Laminar Thermal Mixing in Coating Flows

A. Haas�;1, M. Scholle1, N. Aksel1, H.M. Thompson2, R.W. Hewson2, P.H. Gaskell2

1Department of Applied Mechanics and Fluid Dynamics, University of Bayreuth, D-95440 Bayreuth, Germany2School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK�Corresponding author: andre.haas@uni-bayreuth.de

Abstract: Heat transfer in a plane shear owcon�guration consisting of two in�ntely longparallel plates is considered. The upper pla-nar plate drives the ow by a constant veloc-ity, whereas the lower plate is �xed and hasa regular sinusoidally varying pro�le [3]. Inlaminar ows over undulated substrates eddiescan be generated due to the kinematical con-straints; details of the genesis and manipula-tion of which is discussed in [4] and [5].

A closed form analytical solution for thethe velocity �eld, based on lubrication theoryas well as a semi{analytic solution, providedby the application of Ritz's direct method, forthe temperature �eld is derived for the creep-ing ow. Additionally, detailed numerical so-lutions are obtained via a �nite element for-mulation of the geoverning equations for mass,momentum and energy conservation, enablingan exploration of the inertial e�ects.

It is shown that changes in the mean plateseparation, that is the geometry, and the levelof inertia present a�ect the local hydrody-namic ow structure in the form of kinemat-ically and inertially induced eddies, respec-tively. These in turn impact on the locallaminar thermal mixing, and consequently en-hance the global heat transfer. Resulting Nus-selt numbers are reported for a wide range ofPecl�et and Reynolds numbers with agreementbetween the two methods of solution, for thecase of creeping ow, found to be extremelygood.

Keywords: Flow structure, Thermal mixing,Heat transfer, Shear ow, Topography

1 Introduction

Consider, as illustrated in Fig. 1, the case ofsteady, two-dimensional Couette ow of an in-compressible uid con�ned between two in�-nite, horizontally aligned plates, with the mov-ing upper at plate, temperature T0, sepa-rated by a small mean distance from the sta-

tionary lower one which has a regular sinu-soidally varying pro�le and is at temperatureT1, (T1 > T0). The uid properties are takenas those for silicone oil, as used in [6] in theinvestigation of the corresponding isothermal ow problem. The thermal conductivity, �,and speci�c heat, cp, are assumed constant,while the density � and viscosity �, have theform

�(T ) = �0 (1� �T ) (1)

�(T ) = �0 (1� ��T ) (2)

with the non-dimensional constants � and ��

denoting the coe�cients of thermal expansionand thermoviscosity, respectively.

6?A?

6

H

� -�

-

6~z

~x

- ~u(~x;H) = u0

Figure 1: Schematic of the ow geometry

The two coordinates ~x, ~z are scaled by �=2�with � being the wavelength. The velocityis scaled by the lid velocity u0. Thus, thegeometry is characterised by two nondimen-sional parameters, namely the dimensionlessmean plate distance h and the dimensionlessamplitude a. Additionally, the pressure ~p andtemperature ~T are scaled as follows

~p =2��0u0

�p; ~T = T (T1 � T0) + T0 (3)

with 0 � T � 1.

2 Governing equations

Within the framework of the Boussinesq ap-proximation the continuity equation in non-dimensional form simpli�es to

ux + wz = 0 : (4)

Excerpt from the Proceedings of the COMSOL Conference 2009 Milan

By taking � to be su�ciently small the sys-tem can be de�ned in such a way that buoy-ancy e�ects can be neglected and by a properchoice of the uid parameters the stability con-dition for the non ocurrance of natural con-vection can be ful�lled, further details can befound in [3]. In addition thermal expansionplays no part in the ow. Consequently, termsinvolving �T can be neglected in the Navier{Stokes equations

Re [uux + wuz] = �px+2@x [(1� ��T )ux] +

: : : @z [(1� ��T ) (uz + wx)] (5)

Re [uwx + wwz] = �pz+2@z [(1� ��T )wz] +

: : : @x [(1� ��T ) (uz + wx)] : (6)

If dissipation is assumed to be neglegible, seealso [3], the temperature equation reduces tothe following form:

Pe [uTx + wTz]� [Txx + Tzz] = 0 (7)

The Reynolds and the Pecl�et number in theutilised scaling are de�ned as follows:

Re =�0u0�

2��0; Pe =

�u0�0cp2��

: (8)

3 Boundary conditions

In non-dimensional form the spatial locationsof the lower and the upper plates are given byz = �a cosx and z = h, respectively, alongwhich a no-slip condition is applied

u(x;�a cosx) = 0 u(x; h) = 1 (9)

v(x;�a cosx) = 0 v(x; h) = 0 : (10)

For the temperature �eld the correspond-ing upper and lower plate conditions are theDirichlet ones, namely

T (x;�a cosx) = 1; T (x; h) = 0 : (11)

In addition periodic boundary conditions forall �elds to the left and to the right of the owdomain are imposed.

