Using the lubrication approximation we investigate two-dimensional steady flow of a thin film of fluid with temperature-dependent viscosity on a uniformly heated or cooled horizontal cylinder, which may be stationary or rotating about its axis, in the case when the Biot number (a measure of heat transfer at the free surface) is large. We show that the film thickness (but not the fluid velocity) may be obtained from that in the isothermal case by a simple re-scaling

Duffy, B.R.

Wilson, S.K.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryDuffy, B.R. and Wilson, S.K. (2009) Large-Biot-number non-isothermal flow of a thin film on astationary or rotating cylinder. European Physical Journal - Special Topics, 166. pp. 147-150.ISSN 1951-6355Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/EPJ manuscript No.(will be inserted by the editor)Large-Biot-number non-isothermal flow of a thinfilm on a stationary or rotating cylinderB. R. Duffya and S. K. WilsonbDepartment of Mathematics, University of Strathclyde,Livingstone Tower, 26 Richmond Street, Glasgow G1 1XHAbstract. Using the lubrication approximation we investigate two-dimensionalsteady flow of a thin film of fluid with temperature-dependent viscosity on a uni-formly heated or cooled horizontal cylinder, which may be stationary or rotatingabout its axis, in the case when the Biot number (a measure of heat transfer at thefree surface) is large. We show that the film thickness (but not the fluid velocity)may be obtained from that in the isothermal case by a simple re-scaling.1 IntroductionThe behaviour of a thin film of fluid on a cylinder is a problem of great industrial importance,occurring in, for example, heat exchangers, lubrication and coating processes (see, for instance,the comprehensive volume on coating edited by Kistler and Schweizer [1]). Two related paradigmproblems, namely flow of prescribed flux on a stationary cylinder, and flow of prescribed masson a rotating cylinder, have been studied extensively (see, for instance, Nusselt [2,3], Moffatt[4], Johnson [5]). We consider both of these problems in the case when the cylinder is uniformlyheated or cooled, and the viscosity of the fluid is temperature-dependent, in the case when theBiot number (a non-dimensional measure of heat transfer at the free surface) is large.Wilson andDuffy [6] review work on other thin-film flows of a fluid with temperature-dependent viscosity,in all of which the case of large Biot number is included.2 Problem formulationWe consider two-dimensional steady flow of a film of Newtonian fluid of uniform density ρ andtemperature-dependent viscosity µ on a circular cylinder of radius a with its axis horizontal, thecylinder being at a prescribed uniform temperature T0, hotter or colder than the surroundingatmosphere, at the uniform temperature T∞. We consider both purely gravity-driven flow ofprescribed volume flux on the outside of a stationary cylinder (which we will refer to as “Nusseltflow”; cf Nusselt [2,3]) and flow on a cylinder that is rotating about its axis at uniform angularspeed Ω (which we will refer to as “Moffatt flow”; cf Moffatt [4]); in the latter case the fluidmay be on the inside of the cylinder (“rimming flow”) or the outside (“coating flow”). Thesituation is sketched in Fig. 1, drawn for the case of coating flow.Referred to the polar coordinates r, θ shown in Fig. 1, we take the free surface of the fluidto be at r = a + h for either Nusselt flow or coating flow, and at r = a− h for rimming flow,the film thickness being denoted by h = h(θ).a e-mail: b.r.duffy@strath.ac.ukb e-mail: s.k.wilson@strath.ac.uk2 Will be inserted by the editorrFree surfacer = a + hΩ T0T∞Cylinderr = aθgFig. 