the unified integral transforms (unit) algorithm with

Computational Thermal Sciences, 6 (6): 507–524 (2014)

THE UNIFIED INTEGRAL TRANSFORMS (UNIT)ALGORITHM WITH TOTAL AND PARTIALTRANSFORMATIONRenato M. Cotta,1,2,∗ Diego C. Knupp,3 Carolina P. Naveira-Cotta,2

Leandro A. Sphaier,4 & Joao N. N. Quaresma5

1Laboratory of Transmission and Technology of Heat, LTTC, Department of Mechanical Engi-neering, Universidade Federal do Rio de Janeiro, POLI&COPPE/UFRJ, Cidade Universitaria,Cx. Postal 68503, 21945-970, Rio de Janeiro, RJ, Brazil2Laboratory of Microfuidics and Micro-systems, LabMEMS, Department of Mechanical Engi-neering, Universidade Federal do Rio de Janeiro, POLI&COPPE/UFRJ, Cidade Universitaria,Cx. Postal 68503, 21945-970, Rio de Janeiro, RJ, Brazil3Laboratory of Experimentation and Numerical Simulation in Heat and Mass Transfer, LEMA,Dept. of Mechanical Engineering and Energy, Instituto Politecnico, IPRJ/UERJ, Rua Bonfim, 25,Nova Friburgo, RJ, 28624-570, Brazil4Laboratory of Theoretical and Applied Mechanics, LMTA, Mechanical Engineering Department,Universidade Federal Fluminense, PGMEC/UFF, Rua Passo da Patria 156, Bloco E, Sala 216,24210-240, Niteroi, RJ, Brazil5School of Chemical Engineering, Universidade Federal do Para, FEQ/UFPA, Campus Univer-sitario do Guama, 66075-110, Belem, PA, Brazil

∗Address all correspondence to Renato M. Cotta E-mail: [email protected]

The theory and algorithm behind the open-source mixed symbolic-numerical computational code named UNIT (uni-fied integral transforms) are described. The UNIT code provides a computational environment for finding solutions oflinear and nonlinear partial differential systems via integral transforms. The algorithm is based on the well-establishedanalytical-numerical methodology known as the generalized integral transform technique (GITT), together with themixed symbolic-numerical computational environment provided by the Mathematica system (version 7.0 and up). Thispaper is aimed at presenting a partial transformation scheme option in the solution of transient convective-diffusiveproblems, which allows the user to choose a space variable not to be integral transformed. This approach is shown tobe useful in situations when one chooses to perform the integral transformation on those coordinates with predominantdiffusion effects only, whereas the direction with predominant convection effects is handled numerically, together withthe time variable, in the resulting transformed system of one-dimensional partial differential equations. Test cases areselected based on the nonlinear three-dimensional Burgers’ equation, with the establishment of reference results for spe-cific numerical values of the governing parameters. Then the algorithm is illustrated in the solution of conjugated heattransfer in microchannels.

KEY WORDS: integral transforms, hybrid methods, symbolic computation, Burgers’ equation, conjugatedheat transfer, microchannel flow

1940–2503/14/$35.00 c⃝ 2014 by Begell House, Inc. 507

508 R.M. Cotta et al.

NOMENCLATURE

Dh hydraulic diameter in the conjugated heat w transient operator coefficient in generaltransfer problem problem; heat capacity in the conjugated

d dissipation operator coefficient heat transfer problem, Eqs. (11)g nonlinear source term x position vector in the general problemK diffusion operator coefficient formulationLe distance from the channel centerline to the x longitudinal coordinate in problem (9)

external face of the channel wall in the Y dimensionless transversal coordinate in theconjugated heat transfer problem conjugated heat transfer problem

Lf channel height in the conjugated heat transfery transversal coordinate in problemsproblem (9) and (11)

Lw channel width in the conjugated heat transferZ dimensionless longitudinal coordinate in theproblem conjugated heat transfer problem

M number of subregions in semianalytical and z transversal coordinate in problem (9);Gaussian integration longitudinal coordinate in problem (11)