4 Semi{analytical solution

For simpli�cation a creeping ow is consid-ered, Re ! 0 and thermoviscous couplingterms involving �� are neglected, leading to anunilaterally coupled problem and the hydrody-namic �eld can be solved seperately from thetemperature �eld.

4.1 Hydrodynamic �eld

Invoking the lubrication approximation re-duces the hydrodynamic problem to theReynolds' equation, a single ordinary di�er-ential equation. Its closed form analytic solu-tion leads to the following explicit expressionfor the streamfunction [2],

hZ2=h2 � 4a2 + 3a2Z

2h2 + a2+a

h(Z � 1) cosx ;

(12)by introducing

Z =z + a cosx

h+ a cosx: (13)

as a new coordinate. The coordinate transfor-mation (x; z) ! (x; Z) maps the ow domainof interest sketched in Fig. 1 to a rectangulardomain [0; 2�] � [0; 1]. In the new coordinatesystem, the lubrication solution for the streamfunction can be written as a non-orthogonalseries expansion

= (x; Z) =

n=1Xn=�1

n(Z)e�inx (14)

with the coe�cients

0 =

�h2 � 4a2

�hZ2 + 3ha2Z3

2h2 + a2(15)

�1 = �a

2Z2 (Z � 1) ; (16)

giving an excellent approximation for small di-mensionless amplitudes to a � 1=2.

4.2 Temperature �eld

The essence of the analysis is the constructionof an analogous series representation for tem-perature as for the streamfunction (14), i.e.

T = (x; Z) =

n=NXn=�N

Tn(Z)e�inx (17)

with N 2 N. It can be shown, see [3] thatequation (7) results from variation of the func-tional

I =

�Z��

1Z0

�� Pe T (x; Z) (u � r)T (�x; Z)

+rT (x; Z) �rT (�x; Z)�(h+a cosx) dZ dx

(18)

with respect to the temperature, provided thatthe velocity �eld is symmetric. For isother-mal creeping ow given by equation (12) theseconditions are ful�lled. Use of the above varia-tional formulation is advantageous since Ritz'sdirect method can be applied which representsan e�cient means of solution [3].

5 Finite element solution

Equations (4)-(7) and the associated bound-ary conditions (9)-(11) were solved numer-ically using COMSOL Multiphysics withthe comprised Fluid Dynamics/IncompressibleNavier{Stokes and Heat Transfer/Convectionand Conduction application modules.

A non-uniform mesh comprised of triangu-

lar elements clustered towards the lower platewas used to discretise the ow domain, em-ploying second order interpolation functionsfor velocities and temperature and �rst orderinterpolation for pressure. The resulting sys-tem of equations was solved iteratively usinga form of the damped Newton method as de-scribed in [1]. The problem was programmedin the MATLAB environment to allow for the exible control of geometric and uid param-eters. A variety of mesh densities was exam-ined to establish the number and distributionof elements required to guarantee mesh inde-pendent solutions for the parameter range in-vestigated. For a typical ow geometry witha = 1=2 and h = 3=4 the number of ele-ments required to ensure mesh independencewas found to be 275710.

semi{analytical numerical

streamlines

Pe ! 0

Pe = 10

Pe = 100

Pe = 300

Figure 2: Flow structure (streamlines) and corresponding temperature �eld (isotherms) transition withincreasing Pe, obtained semi-analytically (left) and numerically (right), for the case a = 1=2, h = 3=4 [3].

6 Results and discussion

Results for creeping ow and ow with inertiaare presented independently of each other.

6.1 Creeping ow

6.1.1 Temperature �eld

Fig. 2, for the case a = 1=2 and a mean plateseparation of h = 3=4. These were obtained:

(i) using the semi-analytical approach, withN = 4 modes for the series (17); (ii) numeri-cally as described above. Agreement betweenthe two sets of results is seen to be remarkablyclose, with the streamline plots showing thatthe geometry as speci�ed results in the pres-ence of a large symmetric eddy while the na-ture of the corresponding temperature �eld isPecl�et number dependent. When Pe = 0 theproblem is one of pure heat conduction andthe temperature �eld is symmetric as shown.Symmetry is, however, soon lost due to thepresence of convection as demonstrated forthe case Pe = 10. As Pe is increased fur-ther a point is soon reached, when for su�-ciently large values, see for example the casePe = 100, the asymmetry present becomespronounced as warmer uid is transported up-wards on the left side of the domain with acorresponding downward movement of colder uid on the right. For Pecl�et numbers of ap-proximately 300 and greater, the isothermsmimic the corresponding streamline patternwith laminar thermal mixing occurring.

