Research Article Process Parameter Identification in Thin Film...

Research ArticleProcess Parameter Identification in Thin Film FlowsDriven by a Stretching Surface

Satyananda Panda,1 Mathieu Sellier,2 M. C. S. Fernando,3 and M. K. Abeyratne3

1 Department of Mathematics, National Institute of Technology Calicut, Kerala 673601, India2Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand3Department of Mathematics, University of Ruhuna, 81000 Matara, Sri Lanka

Correspondence should be addressed to Satyananda Panda; [email protected]

Received 25 January 2014; Revised 10 June 2014; Accepted 12 June 2014; Published 21 July 2014

Academic Editor: Francisco Chinesta

Copyright © 2014 Satyananda Panda et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The flow of a thin liquid film over a heated stretching surface is considered in this study. Due to a potential nonuniform temperaturedistribution on the stretching sheet, a temperature gradient occurs in the fluid which produces surface tension gradient at the freesurface of the thin film. As a result, the free surface deforms and these deformations are advected by the flow in the stretchingdirection. This work focuses on the inverse problem of reconstructing the sheet temperature distribution and the sheet stretch ratefrom observed free surface variations.This work builds on the analysis of Santra and Dandapat (2009) who, based on the long-waveexpansion of the Navier-Stokes equations, formulate a partial differential equation which describes the evolution of the thicknessof a film over a nonisothermal stretched surface. In this work, we show that after algebraic manipulation of a discrete form of thegoverning equations, it is possible to reconstruct either the unknown temperature field on the sheet and hence the resulting heattransfer or the stretching rate of the underlying surface.We illustrate the proposedmethodology and test its applicability on a rangeof test problems.

1. Introduction

The analysis of thin film flow and heat transfer over a stretch-ing surface has been a subject of fundamental importanceas it is relevant to several industrial applications such asmetal and polymer extrusion, continuous casting, drawing ofplastic sheets, or cable coatings to name a few.This industrialcontext has drawn fluid dynamists and applied mathemati-cians alike to study this problem from a more canonicalangle. The first important contribution to the understandingof this problem is the work of Wang [1] who formulated amathematical model and developed a solution strategy basedon the homotopy analysis method (HAM). The problem hasbeen revisited several times since this seminal work with theinclusion of additional physics or more complex rheology.Andersson et al., for example, extended Wang’s contributionby analyzing the associated heat transfer problem [2, 3], whileNoor and Hashim built on the work of Dandapat et al. [4, 5]to consider the thermocapillary andmagnetic field effects [6]

and Aziz and Hashim that of viscous dissipation [7]. Khan etal. focused on the effect of the temperature-dependency onthe viscosity and thermal conductivity on the flow in the film[8] andAndersson et al. extended the standard formulation topower-law fluids [9]. A common feature of the literature citedabove is that it is implicitly assumed that the film thicknessis uniform in the domain, a required assumption to enablethe similarity transformation which reduces the set of partialdifferential equations to a more tractable one of ordinarydifferential equations.

Recognizing the restrictions of the plane interfaceassumption, Dandapat and co-workers were the first toextend the formulation to account for local deformation ofthe free surface in [10, 11].The authors exploit the slendernessof the flow domain to derive a long-wave approximation oftheNavier-Stokes equations and solve the resulting governingequation using the matched asymptotic method. Lately, thiswork was extended to include the heat transfer problem [12].In this work, the nonuniform temperature distribution at

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014, Article ID 485431, 12 pageshttp://dx.doi.org/10.1155/2014/485431

2 International Journal of Engineering Mathematics

the stretching sheet induces an inhomogeneous temperaturefield in the film.Consequently, a surface temperature gradientdevelops at the film free surface. As a result of the surfacetension gradients, the film thickness varies along the flow andthese deformations are advected in the stretching direction.

This work focuses on the flow of a thin liquid film overa heated stretching surface. The aforementioned literatureprovides solid modeling foundations upon which one canbuild to indirectly infer process parameters difficult tomeasure in practice. More specifically, since the developedmodels correlate the film thickness to the sheet stretch rateand/or temperature through a set of differential equations,it is only natural to wonder whether the knowledge of thefilm thickness variation allows the reconstruction of eitherthe sheet temperature or the sheet stretch rate. It is preciselythis problemwhich this paper tackles.Themotivation for thiswork is to demonstrate theoretically that it is possible to infereither the stretching rate or the surface temperature fromthe knowledge of the free surface evolution. This could haveinteresting practical outcome such as indirectly inferring theheat transfer involved in the process. The solution strategy tosolve this inverse problem is inspired from the recent workof Sellier and Panda [13–15], which proposed a strategy toreconstruct an unknown field from free surface data in thinfilm flows. The general idea is to algebraically manipulatethe field equation in order to obtain an explicit differentialequation governing the inverse problem. This solution canthen be solved using standard numerical techniques. Thisconcept is applied for the first time here to a transient prob-lem. The simplicity of the proposed approach contrasts withthe traditional way of dealing with such inverse problemswhich rely on pde-constrained optimization framework, [16].

