B.R. Duffy and H.K. Moffatt- Flow of a viscous trickle on a slowly varying incline

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THE CHEMICAL

ENGINEERING

JOURNALThe Chemical Engineering Journal 60 ( 1995) 141-146

Flow of a viscous trickle on a slowly varying incline

B.R. Duffy a, H.K. Moffatta Department of Mathematics, University of Strathclyde. Livingstone Tow er, 26 Richmond St , Glasgow GI I X H , U K

Department of Applied M athematics and Theoretical Physics, Silver St , Cambridge CB3 9EW, U K

Received 30 January 1995; accepted 8 Ma y 1995

Steady flow of a thin trickle of viscous fluid down an inclined surface is considered, vi a a thin-film approximation.Results obtained for

down the topside or underside of an inclined plane are used in a simple way to approximate flow down a non-plane surface.

Viscous flow; Flow ; Thin film approximation

Introduction

Many viscous flow problems of practical importance

e surfaces whos e effects c ontribute significantly

attention is that of the ‘draining’ of viscous films down

l run dow n

a uniform ‘trickle’; if the film is too wide,

each of which will run dow n the surface as a

ckle. W e consider the steady behaviour of such a ‘trickle’

(wh ich w e take to be supplied at a prescribed

ume f lux). W e ado pt a very simple approach to the prob-

-film approxim ation. This lead s to some

trickle on an inclined plane, and

ts are late r used in a rather crud e description of

W e consider only stead y flows, with fixed contact lines, so

culties associated with m oving contact lines are[ 11 , and the many references therein).

Unsteady flows have been considered (within a thin-film

theory) by, for example, Huppert [21 , Schwartz [31, Lister

[41 and Moriarty et al. [51.

2. A uniform rivulet

Consider first the flow of a uniform rivulet down an

inclined solid plate ( se e Fig. 1) .Aspects of this problem have

been considered by Towel1 and Rothfeld [6 ] and by Allen

and Biggin [71, so som e of our results parallel so me of theirs,

though we obtain results in a form m ore useful later.

Suppose Newtonian fluid, of constant density p and vis-

cosity p, is undergo ing steady rectilinear flow in th e form of

a filament, down a plate inclined at an angle a to the hori-

zontal. Referred to Cartesian coordinates O q z as indicatedin Fig. 1, the velocity will be of the form u = u ( y , z) i , and

the Navier-Stokes equ ation s redu ce to

O = - px+pg sin a+p(u,,,+uu,),

free surface

z =M Y )\ t Z

- a 0 U

Fig. 1.A trickle of viscous liquid, of width 2a and maximum depth h,, flowing down a flat plate inclined at an angle (Y to the horizontal.

www.moffatt.tc



142 B.R. D u f i , H.K. Moffu tt/T he Chemical Engineering Journal 60 (1995) 141-146

o = - p y , o = -p , -pg cos a ( 1 )

(subscripts x, and z denoting partial dif feren tiation ). In a

‘thin-film’ theory, these e quation s approxim ate to

0= - p x +pg sin a +pZz,

to be integrated su bject to the boundary conditions

u =O on z = O

p - p a = - h” and u,=O on z = h ( 3 )

h = O a n d h ’ = T t a n p a t y = $ - a

Here z =h( y) is the free-surface profile, a prime denotes d /

dy , p is the pressure in the liqu id, pa s atmospheric pressure,a is the sem i-width of the trickle, g is the gravitational accel-

eration, y is the coefficient of surface tension, and p is the

contact angle at the three-phase contact line. We have

neglected the viscosity of the air above the liquid, and we

have taken the surface curvature to be h”. Also we have taken

y and p to be prescribed constants, nominally with p small.

A constant y means that any surface-tension-driven effectsassociated with surfactants or differential heating are ignored.

A constant p means that any contact-angle hysteresis is

ignored; this is probably reason able for these rivulets.

