Applied Mathematical Modelling 35 (2011) 1264–1270

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Applied Mathematical Modelling

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

An investigation on the influence of slot injection/suction on thespreading of a thin film under gravity on a rotating disk

N. Modhien, E. Momoniat ⇑Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of theWitwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 April 2009Received in revised form 10 August 2010Accepted 1 September 2010Available online 9 September 2010

Keywords:Slot injectionSlot suctionThin filmGravityRotation

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.09.003

⇑ Corresponding author.E-mail addresses: Ebrahim.Momoniat@wits.ac.za

We investigate the influence of slot injection/suction on the axisymmetric spreading of athin film under the influence of gravity and rotation. The effects of surface tension areignored. We allow a very thin film to precede the bulk of the fluid to overcome the singu-larity which arises as a consequence of applying the no-slip boundary condition. We showhow the width of the slot and magnitude of the injection/suction influences the height ofridges and depth of cavities on the profile of the free surface of the thin film. Rotationincreases the depth of the cavities and the height of the ridges as compared to the effectsof gravity alone. The presence of rotation also results in the formation of a breaking wave.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we investigate the influence of slot injection/suction on thin film flow driven by gravity on a rotating sub-strate. The slot is a porous circular disk on a solid substrate. Emslie et al. [1] are amongst the first authors to investigate thespreading of a thin film of Newtonian fluid on a rotating substrate. Emslie et al. [1] have shown that if the initial profile of thefilm on the rotating substrate is monotonically decreasing, the profile of the free surface will develop into a breaking wave.Spin coating is a well known process used in industry for coating integrated circuits, optical mirrors, color television screensand magnetic disks for data storage [2–9]. Oron et al. [10] and Myers [11] discuss the wide range of applications of thin filmflow. The work done by Emslie et al. [1] has been extended to non-Newtonian fluids [12], boundary conditions affecting thefree surface of the thin film [13–16] and boundary conditions on the surface of the rotating disk [14,17–19]. The effect of theCoriolis force have been investigated by Momoniat and Mason [20] and Myers and Charpin [21]. Myers and Charpin [21]show that by a suitable choice of scaling for the horizontal velocity the effects of the Coriolis force can be ignored.

The spreading of a viscous fluid on a porous base [22,23] has application in the spreading of water on a textile fabric or onthe surface of a powder, in ink-jet printing and in the analysis of runoff of rainwater over soils. The effect of uniform injec-tion/suction on the gravity driven spreading of a thin film on a porous base has been considered by Mason and Momoniat[24]. The authors solve the resulting nonlinear equation with a source term using the Lie group approach. The form of thenonlinear source term, modelling the injection/suction, is assumed to depend on the height of the free surface and then onthe gradient of the free surface. The evolution of a drop with a finite contact angle is considered. The authors show that thethickness of the drop always decreases. The injection/suction affects the radius at the base of the drop. Momoniat and Mason[25] extend the work done in Mason and Momoniat [24] by including surface tension effects. The resulting fourth-order

. All rights reserved.

, Ebrahim.Momoniat@gmail.com (E. Momoniat).

N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270 1265

nonlinear partial differential equation is solved numerically. The Lie group method is used to show that the nonlinear sourceterm, modelling the injection/suction, depends on the height of the free surface. Schwartz and Michaelides [26] include theeffects of surface tension when analysing the flow of a Newtonian fluid injected into a hole onto an inclined plane leadingeventually to a long rivulet of fluid. Momoniat et al. [27] consider the effects of gravity and surface tension on the spreadingof a thin film with a single slot.

The paper is set out as follows: In Section 2 we derive the model equation. In Section 3 we consider numerical solutions ofthe model equation when rotation alone is driving the flow. In Section 4 we consider numerical solutions of the model equa-tion including the effects of both gravity and rotation. Concluding remarks are made in Section 5.

2. Derivation of the model equation

To derive the model equation the standard lubrication equations [28] are used. We assume that the surface of the rotatingdisk is smooth and rotates about a vertical axis with constant angular velocity X. The rotating disk is assumed to have aninfinite radius to avoid edge effects and the complicated ‘tea-pot effect’ [17,18,6]. The thin film is axisymmetric about theaxis of rotation, this implies that @/@h (�) = 0. We also assume that [29,30,21]

ho

Lc� 1; Re

ho

Lc

� �2

� 1; ð2:1Þ

where h0 is the characteristic thickness of the thin film, Lc is the characteristic length of the film, m is the kinematic viscosityand Re = U Lc/m is the Reynolds number.