The following section briefly describes the mathematicalmodel and numerical solution procedure used to computethe free surface evolution. This section is followed by thedevelopment of the inverse problem solution methodology.Some examples illustrating the success of the proposedapproach are presented in the penultimate section and finallyconcluding remarks are drawn in the final section.

2. Description of the Direct Problem

2.1. Mathematical Model. We consider here an unsteadyliquid film which lies over a horizontal plane heated elasticsheet as shown in Figure 1. Because surface tension tendsto be a monotonically decreasing function of temperature,temperature gradients at the stretching sheet surface inducesurface tension variations at the free surface. As regions ofhigh surface tension tend to “pull” on regions of lower surfacetension, convective flow cells develop which deform the freesurface. This deformation is advected by the flow in thestretching direction.The question this work addresses is howthe temperature field at the sheet surface and the stretchingvelocity can be reconstructed from a known deformation ofthe free surface.

This study is based on the model developed by Santraand Dandapat [12] using the long wave theory. This modelis briefly described here for completeness. The elastic sheet

Free surface Gas

Liquid

x

z

h0

h(x, t)

g

Heated stretching substrate

u = U(x)u = U(x)

0

qi

L

Figure 1: Sketch of flow geometry.

lies at 𝑧 = 0 and the liquid gas interface lies at 𝑧 = ℎ(𝑥, 𝑡),where the 𝑥-axis is directed along the stretching sheet and the𝑧-axis is normal to the sheet in the outward direction towardthe fluid. Gravity acts along negative 𝑧-direction. Further, thesurface at 𝑧 = 0 starts stretching from rest and within a veryshort time attains the stretching velocity 𝑢 = 𝑈(𝑥).The elasticsurface is heated with its temperatureΘ a function of 𝑥 aloneand the ambient gas phase is at constant temperature 𝑇

𝑎.

The fluid is assumed to be incompressible andNewtonianwith constant kinematic viscosity ], density 𝜌, specific heat 𝑐

𝑝,

and thermal conductivity 𝑘.The surface tension of the liquid-gas interface decreases linearly with temperature according to

𝜎 = 𝜎𝑎− 𝛾 (𝑇

𝑖− 𝑇𝑎) , (1)

where 𝑇𝑖is the inter-facial temperature, 𝜎

𝑎is the surface

tension at 𝑇 = 𝑇𝑎, and 𝛾 is a positive constant specific to the

fluid.Moreover, Newton’s law of cooling describes the convec-

tive heat flux 𝑞𝑖at the interface by

𝑞𝑖= −𝑘∇𝑇 ⋅ n = 𝛼 (𝑇 − 𝑇

𝑎) , (2)

where 𝑇 is the temperature in the fluid, n is the unit normalvector on the interface, and 𝛼 is the rate of heat transfer fromthe liquid to the ambient gas phase.

Following the analysis of Santra and Dandapat [12] basedon the the lubrication equation, the dimensionless evolutionequation for film thickness ℎ(𝑥, 𝑡) is given by

𝜕ℎ

𝜕𝑡

+

𝜕𝐹 (ℎ)

𝜕𝑥

= 0, (3)

where 𝐹(ℎ) is the flow discharge given by

𝐹 (ℎ) = (𝑈ℎ) + 𝜖 {−

1

3

(𝑈𝑈𝑥ℎ3

) −

Fr3

(ℎ3

ℎ𝑥) +

𝑆

3

(ℎ3

ℎ𝑥𝑥𝑥)

−

Mw2

(ℎ2

(

Θ

1 + Biℎ)

𝑥

)}

International Journal of Engineering Mathematics 3

+ 𝜖2PrMw {

Bi6

(

𝑈𝑥Θℎ3

(1 + Biℎ)3)

𝑥

ℎ2

+

1

12

(

𝑈Θ𝑥(3 + Biℎ) ℎ2

(1 + Biℎ)2)

𝑥

ℎ2

} .

(4)

Here the subscript 𝑥 means partial differentiation withrespect to 𝑥. The dimensionless numbers are

(i) the Froude number Fr = 𝑔ℎ3

0/]2 that expresses the

ratio of inertia to body forces;(ii) the Prandtl number Pr = 𝜌𝑐

𝑝]/𝑘 that represents the

ratio of momentum to thermal diffusivity;(iii) the Marangoni number Mw = ℎ0𝛾(𝑇𝑆0 − 𝑇𝑎)/𝜌]

2 thatcharacterizes the relation between the temperaturedependent surface tension and viscous forces;

(iv) the Biot number Bi = 𝛼ℎ0/𝑘 that compares the rela-

tive magnitudes of resistances to internal conductionand surface convection;

(v) the dimensionless number S = 𝜖2𝜎𝑎ℎ0/𝜌]2 is known

as surface tension parameter and 𝜖 = ℎ0/𝐿 is the

aspect ratio.

In the above expressions, ℎ0and 𝐿 are characteristic length

scales in the vertical and horizontal directions, respectively.The symbol 𝑇

𝑆0stands for the sheet temperature at the origin.