For brevity in writing the solution we indicate the threecases 0< a< 7 ~ 1 2 , = 7712 and rr/2 <a< rby ( i ) , ( i i ) and(iii ), respectively. Then the solution is as follows. The veloc-

ity is

pg sin au ( y , 2 ) =- 2hz- ’)

2 P

the free-surfa ce velocity us ( :=u(y , h ) ) i s

and the pressure is

p ( z >=pa-pgz c os a f t a n p J (y p g Ic o s a I )

coth B (i )

cot B ( i i i )

( 4 )

where

B = a ( p g I c o s a l l y ) ” ’ ( for a # r r / 2 ) (7 )

CO S B[- CO S B(iii)

. sin B

where

& = y l a ( - 1<& 1 ) ( 9 )

and the maximum depth h, of the liquid (given by

h, :=h(0) ) satisfies

tanh 1 B ( i )

tan 4 B (i i i )

PIc: a 1 ) ’ 2 hm -{ B(ii)

tan p

with interpretations for case (ii ) as above. Note that the scales

of h, and a in Eqs. (10) and (7) differ essentially by thesmall factor tan p (an d indeed in case (i i) h,/a = 1 tan p )this reflects the fact that the depth of the film is much less

than its width.

The solution is physically sensible only where h ( y )z0.

In case (iii) , h( y) in Eq. (8) is positive for all y satisfying

- <y <a only if B is restricted by 0<B < r r ; the physical

meaning of this upper limit is discussed below.

The volume flux of liquid running d ow n the plate is

a h ( v )_

Q =lu d z d y

- a 0

and with the nondimensionalization

- 9pp g cos2 aQ (for a# d 2 )’= tan3 p sin a

we have

Q= F(B) (13)

where

15B coth3 B- 15 c o t h 2 B -9 B c ot h B + 4 ( i )

F ( B ) = 1 2@ / 35 ( ii )

- 1 5 B c o t 3 B + 1 5 c o t 2 B - 9 B c o t B + 4 (iii)

( 14 )

with case (i i) interpreted as above. The m ean flow speed a( :=Q/Y-,h(y) dy ) satisfies

{

(B coth B- 1 ) - ’ ( i )

i = F ( B ) X ( i i )

( 1-B cot B) - I ( i i i )tan’pl tan a1

B ( > 0) is a Bond num ber for the flow. For a=

d 2 e have (15)- ,B = 0; nonetheless case (ii ) m ay be included formally in Eq.

( 6 ) , the interpretation being that factors of cos a are can-

celled before the limit a+ r/2 is taken.

The function Q =F ( B ) is plotted in Fig. 2 for the three

cases, for the physically relevant domain B > 0, Q>0, an dB < r in case (ii i) . n each case , if Q is prescribed then B is

determined uniquely, and hence a, h ( y ) , p and u(y, z ) mayhe free-surfa ce profile z = h( y) is given by



B.R. D u f i , H . K . Moffatt/ The Chemical Engineering Journal 60 (199.5)141-146 143

Fig, 2. The nondimensional flux 0 as a function of Bond number B in the

three cases (i), ( i i ) , (i i i) .

be foun d. Thu s the non-dimensional parameter Q essentially

determ ines the cross-sectional sha pe of the trickle; the para-

meters ( y/pg I cos a ) I’ and y l p ( together wi th a an d p )then determine the scales of the width, depth and speed. [T he

relationship between Q an d a, involving the parameters p, g ,

y , p, a and p, may be reduced by dimensional analysis to a

relationship between five dimensio nless parameters, say

Qp2sy lp3 , ’pgl y, py?’lgp4, a an d p; in the present approx-

imation this reduces dramatically to the relation (13)between just two parameters, Q and B.]

In cases (i) and (i i i) we have Q(B) - 12B4/35 and

h,-ia tan p as B+O. Also in case (i) we have

Q ( B ) 6B - 11 and A,- (y /p g cos a )”’ tan p as B - t w(and in practice these asymptotic forms are accurate for

B 24) .Result ( 27 ) of Towel1 and Rothfeld [61 (fo r a ‘wideflat rivulet’) corresponds here to case ( i) with B +W.