The lubrication approximation to the Navier–Stokes equations when considering the effects of gravity and rotation aregiven by (see, e.g. [21])

@p@r¼ @

2v r

@z2 þ r; ðaÞ @p@z¼ �B; ðbÞ 1

r@

@rðrv rÞ þ

@vz

@z¼ 0 ðcÞ; ð2:2Þ

where vr is the velocity component in the radial direction and vz is the velocity component normal to the rotating plane. TheBond number is given by B ¼ qgh2

0=ðlUÞ. The system (2.2) is solved subject to the following boundary conditions: On thesurface of the disk at z = 0 we have the no-slip boundary condition

v rðr;0; tÞ ¼ 0: ð2:3Þ

A normal velocity, ~vnðrÞ, at the surface of the disk is induced by the injection/suction. We thus have

vzðr;0; tÞ ¼ ~vnðrÞ: ð2:4Þ

At the free surface of the fluid, z = h(r, t), the kinematic condition holds which ensures that fluid particles on the free surfacemust remain on the free surface, i.e.

vzðr; h; tÞ ¼@h@tþ v rðr;h; tÞ

@h@r: ð2:5Þ

The pressure on the free surface z = h(r, t) depends on atmospheric pressure p0:

pðr; h; tÞ ¼ p0: ð2:6Þ

On the free surface we neglect the effects of surface shear

@v r

@z

����z¼hðt;rÞ

¼ 0: ð2:7Þ

By integrating the continuity Eq. (2.2 c), subject to boundary conditions (2.4) and (2.7), we can eliminate vz(r,h, t) from thekinematic condition (2.5) to give

@h@tþ 1

r@

@r

Z hðt;rÞ

0rv rðr; z; tÞdz ¼ ~vnðrÞ: ð2:8Þ

Integrating (2.2b) subject to (2.6) we obtain the leading order pressure term for the thin film

pðr; h; tÞ ¼ p0 � Bðz� hÞ: ð2:9Þ

Integrating (2.2a) with respect to z subject to (2.3) and (2.7) we find that

v rðr; z; tÞ ¼12

@p@r� r

� �z2 � 2hz� �

: ð2:10Þ

Finally, by substituting (2.10) into (2.8) we obtain the expression for the film height

1266 N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270

@h@t¼ 1

3r@

@rBrh3 @h

@r� r2h3

� �þ ~vnðrÞ: ð2:11Þ

By dividing through by the Bond number, B, we obtain the free surface equation modelling the spreading of the free surfaceof a thin film under gravity and rotation with blowing or suction at the base given by

@h@~t¼ 1

3r@

@rrh3 @h

@r� kr2h3

� �þ vnðrÞ; ð2:12Þ

where

k ¼ ðFr=RoÞ2 ð2:13Þ

is a ratio of the Froude and Rossby number and

~vnðrÞ ¼ BvnðrÞ; ~t ¼ Bt ð2:14Þ

are the scaled normal velocity and scaled time. The constant k measures the ratio of the rotational speed of the disk and grav-itational acceleration.

The function vn(r) models the injection/suction. The slot has a width L and is positioned at the radial position r0. We con-sider injection/suction on an interval of size L starting at r = r0. A suitable function is given by Roy et al. [31]

vnðrÞ ¼A sin pðr�r0Þ

L

h i; r0 6 r 6 r0 þ L;

0 r elsewhere:

(ð2:15Þ

A > 0 represents the amplitude of the injection and A < 0 the amplitude of the suction.We consider the initial curve

hðr;0Þ ¼ expð�r2Þ: ð2:16Þ

We chose this initial profile to overcome the contact line singularity without recourse to a precursor film [13,21,32–34].Numerical solutions are obtained using pdepe in MATLAB. In the figures the position of the slot is indicated by the symbol*. We solve the model Eq. (2.12) subject to the boundary conditions [21]

@hð0; tÞ@r

¼ 0; hð1; tÞ ¼ 0: ð2:17Þ

We have dropped the overhead tilde from the scaled time for convenience.

3. Single slot injection/suction for rotation driven spreading

In this section we consider only the effects of rotation. This allows us to compare our results to the work undertaken byEmslie et al. [1]. Rotation dominated flows are interesting from a mathematical perspective. In reality, the breaking waveprofiles that occur in rotation dominated flows are curtailed by surface tension effects.