At the origin, we apply the symmetry conditions ℎ𝑥= 0

and ℎ𝑥𝑥𝑥

= 0 and at the other end of the domain; weassume that the same sheet temperature profiles and the sheetstretching rate continues beyond the computed domain. Wealso assume that the gradient of the free surface extends outof the computational domain.These boundary conditions areconsistent with those mentioned in [12].

2.2. Numerical Solution Procedure. There are many differentnumerical methods that have been presented in the past forthis kind of analysis [12, 17] and in this paper, we mostlyfollow the finite volume technique described in Sellier andPanda [14] on a uniform grid system with implicit fluxdiscretization.

We discretize our flow domain into the uniform grid andthe flow variable ℎ, the steady stretching velocity𝑈, and sheettemperature Θ are located at the cell centers as shown inFigure 2. Let 𝑁 spatial grid points 𝑥

1< 𝑥2< ⋅ ⋅ ⋅ < 𝑥

𝑁

be equally distributed over the domain [0, 𝐿] with spatialincrement Δ𝑥

𝑖= 𝑥𝑖+1− 𝑥𝑖= 𝐿/(𝑁 − 1), 𝑖 = 1, 2, . . . , 𝑁 − 1.

The numerical solution is sought at the discrete time levels𝑡𝑛

, 𝑛 = 0, 1, 2, . . . with time step Δ𝑡𝑛+1 = (𝑡𝑛+1 − 𝑡𝑛).To set up the discrete equation, the function ℎ is approxi-

mated over the cell [𝑥𝑖, 𝑥𝑖+1]. That is the cell average of ℎ over

[𝑥𝑖, 𝑥𝑖+1] is denoted as

ℎ𝑛

𝑖:=

1

Δ𝑥𝑖

∫

𝑥𝑖+1

𝑥𝑖

ℎ (𝑥, 𝑡𝑛

) 𝑑𝑥. (5)

For the given solution ℎ𝑛𝑖, the solution at the next time

level 𝑡𝑛+1 is obtained by integrating (3) over the space and

x1 x2 xi xi+1 xN−1 xN

tn

tn+1

Θi

Ui

Δxi

Δtn+1

hn+1

i

· · · · · ·

Figure 2: Typical grid used for the finite volume discretizations.Blacked dots and dashed vertical lines represent the nodal pointsand the cell faces, respectively.

time intervals [𝑥𝑖, 𝑥𝑖+1] × [𝑡𝑛

, 𝑡𝑛+1

]. This yields the followingdiscrete equation:

(ℎ𝑛+1

𝑖− ℎ𝑛

𝑖) Δ𝑥𝑖+ (𝐹𝑛+1

𝑖+1/2− 𝐹𝑛+1

𝑖−1/2) Δ𝑡𝑛+1

= 0, (6)

for nodes 𝑖 = 1, 2, . . . , 𝑁 − 1, where the discrete flux function𝐹𝑛+1

𝑖+1/2is given by

𝐹𝑛+1

𝑖+1/2:= 𝐹 (𝑥

𝑖+1, 𝑡𝑛+1

) , (7)

where face values are evaluated using linear interpolationfrom nodal values and gradients using forward differences.For example,

ℎ (𝑥𝑖+1, 𝑡𝑛+1

) =

1

2

(ℎ𝑛+1

𝑖+1+ ℎ𝑛+1

𝑖) ,

ℎ𝑥(𝑥𝑖+1, 𝑡𝑛+1

) =

1

Δ𝑥𝑖

(ℎ𝑛+1

𝑖+1− ℎ𝑛+1

𝑖) .

(8)

Similar expressions can easily be derived for other terms.Thefollowing second-order accurate approximation of the third-order derivative is used:

ℎ𝑥𝑥𝑥

(𝑥𝑖+1, 𝑡𝑛+1

) =

1

Δ𝑥3

𝑖

(ℎ𝑛+1

𝑖+2− 3ℎ𝑛+1

𝑖+1+ 3ℎ𝑛+1

𝑖− ℎ𝑛+1

𝑖−1) . (9)

The first and the last nodes are located at the boundaryand special treatment is required there. Therefore, we applythe boundary conditions in the first node and at the lastnodes. In brief, at the first node we have

ℎ𝑥(𝑥1, 𝑡𝑛+1

) = 0,

ℎ𝑥𝑥𝑥

(𝑥1, 𝑡𝑛+1

) = 0,

ℎ𝑥𝑥𝑥

(𝑥2, 𝑡𝑛+1

) =

1

Δ𝑥3

𝑖

(ℎ𝑛+1

3− 4ℎ𝑛+1

2+ 3ℎ𝑛+1

1) ,


0 2 4 6 8 10

1

0.8

0.6

0.4

0.2

0

x

Θ

(a)

0 2 4 6 8 10

1

0.8

0.6

0.4

0.2

0

x

h

t = 1

t = 2

t = 3

t = 20

ComputedSantra et al. (2009)

(b)

Figure 3: Results for the temperature and stretching velocity profile given by (11) and with Pr = 1.0, Fr = 2, Bi = 1, Mw = 4, S = 2, and𝜖 = 0.05: (a) temperature profile; (b) evolution of free surface profile at different time from 𝑡 = 1 to 𝑡 = 20 with the increment of time step 1.