6.1.2 Global heat transport

A measure for the global heat transport isprovided by the Nusselt number Nu which isfound from the temperature �eld as

Nu = �h

2�

�Z��

Tz��z=h

dx (19)

It represents the non-dimensional globalheat ux, scaled with the corresponding heat ux � (T1 � T0) =h for Couette ow betweenparallel at plates �xed at the same referencetemperatures and in which case Nu = 1.

The Nusselt number was calculated bothsemi-analytically and numerically for the caseof a bottom plate with amplitude a = 1=2 andfor di�erent mean plate separations, the re-sults of which are plotted against Pecl�et num-ber in Fig. 3. Values of h were chosen insuch a way that four qualitatively di�erentcases could be investigated: one without aneddy present (h = 7=4), a second exhibitinga small eddy (h = 1), the third with a largeeddy present (h = 3=4), and a fourth case inwhich the ow has all but degenerated to thatof a driven-cavity like ow consisting almostentirely of a large single eddy (h = 3=5).

0 100Pe

200 3001:0

1:4

1:8

Nu

2:2

2:6

h = 7=4

h = 1

h = 3=4

h = 3=5

Figure 3: Global heat transport depicted as plotsof Nusselt number vs. Pecl�et number for the caseof a lower plate with amplitude a = 1=2 andfour di�erent mean plate separations. The up-per curves represent the results obtained semi{analytically [3].

In all cases the analytical and numericalresults are in very good agreement with eachother. Compared to the case of parallel plates(Nu = 1), it can be seen that there is alreadyan improvement in the global heat transportfor Pe ! 0; that is, even in the case of pureheat conduction it is observed that Nu > 1 asa consequence of the geometry.

For Pe > 0 an additional increase in Nu ,due to convection, is apparent. This e�ect,however, depends signi�cantly on the eddysize: for the case h = 7=4, one without aneddy, only a tiny improvement in the heat uxis observed, whereas the curve correspondingto the ow containing a small eddy (h = 1)reveals a distinct gradual increase of the Nus-selt number with increasing Pe . This e�ectbecomes even more pronounced the larger theeddy (h = 3=4) and especially so in the caseof a driven-cavity like ow (h = 3=5).

6.1.3 Thermal feedback on the ow

Thermal feedback due to thermoviscosity� = �(T ) is investigated numerically, withbuoyancy and thermal expansion e�ects dueto � = �(T ) neglected, as discussed in Sec-tion 2. In Fig. 4 results are shown for thecase a = 1=2, h = 3=4, a Pecl�et numberPe = 100 and a thermoviscous coupling with�� = 1=3 corresponding to a temperature-dependent viscosity given by

�(T ) = �0

�1�

T

3

�(20)

Note that this choice of temperature depen-dence leads to a uid viscosity that is 50%

larger at the colder plate than at the warmerone. Hence, a strong thermoviscous couplingis implicit in the calculations. For comparisonpurposes streamlines and isotherms are shownfor the case of constant viscosity. Shown also isthe constant viscosity case when the tempera-tures of the plates are reversed that is, the topplate is hotter than the bottom one. The sur-prising result which emerges from this �gure isthat, even in the case of strong thermoviscous

coupling, the feedback e�ect of the temper-ature on the ow is negligibly small. More-over, by comparing the resulting temperature�elds there is no discernible di�erence, even inthe case when the temperatures of the upperand lower plates are reversed. Accordingly,this result supports the a priori assumptionmade when formulating the semi{analyticalmethod of solution that the ow and temper-ature �elds can be decoupled.

(a)

(b)

(c)

Figure 4: Numerically calculated streamlines (left) and corresponding isotherms (right) for a ow geometrywith a = 1=2 and h = 3=4, and Pe = 100: (a) constant viscosity; (b) viscosity according to equation (20);(c) as (a) but with the upper and lower plate temperatures reversed [3].