A first order quasi-linear partial differential equation models the evolution of the free surface profile under the effect ofrotation only with slot injection/suction and is given as

@h@t¼ 1

3r@

@rð�kr2h3Þ þ vnðrÞ; ð3:1Þ

We consider the effect of the slot on the wave breaking process. If we let k = 1 in Eq. (3.1), where the wave breaking occurs,we obtain a first order quasi-linear partial differential equation

@h@tþ rh2 @h

@r¼ �2

3h3 þ vnðrÞ: ð3:2Þ

The problem is formulated as a Cauchy initial value problem. In 3-dimensional space (r, t,h), we let r denote the parameteralong the initial curve

hðr;0Þ ¼ expð�r2Þ ð3:3Þ

and let s denote the parameter along the characteristic curve. We then have

t ¼ tðr; sÞ r ¼ rðr; sÞ h ¼ hðr; sÞ: ð3:4Þ

The initial conditions at s = 0 are given by

tðr;0Þ ¼ 0 rðr;0Þ ¼ r hðr;0Þ ¼ expð�r2Þ: ð3:5Þ

The differential equations of the characteristic curves are given by

h

Fig. 1

h

N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270 1267

dtds¼ 1; ð3:6Þ

drds¼ rh2

; ð3:7Þ

dhds¼ �2

3h3 þ vnðrÞ: ð3:8Þ

We solve the system of differential Eqs. (3.6)–(3.8) numerically subject to initial conditions (3.5) by using NDSolve inMATHEMATICA.

We firstly consider the case

vn ¼ 0: ð3:9Þ

When no fluid is injected into the slot we obtain the equation first derived by Emslie et al. [1]. In Fig. 1 we plot the numericalsolution of the system (3.6)–(3.8) for vn = 0. The graph of h against r is plotted for a range of t values as well as the projectionof the characteristic curves on the (r, t)-plane. We notice that the surface profile flattens and decreases and eventually steep-ens and has the formation of a breaking wave. The breaking time and position can be found by considering the minimumpoint on the envelope where the characteristic curves emanating from the r-axis intersect.

In Fig. 2 we plot the effects of suction on the surface profile. We find that the effects of the suction is to create a cavity inthe surface profile. The minimum point on the envelope in the projection of the characteristics has increased when comparedto Fig. 1 indicating that the wave breaks at a later time when compared to the case with vn = 0. We can therefore concludethat suction has delayed the formation of a breaking wave on the free surface.

In Fig. 3 we plot the effects of injection on the surface profile. We note that the injection has caused the formation of aridge in the vicinity of the slot. The minimum point on the envelope in the projection of the characteristics has decreasedwhen compared to Fig. 1 indicating that the breaking wave forms at an earlier time when compared to the case withvn = 0. We can conclude that slot injection has caused the breaking wave to form earlier.

t 0,1,2,3,4

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

r

vn 0

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

r

t

vn 0

. Plot of the numerical solution of the system (3.6)–(3.8) for vn = 0 at t = 0, 1, 2, 3, 4 and a projection of the characteristics in the (r � t) plane.

t 0,1,2,3,4

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

r

A 0.1, r0 0.5, L 0.5

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

r

t

A 0.1, r0 0.5, L 0.5

Fig. 2. Plot of the numerical solution of the system (3.6)–(3.8) for A = �0.1 and A = 0.1 at t = 0, 1, 2, 3, 4.

t 0,1,2,3,4

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

r

h

A 0.1, r0 0.5, L 0.5

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

r

t

A 0.1, r0 0.5, L 0.5

Fig. 3. Plot of the numerical solution of the system (3.6)–(3.8) for A = 0.1 and A = 0.1 at t = 0, 1, 2, 3, 4.

1268 N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270

In this section we have considered the effects of the slot on the formation of a breaking wave on the free surface profile.The graphs of h against r were plotted as well as the projection of the characteristics on the (r, t)-plane to further illustrate theeffects. Increasing the slot width for injection increases the magnitude of the ridges and causes a corresponding decrease inthe breaking time. Increasing the slot width for suction increases depth of the cavities and results in a corresponding increasein the breaking time. In the next section we consider the effects of slot injection/suction when the effects of both gravity androtation are considered.

4. Single slot injection/suction with the effects of both gravity and rotation

In this section we solve (2.12) numerically. We firstly consider the case

Fig. 4.shown

vn ¼ 0: ð4:1Þ

In Fig. 4 we plot the numerical solution of (2.12) for vn = 0. The magnitude of rotation is varied where k = 0, k = 0.5 and k = 1.When k = 0 we have no rotation and gravity is the driving force. We notice from Fig. 4 that when we have both rotation andgravity as the driving forces, the surface profile becomes flatter and decreases, however as we increase the effect of rotationwe notice that the surface profile steepens and we have the formation of a breaking wave.

