In this paper we investigate the locally unidirectional flow of a thin liquid layer confined to an inclined channel driven simultaneously by a surface tension gradient

Emilia Borşa

Titus Petrila

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 32 (2003) no. 2, pp. 203–207ictp.acad.ro/jnaatON THE FLOW OF A THIN LIQUID LAYER IN AN INCLINEDCHANNEL OF TRIANGULAR TRANSVERSE SECTIONDRIVEN BY A SURFACE TENSION GRADIENT.NUMERICAL AND ASYMPTOTIC ANALYSISTITUS PETRILA∗ and EMILIA BORŞA†Abstract. In this paper we investigate the locally unidirectional flow of a thinliquid layer confined to an inclined channel driven simultaneously by a surfacetension gradient.MSC 2000. 76D08, 76D45, 76M55.Keywords. Thin film approximation, surface tension gradient.1. INTRODUCTIONWe consider the steady behavior of a thin layer (trickle, rivulet). We followthe lubrication approximation to obtain an analytical solution for a rivuletof Newtonian fluid moving down a slowly varying substrate. We study thespecial case when the transversal sectional profile of the rivulet is triangularand we include the effects of a surface tension gradient. We are interested inthe study of the Marangoni effect. We take into account a non-zero tangentialstress boundary condition. The extra terms introduced by this non-zero shearstress have a reasonable physical significance.2. PROBLEM FORMULATIONWe choose Cartesian system Oxyz, with the x axis in the direction of flowand the y axis transverse to the direction of flow and so the fluid velocity isgiven by −→u = u(y, z)−→i .Here (L): z = (H+h)(y) denotes the unknown transverse profile of the freesurface, (S): z = H(y) denotes the known transverse profile of the substrateand h is thickness of the layer.We consider the unidirectional flow of a thin layer of a constant width aof Newtonian fluid with constant density ρ and constant viscosity µ down asubstrate, which is inclined at an angle ϕ, 0 ≤ ϕ ≤ π/2, to the horizontal.∗“Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, str. M. Ko-gălniceanu 1, 3400 Cluj-Napoca, Romania, e-mail: tpetrila@cs.ubbcluj.ro.†Department of Mathematics, University of Oradea, Romania, e-mail:borsavasile@yahoo.com.204 Titus Petrila and Emilia Borşa 2Employing the familiar lubrication approximation [1], [7] the governingequations for this unidirectional flow are simply−µuzz = ρg sinϕ− px,py = 0,pz = −ρg cosϕ.They are to be integrated subject to the boundary conditionsu = 0, on z = H(y)ux = 1µΓx and p− p0 = −Γ̄hyy, on z = (H + h)(y)h = 0 and hy = ∓β, on y = ±a.Here p = p(y, z) denotes the pressure in the fluid, p0 is the constant pressurein the ambient gas above the fluid, Γ̄ is the references value of surface tension,g is the acceleration due to gravity, Γx is the surface tension gradient, and βis constant contact angle between the fluid and the substrate at the unknownposition of the three-phase contact line.Since the flow is unidirectional, kinematic condition is identically satisfied.With the respect to the angle ϕ to the horizontal we consider the case0 < ϕ < π/2. In this case with a triangular section we have H(y) = A |y|.Then the solution is as follows:• the velocityu(y, z) = ρg sinϕ2µ(z −A|y|)(2h+A |y| − z)+ 1µΓx(z −A|y|),• the pressurep(z) = p0 − ρgz cosϕ+ β(Γ̄ρg| cosϕ|)1/2 coth ((ρg|cosϕ|Γ̄ )1/2a), 0 < ϕ < π/2,• the free-surface profile z = (H + h)(y), with H(y) = A |y|, is given by(H + h)(y) = β(Γρg|cosϕ|)1/2 ch( ρg|cosϕ|Γ̄ )1/2a−ch( ρg|cosϕ|Γ̄ )1/2ym sh(ρg|cosϕ|Γ̄)1/2a, 0 < ϕ < π/2,where we have introduced the notation m =√|cosϕ| .The volume flux of liquid running the substrate isQ = ρg sinϕ3∫ a−ah3(y)dy + 12µΓx∫ a−ah2(y)dy.At this point it is convenient to introduce appropriate nondimensional vari-ables defined by:x = 1β(Γ̄ρg)1/2x∗, y =(Γ̄ρg)1/2y∗, z = β(Γ̄ρg)1/2z∗, u = β2Γ̄µ u∗,Q = β3Γ̄2ρgµ Q∗, h = β(Γ̄ρg)1/2h∗, H = β(Γ̄ρg)1/2H∗, a =(Γ̄ρg)1/2a∗, Γ = Γ̄ Γ∗3 On the flow of a thin liquid layer in an inclined channel 205and so, dropping the star superscripts, we yield the following solution for Q:Q == sinϕ316m4[A3(12ma cth3ma− 24 cth2ma− 18ma cth2ma− 18m2a2 cth2ma+ 18m cth2ma− 12m3a3 cthma+ 72ma cthma+ 18m2a cthma+ 18ma+ 18m+ 3m4a4 − 44)+ 6A2(6ma cth3ma− 15 cth2ma+ 9ma cth2ma− 6m2a2 cth2ma+ 18 cth2ma− 3m cth2ma+ 2m3a3 cthma− 12ma cthma− 3m2a cthma− 9ma− 3m+ 6m2a2 − 2)+ 2A(− 18ma cth3ma+ 18 cth2ma− 27ma cth2ma+ 9m2a2 cth2ma+ 27ma+ 9m2a2 − 3)+ 2(6ma cth3ma− 15 cth2ma+ 9m2a2 cth2ma− 9ma+ 4)]+ 12Γx13m3[A2(9ma cth2ma+ 6m2a2 cthma+ 6 cthma+ 2m3a3 − 3ma)+ 3A(2m2a2 cth2ma+ 2 cthma− 6ma cth2ma+ 2ma)+ 3(3ma cth2ma− 3 