International Journal of Multiphase Flow 130 (2020) 103354

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International Journal of Multiphase Flow

journal homepage: www.elsevier.com/locate/ijmulflow

Linear stability analysis of evaporating falling liquid films

Hammam Mohamed, Luca Biancofiore ∗

Department of Mechanical Engineering, Bilkent University, Bilkent, 06800, Ankara,Turkey

a r t i c l e i n f o

Article history:

Received 15 January 2020

Revised 14 May 2020

Accepted 19 May 2020

Available online 23 May 2020

Keywords:

Falling liquid films

Hydrodynamic stability

Phase change

Thermocapillarity

Thermal instabilities

a b s t r a c t

We consider the linear stability of evaporating thin films falling down an inclined plate. The one sided-

model presented first by “Burelbach, J.P., Bankoff, S.G., Davis, S.H., 1988, Nonlinear stability of evaporat-

ing/condensing liquid films, Journal of Fluid Mechanics 195, 463–494. ” was implemented to decouple

the dynamics of the liquid than those of the vapor at the interface, at which the evaporation is modeled

based on a thermal equilibrium approach. We consider the base state solution derived by “Joo, S., Davis,

S.H., Bankoff, S., 1991, Long-wave instabilities of heated falling films: two-dimensional theory of uniform

layers, Journal of Fluid Mechanics 230, 117–146. ” which is based on the slow evaporation assumption. In

previous works, only low dimensional models. i.e. the long wave theory, have been analysed for evapo-

rating liquid films. Conversely in this paper, we extend the Orr-Sommerfeld eigenvalue problem for a film

falling down a heated wall to include evaporation effects namely, vapor recoil and mass loss. As expected,

we observe that the long wave theory fails in predicting the correct behavior when the inertia is strong

or the wavenumber k is large. We confirm that the instability induced by vapor recoil (E-mode) behaves

in a similar fashion to the instability due to the thermocapillary effect (S-mode). Both the S-mode and

the E-mode can enhance each other, specially, at low Reynolds numbers Re . Moreover, we examine the

perturbation energy budget to have an insight into the instabilities mechanism. We show that the pres-

ence of evaporation adds a new term corresponding to the work done by vapor recoil at the interface

(VRE). We also find that the main contributor to the perturbation kinetic energy in the unstable E-mode

is the work done by shear stress while VRE is negligible, unless Re

2 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Fig. 1. Schematic diagram of an evaporating thin film flowing down an inclined

surface. The local film thickness is a function of space and time h ( x, z, t ), and h̄ is

the mean film thickness.

b

S

t

p

l

s

w

f

j

v

s

o

a

t

a

w

m

i

c

B

s

d

s

m

t

a

r

2

c

A

e

y

r

o

h

f

a

T

c

wave approximation. They obtained a quadratic equation for the

critical Reynolds number, where one root represents the modified

H-mode by thermocapillary effects, while the other root repre-

sents the thermocapillary instability. They also showed that as the

Marangoni number increases, the two unstable regions approach

each other, until the flow is unstable for all Reynolds numbers.

Moreover, Goussis and Kelly (1991) classified the instabilities due

to thermocapillary effects as two types, (i) the long wave instabil-

ity (S-mode) at small wavenumbers which is due to the modifi-

cation of the surface tension due to the temperature gradient at

the interface as it deforms, and (ii) a short wave type instability

in the shape of convective rolls (P-mode) at finite wavenumbers,

which results from the interaction between the temperature gradi-

ent across the film with the base state velocity.

When the liquid is volatile, evaporation across the interface

causes a discontinuity in the fluid velocity and the linear mo-

mentum due to the rapid change in the fluid density, and since

the vapor density is much smaller than the liquid density, the

vaporized particles at the interface accelerate dramatically, caus-

ing a back reaction called vapor recoil . The effect of phase change

on the stability of falling films was studied by many authors e.g.

Bankoff (1971) and Spindler (1982) . Palmer (1976) studied the lin-

ear stability of rapidly evaporating films and found that vapor re-

coil has a significant destabilizing effect on the hydrodynamic in-

stability. Plesset and Prosperetti (1976) and Palmer (1976) derived

a constitutive equation to relate the mass flux across the inter-

face and the interfacial temperature based on the kinetic theory.

Moreover, Burelbach et al. (1988) considered a horizontal volatile

film, and derived the one-sided model which assumes that the va-

por is mechanically and thermally passive which allows the sep-

aration of the dynamics of the liquid than those of the vapor.

They also implemented a long wave approximation to derive a

Benney type evolution equation describing the effects of thermo-

capillary, evaporation, surface tension and intermolecular forces on

the dynamics of the film. Joo et al. (1991) extended the work of

Burelbach et al. (1988) to include the effect of gravity and studied

the non-linear evolution of evaporating falling liquid films. Both

authors utilized the evolution equation to study the linear sta-

bility of these films and showed that instability mode induced

by vapor recoil behaves similarly to that caused by thermocapil-

lary effect (S-mode). The effect of non-volatile dissolved surfac-

tant on the linear stability of evaporating films was discussed by

Danov et al. (1998) for long wave disturbances.

In more recent work, Shklyaev and Fried (2007) generalized the

one-sided model by Burelbach et al. (1988) by accounting for the

energy flux across the interface and the effective pressure. They

studied the linear stability of evaporating thin films by applying

a long wave approximation and showed that these two effects

are significant for liquids with relatively fast evaporation such as

molten metals, while the model reduces to Burelbach et al. (1988) ’s

one for liquids with slow evaporation, such as water or ethanol.

Sultan et al. (20 04, 20 05) derived a two-sided model that takes

the dynamics of the vapor into account. They studied the lin-

ear stability of flat films in the case of diffusion-limited evapora-

tion. Mikishev and Nepomnyashchy (2013) revisited the problem of

evaporating films with presence of insoluble surfactants on the in-

terface and studied the linear stability of long waves disturbances

for nonequilibrium and quasi-equilibrium evaporation.

For non-volatile liquid films, there exist higher order models

that overcome the small Reynolds number limitation of the long

wave theory. One example is the integral boundary layer approxi-

mation (IBL) which was formulated by Shkadov (1967) for isother-

mal films and then extended by Kalliadasis et al. (2003) for heated

films. The IBL achieves better results in the region of moder-

ate Reynolds numbers but suffers from a 20% error in predicting

the threshold of the instability. This deficiency was later cured

y Trevelyan and Kalliadasis (2004) , Ruyer-Quil et al. (2005) and

cheid et al. (2005) who introduced ”weighted residual models”

hat have a better agreement with the Orr-Sommerfeld eigenvalue

roblem.

It is evident at this stage that most of the contributions to

inear stability analysis of liquid films are based on low dimen-

ional models. Especially when it comes to volatile liquid films

hich are only modeled by long wave approximation that suf-

ers from severe limitations on inertia and wavenumber. The ob-

ective of this work is to derive the Orr-Sommerfeld (OS) eigen-

alue problem for evaporating falling films. This will allow us to

tudy the instability induced by evaporation effects (E-mode) with-

ut the restrictions implied by the long wave approximation. We

lso utilize the OS problem in studying the interaction between

he different instability modes (H-mode, S-mode and E-mode) for

wide range of Reynolds numbers and wavenumbers. Moreover,

e analyze the perturbation energy budget in order to define the

echanisms responsible for the evaporation instability. This paper

s structured as follows. In Section 2 we present the mathemati-

al formulation of the problem based on the one-sided model by

urelbach et al. (1988) and we obtain the time-dependent base

tate under the assumption of slow evaporation. In Section 3 , we

erive the general Orr-Sommerfeld eigenvalue problem for both

treamwise and transverse direction, but focusing only on the for-

er one in the following parts of the paper. The effect of evapora-

ion on the temporal stability and the perturbation energy budget

re examined in Section 4 . Finally, we briefly discuss and summa-

ize the results in Section 5 .