For case (ii ) (i.e. for a vertical wall) the above results are,

in a more explicit form,

Y tan pp ( z ) =pa+ - t an p , h=- a ’ - y ’ ) ,

a 2a

2a’pg tan’ p35P

U=

In an exac t analysis the free surface h ( y ) n this case would

be an arc of a circle; the approx imation to this in Eq. ( 16) is

correct to O( ’ ) . The expression for a in Eq. ( 16) (which

is independent of 7 ) agrees with results (24) and (25) of

[ 6 ] (but note that Eq. (2 5) in [6 ] should bef ( 6 )- 6831

35) .In case (iii) we have a - rr( y l p g I co s a I ) I 2 ,

Q (B ) - 1 5 ~ l ( ~ - B ) ~nd hmla tanp -2 / [ . r r ( . r r -B)] as

B + .rr, this breakdo wn in the solution as B+ corresponds

to the non-existence of a solution to the problem of a static

two-dimensional drop hanging from a horizontal support

when the mass of the drop is so large that the surfac e tension

cannot hold it up. (Actually our ‘thin-film’ approximation

would be invalid well before this breakdown could occur,since h , as B+T nevertheless eve n in a fully nonlinear

analysis one would expect a solution to exist only for

restricted B. Stability considerations would further restrictthe allowed values of B.) In addition, since the minimum

pressure in the liquid (occurring at the plate z = 0 in case

(i i i)) must exceed the vapour pressure p v ( < p , ) of the

liquid, B must also satisfy 0<B <B , <T , where cot Bo=

- p , p v ) / [ ( ypg I cos a I ) I’ tan p ] (so that Bo > d 2 ) .W e have taken p to be constant (i.e. we have ignored any

contact-angle hysteresis). In reality p may vary somewhat

along a contact line; however at an anchored (i.e. fixed)

three-phase line it is expected that pr< p < Parwhere p, and

p, are constants (the receding and advancing contact angles).

If p does vary along the c ontact line of th e rivulet then the

flow will not be truly unidirectional; however the above the-

ory may still be approximately co rrect if p varies only slowly.

Then the cross section of the filament will vary s lowly withdistance d own the plane: an increase in p (with Q fixed) will

be associated with a decrease in Q (since Q - ta n p ) 3 ) ,

with an increase in h, (since h,- (tan p ) ‘I4), ith a

decrease in B (since B - ta n p ) 3 / 4 ) , and therefore with a

decrease in a.

3. Flow of a trickle down a surface of slowly varying

slope

Nusselt, in his classic papers [ 8 ] , gave, amon gst many

other things, the ‘thin-film’ description of the steady flow of

viscous liquid round a circular cylinder of large diameter,

with its axis horizontal. H e made a ‘quasi-static’ assumption

that, at each station round the cylind er, the liquid depth and

velocity have forms appropriate to flow down aflat plate

inclined at the local value of the cylin der’s slope. He treated

films of effectively infinite lateral extent (in the direction of

the cylinder’s generator s), so that there are no contact lines;

also he neglected surface tension. Nusselt w as primarily con-

cerned with the thickening of a film (of water) due to con-

densation (of steam) ; if such condensation is ignored, his

solution fo r the liquid depth h, he velocity co mpon ent U downthe line of greatest slop e and the s urface velocity U, = U =,,)

may be written

pg sin a(2hz- ’ ) ,

pg sin a 2 P

Here Q 2 s the prescribed volume flux (pe r unit wid th) round

the cylinder, z is a local normal coordina te, and a is the localangle of slope of the cylinder’s surface ( with 0<a < T ) he

quantities U , h and U, thus depend on distance s measured

down the surface, this dependence arising via the slow vari-



144 B.R. D U B , .K . Moflatt/The Chemical Engineering Journal 60 ( 1995)141-146

ation of a with s. (I n fact, Nusselt’s results also provide the

solution for flow down surfaces of more general sh ape, pro-

vided that the surface slope varies sufficiently slowly, and

that 0<a<T . )

In a som ewh at similar way w e consider the flow of a trickle

of viscous liquid dow n a solid cylindrical surface with hori-

zontal generators an d with slowly varying slope; mor e pre-

cisely, we take the width of the trickle to be much less than

the radius of curvatu re of the solid sur face. Following Nusselt

we mak e a ‘quasi -static’ assumption that at each station thetrickle attains an equili brium sh ape and velocity appropriate

to flow down a plate inclined at the local slope a (with

0< a < T ) Thus at each station the solution is simply the

uniform rivulet presented in Section 2. Since a is now a

variable (depending on distance s measured down the solid

surface) , it is more suitable to nondimensionalize as1/ 2 112

c o t p h , y = ( $ y ,

9 w g Q

tan3 p

the new

sin a.Then the relationship Q = F ( B ) in Section 2 becomes

earlier ones B and Q

= F ( b\/ lcoscul)q cos2 a

si n a

Since F ( . ) is strictly monotonic in its argument we may

write formally

with

b = ( 3 5 q / 1 2 ) 1 / 4 ( b , , say) , for a = ~ / 2 ( 2 1 )