(

Θ

1 + Biℎ)

𝑥

(𝑥1, 𝑡𝑛+1

)

=

2

Δ𝑥𝑖

((

Θ

1 + Biℎ)

𝑛+1

𝑥1+Δ𝑥1/2

− (

Θ

1 + Biℎ)

𝑛+1

𝑥1

) ,

(

𝑈𝑥Θℎ3

(1 + Biℎ)3)

𝑥

(𝑥1, 𝑡𝑛+1

)

=

2

Δ𝑥𝑖

((

𝑈𝑥Θℎ3

(1 + Biℎ)3)

𝑛+1

𝑥1+Δ𝑥1/2

− (

𝑈𝑥Θℎ3

(1 + Biℎ)3)

𝑛+1

𝑥1

) ,

(


(1 + Biℎ)2)

𝑥

(𝑥1, 𝑡𝑛+1

)

=

2

Δ𝑥𝑖

((


(1 + Biℎ)2)

𝑛+1

𝑥1+Δ𝑥1/2

− (


(1 + Biℎ)2)

𝑛+1

𝑥1

) ,

(10)where the value of the functionsΘ,Θ

𝑥,𝑈, and𝑈

𝑥are known

at the cell face 𝑥1. Similarly, we approximate the discrete

values at the last node.Equation (6) describes an implicit time discretization

scheme. Since the governing equation is nonlinear, a systemof nonlinear algebraic equations needs to be solved at eachtime step. We use fsolve in MATLAB for this purpose. Agood initial starting guess is required to solve the nonlinearequations. A reasonable initial guess for the free surface ischosen to be unity throughout the discrete domain at the firsttime step. The solution from the previous time step can be

used otherwise. Convergence is usually achieved in less than10 iterations and the convergence criterion is that the normof the residuals should be less than 10−7.

To demonstrate the successful implementation of theproposed finite volume algorithm, the numerical resultsobtained with the proposed algorithm are compared to thoseobtained by Santra and Dandapat, [12]. Specifically, theresults of Figure 7 in [12] are reproduced here for validationpurpose. For this benchmark case, the following temperaturedistribution and stretching velocity profiles are imposed:

Θ (𝑥) = 0.5 [1.1 + tanh {35 (2.5 − 𝑥)10

}] ,

𝑈 (𝑥) = (0.1) 𝑥.

(11)

The temperature profile is illustrated in Figure 3(a). Theother nondimensional parameters used in the simulation arereported in the figure caption.The simulation was performedusing 150 grid points and time steps of 1.0 (dimensionlessunits).

Figure 3(b) illustrates the evolution of free surface atdifferent times. The observed results replicate well thoseobtained by Santra and Dandapat in [12], and therebyvalidating the formulation and implementation of the pro-posed numerical scheme. It is clear from Figure 3(b) thatthermocapillary deforms the free surface during the earlystages and this deformation is advected downstream by thestretching sheet. To clearly see the motion of the free surfacedisturbance, the local maximum on each curve is identifiedwith star.


1.4

1.2

1

0.8

0.6

0.4

0.2

0

U

U = 0.6(0.1x + 0.01x2)

U = 0.75(1 − (0.1x − 1)2)

U = 0.1x

0 2 4 6 8 10

x

(a)

0 2 4 6 8 10

x

h

Santra et al. (2009)U = 0.6(0.1x + 0.01x

2)

U = 0.75(1 − (0.1x − 1)2)

U = 0.1x

1

0.95

0.9

0.85

0.8

0.75

0.7

(b)

Figure 4: Results for the different stretching velocity profile with temperature profile given by (12) and with Mw = 2, S = 2, Bi = 1, Pr = 1,Fr = 2, and 𝜖 = 0.05: (a) stretching velocity profile; (b) free surface profile.

The effect of stretching velocity distribution on filmheight is demonstrated next.The following Gaussian temper-ature profile

Θ (𝑥) = 1 − 𝑒−𝑥2/33 (12)

is imposed at the sheet and three different stretching velocitydistributions are considered:

𝑈 (𝑥) = 0.6 (0.1𝑥 + 0.01𝑥2

) ,

𝑈 (𝑥) = 0.75 (1 − (0.1𝑥 − 1)2

) ,

𝑈 (𝑥) = 0.1𝑥.

(13)

The free surface is plotted in Figure 4(b) for the differentstretching velocities (illustrated in Figure 4(a)) at a fixedtime of 𝑡 = 2 and the parameters reported in the figurecaption. Visual inspection confirms that the computed freesurface profiles match well with those computed by Santraand Dandapat in Figure 3 of [12] further demonstrating theaccuracy of the proposed numerical approach.

3. The Inverse Problem

The evolution of the free surface provides valuable infor-mation on the flow process parameters. Such informationcan be used to infer unknown conditions otherwise difficultto measure. In the following, we will consider two inverseproblems which can be solved using the same conceptualidea.

