shown by Yih (1963) t h a t one can generate a solution to the falling film stability problem of the form m
Large Reynolds Number Asymptotic Forms m
a. Neutral Stability. If one examines the results of the numerical solution to eq 1 which were cited by Anshus and Goren (1966), one notes that the neutral stability wave number appears to increase as the Reynolds number increases. Further, it also appears that as the Reynolds number gets large the real part of the nondimensional wave velocity, CR, approaches ’/2 from above. (Physically this is equivalent to the wave velocity decreasing toward the surface fluid velocity.) Consideration of these two facts leads one to search for a neutral stability curve (CI = 0) for Re -P such that, a aoRen, C’R coRePm,where m and n are both positive. Upon substitution of the assumed forms into the Orr-Sommerfeld equation and boundary conditions, one concludes that the trivial solution, 6 = 0, would be satisfactory except near the free surface y = 0, where we insist that + ( O ) = 1. One is therefore led. to construct a matched asymptotic solution to the Orr-Sommerfeld equation where the “inner” region has thickness of order Re-L near the free surface and the “outer” region consists of the remainder of the flow field. The details of the technique of matched asymptotic espansions are covered in numerous publications (see, e.g., Van Dyke, 1964) and will not be discussed here. It suffices to say that the Reynolds number should not appear in the equation or boundary condition of the asymptotic forms. It is found that the present problem requires only the idea of inner and outer expansion to obtain a great deal of information; the lowest-order term of the inner solution is all that is developed. The generation of additional terms would entail much work with little additional information to be gained. I n keeping with the above ideas and following standard practice (e.g., Vidal and Acrivos, 1966) a “stretched” coordinate y* = Re‘y is defined and, together with the assumed forms for a and c’, is substituted into the Orr-Sommerfeld equation and boundary conditions (eq 1). (The primes here denote differentiation with respect to y*.) 6”” - 2a026”Re2n--2k + a04+Re47L-4k =
-
-
iao[ - 3/*y*2(+"Rei +n-4k - ao2+Rel+3n -6k 1 Ca(+”Rel+n--m-Zk
- a02+Re1+3n-m-4k
)
4 +0, 6’+0 as y * + m +”
+ (a02Re2n-2k
-
!!Ren-Zk)+ co
-
+ 3#Re1+n-4k1
(2)
(2i and ii)
c
arc,
= r=O
T h a t this can be done, although it cannot be done in a fised wall parallel flow, is due to the fact that the eigenvalue problem is retained (and this through the third boundary condition) upon setting O( = 0 in the equation and boundary conditions. This feature of boundary condition 2iii is therefore considered essential in any form of the falling film stability problem. Tjtilizing conditioiis I , 11, and 111,it is perfectly straightforward (but somewhat tedious in detail) to conclude that the only possible values that the exponents can assume are IC = n = l / a , rn = 2/a, L e . , that the asymptotic forms of wave number and velocity are a aORe113, c coRe-’I3 and that the thickness of the inner region is O(Re-’/’)). The asymptotic equations for the neutral stability eigenvalue problem a t large Reynolds number then become, with ax = a O ~ R e 1 /cg 3, = and for the inner region, y * = yRe’I3
-
6”” - 2ad29”
+ a04+
-
=
iao[- (CO
+ 3 / ~ ~ * 2 ) ( -6 ”
ao2+)
+ +0, 4‘ +0 as y* +
6”’
+ (iaoCo - 3 a 0 2 ) ~+’ co
iff0 -
(3 cot p
+ 341
(3)
(3i and ii)
x
+ aO2{)+= 0 a t y* = 0
(3iv)
For the outer region + = 0. This condition is valid for all of the large Reynolds number forms described below. For neutral stability COI =
0
(4)
Equations 3 and 4 represent the large Reynolds number asymptotic form of the neutral stability eigenvalue problem. Equations 3 constitute a comples eigenvalue problem from which, using eq 4, the t\vo real quantities a o ( { , { * , p ) and COR(^,{*,^) can be found. Solution to this problem is still, unfortunately, difficult since the equation has one nonconstant coefficient just as the original eq 1 has. The fact that two of the boundary conditions demand that the stream function $I vanish as y* -P m , combined with the fact that two of the fundamental solutions to eq 3 grox without bound as y* + m , makes this a difficult problem to solve numerically. h s shown Ind. Eng. Chem. Fundam., Vol. 1 1 , No.