(and again Eq. ( 2 1 ) is an instance of Eq. ( 2 0 ) , withF ( B ) = 12B4/35 and with a set to ~ / 2fter co s a terms have

xi 4 x i2 c( 3ni41ll

Fig. 3. The nondimensional sem i-widthb(a) f trickles running around the

outside of a circular cylinder, for various values of the nondimensional fluxq . Here a= 0 corresponds to the top of the cylinder and a= T corresponds

to the bottom.

been cancelled). Thus with the nondimensional flux q pre-

scribed, the semi-width,b, of the trickle is determi ned by Eq .( 2 0 ) as a function of the slope a, and hence the complete

solution (valid for 0< a< T ) s determined precisely as in

Section 2, with z and y interpreted as local normal and trans-verse coordinates, respectively, and with a now variable.

Eq. ( 1 9 ) may be written

This leads to an equivalent parametric representation of the

function b (a ) ,which is somewhat easier to use in practice,

namely

( i )( i i )

r - s i n - ’ A (iii)

b = B / ( 1 - A 2 ) 1 ’ 4 ( 2 4 )

A = { [ ( F ( B ) ) 2 4 q2 ] ’2 - F ( B ) / 2q

where

( 2 5 )

with the parameter B satisfying

O < B < m ( i )

B+O ( i i )

O<B<.ir (iii)

Then from Eq. ( 8 ) the cross-sectional profile of the trickle at

each station a is given by

i ( y , a )=

fo r

2

H

( i i )

cos(y{ 1 cos a I ) - os B

d m in B(iii)

q = l O

I

0.01

n14 X I 2 c( 3x84I ll

Fig. 4. he nondimensional maximum depth H ( a) f trickles, as in Fig. 3 .



B.R. D U B , .K . Mo ffa tt/ Th e Chemical Engineering Journal 60 (1995)141-146 145

Fig. 5 . The curves in Fig. 4 rawn on a circular cylinder, indicating the shap e

of film predicted.

- b ( a ) < p < b ( a ) ( 2 8 )

Also the nondimensional maximum depth H ( a ) , efined by

H :=h (0,a) is given by

tanh( ill) / ( 1 - ( i )

H = Lb (ii) ( 2 9 )

{ z t a n ( $ B ) / ( l - A * ) “ ~ (iii)

with bl = ( 3 5 q / 1 2 ) I4.

Appropriate asym ptotic expan sions show that

1 1 a2

4- , H - I + - s a - 0- -

6 a 6

b b13 Tb - b , + - a , H - 1

- ( ),1(718 2 ) 2 7 2 2 - arr

a s a + -

2

and

The lateral extent of the trickle is specified by the co ntact

lines y = k ( a ) ,which vary slowly with s. Figs. 3 and 4

show respectively, for various values of q, the predicted

shapes of the bounding curve b ( a ) and of the maximum

depth H ( a ) for the case of flow round the outer surface of acircular cylinder with horizontal axis (so that s=CUR, here

R is the radius of the cylinde r). For q less than about 0.1 the

cross-sectional profile is rather uniform around the cy linder,except near a = 0 an d a = rr . For larger q there is more vari-

ation. Fig. 5 show s a cross-sectional view of the cylin der and

films (the film thickness being exaggerated for clarity ), and

Fig. 6 shows (fo r the case q = 1 ) exam ples of the film profiles

at various stations a around the cylinder. Note that, unlike

Nusselt’s solution (1 7) , this solution does not have top-to-

bottom symm etry, i.e. a profile at a station a on the ‘topside’

is different from the profile at the corresponding station

( r r - a ) on the ‘underside’. (Th is may be see n from the fact

that the forms of F fo r a < rr /2 an d a > rr /2 in Eq. ( 14) are

different,

This approximate solution breaks down near the top

(a=O) and the bottom ( a= r ) of the cylinder, as does

Nusselt’s solution for the ‘multi-tube’ case. Nusselt inter-preted the infinite values of h in his solutio n as representing

fluid falling onto or falling off the cyl inder, at a = 0 an d a= T,

respectively; in our solution, an infinite h occurs at a= r r

0.8 1~ a = x / 6

113 1/3

b -n - ( 1 5 ~ ( ~ -) , H - ( 84 )1 5 ~ (- )

as (Y+T ( 3 2 )