. Problem formulation

Fig. 1 illustrates an evaporating thin film falling down an in-

lined wall at an angle β with respect to the horizontal direction. three-dimensional (3D) Cartesian coordinate ( x, y, z ) is consid-

red, where x is in the streamwise direction (direction of the flow),

is the coordinate normal to the wall, and z is the spanwise di-

ection. The wall is fixed at y = 0 , while the interface is a functionf space and time located at y = h (x, z, t) . The wall is uniformlyeated to a fixed temperature T w . The liquid is volatile and there-

ore mass flux across the interface is present ( J ). The vapor is fixed

t a constant temperature and pressure T ∞ and P ∞ , respectively.his section will present the system of equations and boundary

onditions governing the flow illustrated in Fig. 1 . Additionally, a

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 3

p

s

e

2

e

∇

∂

∂

w

y

(

a

s

a

u

T

m

a

t

v

m

v

i

−

w

t

t

t

PPP

w

z

w

s

a

J

w

d

J

w

h

f

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∂

∂

∂

∂

P

a

u

T

t

−+

−

J

K

d

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M

roper scaling which will lead to the dimensionless parameters is

uggested. A base state solution is presented based on the slow

vaporation assumption.

.1. Governing equations

The governing equations, namely, 3D Navier-Stokes and energy

quation for the film illustrated in Fig. 1 are as follows:

· � u = 0 , (1a)

t � u + � u · ∇ � u = −ρ−1 ∇p + ν∇ 2 � u + � g , (1b)

t T + � u · ∇ T = κρ c p

∇ 2 T , (1c) here � u = (u, v , w ) is the fluid velocity in the directions x, , and z, p is pressure, � g is the gravitational body force � g = g sin β, −g cos β, 0 ), T is temperature, κ is thermal conductivity,nd c p is the specific heat. The governing equations ( 1a –1c ) are

ubjected to no-slip, no-penetration and fixed temperature bound-

ry conditions at the heated wall ( y = 0 ): � = 0 , and (2a)

= T w . (2b) With regards to the interface boundary conditions, we imple-

ent the one-sided model derived by Burelbach et al. (1988) by

ssuming that the density, viscosity and thermal conductivity of

he vapor are much less than those of the liquid. Therefore the

apor above the free surface is considered mechanically and ther-

ally passive unless it is multiplied by a large value of the vapor

elocity. The normal stress boundary condition at the vapor-liquid

nterface is:

J 2

ρ(v ) − ( P P P · � n ) · � n = 2 Hσ (T ) , (3a)

here the first term represents the vapor recoil effect with ρ( v ) ishe vapor density, P P P is the stress tensor, � n is a unit vector normal

o the interface, and H is the mean curvature of the interface. The

angential stress boundary conditions are:

· � n · � τi = ∇ s σ · � τi , i = 1 , 2 (3b) hich balances the tangential shear stress in the directions ( x,

) with the surface tension gradient at the liquid-vapor interface,

here � τi are unit vectors tangential to the interface and ∇ s is aurface gradient. The kinematic boundary condition can be written

s:

= ρ( � v − � v s ) · � n , (3c) here � v s is the interface velocity. Finally, the energy boundary con-

ition is:

[ L + 1

2

[J

ρ(v )

]2 ] = −κ∇T · � n , (3d)

here L is the latent heat of vaporization. The left hand side shows

ow the heat conducted across the film is consumed at the inter-

ace. First term represents the heat used to vaporize the liquid par-

icles, while the second term shows the kinetic energy given to the

apor particles. In order to relate the mass flux across the inter-

ace to the interfacial temperature, we implement the constitutive

quation ( Palmer, 1976 ):

= (

αρ(v ) L

T 3 2

)(M w

2 πR g

) 1 2

(T s − T ∞ ) , (3e)

∞

here T s is the interface temperature, α is the accommodation co-fficient, M w is the molecular weight, and R g is the universal gas

onstant.

.2. Non-dimensional and scaled parameters

The governing equations and boundary conditions (1) - (3) are

caled using the following length and viscous scales where the

tarred quantities are non-dimensional:

(x, y, z) → (x ∗, y ∗, z ∗) ̄h , (u, v , w ) → (u ∗, v ∗, w ∗) νh̄

, t → h̄ 2

νt ∗,

→ T s + T ∗T , p → ρν2

h̄ 2 p ∗, J → κT

h̄ L J ∗.

hese scales were shown to be suitable for isothermal films by

illiams and Davis (1982) , and also for horizontal heated films

ith evaporation by Burelbach et al. (1988) . The scaled governing

quations are as follows (with omitting the stars):

x u + ∂ y v + ∂ z w = 0 , (4a)

t u + u∂ x u + v ∂ y u + w∂ z u = −∂ x p + ∂ xx u + ∂ yy u + ∂ zz u + Re, (4b)

t v + u∂ x v + v ∂ y v + w∂ z v = −∂ y p + ∂ xx v + ∂ yy v + ∂ zz v − Ct, (4c)

t w + u∂ x w + v ∂ y w + w∂ z w = −∂ z p + ∂ xx w + ∂ yy w + ∂ zz w, (4d)

(∂ t T + u∂ x T + v ∂ y T + w∂ z T ) = ∂ xx T + ∂ yy T + ∂ zz T , (4e) long with the scaled wall boundary conditions at y = 0 : = v = w = 0 , (5a)

= 1 , (5b) he scaled interface boundary conditions at y = h (x, z, t) are:

p = 3 2

V r J − ( 3� − (2 M/P ) T ) (∂ xx h [1 + (∂ z h ) 2 ] + ∂ zz h [1 + (∂ x h ) 2 ]

− 2 ∂ x h∂ z h∂ xz h )/n 3

+ 2 n 2

[∂ x u (∂ x h )

2 + ∂ z w (∂ z h ) 2 + ∂ x h∂ z h (∂ x w + ∂ z u ) − ∂ x h (∂ y u + ∂ x v ) − ∂ z h (∂ z v + ∂ y w ) + ∂ y v ] (6a)

2 M

P [ ∂ x T − ∂ x h∂ y T ] /n = 2 ∂ x h (2 ∂ y v − 2 ∂ x u )

[1 − (∂ x h ) 2 ](∂ y u + ∂ x v ) − ∂ z h (∂ x w + ∂ z u ) − ∂ x h∂ z h (∂ z v + ∂ y w ) (6b)

2 M

P [ ∂ z T − ∂ z h∂ y T ] /n = 2 ∂ z h (2 ∂ y v − 2 ∂ z w )

+ [1 − ( ∂ z h ) 2

]( ∂ z v + ∂ y w ) − ∂ x h ( ∂ x w + ∂ z u )

− ∂ x h∂ z h ( ∂ y u + ∂ x v ) EJ = [ v − ∂ t h − u∂ x h − w∂ z h ] /n, (6c)

+ E 2

D 2 L J 3 = [ ∂ x h∂ x T − ∂ y T + ∂ z T ] /n, (6d)

J = T . (6e) The non-dimensional constitutive equation (6e) governs the

egree of non-equilibrium at the interface through the non-

quilibrium number K . It linearly approximates the vapor pres-

ure force for mass transfer across the interface Oron et al. (1997) .

oreover, when the liquid is non-volatile ( E = 0 ), K represents the

4 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Table 1

Definition and physical meaning of the scaling parameters.

Parameter Definition Physical meaning

Reynolds number ( Re ) h̄ 3 g sin β/ν2 Ratio of inertia to viscous forces

Inclination number ( Ct ) h̄ 3 g cos β/ν2 Measure of the hydrostatic pressure

Kapitza number ( �) σo ̄h / 3 ρν2 Ratio of the surface tension force to the inertia

Prandtl number ( P ) νc p ρ/ κ Ratio of the momentum diffusivity to the thermal diffusivity

Marangoni number ( M ) γT ̄h / 2 μκ Ratio of the surface tension gradient force to the inertia

Evaporation number ( E ) κT / ρL ν Ratio of the viscous time scale to the evaporative time scale

Density ratio number ( D) 3 ρv /2 ρ Ratio between the liquid and vapor densities Vapor recoil number ( V r ) E

2 / D Measure of the vapor recoil force Latent heat number ( L ) 8 ̄h 2 L/ 9 ν2 Measure of the latent heat Non-equilibrium number ( K ) (κT 3 / 5 s /αh̄ ρ

(v ) L 2 )(2 πR g /M w ) 0 . 5 Measure of the degree of non-equilibrium at the interface

m

t

o

e

A

t

h

J

U

P

c

a

inverse of the Biot number ( κ g /2 αν2/3 ), where it becomes a mea-sure of the heat transfer from the liquid to the vapor. The reader

is referred to Table 1 for the dimensionless groups showing in Eqs.