3.1. Temperature Reconstruction from Two Free Surface Snap-shots and Prescribed Stretching Velocity. This first inverse

problem consists in inferring the temperature of the stretch-ing sheet from known free surface deformation data anda known stretching velocity distribution over the sheet.The inverse counterpart of the above problem is discussedassuming the transient solution for ℎ exist. More specifically,it is assumed that two free surface profiles at two successivetime steps are available. In the following, we reconstruct thesheet temperature distribution Θ from the film thicknessprofiles obtained by solving the direct problem up to aspecific time level 𝑡 with a prescribed stretching velocity.The main idea behind the reconstruction strategy is thatthe governing equations, (3) and (4), hold equally well forthe forward or the inverse problem. Thus, we can obtain anexplicit partial differential equationwhich “governs” what thetemperature distribution needs to be to result in a given filmthickness distribution variation.Therefore, unlike traditionalinverse problem methods which tend to rely on recasting thereconstruction problem into an optimization one wherebyone has to infer the “optimal” temperature distribution whichminimizes a given functional, the proposed methodologyonly involves finding the solution of a partial differentialequation. Like the direct problem, we make use of the thesame implicit discretized equation (6). Recall the equation as

(𝐹 (𝑥𝑖+1, 𝑡𝑛+1

) − 𝐹 (𝑥𝑖, 𝑡𝑛+1

)) Δ𝑡𝑛+1

= − (ℎ𝑛+1

𝑖− ℎ𝑛

𝑖) Δ𝑥𝑖.

(14)

This nonlinear discrete equation could equally be written as

L𝑖(hn+1, hn,Θ,U) = 0, (15)

where the bold font indicates the vector of discrete values.Theexplicit form of the operator L

𝑖is lengthy and not reported

here. With the governing equation in this form, it becomes


0 2 4 6 8 10

x

t = 0.3 (no noise)t = 0.3 (3% noise)t = 0.3 (5% noise)

0.754

0.752

0.75

0.748

0.746

0.744

0.742

0.74

h

(a)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Θ

ActualReconstructed (no noise)

Reconstructed (3% noise)Reconstructed (5% noise)

0 2 4 6 8 10

x

(b)

Figure 5: Results for the sine sheet temperature profile, that is,Θ(𝑥) = 0.6+ 0.5 sin(2𝜋𝑥/10), and with Pr = 1, S = 2, Fr = 2, Bi = 1, Mw = 10,𝜖 = 0.05, and stretching velocity 𝑈 = 𝑥: (a) free surface profiles with and without noise; (b) actual and reconstructed temperature profiles.

apparent that this equation describes three possible problems.The usual forward problem, as it is usually described, isrecovered if we assume that hn, Θ, and U are known andhn+1 is sought. The two inverse problems of interest here arerecovered if

(i) hn+1, hn, and U are known andΘ is sought;(ii) hn+1, hn, andΘ are known and U is sought;

Irrespective of the inverse problem considered, (15) is animplicit equation for the temperature distribution (given thestretching rate), or the stretching rate (given the temperaturedistribution). Boundary conditions similar to the ones usedfor the forward problem are implemented in the inverseproblem. Equation (15) is solved implicitly in the scientificcomputing programMATLAB using the fsolve routine for thereconstruction ofΘ from the knowledge of the film thicknessdistribution at two successive time steps and the temperatureat the end points of the flow domain. An initial guess of zerotemperature distribution, that is, Θ(𝑥) = 0, is considered forthe implicit solver.

For the first test case, the artificial “experimental” freesurface data is obtained for the prescribed linear stretchingvelocity𝑈(𝑥) = 𝑥 and the following sine temperature profile:

Θ (𝑥) = 0.6 + 0.5 sin(2𝜋𝑥10

) . (16)

Additional noise with a standard uniform distribution isadded to the obtained film thickness profile to replicate thelikely measurement uncertainty. The amplitude of the addednoise was either 3% of the total film thickness variation or5%.Without any prior processing of the noisy data, the recon-struction algorithm invariably failed. However, the algorithm

was successful if a cubic spline was first fitted through thenoisy film thickness variation.TheMATLAB function csaps()was used for this purpose with a smoothing parameter setto 0.9, [18]. The nondimensional parameters used in thesimulation are reported in the figure caption. The numberof grid points is 150 and time steps of 0.03 are used in thesimulation. The corresponding free surface profile with andwithout noise at 𝑡 = 0.3 is shown in Figure 5(a). Figure 5(b)shows the reconstructed and the actual temperature pro-files. Without noise, these two profiles are indistinguishablethereby validating the conceptual reconstruction idea and theimplementation of the algorithm. Given the smoothness ofthe actual temperature profile, it comes as little surprise thateven with the addition of 3% or 5% noise on the free surfacedata, the reconstructed temperature profile still is in goodagreement with the actual one.This highlights the robustnessof the proposed methodology even in the presence of noise.