1. Geometry of the problem: two-dimensional steady flow of a thin film of viscous fluid on aheated or cooled cylinder (drawn for the case of coating flow).The fluid velocity u, pressure p and temperature T , all functions of r and θ, are governed bythe familiar mass-conservation, Navier–Stokes and energy equations. On the cylinder r = a thevelocity u satisfies the no-slip and no-penetration conditions, and the temperature is T = T0(a prescribed constant). On the free surface r = a ± h the usual normal and tangential stressbalances and the kinematic condition apply. Moreover, we shall take the Biot number (definedby B = ǫaαth/kth, where the constants kth and αth denote the thermal conductivity of the fluidand a surface heat-transfer coefficient) to be large, so that the free surface is at the same uniformtemperature as the atmosphere, that is, T = T∞ on the free surface r = a ± h. A large Biotnumber is certainly feasible physically; for example, for a film of water the parameter valuesǫ = 10−1, a = 0.05m, kth = 0.604 kgm s−3 K−1 and αth = 103 Wm−2 K−1 lead to B ≃ 10.The analysis is valid for an arbitrary dependence of viscosity on temperature, that is, forany viscosity function µ(T ); without loss of generality we define µ(T ) to satisfy µ = µ0 anddµ/dT = −λ at T = T0, where λ is a positive constant.We will consider only thin films, whose aspect ratio ǫ (defined by ǫ = (µ0U0/ρga2)1/2, whereU0 is a characteristic velocity defined by U0 = (ρgQ2/µ0)1/3 for Nusselt flow and U0 = aΩ forMoffatt flow) satisfies ǫ≪ 1. We scale the system by writingr = a(1 ± ǫy∗), h = ǫah∗, u = U0u∗, v = ǫU0v∗, p− pa = ±ǫρgap∗,Q = ǫU0aQ∗, M = ǫρa2M∗, T = T∞ + (T0 − T∞)T∗, µ = µ0µ∗,}(1)where Q is the azimuthal volume flux of fluid per unit axial length crossing a station θ =constant,M is the mass per unit axial length of fluid on the cylinder, pa is atmospheric pressure,g is gravitational acceleration, and we have written u = ueθ + vey, where eθ and ey denoteunit vectors in the directions of increasing θ and y, respectively; moreover, in the scaling for rand p the + sign refers to Nusselt flow or coating flow, and the − sign to rimming flow, so thatin this representation the fluid occupies 0 ≤ y∗ ≤ h∗(θ) for −π < θ ≤ π. For clarity the starsuperscripts on non-dimensional variables will be dropped from now on.At leading order in ǫ the governing equations giveuθ + vy = 0, (µuy)y = cos θ, py = − sin θ, Tyy = 0, (2)together with the boundary conditionsu = U, v = 0 and T = 1 on y = 0 (3)(where U = 0 for Nusselt flow and U = 1 for Moffatt flow) anduy = 0, p = 0 and T = 0 on y = h. (4)Will be inserted by the editor 3Therefore the solution for u, p and T isu = U − h2 cos θ∫1TT˜µ(T˜ )dT˜ , p = sin θ(h− y), T = 1−yh. (5)The mass M of fluid per unit axial length is given byM =∫ pi−pih dθ, (6)and the azimuthal volume flux of fluid per unit axial length isQ =∫ h0u dy = Uh−cos θ3fh3, (7)where f = f(h) (> 0) is the fluidity of the fluid film, defined byf =3h3∫ h0∫ y0h− y˜µ(T (θ, y˜))dy˜ dy =3h3∫ h0(h− y)2µ(T (θ, y))dy. (8)With µ(T ) prescribed, the film thickness h = h(θ) is determined in terms of the flux Qby the algebraic equation (7). Since the flow is steady, Q is constant in Moffatt flow and ispiecewise constant in Nusselt flow; moreover, in Nusselt flow Q is prescribed, and so (6) servesonly to determine the mass M (if desired), whereas in Moffatt flow it is M that is prescribed,and (6) must be included in order for the value of Q (and hence the form of h) to be determined.Once h is known the solution for u, p and T is given explicitly by (5). The fact that h, u andT depend on θ only through cos θ shows that the flow has top-to-bottom symmetry, at leastfor the simple type of solution considered here. Putting θ = π in (7) we note that Q > 0 inMoffatt flow; however, in Nusselt flow we have Q > 0 on the left-hand side of the cylinder (i.e.12π < |θ| ≤ π), and Q < 0 on the right-hand side (i.e. |θ| < 12π).2.1 Isothermal flowThe fluidity f is defined in such a way that if the cylinder is at the same temperature as theatmosphere, so that the fluid film is isothermal (T = T0 = T∞) and the viscosity is constant(µ ≡ 1), then f = 1. In that case we recover the isothermal problems in which h = h0, u = u0,Q = Q0 and M = M0, whereu0 = U −cos θ2(2h0 − y)y, Q0 = Uh0 −cos θ3h30, M0 =∫ pi−pih0 dθ. (9)In isothermal