(4) - (6) and their physical interpretation.

Worth mentioning that the evaporative time scale t E =ρh̄ 2 L/κT is not utilized explicitly but instead it shows up inthe scaled boundary conditions within the evaporation number

E . Moreover, we assume that the evaporative time scale is much

longer than the viscous time scale and, therefore, E is assumed to

be small as in Burelbach et al. (1988) .

2.3. Base state solution

The system of governing equations and boundary conditions

Eqs. (4) - (6) has a base state solution analogous to a spatially uni-

form and time dependent flow. Burelbach et al. (1988) obtained

the base state solution for evaporating horizontal films by ex-

panding the system in powers of the evaporation number E under

the assumption of slow evaporation, then taking the leading order

as the base state solution. This can be justified by the fact that

for slow evaporation, the base state develops in time and space

Fig. 2. (a) Film height, (b) mass flux across interface, (c) temperature gradient across the

T s is the interface temperature, T w is the wall temperature and t D is the dry out time.

uch slower than the exponentially growing/decaying perturba-

ions ( Shklyaev and Fried, 2007 ). By following the same procedure

f Burelbach et al. (1988) , we obtain the base state solution for

vaporating falling films by including gravitational effects (refer to

ppendix A for derivation details). The base state solution is ob-

ained as follows:

¯ (t) = −K +

(K 2 + 2 K + 1 − 2 Et

) 1 2

, (7a)

̄(t) = (

K 2 + 2 K + 1 − 2 Et )− 1 2

, (7b)

¯ (y, t) = Re y

(h (t) − y

2

), (7c)

̄(y, t) = Ct (

h (t) − y )

+ 3 2

V r ̄J 2 , and (7d)

¯ (y, t) = 1 − J̄ y. (7e)Fig. 2 shows the behavior of the base state for three different

ases. Starting with the non-volatile case ( K −1 = 0 ), the temper-ture gradient across the film is zero, and therefore there is no

film and (d) horizontal velocity profile, for the base state through time. Note that

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 5

e

F

t

N

t

t

d

t

m

t

a

p

c

t

m

p

3

s

e

p

e

K

a

E

c⎧⎪⎨⎪⎩w

c

o

t

s

v

r

r

E

t

i

w

φ

τ

a

φ

[

D

J

fi

i

p

G

s

e

t

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B

fi

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τ

η

[

D

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b

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vaporation. The film thickness remains constant as illustrated in

ig. 2 (a), while the velocity has a parabolic profile ( Fig. 2 (d)). Note

hat the classical Nusselt film solution for isothermal falling films

usselt (1916) can be retained by setting ( E = 0 , K −1 = 0 ). As forhe non-equilibrium case, where the parameter K is finite ( K � = 0),he film thickness decreases with time until it reaches zero at the

ry out-time ( t D = (1 + 2 K) / 2 E). Moreover, the interface tempera-ure T s approaches the wall temperature T w as the film thins. The

ass flux increases with time and its maximum is at the dry out

ime. For the quasi-equilibrium case ( K = 0 ), dry out occurs within shorter time ( t D = 1 / 2 E) than in non-equilibrium case. The tem-erature difference between the interface and the wall is constant,

onsequently the increase in the heat flux across the film is higher

han that in the previous scenario ( K � = 0), this explains why theass flux is higher in this case. The velocity maintains its parabolic

rofile in all the three scenarios.

. Linear stability analysis

Long wave theory has been extensively used in the literature to

tudy the linear stability of evaporating falling liquid films, see for

xample Joo et al. (1991) and Tiwari and Davis (2009) . In this pa-

er, we use a different approach by extending the Orr-Sommerfeld

igenvalue problem for thin films flowing over a heated wall

alliadasis et al. (2011) to include evaporation effects. This is

chieved by considering the stability of the base state solution in

qs. ( 7a - 7e ) with respect to infinitesimal perturbations, then de-

omposing the perturbation into normal modes as follows:

˜ v ˜ T ˜ h ˜ J

⎫ ⎪ ⎬ ⎪ ⎭ =

⎧ ⎪ ⎨ ⎪ ⎩

φ(y ) τ (y ) ηJ

⎫ ⎪ ⎬ ⎪ ⎭ exp[ i ( � k · � x − ωt)] , (8)

here � x = (x, z), � k = ( k x , k z ) is the complex wavenumber, ω is theomplex angular frequency and i is the imaginary unit. We limit

ur study to temporal stability analysis, where the evolution of dis-

urbance in time is concerned. The disturbance wavenumber � k is

et to be real, then the problem is solved for the complex eigen-

alue c . The imaginary part of ω presents the temporal growthate ω i (subscript i is used to denote imaginary, while subscript denotes real), while c r = ω r /k is the phase velocity. Substitutingqs. 8 into the linearized perturbation equations (Eqs. 24 ) leads to

he generalized Orr-Sommerfeld eigenvalue problem for evaporat-

ng falling films (refer to Appendix B for detailed derivation):

(D 2 − k 2 ) 2 φ + i [(ω − k x ̄U )(D 2 − k 2 ) + k x D 2 Ū ] φ = 0 , (9a)

(D 2 − k 2 ) τ + P [ D ̄φ − i (ω − k x ̄U ) τ ] = 0 , (9b) ith boundary conditions at the wall

(0) = Dφ(0) = 0 , (10a)

(0) = 0 , (10b) nd at the free surface

(h ) + i 1 2 η(2 ω − 2 ̄U (h ) k x ) = 0 , (11a)

(D 2 − 3 k 2 ) + i (

ω − Ū (h ) 2

k x

)]Dφ(h )

= ηk 2 [

Ct + (

3� − 2 M P

̄(h ) )

k 2 ]

+ 3 k 2 V r ̄J J , (11b)

(D 2 + k 2 ) φ(h ) + 2 M [ ηD ̄(h ) + τ (h )] k 2 + ik x ηD 2 Ū (h ) = 0 , (11c)

P

τ (h ) + J = 0 , (11d)

= τ (h ) K

+ D ̄K

η. (11e)

Even though the Squire’s theorem does not apply for heated

lms, the streamwise perturbations are the most unstable us-

ng numerical integration of the 3D Orr-Sommerfeld eigenvalue

roblem, at least for small wavenumbers ( Smith and Davis, 1983 ;

oussis and Kelly 1991 ). We also found that evaporation has no

ignificant effect on the unstable mode induced by thermocapillary

ffect in the transverse direction (P-mode), therefore we consider

he limiting case of streamwise perturbations ( k x = k, k z = 0 ) only. The streamfunction is utilized as follows for convenience:

(y ) = −ik x ϕ(y ) . (12) y substituting in the system (Eqs. 9 - 11 ) and writing ω = kc, thenal form of the Orr-Sommerfeld eigenvalue problem for stream-

ise perturbations then becomes:

(D 2 − k 2 ) 2 ϕ + ik [(c − Ū )(D 2 − k 2 ) + D 2 Ū ] ϕ = 0 , (13a)

(D 2 − k 2 ) τ + P ik [ D ̄ϕ + (c − Ū ) τ ] = 0 , (13b)

(0) = Dϕ(0) = 0 , (13c)

(0) = 0 , (13d)

= ϕ(h ) [ c − Ū (h )] , (13e)

(D 2 − 3 k 2 ) + ik (c − Ū )] Dϕ(h ) − iηk [

Ct + (

3� − 2 M P

̄)

k 2 ]

−3 ikV r ̄J J = 0 , (13f)

(D 2 + k 2 ) ϕ(h ) + ik 2 M P

[ ηD ̄ + τ (h )] + D 2 Ū η = 0 , (13g)

τ (h ) + J = 0 , (13h)

= τ (h ) K

+ D ̄K

η. (13i)

The system of Eqs. 13 represents an eigenvalue problem that

as solved numerically using spectral collocation method based on

hebyshev polynomials ( Boomkamp et al., 1997 ; Trefethen, 20 0 0 ;

chmid and Henningson, 2012 ).

In particular, we used the MATLAB built-in function “eig”, based

n the QZ algorithm ( Moler and Stewart, 1973 ). Finally, the pertur-

ation amplitude ( φ, τ , η, J ), useful for the perturbation energynalysis conducted in Section 4.3 , can be obtained from the eigen-

alue problem solution.