The ability of the proposed algorithm to reconstructtemperature profiles with a much steeper variation is testedin the next test case. The artificial temperature profile givenby a smoothed step temperature function, that is,

Θ (𝑥) = 0.5 [1.1 + tanh {35 (2.5 − 𝑥)10

}] . (17)

As in the previous case, 3% and 5% noise was then addedto the film thickness profiles. Cubic splines were then fittedthrough these free surface profiles and used as input to thereconstruction algorithm. The resulting free surface profileswith and without noise at 𝑡 = 20 can be seen in Figure 6(a).It is apparent from these profiles that the film thicknessvariations occur on a much shorter length scale which willlead to a more difficult test case for the reconstructionalgorithm, as one would expect. Reconstruction results can


0.154

0.152

0.15

0.148

0.146

0.144

0.142

0.14

0.138

0.136

0.134

h

t = 20 (no noise)t = 20 (3% noise)

t = 20 (smoothed curvefor 3% noise)t = 20 (5% noise)

0 2 4 6 8 10

x

(a)

1.2

1

0.8

0.6

0.4

0.2

0

Θ



0 2 4 6 8 10

x

−0.2

(b)

Figure 6: Results for the smooth step sheet temperature profile given in (11) and with Pr = 1, S = 2, Fr = 2, Bi = 1, Mw = 4, 𝜖 = 0.05, andstretching velocity 𝑈 = 0.1𝑥: (a) free surface profiles; (b) actual and reconstructed temperature profiles with or without added noise on theinput data.

be seen on Figure 6(b) without noise, for 2% added noise,and for 5% added noise. Without noise, it is apparent thatthe actual and reconstructed temperature profiles are strictlyidentical again confirming that it is possible to reconstruct thesurface temperature from the knowledge of two successivefree surface profiles and the stretching velocity. It is also clearthat the actual and reconstructed temperature profiles are stillin reasonable agreement when noise is added to the inputdata. The reconstruction algorithm only mildly amplifies theinput noise. The quality of the reconstruction for this secondcase is not as good as for the first one andone can attribute thisto the difficulty of fitting a good representative cubic splinethrough rapidly varying data. Hence, one would expect thisreconstruction algorithm to work best when the unknowntemperature profile does not vary too rapidly.

In the inevitable presence of noise in an experimentalsystem, the natural procedure is to run several realizationsof the same experiment and average the output data. Thisis precisely what we proceeded to do in order to checkwhether, as the number of realization increases, the agree-ment between the reconstructed and actual temperatureprofile improves, as one would expect. Results are reportedin Figure 7 for the 5% noise case for which the agreementbetween the actual and reconstructed temperature profileswas the poorest. In effect, we ran numerous reconstructionwith 5% noise added to the film thickness and averaged thereconstructed temperature profiles. Figure 7(a) shows tworealizations of the reconstructed temperature profiles (Real-izations 1 and 2) which are rather poor. However, averagingthe reconstructed temperature profile over 50 realizationsyields a very good agreement with the actual distribution,as anticipated. Figure 7(b) shows the norm of the difference

between the actual and reconstructed temperature profiles. Itis clear from this graph that the norm decreases very rapidlyfor the first few realizations with very marginal reductionof this norm between 10 to 50 realizations. This indicatesthat, in practice, few realizations of the experiment would benecessary in order to obtain a reliable reconstruction of thetemperature distribution.

To explore the effects of the Prandtl number (Pr), theBiot number (Bi), and the surface tension parameter (S) onthe reconstruction algorithm the algorithm was run for fourdifferent scenarios where Fr = 1, M = 4, and 𝜖 = 0.05

are fixed and the Biot number, Prandtl number and surfacetension parameter vary, that is, Bi ∈ {0.1, 1, 2}, Pr ∈ {1, 5},and S ∈ {1, 2}. A nonlinear parabolic (concave) profile isconsidered for the stretching velocity that is, 𝑈 = 0.6(0.1𝑥 +

0.005𝑥2

) for all four cases. Considering Figures 8(b), 9(b),and 10(b), it can be seen that the reconstruction algorithmrecovers the temperature profile extremely well without noiseeven if the imposed stretching velocity is nonlinear. For2% and 5% added noise, the quality of the reconstructionis still reasonable. The corresponding free surface profilescan be seen in Figures 8(a), 9(a), and 10(a). Slightly largerdiscrepancies can be observed when the surface tensionparameter is increased from S = 1 to S = 2, see Figure 11(b).

3.2. Stretching Velocity Reconstruction from Two Free SurfaceSnapshots and Prescribed Temperature Distribution. It is clearfrom the work of [10–12] that the stretching velocity playsan important role in the evolution of the thin film height.In addition to the Marangoni effect, the type of stretchingapplied at the bottom sheet mainly influences the film profile.In the following analysis, we discuss how to reconstruct


1.2

1

0.8

0.6

0.4

0.2

0

Θ

Actual

0 2 4 6 8 10

x

−0.4

−0.2

Reconstructed (basedon 50 realizations)

Realization 1

Realization 2

(a)

0 10 20 30 40 50

3.5

3

2.5

2

1.5

1

0.5

0

Number of realizations

Erro

r nor

m(b)

Figure 7: Effect of the number of realizations on the reconstructed temperature profile: (a) individual and averaged reconstructed temperatureprofiles; (b) norm of the error between the actual temperature profiles and the one reconstructed using multiple realizations.