6.2 Flow with �nite Reynolds number

6.2.1 E�ect of increasing inertia

Numerical solutions were obtained, see Fig.6, for Stokes ow and at Reynolds numberRe = 100 for the case a = 1=2 and the samefour mean plate separations as considered inFig. 3, and with Pe = 100. The streamlineplots to the left, for Stokes ow, show the in u-ence of the mean plate separation, that is thegeometry, on the eddy structure present; thoseon the right, for the case Re = 100, revealthat the presence of inertia can lead to botheddy generation and to increased asymmetryof an existing eddy structure. The shift in thevortex core, in the case of the latter, has con-sequences for the corresponding temperature�eld: in that the area where the heat transportdue to convection is signi�cant is also shiftedto the right.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 50 100 150 200 250 300 350

Re = 10Re = 100Re = 300 ?Re

?Re

6Re

6Re

Pe

Nu

(a)

(b)

(c)

(d)

Figure 5: Global heat transport depicted as plotsof Nusselt number vs. Pecl�et number, at three dif-ferent Reynolds numbers, for a lower plate withamplitude a = 1=2 and the four mean plate sepa-rations (a){(d) used in Figure 3, see [3].

6.2.2 Global heat transport

Using equation 19 the Nusselt number was cal-culated for the case of a lower plate with am-plitude a = 1=2, for the same four mean plateseparations. This was done at three di�er-ent Reynolds numbers Re = 10; 100; 300 forPecl�et numbers between 0 and 350. The re-sulting curves of Nu versus Pe are shown inFig. 5. Compare these with those in Fig. 3for the corresponding creeping ow problem.The e�ect of inertia is qualitatively di�erentfor the four geometries considered: in the twocases (d) and (c) commensurate with a smallermean plate separation, h = 3=5 and h = 3=4,respectively, the global heat transport is re-duced with increasing inertia, whereas in the

two cases (b) and (a) with a larger mean plateseparation, h = 1 and h = 7=4 respectively,the opposite occurs and it is enhanced. Theexplanation for this qualitative di�erence isfound by examining the underlying ow struc-tures. In the case h = 3=4, for instance, thecorresponding velocity �eld shown in Fig. 6reveals a shift to the right of the vortex core.Therefore, the area over which convective heattransport is relevant is reduced. The samequalitative e�ect is found for h = 3=5, sincein this case too there is a large eddy in the ow. In contrast, for h = 1 there is only asmall eddy present, while for h = 7=4 there isno eddy present at assisting the transport ofheat which shows as a corresponding increasein the Nusselt number.

(d)

(c)

(b)

(a)

Re ! 0 Re = 100

Figure 6: Numerically calculated streamlines (left) and corresponding isotherms (right) for a ow geometry

with a = 1=2 and h = 3=4, and Pe = 100: (a) constant viscosity; (b) viscosity according to equation (20);(c) as (a) but with the upper and lower plate temperatures reversed [3].

7 Conclusion

The subtle interplay that exists between theglobal transfer of heat and the underlying ow structure and hence local laminar thermalmixing in the case of shear ow between tworigid surfaces at di�erent �xed temperatures {the hot, upper one, planar; the lower, coolerone, varying sinusoidally { a small mean dis-

tance apart, has been explored both analyti-cally and numerically. For creeping ow condi-tions and varying Pecl�et number, the thermal�eld is found to be asymmetric for all values ofthe Pecl�et number other than the limiting con-ditions of zero and in�nity, at which extremesthe corresponding thermal �eld is symmetric.

Global heat transport, in the form of plotsof Nusselt number against Pecl�et number, is

investigated with agreement between predic-tions from analysis and ones obtained numer-ically seen to be extremely good, particularlyfor higher values of mean plate separation. Itis found that compared to the case when bothtop and bottom plates are at there is an im-provement to be seen in global heat transportas a consequence of the geometry, even forthe case of pure heat conduction (Pe ! 0),but which depends signi�cantly on the sizeof the underlying eddy structure present asthe Pecl�et number is increased and convectionplays a more signi�cant role.

In the case of non-creeping ow, the e�ectof increasing inertia on both the temperature�eld and global heat transport is revealed, theobvious one being to skew the underlying eddystructure { for moderate Reynolds numbers,Re = 100, this takes the form of a shift to theright of the vortex core. The consequence forthe corresponding temperature �elds is thatthe region where heat transfer due to convec-tion is signi�cant is also skewed in the samedirection. The results obtained for global heattransfer expose the interrelationship betweeneddies induced kinematically and due to iner-tia, in that inertia can result in the creation ofan eddy or enlarge an existing eddy, for a givenlower plate pro�le and mean plate separation,that is �xed ow geometry. Indeed the presentwork suggests that for a given sinusoidal vari-ation of the lower plate it should be possible,from a practical standpoint, to �nd a criti-cal mean plate separation for which Reynoldsnumber e�ects on the global heat transfer are

minimised.

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