. Results and discussion

We study the temporal stability of evaporating falling liquid

lms using the OS eigenvalue problem presented in Section 3 . The

S problem model alongside the numerical procedure is validated

or several cases in Section 4.1 . Then, we discuss the different in-

tability modes and study the effect of adding evaporation to the

roblem in Section 4.2 . Finally, Section 4.3 presents a detailed en-

rgy perturbation analysis in which we study the instability mech-

nisms.

6 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Fig. 3. (a) Growth rate comparison between the current OS solver (solid line) and LW approximation (circles) for β = 15 ◦, � = 15 , P = 1 , and K = 1 . (b) Neutral curve comparison between the current OS solver (solid line) and OS eigenvalue problem solved by Kalliadasis et al. (2011) (circles) for β = 15 ◦ .

Fig. 4. Growth rate comparison between the current OS solver (solid line) and LW approximation (circles) for k = 0 . 001 , β = 15 ◦, � = 15 , P = 1 , and K = 0 . 1 . (a) Growth rate versus Reynolds number for different values of V r and (b) growth rate as a function of time for E = 0 . 015 ..

b

�

l

(

t

e

m

v

a

(

i

4

p

o

o

t

u

4

e

F

i

t

s

4.1. Validation

The OS model and the numerical scheme are validated by com-

paring the results against several benchmarks in the literature. We

compare our results against those of the long wave (LW) theory

derived by Joo et al. (1991) for evaporating falling liquid films. The

following relationship was obtained by Joo et al. (1991) when Re is

small and k → 0: ˙ H

H = ω a (t) + ω m,a (t) − ikc a (t) , (14)

where ω a ( t ) is the temporal growth rate, c a ( t ) is the phase speedand, finally, the term ω m,a ( t ) is an additional real term which doesnot contribute to the growth rate. It only relates the initial per-

turbation amplitude to the decreasing thickness of the film and

does not effect the exponential instability Joo et al. (1991) . This

was also implemented in the OS model by ignoring the mass loss

term in the perturbation kinematic boundary condition Eq. (26a) ,

see Appendix B for more details.

First, we validated our model for heated non-volatile films

where thermocapillarity is taken into account, while evaporation

effects are ignored. The temporal growth rate ω i as a functionof the Reynolds number Re for a small wavenumber k = 0 . 001 isshown in Fig. 3 (a), excellent match is found between the two mod-

els for isothermal case (blue line), and also when the Marangoni

effect is added. Our model was also validated for larger k and Re by

replicating the neutral curves obtained by Kalliadasis et al. (2011) ,

see Fig. 3 (b). In order to match the Nusselt film scaling used in

Kalliadasis et al. (2011) which is based on the Reynolds num-

er, the non-dimensional parameters have to be chosen as follows,

= �∗Re 0 . 3 / 3 and M = M ∗Re 0 . 3 / 2 P, where �∗ = 250 . When the liquid is volatile, the only benchmark available in the

iterature to our knowledge is a stability analysis for long waves

Joo et al., 1991 ). Therefore, we only could validate our model in

he region of small k and Re . We first consider the vapor recoil

ffect with the absence of mass loss effect ( ̄h = 1 ). Again the twoodels agree very well in the small ( k − Re ) region for different

alues of the parameter V r , as shown in Fig. 4 (a). The results also

gree with the long wave theory in terms of the mass loss effect

̄h changes with time) when vapor recoil is constant ( V r = 2 . 25 ), ast can be seen from Fig. 4 (b).

.2. Temporal stability analysis

We examine several instability modes induced by different

hysical effects. We focus on the instability mode induced by evap-

ration effects in particular. Moreover, we discuss the interaction

f the instability modes when evaporation is present. The parame-

ers for the upcoming results are β = 15 ◦, � = 15 , P = 1 , K = 0 . 1 ,nless specified otherwise.

.2.1. Hydrodynamic instability (H-mode)

In the absence of any thermal effects, only one unstable mode

xists, the well known H-mode ( Kalliadasis et al., 2011 ), shown in

ig 5 . This mode is due to gravity effects induced by the surface

nclination. For small Reynolds number Re , the inertia is weak and

he hydrostatic pressure is stabilizing the flow. The inertia becomes

tronger as the Reynolds number increases, and at some point it

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 7

Fig. 5. Contours of the growth rate in the Re - k plane showing the H-mode for different Kapitza number � and inclination angle β .

o

R

i

g

p

s

s

5

n

F

fi

c

d

c

a

F

4

u

c

F

t

f

p

f

t

n

m

c

4

e

F

O

(

f

k

t

f

r

2

b

T

i

f

w

i

r

f

f

J

a

e

t

o

t

i

i

w

E

vercomes the hydrostatic pressure and destabilizes the flow for

e > 5 / 2 cot (β) as shown in Fig. 5 (a). The effect of decreasing the surface tension force by decreas-

ng the Kapitza number is shown in Fig. 5 (b). The unstable re-

ion expands along the wavenumber axis since short wavelength

erturbations are destabilized due to the decrease in surface ten-

ion. However, perturbations in the limit k → 0 are unaffectedhowing that the critical Reynolds number remains the same Re c = / 2 cot (β) . Furthermore, the consequence of changing the incli-ation angle β on the H-mode is easily observed by comparingigs. 5 (a) and 5 (c). As β increases, the destabilizing streamwiselm inertia increases, while the stabilizing hydrostatic pressure de-

reases. This yields to expanding the H-mode region, and also to

ecrease the critical Reynolds number Re c . We also recovered the

lassical result by Benjamin (1957) , where the film is unstable for

ll Reynolds numbers at β = 90 ◦ in the limit k → 0 as shown inig. 5 (d).

.2.2. Thermocapillary instability (S-mode)

Next, thermal effects are taken into consideration while the liq-

id is still non-volatile. For small M , a second unstable mode asso-

iated with the Marangoni effect appears for small Re as shown in

ig. 6 . This mode is called S-mode, which is a result of the surface

ension gradient forces due to the temperature gradient along the

ree surface. As Re increases, the film thickens causing the disap-

earance of the S-mode due to the stronger hydrostatic pressure

orce. Additionally, we observe by comparing Figs. 6 (a) and 6 (b)

hat the H-mode is further destabilized, since the critical Reynolds

umber diminishes. More interestingly as M increases, the two

odes combine into one unstable region, thus showing that they

an support each other, see Fig. 6 (c) ( Kelly and Davis, 1986 .

.2.3. Evaporation induced instability (E-mode)

The results presented in this section are associated with the

vaporation effect on the temporal instability of falling liquid films.

irst, we compare the effect of vapor recoil between the extended

rr-Sommerfeld eigenvalue problem and the long wave theory

Joo et al., 1991 ). Fig. 7 shows the temporal growth rate ω i as aunction of k for different values of Re . The two models agree when

is small and inertia is weak ( Re = 1 ). However, as Re increases,he agreement between the two models degrades as the inertial

orces become significant, where the error in the maximum growth

ate predicted by the long wave theory is approximately 60% and

50% for Re = 10 and Re = 15 , respectively. The cut-off wavenum-er is another benchmark to compare between the two models.

he long wave theory also fails with showing an increase of 15%

n the cutoff wave number obtained by the Orr-Sommerfeld model

or Re = 10 and an increase of 30% for Re = 15 . Hence, the longave theory significantly overpredicts the instability, demonstrat-

ng how the Orr-Sommerfeld problem provides a much more accu-

ate model to study the effect of evaporation on the instability in

alling liquid films.

The instability induced by vapor recoil behaves in a similar

ashion to the one induced by thermocapillarity as suggested by

oo et al. (1991) . Fig. 8 (a) shows that a new unstable region exists

t small Re for small V r (E-mode), while the unstable H-mode is

xtended. As we move along the Re axis, the E-mode disappears as

he hydrostatic pressure increases. Moreover after a critical value

f V r , the two modes combine forming one unstable region as ob-

ained in Fig. 8 (c).

Furthermore, the effect of mass loss (film thinning) is exam-

ned. If we compare the growth rate contours for a thinning film

n figs. 9 (b) and 9 (c) against a constant thickness film in fig. 9 (a),

e observe that the H-mode shrinks as the film thins, while the

-mode region is expanded. The former trend is intuitive because

8 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Fig. 6. Contours of the growth rate in the ( Re - k )-plane for different Marangoni numbers.