In Fig. 5 the surface profile is plotted for varying magnitudes of k. The position of the slot is fixed at r0 = 0.5, the width ofthe slot is L = 0.25 and the magnitude of injection is given by A = 0.5. We note that increasing the effect of rotation decreasesthe height of the free surface profile and increases the height of the ridges with higher amplitudes being created for greatervalues of suction.

In Fig. 6 we fix the position of the slot at r0 = 0.5, the width of the slot to L = 0.25 and the suction to A = � 0.5. We plot thefree surface profile for varying magnitudes of rotation. In Fig. 6 we note that an increase in the magnitude of rotation

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

h

r0=0.5, L=0.25, A=0

Initial Profile

λ=1

λ=0.5

λ=0

Plot of the numerical solution of (2.12) for vn = 0 and the magnitude of rotation is varied. We chose k = 0, k = 0.5 and k = 1. The solution curves areat t = 1.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

h

r0=0.5, L=0.25, A=0.5

Initial Profile

λ=1

λ=0.5

λ=0

Fig. 5. Plot of the numerical solution of (2.12) where r0 = 0.5, L = 0.25, A = 0.5. The magnitude of the rotation is varied. We chose k = 0, k = 0.5 and k = 1. Thesolution curves are shown at t = 1.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

h

r0=0.5, L=0.25, A=−0.5

λ=0

λ=0.5

λ=1

Initial Profile

Fig. 6. Plot of the numerical solution of (2.12) where r0 = 0.5, L = 0.25, A = �0.5. The magnitude of the rotation is varied. We chose k = 0, k = 0.5 and k = 1. Thesolution curves are shown at t = 1.

N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270 1269

decreases the height of the surface profile and increases the depth of the cavities. The greater the amount of the fluid beingsucked out of the slot the greater the depth of the cavities.

Increasing the width of the slot causes an increase in the magnitude of the cavities and ridges for injection and suctionrespectively. From the work done in this section we can conclude that increasing the effects of rotation decreases the heightof the free surface profile. By increasing the width of the slot, we observed an increased height of the ridges and depth of thecavities forming.

5. Concluding remarks

In this paper we have considered the influence of slot injection/suction on the formation of a breaking wave in rotationdominated flow. We note that suction decreases the mass of the fluid and delays the formation of a breaking wave whencompared to the case with no suction. Injection increases the mass of fluid on the rotating plane and causes the breakingwave to form earlier than the case with no slot. The position of the slot and the width of the slot can prevent the onsetof a breaking wave.

We have also considered the spreading of a thin film under gravity and rotation. As injection is increased so too are theheight of the ridges. Increasing the effect of suction increases the depth of cavities. However rotation decreases the height ofthe free surface profile faster as compared to the effects of gravity alone (see [27]). As we increase the effects of rotation, wehave the formation of a breaking wave for both slot injection and suction.

In this paper we have shown the effective use that can be made by a slot in forming drop profiles that have very sharpedges. In the industrial process of spin coating a flat drop profile is preferred. In the spreading of thin drops on a porous sur-face, the final drop profile is not necessarily flat. In the dyeing of a fabric or in ink-jet printing the edge of the drop needs to be

1270 N. Modhien, E. Momoniat / Applied Mathematical Modelling 35 (2011) 1264–1270

well defined to show the outline of a pattern on the fabric or the edge of a character on a printed sheet. The drop surface isusually curved to reflect the most light and create a well defined character or pattern. In this paper we have shown how slotscan produce well defined, sharp edges, by stopping the rotation just before breaking. We believe the use of slots in thin filmcoating is both useful and worth further investigation.

In this investigation we have excluded the effects of surface tension. Surface tension plays an important part in edge ef-fects, especially in the industrial process of spin coating. This will modify the sharp edges we observe in this analysis. Myersand Charpin [21] have shown that when considering the effects of gravity, rotation and surface tension that capillary ridgesform at the edge of the fluid. The contact angle, the angle made by the thin film with the solid substrate, is also affected bythe inclusion of surface tension. Surface tension effects will form the basis of future work into this problem.

Acknowledgments

EM acknowledge support received from the National Research Foundation of South Africa under Grant No. 2053745. NMacknowledge support received from the National Research Foundation of South Africa under Grant No. 66528. Opinions ex-pressed and conclusions arrived at, are those of the author and are not necessarily to be attributed to the NRF.

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