0.95

0.94

0.93

0.92

0.91

0.9

0.89

h

t = 1 (no noise)t = 1 (3% noise)t = 1 (5% noise)

0 2 4 6 8 10

x

(a)

1.2

1

0.8

0.6

0.4

0.2

0

Θ



0 2 4 6 8 10

x

−0.2

(b)

Figure 8: Results for the temperature profile Θ(𝑥) = 1 − 𝑒−𝑥2/33 at time 𝑡 = 1 with Bi = 0.1, Pr = 1, and S = 1: (a) free surface profiles with

and without noise; (b) actual and reconstructed temperature profiles.

the stretching velocity from given free surface data andtemperature distribution at the surface. For this purpose,we first solve the forward problem for the film thicknessprofile up to a particular time for a given temperaturedistribution and stretching sheet velocity. To reconstruct thestretching velocity profile, we consider the same governingequations: (3) in continuous form and (6) in discretized form.In order to solve the nonlinear algebraic equation (6) for

the stretching velocity 𝑈 from the known two successivefree surface profiles and the temperature function Θ, we firstprescribe the stretching sheet velocity at the boundary pointsand then solve the equation using MATLAB fsolve routine.It is observed that for any nonzero constant initial guess, theMATLAB solver converges within less than 10 iterations.

For the first case, the film thickness is obtained up to aparticular time of 𝑡 = 2 for the stretching velocity 𝑈(𝑥) =


0.95

0.94

0.93

0.92

0.91

0.9

0.89

h


0 2 4 6 8 10

x

(a)

1.2

1

0.8

0.6

0.4

0.2

0

Θ



0 2 4 6 8 10

x

−0.2

(b)

Figure 9: Results for the temperature profile Θ(𝑥) = 1 − 𝑒−𝑥2/2 at time 𝑡 = 1 with Bi = 2, Pr = 1, and S = 1: (a) free surface profiles with and

without noise; (b) actual and reconstructed temperature profiles.

0.95

0.94

0.93

0.92

0.91

0.9

0.89

h


0 2 4 6 8 10

x

(a)

1.2

1

0.8

0.6

0.4

0.2

0

Θ



0 2 4 6 8 10

x

−0.2

(b)

Figure 10: Results for the temperature profileΘ(𝑥) = 1 − 𝑒−𝑥2/33 at time 𝑡 = 1 with Bi = 2, Pr = 5, and S = 1: (a) free surface profiles with and


0.6(0.1𝑥 + 0.01𝑥2

), and temperature profile is given by (12).Theother flowparameters used in the simulation are reportedin the figure caption. The transient solution is obtained bysolving (6), for 150 grid points and time steps of 0.1. Thefilm thickness at 𝑡 = 2 is shown in Figure 12(a) withoutnoise and with 3% and 5% noise. Figure 12(b) shows thereconstructed and actual stretching velocity profiles. In the

absence of noise, the perfect agreement between the twoprofiles confirms that the reconstruction algorithm is able torecover the stretching velocity equally well as it could recoverthe substrate temperature profile. The addition of up to 5%noise still lead to a reasonable reconstruction.

For the second case, the film thickness is obtained up toa particular time of 𝑡 = 1 for the nonmonotonic stretching


0.95

0.94

0.93

0.92

0.91

0.9

0.89

h


0 2 4 6 8 10

x

(a)

1.2

1

0.8

0.6

0.4

0.2

0

Θ



0 2 4 6 8 10

x

−0.2

(b)

Figure 11: Results for the temperature profile Θ(𝑥) = 1 − 𝑒−𝑥2/2 at time 𝑡 = 1 with Bi = 1, Pr = 1, and S = 2: (a) free surface profiles with and


0.95

h


0 2 4 6 8 10

x

0.9

0.85

0.8

0.75

0.7

(a)

1.4

1.2

1

0.8

0.6

0.4

0.2

0



0 2 4 6 8 10

x

U

(b)

Figure 12: Results for the nonlinear stretching velocity 𝑈(𝑥) = 0.6(0.1𝑥 + 0.01𝑥2), temperature profile given by (12) and with Pr = 1, S = 2,Fr = 2, Bi = 1, Mw = 2, and 𝜖 = 0.05: (a) free surface profiles with and without noise; (b) actual and reconstructed stretching velocity profileswith or without added noise on the input data.

velocity𝑈(𝑥) = 0.92542{−0.535261+0.5𝑥+𝑒[−0.025(𝑥−5)2]

} andtemperature profile Θ(𝑥) = 𝑒−𝑥

2/33. The other flow parame-

ters used in the simulation are reported in the figure caption.The simulation results for reconstructed surface velocitywithout surface tension (S = 0) and with surface tension(S = 0.1) are given in Figures 13(b) and 14(b), respectively.

In the absence of surface tension the reconstruction algo-rithm is able to recover the stretching velocity profile. It can beseen that with the inclusion of surface tension the algorithmperfectly reconstruct the sheet velocity in the absence of noise(Figure 14(b)). With the addition of noise the reconstructedresults differ mildly from the actual one.