Fig. 7. Temporal growth rate ω i in versus the wavenumber k . We compare the OS

model (solid line) and LW expansion (dashed line) for different values of Reynolds

number when V r = 4 .

e

E

s

F

t

t

H

b

t

s

fi

4

l

p

i

o

g

b

T

R

1

v

d

c

f

c

f

e

t

as the film thins, the viscous forces become more dominant and

therefore more stabilizing. At the same time, the evaporation rate

becomes higher as the film thins, and thus the vapor recoil effect

is stronger enhancing the vapor recoil instability.

The combined effect of Marangoni (S-mode) and vapor recoil

(E-mode) is studied for two different values of the Reynolds num-

ber. Fig. 10 (a) shows the growth rate along the wavenumber for

Re = 0 . 5 . In the isothermal case the film is always stable at smallRe . The red (magenta) line shows the growth rate due to the in-

troduction of the Marangoni effect (vapor recoil) only. The combi-

nation of the effects of vapor recoil and thermocapillarity results

in a significant increase in (i) the growth rate and (ii) the band

of unstable wavenumbers. This indicates that two effects enhance

Fig. 8. Contours of the growth rate in the Re - k

ach other. At Re = 15 , Fig. 10 (b) shows how the S-mode and the-mode (and their superposition) support the H-mode as the in-

tability is enhanced for all the wavenumbers.

The effect of the non-equilibrium parameter K is displayed in

ig. 11 . For K = 0 , which represents the quasi-equilibrium case,he temperature gradient across the film is constant (see Fig. 2 (c)),

herefore the trough of a wave will experience higher mass fluxes.

ence the trough will have higher vapor recoil than a crest, desta-

ilizing the film. As the parameter K increases, corresponding to

he non-equilibrium cases, the film becomes less volatile. Con-

equently the vapor recoil effect is weaker, thus stabilizing the

lm.

.2.4. Phase speed analysis

The influence of vapor recoil and mass loss on the abso-

ute/convective nature of the instability is another important as-

ect to take into consideration. Despite a detailed spatio-temporal

nstability analysis would be required to carefully study the nature

f the instability, our model can be utilized to get some hints re-

arding this aspect by examining the phase speed of the pertur-

ation c r ( Biancofiore and Gallaire, 2012; Biancofiore et al., 2015 ).

he effect of vapor recoil on the phase speed in the E-mode (for

e = 0 . 5 ) and H-mode (for Re = 15 ) is shown in Figs. 12 (a) and2 (b), respectively. For both modes, the phase speed drops as the

apor recoil increases indicating that the vapor recoil tends to slow

own the instability and then, probably, hindering its convective

haracter for the given set of parameters. As for the mass loss ef-

ect, Fig. 12 (c) shows the phase speeds in the E-mode range. This

ould mean that the absolute nature of the instability would be

avored for this parameter set. Finally, the mass loss effect influ-

nces the H-mode by making it less convective as the decrease in

he phase speed could indicate, see Fig. 12 (d).

plane for different vapor recoil numbers.

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 9

Fig. 9. Contours of the growth rate in the Re - k plane for V r = 2 . 25 for different film thickness values.

Fig. 10. Temporal growth rate ω i in terms of the wavenumber k for different combinations of V r and M when (a) Re = 0 . 5 and (b) Re = 15 .

Fig. 11. Temporal growth rate ω i against the wavenumber k for different values of

the parameter K , when Re = 0 . 1 , and V r = 4 .

4

w

t

g

a

a

t

t

I

r

s

t

e

s

a

i

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p

t

n

4

t

I

h

o

K

e

p

p

t

l

E

t

K

.2.5. Discussions on the frozen interface assumption

The frozen interface assumption is an important aspect of this

ork. It is assumed that the instability timescale is much shorter

han the evaporative timescale, which means that the perturbation

rows/decays much faster than the film decaying thickness. This

ssumption is represented by the ratio E / ω i which must be rel-tively small. Burelbach et al. (1988) listed the physical parame-

ers and the dimensionless numbers for evaporating layers of wa-

er and ethanol in order to approximate the dimensionless groups.

n this paper we have chosen the parameters sets in the physical

ange proposed by Burelbach et al. (1988) to have the ratio E / ω i mall enough (maximum ~ 0.1). However, it should be noted that

his ratio can be larger by implementing different sets of param-

ters still in the physical range. Therefore, the frozen interface as-

umption would fail in these cases. In our work, this assumption is

lso more stringent than using the long wave assumption since the

nstability in low dimensional models is overestimated for large

avenumbers and Reynolds numbers (see Fig. 7 ). In this range of

arameters, then, the ratio ω i / E for the long wave theory is smallerhan for the OS problem and the frozen interface assumption erro-

eously seems to be respected for a larger range of parameters.

.3. Perturbation energy analysis

We conduct a perturbation energy analysis to understand how

he various physical mechanisms contribute to the instability.

t also elaborates on how the discussed instability modes en-

ance each other. The equation governing the energy budget

f the perturbation was first derived for isothermal films by

elly et al. (1989) . Thermocapillarity was later included in the

quation by Goussis and Kelly (1991) . Here, we follow the same

rocedure to include evaporation. First, we sum Eq. 24 (b) multi-

lied by ˜ u and Eq. 24 (c) multiplied by ˜ v . Afterwards we integratehe resulting equation across the film thickness and over the wave-

ength λ = 2 πk

,

1

2 λ

∫ h 0

∫ λ0

(∂ t + Ū ∂ x )( ̃ u 2 + ̃ v 2 ) dx dy = − 1 λ

∫ h 0

∫ λ0

D ̄U ˜ u ̃ v dx dy

− 1 λ

∫ h 0

∫ λ0

( ̃ u ∂ x + ̃ v ∂ y ) ˜ p d x d y

+ 1 λ

∫ h 0

∫ λ0

[ ˜ u 2 (∂ xx + ∂ yy ) + ̃ v 2 (∂ xx + ∂ yy )

] d x d y. (15)

q. (15) can be simplified by using the continuity equation and in-

erface boundary conditions to the final form:

IN + STE + HYD = REY + SHE + DIS + MAR + VRE (16)

10 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Fig. 12. Phase speed c r versus wavenumber k for (a) Re = 0 . 5 with no mass loss, (b) Re = 15 with no mass loss, (c) Re = 0 . 5 , V r = 2 . 25 and E = 0 . 015 (d) Re = 15 and V r = 2 . 25 and E = 0 . 015 .

r − Uc r −

e

W

t

d

V

e

r

a

e

a

t

s

o

d

u

e

where,

KIN = 1 2 λ

d

dt

∫ λ0

∫ h 0

( ̃ u 2 + ̃ v 2 ) dy dx,

HYD = Ct λ

∫ λ0

˜ v | h ( ̃ h ) dx,

DIS = − 1 λ

∫ λ0

∫ h 0

[2(∂ x ̃ u ) 2 + (∂ y ̃ u + ∂ x ̃ v ) 2 + 2(∂ y ̃ v ) 2 ] dy dy,

MAR = −2 M P λ

∫ λ0

[

̄(h ) ̃ v | h ∂ xx ̃ h + ˜ u | h (D ̄(h ) ̃ h + ∂ x ̃ T )

] dx,

The perturbation velocity, temperature and surface deflection

are defined as follows:

˜ u = [ Dϕ r cos (θ ) − Dϕ i sin (θ )] e kc i t (18a)

˜ v = [ ϕ r cos (θ ) + ϕ i sin (θ )] k e kc i t (18b)

˜ T = [ τr cos (θ ) + τi sin (θ )] e kc i t (18c)

˜ h = [ ηr cos (θ ) + ηi sin (θ )] e kc i t (18d)where, θ = k (x − c r t) and ηr and ηi are defined as follows:

ηr = ϕ r (h )(c r − Ū (h )) + ϕ i (h ) c i (c r − Ū (h )) 2 + c 2 i

, and ηi = ϕ i (h )(c

((

The terms on the left hand side in Eq. 16 show how the total

energy available is redistributed to the disturbance. KIN represents

the rate of change of the kinetic energy of the disturbance, while

STE and HYD represent the rate of work done to face the stabilizing

STE = −3 �λ

∫ λ0

[ ̃ v | h (∂ xx ̃ h )] dx,

SHE = −D 2 Ū

λ

∫ λ0

˜ u | h ̃ h dx,

REY = − 1 λ

∫ λ0

∫ h 0

˜ u ̃ v D ̄U d y d x,

VRE = −3 V r J̄ K λ

∫ λ0

˜ v | h [

˜ T | h + D ̄| h ̃ h ]

dx.