cthma−ma)].3. NUMERICAL RESULTSIf H(y) = A |y| and 0 ≤ ϕ < π/2 then h(y) = (1−A sgn y)f(y)+A(a−|y|),where the function f = f(y) is defined by f(y) = coshma−coshmym sinhma , if 0 ≤ ϕ <π2 .We calculated the values of the volume flux for the next numerical values:2a = 0.0387(cm), m = 0.8752396489, A = 0.0517, ϕ = 40◦.If Γx = 30.888689(mN/m) then we obtain for the volume flux the numericalvalue Q∗ = 0.0006.If Γx = 30.88869(mN/m) then Q∗ = 0.0008.In this case we compare the value of the volume flux given by [4], namelyQ∗ = 0.0007, with our values. We observe that the error is O(10−4).4. ASYMPTOTIC ANALYSIS FOR H 6= 0A) The limit a→ 0.In the limit of a narrow rivulet, a→ 0, gravity effects become insignificantand the behavior of Q depends only on the local behavior of H near y = 0.206 Titus Petrila and Emilia Borşa 4In particular, if H(y) = O(y), as a → 0, then h(y) ∼ a2−y22a , as a → 0, andhence Q ∼ 4a4105 sinϕ+ Γxa315 , in agreement with the corresponding result in thespecial case H ∼ 0.Alternatively, if H(y) ∼ A |y|, as y → 0, thenh(y) ∼ (1−A sgn(y))a2−y22a +A(a− |y|), as a→ 0,and the henceQ ∼ a4420 sinϕ(147A3 + 174A2 + 77A+ 16)+ 12Γxa330(28A2 + 25A+ 8).B) The limit ϕ→ 0+.In the limit ϕ→ 0+ solutions satisfying Q = Q̄ are possible only if a→∞(i.e., if the rivulet becomes infinitely wide).When 0 ≤ ϕ ≤ π/2 the behavior of Q as a → ∞ depends on the behaviorof H(y), as |y| → ∞. If H(y) = A |y|, with A 6= 0, then h(y) ∼ A(a− |y|) andhenceQ ∼ a46 sinϕA3 + Γx a33 A2, as a→∞.5. DISCUSSIONIf H(y) = A |y| then the profile is always convex when A < 1, and alwaysconcave when A > 1.The case A = 1 is exceptional. In this case (H + h)(y) = H(a) = a, i.e, thetransverse rivulet profile is identically flat for all values of ϕ. As a consequence,the exact expression for Q is particulary simple:Q = 207a4210 sinϕ+ Γx61a360 .If Γx = 0 then we can write an explicit expression for a, namelya =(210Q207 sinϕ)1/4.6. CONCLUSIONSIn the lubrication approximation we have considered the flow in an inclinedchannel of triangular section. Consequently, the flow is driven simultaneouslyby gravity and a surface tension gradient. This gradient implies a non-zerotangential stress boundary condition (Marangoni effect). We have estimatedthe respond of the fluid to such stress. The extra terms that correspond to thisboundary condition are in perfect agreement with the physics of the flow. Morethan that, in some particular situations we have estimated them numerically.There is a fairly good agreement between our computed values of the volumeflux of the fluid and those reported in [4].5 On the flow of a thin liquid layer in an inclined channel 207REFERENCES[1] Acheson, D. J., Elementary Fluid Dynamics, Oxford University Press, Oxford, UnitedKingdom, pp. 238–259, 1990.[2] Allen, R. F. and Biggin, C. M., Stability regions of nonlinear autonomous systems,IEEE Trans. Aut. Control, 33, pp. 16–27, 1988.[3] Braun, R. F. and Murray, B. T., Lubrication theory for reactive spreading of a thindrop, Phys. Fliuds, 7, 1995.[4] Chifu, E., Gheorghiu, C. I. and Stan, I., Surface mobility of surfactant solution.Numerical analysis for the Marangoni and gravity flow in a thin liquid layer of triangularsection, Rev. Roumaine Chim., 29, pp. 31–42, 1984.[5] Duffy, B. R. and Moffatt, H. K., Flow of a viscous trickle on a slowly varying, TheChemical Engeneering Journal, 60, pp. 141–146, 1995.[6] Duffy, B. R. and Wilson, S. K., A third-order differential equation arising in thin-film flows and relevant to Tanner’s law, Appl. Math. Lett., 10, no. 3, pp. 63–68, 1997.[7] Ockendon, H. and Ockendon, J. R., Viscous Flow, Cambridge University Press,1995.[8] Towel, G. D. and Rothfeld, L. B., Hydrodynamics of rivulet flow, AICHE. J., 12,1972.[9] Wilson, S. K. and Terrrill, E. L., The dynamics of planar and axisymmetric holesin thin fluid layers, in First European Coating Symposium on The Mechanics of ThinFilm Coatings, Leeds University, U.K., edited by Gaskell, P. H., Savage, 1995.[10] Young, G. W. and Davis, S. H., Rivulet instabilities, J. Fluid Mech., 176, no. 1, 1987.Received by the editors: May 25, 2000.

On the flow of a thin liquid layer in an inclined channel of triangular transverse section driven by a surface tension gradient. Numerical and asymptotic analysis

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On the flow of a thin liquid layer in an inclined channel of triangular transverse section driven by a surface tension gradient. Numerical and asymptotic analysis

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