1.1

1

0.9

0.8

h


0 2 4 6 8 10

x

(a)

0.8

0.6

0.4

0.2

0

U



0 2 4 6 8 10

x

(b)

Figure 13: Results for the nonmonotonic stretching velocity 𝑈(𝑥) = 0.92542{−0.535261 + 0.5𝑥 + 𝑒[−0.025(𝑥−5)

2] with temperature profile

Θ(𝑥) = 𝑒−𝑥2/33 and Pr = 1, S = 0, Fr = 2, Bi = 1, Mw = 2, 𝜖 = 0.05: (a) free surface profiles with and without noise; (b) actual and

reconstructed stretching velocity profiles with or without added noise on the input data.

h


0 2 4 6 8 10

x

1.1

1

0.9

0.8

0.7

(a)

0.8

0.6

0.4

0.2

U



00

2 4 6 8 10

x

(b)

Figure 14: Results for the nonmonotonic stretching velocity 𝑈(𝑥) = 0.92542{−0.535261 + 0.5𝑥 + 𝑒[−0.025(𝑥−5)

2] with temperature profile

Θ(𝑥) = 𝑒−𝑥2/33, Pr = 1, S = 0.1, Fr = 2, Bi = 1, Mw = 2, and 𝜖 = 0.05: (a) free surface profiles with and without noise; (b) actual and

reconstructed stretching velocity profiles with or without added noise on the input data.

4. Concluding Remarks

Asolution strategy is presented in thiswork to reconstruct thesurface temperature profile or the surface stretching rate fromthe knowledge of the thickness at a given time and its rate ofchange. The main idea is simply based on a rearrangement

of the equations derived by Santra and Dandapat in [12]for nonisothermal thin liquid films on stretching surfaces.This rearrangement of the governing equations leads toan explicit partial differential equation which governs theinverse problem. This allows to solve the inverse problemin “one shot” which is a considerable advantage compared


to the traditional way of approaching such inverse problemsbased on pde-constrained optimization theory. We havedemonstrated that the algorithm is quite robust in the sensethat noise in the input data is not too amplified provided theinput data is smoothed using a cubic spline. Whilst the focusof this paper is on the presentation of the conceptual idea,future work will assess the feasibility and practicality of thesuggested approach using true experimental data.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

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[2] H. I. Andersson, J. B.Aarseth, andB. S.Dandapat, “Heat transferin a liquid film on an unsteady stretching surface,” InternationalJournal of Heat andMass Transfer, vol. 43, no. 1, pp. 69–74, 2000.

[3] I.-C. Liu and H. I. Andersson, “Heat transfer in a liquid film onan unsteady stretching sheet,” International Journal of ThermalSciences, vol. 47, no. 6, pp. 766–772, 2008.

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[7] R. C. Aziz and I. Hashim, “Liquid film on unsteady stretchingsheet with general surface temperature and viscous dissipation,”Chinese Physics Letters, vol. 27, no. 11, Article ID 110202, 2010.

[8] Y. Khan, Q. Wu, N. Faraz, and A. Yildirim, “The effects ofvariable viscosity and thermal conductivity on a thin film flowover a shrinking/stretching sheet,” Computers & Mathematicswith Applications, vol. 61, no. 11, pp. 3391–3399, 2011.

[9] H. I. Andersson, J. B. Aarseth, N. Braud, and B. S. Dandapat,“Flow of a power-law fluid film on an unsteady stretchingsurface,” Journal of Non-Newtonian FluidMechanics, vol. 62, no.1, pp. 1–8, 1996.

[10] B. S. Dandapat, A. Kitamura, and B. Santra, “Transient filmprofile of thin liquid filmflowon a stretching surface,”Zeitschriftfur angewandteMathematik und Physik ZAMP, vol. 57, no. 4, pp.623–635, 2006.

[11] B. S. Dandapat and S. Maity, “Flow of a thin liquid film onan unsteady stretching sheet,” Physics of Fluids, vol. 18, no. 10,Article ID 102101, 7 pages, 2006.

[12] B. Santra and B. S. Dandapat, “Unsteady thin-film flow over aheated stretching sheet,” International Journal of Heat and MassTransfer, vol. 52, pp. 1965–1970, 2009.

[13] M. Sellier, “Substrate design or reconstruction from free surfacedata for thin film flows,” Physics of Fluids, vol. 20, no. 6, ArticleID 062106, 2008.

[14] M. Sellier and S. Panda, “Surface temperature reconstructionbased on the thermocapillary effect,”TheANZIAM Journal, vol.52, no. 2, pp. 146–159, 2010.

[15] M. Sellier and S. Panda, “Inverse temperature reconstruction inthermocapillary-driven thin liquid films,” International Journalof Numerical Analysis andModeling B, vol. 3, no. 3, pp. 285–296,2012.

[16] O. Pironneau, Optimal Shape Design for Elliptic Systems,Springer, New York, NY, USA, 1984.

[17] Y. Ha, Y.-J. Kim, and T. G. Myers, “On the numerical solution ofa driven thin film equation,” Journal of Computational Physics,vol. 227, no. 15, pp. 7246–7263, 2008.

[18] The MathWorks Inc., MATLAB R2011b Documentation, TheMathWorks Inc., 2011.

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