¯ (h )) − ϕ r (h ) c i

Ū (h )) + c 2 i

.

ffects of the surface tension and hydrostatic pressure, respectively.

ith regards to the terms on the right hand side, they represent

he rate of change of the available energy. SHE represents the work

one by perturbation shear stress and is always positive. MAN and

RE have destabilizing nature since they show the rate at which

nergy is transferred to the perturbation by Marangoni and vapor

ecoil forces, respectively. REY is the work done by Reynolds stress

nd is negligible in the range of Reynolds numbers under consid-

ration. Finally DIS is the rate of energy dissipation by viscosity

nd is always negative. Consequently, the total energy available for

he perturbation is the net sum of DIS, SHE, MAN, and VRE. If the

um is positive, the flow is unstable and the disturbance grows,

therwise the flow is stable and the disturbance is damped. The

ifferent terms in the energy balance equation are evaluated by

sing the Orr-Sommerfeld problem solution combined with the

igenfunctions ϕ( y ), τ ( y ), and the complex phase speed c to find

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 11

Table 2

Numerical and analytical solution of normalized SHE, MAR and VRE for different values of k and Re = 1 . .

k SHE ∗a SHE ∗ % Error MAR ∗a MAR

∗ % Error VRE ∗a VRE ∗ % Error

0.001 0.9999 0.9999 0 6.251e-05 6.249e-5 0.0320 5.598e-05 5.599e-04 0.0179

0.005 0.9971 0.9975 0.0401 0.001646 0.001640 0.3645 0.001402 0.001396 0.4208

0.01 0.9892 0.9900 0.0809 0.006617 0.006509 1.6322 0.005636 0.005543 1.6359

0.05 0.6999 0.7990 14.1663 0.1971 0.1322 32.9306 0.1679 0.1123 33.1130

0.1 1.9926 0.4960 75.3739 1.9659 0.3341 83.0017 1.6744 0.2823 83.1366

t

t

f

e

T

h

A

t

S

fi

e

w

c

a

a

S

s

e

m

f

t

i

l

S

w

r

p

4

t

s

n

i

S

l

t

a

fl

t

o

d

a

b

4

e

d

c

g

w

b

r

s

K

o

t

4

p

t

S

i

i

c

d

O v

c

4

m

i

b

f

b

e

d

e

s

w

t

a

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o

b

i

i

c

i

c

i

f

t

c

a

c

s

S

s

w

i

t

he perturbation quantities ( Eqs. 18 ). Then, the perturbation quan-

ities are substituted in the energy terms and integration is per-

ormed using the trapezoidal rule when required.

An analytical solution of the different terms in the kinetic en-

rgy rate balance ( Eq. 16 ) can be obtained for long waves ( k → 0).his was done for isothermal films by Kelly et al. (1989) , and

ere we extend it to include heating and evaporation effects, see

ppendix C for the complete results. Note that we indicate the

erms obtained asymptotically for k → 0 with the index a , e.g.HE a for the work done by the shear stress. One evident bene-

t of this method is having a simple analytical solution for the

nergy balance equation near the instability threshold. Moreover,

e validate our numerical procedure by comparing the numeri-

al and analytical values of the different terms of the energy bal-

nce ( Eq. 16 ) in the limit k → 0. Table 2 shows SHE ∗, MAR ∗nd VRE ∗ (the starred terms indicate normalization with DIS, e.g.HE ∗ = SHE / DIS ) computed both numerically and analytically foreveral k values. Excellent agreement is found when k ≤ 0.01. How-ver, since the analytical solution is accurate at O(k 2 ) the agree-ent becomes weaker for larger wavenumbers.

Following, we analyze the energy budget of the perturbation

or the different instability modes presented earlier. We examine

he behavior of the energy balance terms along Re while averag-

ng along the wavenumber k , for example SHE ∗ is averaged as fol-ows:

HE ∗ = 1

k̄

∫ k̄ 0

SHE ∗dk, (19)

here k̄ was chosen equal to k̄ = 0 . 4 to contain all the unstableange of wavenumbers for all the set of parameters studied in this

aper.

.3.1. Energy analysis of the H-mode

For isothermal falling films, the net available energy for the per-

urbation comes mainly from the work done by the perturbation

hear stress, while the rate of work done by Reynolds stresses is

egligible ( Kelly et al., 1989 ). Fig. 13 (a) shows that SHE ∗

is grow-

ng proportional to the Reynolds number Re > 1. For small Re ,

HE ∗

< DIS ∗

and the flow is stable. As Re increases, SHE ∗

becomes

arger than DIS ∗

and the flow becomes unstable. Fig. 13 (b) shows

he left hand side terms in the kinetic energy balance. These terms

re proportional to the growth rate as they are positive when the

ow is unstable. This means that (i) the kinetic energy of the dis-

urbance grows and (ii) the energy is added to the disturbance to

vercome the stabilizing forces of the surface tension and the hy-

rostatic pressure. Moreover, when Re is small the flow rate is low,

nd HYD ∗

is comparable to STE ∗, however as Re increases HYD

∗

ecomes dominant and STE ∗

becomes negligible.

.3.2. Energy analysis of the S-mode

When the effect of thermocapillarity is included ( M = 20 ), thenergy is transferred to the perturbation through the rate of work

one by the shear stress and thermocapillary forces. Fig. 13 (c)

ompares SHE ∗

and MAR ∗

along Re . For Re 1. Moreover, we

onclude that SHE contributes effectively to the E-mode instability

ue to the fact that VRE ∗

becomes negligible and SHE ∗

becomes

(1) even before the E-mode disappears. But similarly to the pre-

ious case, the E-mode exists only when V r � = 0 showing that theontribution of the vapor recoil still necessary.

.3.4. Energy analysis of the combined S-mode and E-mode

Finally, we revisit the combination of the vapor recoil and ther-

ocapillary forces by using the perturbation energy analysis. The

nteraction between the S-mode and the E-mode is examined first

y analyzing the work done by the vapor recoil or thermocapillary

orces. Fig. 14 (a) shows an increase in MAR ∗

when S-mode is com-

ined with E-mode, while Fig. 14 (b) shows that VRE ∗

is strength-

ned when joining E-mode and S-mode. As both MAR ∗

and VRE ∗

epend on the temperature gradient across the interface which is

nhanced by either vapor recoil or thermocapillary forces is not

urprising that the two modes reinforce each other. Additionally,

e can observe the same trends by analyzing the asymptotic solu-

ion of the energy budget terms. The analytical expressions of MAR

nd VRE ( Eqs. 32 (e) and 32 (f), respectively) share the same exact

erm proportional to MV r showing that the two terms support each

ther equally.

Moreover, the effect of thermal instabilities on the work done

y shear stress is presented in Fig. 14 (c). At large Reynolds, an

ncrease in the term SHE ∗

is observed as either thermocapillar-

ty or vapor recoil is present. This results in decreasing the criti-

al Reynolds number and expanding the H-mode, while, combin-

ng both thermocapillary and vapor recoil will lead to further in-

rease in SHE ∗. This growth of the work done by the shear stress

s explained by the fact that both vapor recoil and thermocapillary

orces tend to push the flow towards the crest of the perturba-

ion, and therefore enhancing the amplitude, which in return in-

reases the shear stress effect at the interface. On the other hand,

decrease in SHE ∗

was observed when Re is small as thermo-

apillarity, vapor recoil or both are added. The same trend is ob-

erved when analyzing the long wave approximation of the term

HE in Eq. 32 (g); negative terms of order O(1 /Re 2 ) , dominating atmall Re , cause the decrease of the work done by the shear stress,

hile at large Re , as these terms become negligible, the other pos-

tive terms take the lead enhancing SHE a . Note that these negative

erms of order O(1 /Re 2 ) are proportional to M 2 , V 2 r and MV r .