N truncation order in eigenfunction expansionn number of coupled potentials in the general Greek Symbols

problem formulation α boundary condition coefficientPe Peclet number in the conjugated heat transferβ boundary condition coefficient

problem θ dimensionless temperature field in theT dimensionless potential in the general conjugated heat transfer problem

problem formulation; potential in Burgers’ µ eigenvaluesequations (9) and (10); temperature field ν diffusion coefficient in Burgers’ equationin the conjugated ϕk source term in boundary condition

t dimensionless time variable ψ eigenfunctionsheat transfer problem, Eqs. (11)

U dimensionless fully developed velocity profileSubscripts & Superscriptsin the conjugated heat transfer problem i order of eigenquantities

u nonlinear function in convection term in k quantity corresponding to the equation of theBurgers’ equation (9); fully developed kth potential in general problemvelocity profile in the conjugated heat - position vector excluding the space variabletransfer problem, Eqs. (11) not to be integral transformed

u0 linear parameter in nonlinear convection term – integral transformin Burgers’ equation, Eqs. (9) ∼ normalized eigenfunction

1. INTRODUCTION

Integral transforms have been successfully used indifferent branches of the physical, mathematical, andengineering sciences for about 200 years. Its introductioncan be attributed to Fourier, after the publication of histreatise on the analytical theory of heat (Fourier, 1822).In essence, Fourier at that time advanced the idea of sepa-ration of variables, so as to handle and interpret the solu-

tions of the newly derived heat conduction equation, afterproposing the constitutive equation known nowadays asFourier’s law. He gave a series of examples before stat-ing that an arbitrary function defined on a finite intervalcan be expanded in terms of a trigonometric series thatis now known as the Fourier series. In an attempt to ex-tend his new ideas to functions defined on an infinite in-terval, Fourier discovered an integral transform and its in-version formula which are now commonly known as the

Computational Thermal Sciences

The Unified Integral Transforms (Unit) Algorithm 509

Fourier transform and the inverse Fourier transform, re-spectively. His work provided the modern mathematicaltheory of heat conduction, but also introduced Fourier se-ries, Fourier integrals, and stated an important result thatis known as the Fourier integral theorem, later rephrasedby Dirichlet (Debnath and Bhatta, 2007).

The method of integral transforms was then widelyused in solving linear partial differential equations(PDEs) of mathematical physics along the followingyears, and the classical treatise of Carslaw and Jaeger(1947) provides a wide collection of solutions obtainedin heat conduction theory by this and the competing an-alytical approaches then available. According to Luikov(1980), father of modern analytical heat diffusion the-ory, as detailed in his most classical work (Luikov, 1968),it was not until the work of Koshlyakov (1936) that anidea was provided in handling nonhomogeneous diffusionequations and boundary conditions by the method of finiteintegral transformations, and the theory of such integraltransforms was developed in detail by Grinberg (1948),who also extended this approach to multilayer prob-lems, considering step variations of the material proper-ties along the transformation coordinate. A very active pe-riod of research on exact analytical solutions of nonhomo-geneous heat and mass transfer problems then followed,when besides the continuous contribution of Luikov andcoworkers, one should certainly include the contributionsof other very prominent researchers, such as Olcer (1964,1967), Ozisik (1968), and Mikhailov (1967, 1972). Thisperiod was so fruitful for analytical heat and mass trans-fer that Luikov himself, in 1974, contacted both Ozisikand Mikhailov regarding the joint publication of bookson heat conduction and convection. However, Luikovpassed away even before the startup of these projects,but Mikhailov and Ozisik finally met each other in 1976and started a new project inspired by Luikov’s sugges-tions, work that would be completed and published onlyin 1984, when most of the available exact solutions ofheat and mass diffusion through integral transforms wereunified in seven different classes of problems and sys-tematically presented in a reference book (Mikhailov andOzisik, 1984). This book was very much influenced by theprevious publications of Mikhailov along a very produc-tive period of more than one decade (including Mikhailov,1967, 1972, 1973a,b, 1975, 1977a,b; Mikhailov andShishedjiev, 1976; Mikhailov and Ozisik, 1980, 1981;Mikhailov et al., 1982; Mikhailov and Vulchanov, 1983),when he challenged the integral transform method to han-dle different classes of unified formulations in heat andmass diffusion. Ozisik also pursued the rewriting of his

1968 textbook during this period (Ozisik, 1968) and pre-pared a fairly complete new work on heat conduction,compiling different analytical, approximate, and numer-ical approaches (Ozisik, 1980).