Nusselt flow (U = 0) we recover Nusselt’s [2,3] solution for h0, namelyh0 =(−3Q0cos θ)1/3, (10)representing a film of fluid incident on the cylinder from above, flowing round the cylinder, andfalling off at the bottom. For isothermal Moffatt flow (U = 1) Moffatt [4] showed that (9b) has acontinuous solution h provided that M satisfies 0 < M0 ≤M0c ≃ 4.44272 (and correspondingly0 < Q0 ≤ Q0c =23), showing that, for example, in coating flow a continuous fluid film can besupported steadily on the cylinder against gravity by rotation of the cylinder only if its massdoes not exceed the critical value M0c. Johnson [5] further showed that discontinuous solutionsthat involve jumps (“shocks”) in the free surface are also possible; these solutions, which arethe only possible steady two-dimensional solutions when M > M0c, are particularly relevantfor rimming flow.4 Will be inserted by the editor2.2 Non-isothermal flowIn the present non-isothermal problem with large Biot number, equation (8) givesf = 3∫ 10T 2µ(T )dT, (11)which shows immediately that in this case f is a constant , independent of θ. Therefore withthe re-scalingh =h0f1/2, Q =Q0f1/2, M =M0f1/2(12)we recover from (5)–(7) the classical isothermal problem (9) for the film thickness h. Thus ourrather surprising conclusion is that the solution h for the non-isothermal Nusselt or Moffattflow may be obtained from that of the corresponding isothermal problem merely by a simplere-scaling. In particular, in Moffatt flow the critical mass is Mc = M0cf−1/2 ≃ 4.44272f−1/2,occurring when Q = Q0cf−1/2 = 23f−1/2. On the other hand, from (5) we note that the velocityprofile in non-isothermal cases is quite different from the semi-parabolic profile in (9) in theisothermal case (and the streamline pattern is correspondingly different). For example, for anexponential viscosity model given (in dimensional terms) byµ(T ) = µ0 exp(−λ(T − T0)µ0)(13)we have µ = exp(V y/h), where V is the non-dimensional thermoviscosity number defined byV = λ(T0 − T∞)/µ0, and the velocity u and fluidity f are found to beu = U −h cos θV 2[(V − 1)h− (V h− V y − h) exp(−V yh)](14)andf =3V 3[(V − 1)2 + 1− 2 exp(−V )]. (15)A related conclusion was arrived at by Wilson and Duffy [6] in their study of the unsteadygravity-driven draining of a thin rivulet of fluid with temperature-dependent viscosity down aheated or cooled substrate; they showed that in the limit of large Biot number the evolutionequation for such a rivulet can be obtained directly from the corresponding equation for anisothermal rivulet by re-scaling time and flux with the fluidity f (and therefore that in steadyflow only the flux need be re-scaled).3 ConclusionsWe investigated two-dimensional steady flow of a thin film of fluid with temperature-dependentviscosity on a uniformly heated or cooled horizontal cylinder, which may be stationary orrotating uniformly about its axis, in the case when the Biot number is large. We showed thatthe solution for the fluid velocity, pressure and temperature is expressible in terms of the filmthickness h, and we derived the algebraic equation (7) that determines h. We showed that thefilm thickness (but not the fluid velocity) can be obtained from that in the isothermal case bya simple re-scaling.References1. S. F. Kistler, P. M. Schweizer (editors), Liquid Film Coating (Chapman & Hall, 1997)2. W. Nusselt, Z. Vereines deutscher Ingenieure 60, 541–546 (1916)3. W. Nusselt, Z. Vereines deutscher Ingenieure 60, 569–575 (1916)4. H. K. Moffatt, J. Me´c. 16, 651–673 (1977)5. R. E. Johnson, J. Fluid Mech. 190, 321–342 (1988)6. S. K. Wilson, B. R. Duffy, J. Engng Maths 42, 359–372 (2002)

Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder

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B. R. Duffy

S. K. Wilson

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http://dx.doi.org/10.1140/epjst/e2009-00896-8

Large-Biot-number non-isothermal flow of a thin film on a stationary or rotating cylinder

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