12 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

Fig. 13. Normalized energy terms along Re for (a,b) isothermal case, (c) including only thermocapillarity ( M = 20 , V r = 4 ), and (d) including only vapor recoil ( M = 0 , V r = 4 ).

Fig. 14. (a,b,c) Normalized energy terms along Re when thermocapillary and vapor recoil instabilities are combined. (d) Energy production terms compared to dissipation in

function of the wavenumber for Re = 0 . 5 .

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 13

a

b

b

u

i

s

d

p

l

S

s

5

w

t

a

a

s

c

c

f

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t

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l

a

w

n

n

w

v

b

i

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a

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b

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l

a

t

i

W

r

c

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a

k

c

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t

(

P

i

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a

b

u

b

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i

f

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i

2

s

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e

a

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o

fi

e

b

b

t

s

D

c

i

C

w

c

v

A

R

p

An alternative way to show the interaction between the S-mode

nd E-mode is comparing the total available energy of the pertur-

ation as a function of the wavenumber for small Reynolds num-

ers. As seen in Fig. 14 (d), the sum of SHE ∗ and MAR ∗ creates thenstable S-mode (red line). On the other hand, if vapor recoil is

ncluded and thermocapillary neglected, the E-mode exists as a re-

ult of the work done by shear stress and vapor recoil SHE ∗+VRE ∗

epicted by the magenta line. When both Marangoni effect and va-

or recoil are present, the total energy, i.e. SHE ∗+MAR ∗+VRE ∗, (yel-ow line) increases significantly to a value much larger than for the

-mode or E-mode alone, showing that the two modes reinforce

ignificantly each other.

. Conclusions

In summary, the linear stability of volatile falling liquid films

as examined using the Orr-Sommerfeld eigenvalue problem. In

his way we can relax the assumption of the long wave theory and

ccurately study the stability of evaporating falling liquid films for

ll the wavenumbers and also for large Reynolds numbers. A one-

ided approach was implemented to model the interface boundary

onditions, where the vapor was considered thermally and me-

hanically passive, with no effects on the dynamics of the inter-

ace. The rate of evaporation across the interface is governed by

he thermodynamic equilibrium, where the parameter K defines

he degree of equilibrium at the interface and thus the volatility of

he liquid. A base state solution was derived based on the assump-

ion of slow evaporation ( E

14 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

A

m

�v ˜ J ,

i

w

p

1

∂

∂

∂

∂

P

a

u

T

a

v

−

0

∂

J

b

c

b

t

A

b

t

k

(

ϕ

c

τ

Appendix A. Base state solution

We follow the same procedure by Burelbach et al. (1988) in de-

riving the base state but including the gravitational effect. The gov-

erning equations and boundary conditions Eqs. (4) - (6) are simpli-

fied as follows to obtained a spatially uniform time dependent base

state v = ∂ x = 0 ∂ t ̄u = ∂ yy ̄u + Re, (20a)

∂ y ̄p = −Ct, (20b)

P ∂ t ̄T = ∂ yy ̄T , (20c)with boundary conditions at the wall (y = 0) ū = 0 , T̄ = 1 , (21a)and at the free surface (y = h̄ (x, t)) : ∂ t ̄h = −J̄ E, (22a)

p̄ = 3 2

V r ̄J 2 , (22b)

∂ y ̄u = 0 , (22c)

J̄ + V r DL J̄ 3 = −∂ y ̄T , (22d)

K ̄J = T̄ . (22e)Since we assumed that evaporation is slow ( E < < 1), the sys-

tem can be expanded in terms of the film evaporation number ( E )

( Burelbach et al., 1988 ). The velocity ū (y, t) , mass flux J̄ (t) , liquid

temperature T̄ (y, t) , and pressure p̄ (y, t) are assumed to be of or-

der unity, while the film thickness h̄ (t) is considered an unspeci-

fied order-one function.

ū = u o + Eu 1 + E 2 u 2 + O(E 3 ) , (23a)

J̄ = J o + EJ 1 + E 2 J 2 + O(E 3 ) , (23b)

̄ = o + E1 + E 2 2 + O(E 3 ) , (23c)

p̄ = p o + E p 1 + E 2 p 2 + O(E 3 ) . (23d)Several approximations are used in order to find the base state

solution ( Burelbach et al., 1988 ):

• Since evaporation is slow, the effect of mass loss in the kine-

matic boundary condition ( Eq. 22a ) is recovered by rescaling

the time on the evaporation scale:

˜ t = tE and ˜ z = z. • The vapor recoil term in the normal stress boundary condition

( Eq. 22b ) is conserved by assuming

V r = O(1) . • The parameter ( L ) is large, and therefore the kinetic energy of

the vapor particles in the energy boundary condition ( Eq. 22d )

is assumed negligible:

L −1 = o(1) . The base state solution in Eqs. 7 is obtained by applying

these approximations, substituting the expansions and taking the

leading-order terms.

ppendix B. Orr-Sommerfeld eigenvalue derivation

The linear stability of the base state with respect to infinitesi-

al perturbations is considered by substituting

= ( ̄U + ˜ u , ̃ v , ˜ w ) , T = ̄ + ˜ T , p = P̄ + ˜ p , h = h̄ + ̃ h , J = J̄ +n the governing equations ( Eqs. 1 ) and boundary conditions, in

hich the “tilde” quantities are the perturbations. The linearized

erturbation equations are obtained by setting ( ̃ u , ˜ v , ˜ T , ˜ p , ˜ h , ˜ J ):

x ̃ u + ∂ y ̃ v + ∂ z ˜ w = 0 , (24a)

t ̃ u + Ū ∂ x ̃ u + D ̄U ̃ v + ∂ x ̃ p − ∇ 2 ˜ u = 0 , (24b)

t ̃ v + Ū ∂ x ̃ v + ∂ y ̃ p − ∇ 2 ˜ v = 0 , (24c)

t ˜ w + Ū ∂ x ˜ w + ∂ z ̃ p − ∇ 2 ˜ w = 0 , (24d)

(∂ t ̃ T + Ū ∂ x ̃ T + D ̄˜ v ) − ∇ 2 ˜ T = 0 , (24e)long with the boundary conditions, at the plate y = 0 ,

˜ = ˜ v = ˜ w = 0 , (25a)

˜ = 0 , (25b)nd at the free surface y = h (x, z, t)

˜ = ∂ t ̃ h + Ū ∂ x ̃ h + E ̃ J , (26a)

˜ p = 2 ∂ ̃ v ∂y

+ Ct ̃ h − (3� − 2 M P

̄(h )) ∇ 2 xz ˜ h + 3 V r ̄J ̃ J , (26b)

D 2 Ū ̃ h = ∂ y ̃ u + ∂ x ̃ v + 2 M P

(∂ x ̃ T + D ̄∂ x ̃ h ) , (26c)

= ∂ z ̃ v + ∂ y ˜ w + 2 M P

(∂ z ̃ T + D ̄∂ z ̃ h ) , (26d)

y ̃ T + ˜ J = 0 , (26e)

˜ =

˜ T

K + D ̄

K ˜ h . (26f)

As discussed in Section 4.1 , the ratio between the initial pertur-

ation amplitude and the decaying thickness of the film does not

ontribute to the temporal growth rate ω( t ). This can be achievedy removing the term proportional to the evaporation number E in

he kinematic boundary condition ( Eq. 26a ).

ppendix C. Long wave solution of the perturbation energy

alance

In order to obtain an analytical solution of the energy balance

erms in Eq. (16) , we first obtain an asymptotic solution of un-

nowns ϕ( y ), τ ( y ) and c in powers of the small wavenumber k Yih, 1963 ):

= ϕ o + ikϕ 1 − k 2 ϕ 2 + O(k 3 ) , (27a)

= c o + ikc 1 − k 2 c 2 + O(k 3 ) , (27b)

= τo + ikτ1 − k 2 τ2 + O(k 3 ) . (27c)

H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354 15

p

T

ϕ

c

τ

w

ϕ

c

τ

w

E

F

ϕ

c

w

A

I

a

i

d

p

u

i

k

w

t

− p

O

K

S

H

R

M

V

S

D

w

E

t

The expansions in Eq. (27) are substituted in the OS eigenvalue

roblem ( Eq. 13 ) and the solution is obtained at different orders.