4, 1972 503
below, however, the equivalent eigenvalue problem for a constant coefficient equation is easily solved. b. Maximum Instability. Of more interest from a pragmatic point of view than the neutral stability wave number are the growth rate, wave number, and wave velocity of the most rapidly growing wave. The growth rate of a n infinitesimal disturbance of wave number a is represented by a c ~ tand, a t a given Reynolds number above the critical Reynolds number Re,, there is some wive number for which the quantity ~ C isI maximum. The wave number of maximum growth rate has been referred to (Chandrasekhar, 1961; Lord Rayleigh, 1916) as that of “maximum instability.” It is the wave number corresponding to the first observable wave on a falling film subject t o a random infinitesimal disturbance. Since CI = 0 a t both a = 0 and a t CY = a ~ , i t follows that a c ~attains its maximum value a t a point a~ where 0 < a~\z< a ~ further, ; ( d a c ~ / d a )= 0 a t a~ (assuming, of course, that CI is a sufficiently smooth function). The locus of points on the a-Re plane satisfying ( d a c ~ / d a )= 0 is thus of interest, as are the wave velocity, CR, and the growth rate parameter, a c ~ ,correspondingly t o this wave number. Development of the asymptotic forms for the three param) ~ exactly that in the previous section, eters ( a , CR, a c ~follows the only difference being that now c has a nonzero imaginary part. Thus the forms are
r, r*,
where aOhf and cOM are functions only of and p. The asymptotic problem reduces to solution of eq 3, together with the condition for maximum instability
which replaces eq 4.Before proceeding further, the physical significance of the predicted asymptotic forms (eq 5 ) will be considered. At first glance, the result that the growth rate parameter (CYCI)decreases as R e + m would seem to contradict physical intuition. Upon closer examination, however, the result can be made sense of. The growth rate is first expressed using dimensional qualities. For film flow on an inclined plane there is a relation between average film thickness and average film velocity: a, = (gh2/3u) (see, e.g., Yih, 1963). With this relation, both Q, and h may be expressed in terms of the Reynolds number and fluid properties: a, = ReZ/’(gv/3)’/’, h = Re’/’ ~ ~ ~ be (39/g)’/’. Hence, the nondimensional quantity ( c Y c I ) may ~ = (ac1)hlRe’/~(g~/9v)’/’t*, where t* is written ( ~ C I ) I \(a,/h)t* the dimensional time. For large Reynolds numbers, where the forms (eq 5 ) may be used, the growth expression is
(7) The maximum growth rate, accordifig to this analysis, approaches a constant value (dependent on p ) for any given fluid as R e m. According to the stability analysis leading to eq 1, the disturbance is assumed to grow in time. Of course, in any experimental study of the falling film, the disturbance grows in distance down the plane. The results of the analysis may be compared with experiment if spatial and temporal growth are equivalent; and they will be equivalent if the characteristic growth time ( t * / a c ~ t )is large compared to the time (AM/ -+
504 Ind. Eng. Chem. Fundam., Vol. 11, No. 4, 1972
C R M ~ , ) for wave travel over one wavelength. That is, the theory will be expected to give worthwhile results if
Assuming that the inequality (8) is satisfied, the result (eq 7) may then be employed for spatially growing disturbances. An infinitesimal disturbance growing on the film will first be detected below the inlet a t a distance, L , the line of wave inception, which is proportional to the characteristic growth time ( t * / ( a c ~ ) ~and t ) , proportional to the fluid velocity, a,; Le., L a ( t * . i L , / ( a c ~ ) h f t ) .As noted above, the average velocity, a,, is proportional to the two-thirds power of the Reynolds number, so that for large Reynolds number where, according to eq 7 , the characteristic growth time is independent of Re, we predict that L 0: Re2/’. The asymptotic result for the wave number, ah1 = ~~~h1Re’/’, may be converted to a form for the wavelength by noting the similar dependence of h upon Reynolds number. Thus (9) The wavelength of maximum instability is therefore predicted to be independent of Reynolds number for a given m . If the theory is correct, the inequality (8) fluid as R e may now be seen to be satisfied for large Re, since the lefthand side is proportional to Re-’l3. The asymptotic form C ’ R S ~= c o ~ ~ R e - * /when ’, converted to the dimensional wave velocity CR*, yields -+
CR* = us
+
‘/3
=
CORM(%)
(”2
Re2/’
+
(9’’ (10)
CORM)
T h a t is, the difference between the wave velocity, CR*, and the surface velocity, u, (both of order Re213)becomes a constant, dependent only on 0and fluid properties as R e + m . large Reynolds Number, Large-[ Asymptote
Having asymptotic expressions for large Reynolds number, one is tempted to follow this with the consideration of large values of the surface tension parameter {. It is to be expected that the wave number of maximum instability, ah1, will decrease as surface tension becomes large. This is so, qualitatively, because the greater the surface tension, the less deformable is the surface and hence the region of instability is confined to very long waves. Also, because e’ = 3//z a t a: = 0 and because C Y ~ I decreases with increasing it is expected that C ’ R J ~ will increase as increases. I n the analysis we start with eq 3 and 6 for the maximum instability and then allow to approach infinity. If it is again required that criteria I, 11, and I11 be met by the asymptotic form, then the assumption that CYOM ao~j--“,C O ~ I COO[*” requires introducing a “compression” (as opposed to a “stretch”) of the inner region in the parameter j-. The inner region is now assumed to have thickness of the order Re-’/’{’, where clearly we must require that the thickness be still very small. It is concluded that the only possible values which may be assumed by m, n, and k are m = k = 1/11, n = 4/11. Thus as Re m and -P m in such a way that Re-’/’j-’’” (f r*) ‘I’
Re,
L.