Thus this solution predicts that b+cc an d H + 1 as a+0,

and that b + r an d H + as a+ T,or any value of q. Also

one can show that dblda < 0 fo r 0< a G ~ l 2nd that db/d a > i b tan a fo r r r / 2 < a< r.

, a = x / 2

t Y

-1.5 -1 -0.5 0 0.5 1 1. 5

Fig. 6 . Examples of film profiles on a circular cylinder, fo r the case q = 1 in

F i g .3 : p l o t o f ( p g / y ) ” % c o t P a s a f u n c t i o n o f ( pg / y ) ”Zyf r om Eq. (27).

at various stations a ound the cylinder. For this value of q , the maximum

film depth H(a) ssentially decreases with a until a=80°, and then

increases; also the maximum width 26(a) ecreases with a ntil a= 1loo,

and then increases.



146 B.R. D u n , H .K . Mo#att/T he Chemical Engineering Journal 60 (1995) 141-146

only, but an infinite b oc curs at a = 0. [Incidentally, for flow

down the inside surface of the same cylinder, Figs. 3 and 4

merely need to be viewed right-to-left, that is, a = 7r s at the

top of the cylinder and a = 0 is at the bottom. T he singularities

in h and in b then occur at the topmost and bottom-most

points, respectively. ]

The asymptotic forms (30 )-(3 2) show that db /

d a = - b I3 / 18 at c u = ~ / 2 nd that dblda- S w as a + ~ ,so there must be a point where d b ld a =O , that is, the trickle

always attains its minimum width in a > ~ 1 2 . n the other

hand H ncreases from its value 1 near a = 0, and satisfies

H = b 1 / 2 a nd d H / d a = b I 3 / 1 2 a t a= 7rt2 (an d approaches,

03 as a+ T ) Thu s, dependin g on the value of b, (i.e. of q )

the liquid depth in 0< a < d 2 may either increase monoton-

ically with.a, or may initially inc rease , then decrease, then

increase again.

Some aspects of these flows accord with common experi-

ence. Simple ‘kitchen’ experiments were performed, with

trickles of syrup ( of width - -15 mm and depth - -8 m m)running round the curved side of a large cooking pot (of

diameter - 0 cm ), lying on its side; the experiments showed

rough ag reem ent with the curves in Figs. 3-5. Better exper-

iments would be needed to test whether the agreement is more

than superficial.

Acknowledgements

This work was supp orted by the Science Research Councilunder grant no. GRlAl5993.4.

References

[ 11 S.H. Davis, Contact line problems in fluid mechan ics, Trans. ASME J.

Appl. Mech. , 50 (1983) 977-982.

[ 2 ] H.E. Huppert, Flow and instability of a viscous current down a slope,

Nature, 300 (1982) 42742 9.

[ 3 ] L.W. Schwartz, Viscous flows down an inclined plane: instability and

finger form ation, Phys. Fluids, A I (3) (1989) 443-445.

[ 4 ] J.R . Lister, Viscous flows down an inclined p lane from point and line

sources, J. Fluid Mech ., 242 (1992) 631-653.

[ 5 ] J.A. Moriarty, L.W. Schwartz and E.O. Tuck, Unsteady spreading of

thin liquid films with small surface tension, Phys. Fluids, A3 (1991)

[6 ] G.D. Towel1 and L.B. Rothfeld, Hydrodynamics of rivulet flow, J. Am.

Inst. Chem. Eng., 12 (1966) 972-980.

[71 R.F. Allen and C.M. Biggin, Longitudinal flow of a lenticular liquid

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B.R. Duffy and H.K. Moffatt- Flow of a viscous trickle on a slowly varying incline

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