he zeroth order solution ( k = 0 ) is: o = y 2 , (28a)

o = Re, (28b)

o = E o Re

y, (28c)

here E o = 2 / (1 + K) 2 . The first order solution reads as follows:

1 = Re y 5

60 − Re y

4

12 + Ct y

3

3 Re − V r y

3

Re (K + 1) 3 , (29a)

1 = 2 Re 2

15 − Ct

3 + MK

P (K + 1) 2 + V r

(K + 1) 3 , (29b)

1 = P (

−Eo y 5

40 −

(D ̄ − Eo

)y 4

12 − Eo y

3

6

) + E 1 y, (29c)

here

1 = 2 (K + 1) 2

(Ct

Re 2 − 1

3

)− 4 K M

P Re 2 ( K + 1 ) 4 + P K

120(K + 1) (10

(4 + 1

K

)D ̄ +

(35 + 13

K

)E 0

)− 6 V r

Re 2 (K + 1) 5 .

inally the second order solution is presented:

2 = (

− Re 2

20160

)y 9 +

(Re 2

2240

)y 8 −

(Ct

1260 + Re

2

630 − V r

420 ( K + 1 ) 3 )

y 7

+ (

Ct

180 + Re

2

360 − V r

60 ( K + 1 ) 3 )

y 6 −(

Ct

60 − V r

20 ( K + 1 ) 3 )

y 5

−(

Re 2

90 − Ct

36 + V r

12 ( K + 1 ) 3 + K M

12 P ( K + 1 ) 2 + 1

6

)y 4 + A

6 y 3 ,

(30a)

2 = Re + 4 Re 3

63 − 10 Ct Re

63 + K M

40 P (K + 1) 2 (

19 Re − 7 P Re 3(K + 1) +

5 K P Re

(1 + K) )

+ V r (

10 Re

21(K + 1) 3 + P Re ( 15 K − 7 ) 120 ( K + 1 ) 4

), (30b)

here

= − 2 Ct 3

+ 2 Ct 2

Re 2 + 18 V r

2

Re 2 ( K + 1 ) 6 − 4 Ct K M

P Re 2 ( K + 1 ) 2 − 2

+ V r (

2

( K + 1 ) 3 − 12 Ct

Re 2 ( K + 1 ) 3 − P ( 15 K − 7 )

20 ( K + 1 ) 4 + 12 K M

P Re 2 ( K + 1 ) 5 )

.

(31)

t is interesting to notice that the long wave approximation of ϕ( y )nd τ ( y ) is not valid when Re = 0 showing how the present scal-ng is not completely suitable for treating films without inertia. A

ifferent type of scaling could avoid this issue.

The long wave solution in Eq. (27) is now substituted in the

erturbation quantities ( Eqs. 18 ) and the energy terms are found

p to O(k 2 ) . Thus, τ 2 ( y ) is not presented here for simplicity sincet shows at higher orders only. Moreover, we need to assume

2 � = O(1) in order to include surface tension. For this reasone consider only one term depending on the surface tension in

he third order of Eq. 27a . Therefore, we can assume that ϕ =

3

ky 3 �/Re with maintaining the same accuracy. The analytical ex-

ressions of the different terms in the kinetic energy balance at

(k 2 ) are listed as follows:

IN a = k 2

3

[4 Re 2

15 − 2 Ct

3 + 2 V r

( K + 1 ) 3 + 2 K M

P ( K + 1 ) 2 − 2 k 2 �

]E 2 ,

(32a)

TE a = 3�k 4 [

4 Re 2

15 − 2 Ct

3 + 2 V r

( K + 1 ) 3 + 2 K M

P ( K + 1 ) 2 − 2 k 2 �

]E 2 ,

(32b)

YD a = C tk 2 [

4 Re 2

15 − 2 C t

3 + 2 V r

( K + 1 ) 3 + 2 K M

P ( K + 1 ) 2 − 2 k 2 �

]E 2 ,

(32c)

EY a = k 2 [

Ct

180 − 31 Re

2

20160 − V r

60 ( K + 1 ) 3 + �

60 k 2

]E 2 , (32d)

AR a = MK P

k 2 [

5

6 ( K + 1 ) 2 − P (15 K − 7)

30 ( K + 1 ) 3 − 2 Ct

Re 2 ( K + 1 ) 2

+ MK P

8

Re 2 ( K + 1 ) 4 + 6 V r

Re 2 ( K + 1 ) 5 − k 2 � 14

Re 2 ( K + 1 ) 2 ]E 2 ,

(32e)

RE a = V r k 2 [

4

5 ( K + 1 ) 3 − P (15 K − 7)

40 ( K + 1 ) 4 − 2 Ct

Re 2 ( K + 1 ) 3

+ 6 V r Re 2 ( K + 1 ) 6

+ 6 M K P Re 2 ( K + 1 ) 5

− k 2 � 6 K + 9 K Re 2 ( K + 1 ) 3

]E 2 ,

(32f)

HE a = 2 + k 2 [

41 Ct

180 + 4321 Re

2

20160 − 2 Ct

2

Re 2 + 20

3

+ MK P

(P(15 K − 7) 30 ( K + 1 ) 3

+ 8 Ct Re 2 ( K + 1 ) 2

+ 1 2 ( K + 1 ) 2

− MK P

8

Re 2 ( K + 1 ) 4 )

+ V r (

3 P(15 K − 7) 40 ( K + 1 ) 4

+ 12 Ct Re 2 ( K + 1 ) 3

− 41 60 ( K + 1 ) 3

− V r 18 Re 2 ( K + 1 ) 6

)

− 24 MV r K P Re 2 ( K + 1 ) 5

+ k 2 �(

− 6 Ct Re 2

+ 7 4

+ 14 K M P Re 2 ( K + 1 ) 2

+ 18 V r Re 2 ( K + 1 ) 3

)

− k 4 9 �2

Re 2

]E 2 , (32g)

IS a = −2 + k 2 [

− 17 Ct 90

− 1249 Re 2

10080 + 4 Ct

2

3 Re 2 − 19

3

+ MK P

(− 2

3 ( K + 1 ) 2 − 4 Ct

Re 2 ( K + 1 ) 2 )

+ V r (

− P(15 K − 7) 20 ( K + 1 ) 4

− 8 Ct Re 2 ( K + 1 ) 3

+ 17 30 ( K + 1 ) 3

+ V r 12 Re 2 ( K + 1 ) 6

)

+ 12 K MV r P Re 2 ( K + 1 ) 5

+ k 2 �(

4 Ct

Re 2 − 11

10 − 12 V r

Re 2 ( K + 1 ) 3 )

+ k 4 6 �2

Re 2

]E 2 ,

(32h)

here E = exp (kc i t) . Note that Ct = Re cot β, then the terms inqs. 32 proportional to Ct

Re 2 and Ct

2

Re 2 are O(Re −1 ) and O(1) , respec-

ively.

16 H. Mohamed and L. Biancofiore / International Journal of Multiphase Flow 130 (2020) 103354

M

NO

P

S

S

S

S

S

S

S

S

S

T

T

T

T

V

V

W

Y

Supplementary material

Supplementary material associated with this article can be

found, in the online version, at 10.1016/j.ijmultiphaseflow.2020.

103354 .

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Linear stability analysis of evaporating falling liquid films1 Introduction2 Problem formulation2.1 Governing equations2.2 Non-dimensional and scaled parameters2.3 Base state solution

3 Linear stability analysis4 Results and discussion4.1 Validation4.2 Temporal stability analysis4.2.1 Hydrodynamic instability (H-mode)4.2.2 Thermocapillary instability (S-mode)4.2.3 Evaporation induced instability (E-mode)4.2.4 Phase speed analysis4.2.5 Discussions on the frozen interface assumption

4.3 Perturbation energy analysis4.3.1 Energy analysis of the H-mode4.3.2 Energy analysis of the S-mode4.3.3 Energy analysis of the E-mode4.3.4 Energy analysis of the combined S-mode and E-mode

5 ConclusionsDeclaration of Competing InterestCRediT authorship contribution statementAcknowledgmentsAppendix A Base state solutionAppendix B Orr-Sommerfeld eigenvalue derivationAppendix C Long wave solution of the perturbation energy balanceSupplementary materialReferences