e
C 0,
s
Summary
=
j-*
The main conclusions to be drawn from the present work on the falling film stability problem are as follows. (a) For any fluid, the growth rate and the wavelength of the most unstable wave approach constant values as Reynolds number becomes large. (b) The point of wave inception, L , increases with Reynolds number with asymptotic form L Re2/’ as R e -+ m . The wave number, wave velocity, and growth rate have, as Re + m , { + m , asymptotic forms given by the first three columns of Table I. The last three columns are results obtained elsewhere by Yih (1963), Benjamin (1957), and Anshus and Acrivos (1967), respectively, and are included here for completeness.
-
X
=
v
=
4
= = =
p
u
1P (-”’)$(y
, surface elasticity parameter
wavelength. dimensional kineinatye viscosity, (L2/T) #(y), disturbance stream function, dimensionless density, X , L3 surface tension, M / T 2
SUBSCRIPTS 0, 00 = refer to asymptotic coefficients, meaning depends on context I = imaginary part &I = refers to maximum amplification S = refers to neutral stability R = real part Literature Cited
Nomenclature
+ i c ~ nondimensional , complex wave velocity
= CR = -
C C’
CR
*
9
Q* h i
L Re Re,
a
aa
US
?!
3 ‘
i 2
= dimensional wave velocity ( L , / T ) = g* sin p ( L / T 2 ) = acceleration of gravity, dimensional
= film thickness, dimensional =
dq
(L!T2)
(L)
line of wave inception, dimensional ( L ) = (aah/v)Reynolds number = critical Reynolds number = 3 / 2 ( 1 - yz), nondimensional undisturbed fluid velocity = dimensional average fluid velocity ( L I T ) = fluid velocity a t surface, dimensional ( L I T ) = nondimensional coordinate normal to surface =
GREEKLETTERS O( = 2nh/h, wave number, nondimensional 0 = angle of inclination from horizontal (0 = n / 2 for vertical film) y = surface concentration of insoluble surface-active material ( M / L 2 ) = ! ! surface tension parameter, dimensionless jP
508
(”>”, sv4
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 4, 1972
Anshus, B. E., Acrivos, A., Chem. Eng. Sei. 22, 389 (1967). Anshus, B. E., Goren, S. L., A.I.Ch.E. J. 12, 1004 (1966). Benjamin, T. B., J. Fluid Mech. 2, 615 (1957). Benjamin, T. B., Arch. Mech. Stosowanej 16, 615 (1964). Binnie, A. X., J . Fluid Mech. 2, 551 (1957). Chandrasekhar, S., “Hydrodynamic and Hydromagnetic Sta.bility,” Oxford University Press, London, 1961. Jones, L. O., Whitaker, S., A.I.Ch.E. J . 12, 525 (1966). ENG.CHEM.,FUNDAM. 9, 107 Krantz, W. B., Goren, S. L., IND. f 1970). \ - -
- I
Krantz, W. B., Goren, S. L., IND.ENG.CHEW,FUNDAM. 10, 91 (1971). Lin, C. C., “The Theor of Hydrodynamic Stability,” p 28, Cambridge Universit Sress, New York, X. Y., 1955. Lord Rayleigh, Phil. d g . 32, 529 (1916). Stainthorpe, F. P., Allen, J. M., Trans. Inst. Chem. Eng. 43, T85 (1965). Tailby, S. R., Portalski, S., Trans. Inst. Chem. Eng. 39, 328 (1961). Tailby, S. R., Portalski, S., Trans. Inst. Chem. Eng. 40, 114 (1962). Vidal, A., Acrivos, A., J . Fluid Mech. 26, 807 (1966). Whitaker, S., IKD. ENG.CHEW,FUNDAM. 3, 132 (1964). Yih, C. S.,Phys. Fluids 6 , 321 (1963). Van Dyke, M.,“Perturbation Methods in Fluid Mechanics,” Academic Press, New York, N.Y., 1964. RECEIVED for review August 5, 1971 ACCEPTED July 31, 1972