Within this same period, Ozisik and Mikhailov, inde-pendently, also perceived the limitations of the integraltransforms method, as it was known at that time, whenthey tried to solve problems with time-dependent bound-ary condition coefficients (Ozisik and Murray, 1974;Yener and Ozisik, 1974) or time-dependent equation co-efficients (Mikhailov, 1975). In these early works, the in-tegral transformation process, due to the time dependenceof the transformation kernel represented by the eigen-values, eigenfunctions, and norms, was not successfulin fully transforming the original PDE and resulted in acoupled infinite ordinary differential system for the trans-formed temperature fields. Nevertheless, the authors werestill able to propose analytical approximations by takingonly a limited number of terms in the coupling terms andthen forcing the simplification to a decoupled system. Inthe work of Ozisik and Murray (1974), the expressiongeneralized integral transform technique was employedfor the first time. This same concept of approximate an-alytical solution was later on employed by Ozisik andGuceri (1977) in the solution of phase change problems,by Bayazitoglu and Ozisik (1980) in the analysis of inter-nal forced convection with axial diffusion effects, by Bo-gado Leite et al. (1980, 1982) in solving moving bound-ary diffusion problems related to the erosion of fusion re-actor walls, and by Cotta and Ozisik (1985) in the solu-tion of transient internal convective heat transfer due towall temperature variations. Such approximate solutions,though elegant and easy to compute, had limitations interms of accuracy, within certain ranges of the involvedparameters and independent variables, and would also re-quire some sort of numerical solution for verification pur-poses. However, in Cotta (1986a) the complete solution ofthe coupled transformed system was achieved, based onthe numerical solution of a truncated version of the trans-formed ordinary differential equation (ODE’ system, asobtained for a diffusion problem with a prescribed mov-ing boundary, associated with oxidation of nuclear fuelrods cladding. The resulting transformed system typicalof such integral transformations is likely to present sig-nificant stiffness, especially for larger truncation orders,but at that time reliable solvers for stiff initial value prob-lems were already available, allowing for error-controlledsolutions of the transformed potentials. Only then wouldthe GITT (generalized integral transform technique) beproposed as a full hybrid numerical-analytical solution of

Volume 6, Number 6, 2014


nontransformable diffusion or convection–diffusion prob-lems. The GITT was later advanced to offer analyticalsolutions of the complete transformed system for lin-ear problems, such as in the transient internal convec-tion problem in the complex domain solved in Cotta andOzisik (1986). In a natural sequence, the previous so-lutions of diffusion problems with variable equation orboundary coefficients were formalized soon afterward,under the new concept of obtaining the solution of thecomplete transformed systems (Cotta, 1986b; Cotta andOzisik, 1987). Clearly, the generalized approach was theninterpreted as the closest in nature to the exact solutionsobtainable by integral transforms in the case of trans-formable problems, although somehow still approximatedue to the truncation of the infinite transformed system.Once the transformed system is numerically solved bycontrolling the relative error in the initial value prob-lem algorithm, one is left with the task of controlling theglobal error of the solution by adequately choosing thesystem size and thus the eigenfunction expansion trunca-tion order.

An avenue of opportunities was then opened and thesuccessive challenges for the GITT extension would formthe basis of a series of theses and papers. Just to name afew contributions within this fruitful period, the analysisof diffusion within irregular domains was soon proposed(Aparecido and Cotta, 1987; Aparecido et al., 1989), andfollowed by the analysis of nonlinear diffusion problems(Cotta, 1990; Serfaty and Cotta, 1990, 1992), conjugatedconvection-conduction problems (Guedes et al., 1989;Guedes and Cotta, 1991), ablation moving boundaryproblems (Diniz et al., 1990), boundary layer equations(Cotta and Carvalho, 1991; Carvalho et al., 1993), Navier-Stokes equations (Perez Guerrero and Cotta, 1992), dry-ing problems (Ribeiro et al., 1993), and natural convec-tion in porous media (Baohua and Cotta, 1993). Basedon the above works and a few others, the first referenceand textbook on the GITT was prepared and publishedin 1993 (Cotta, 1993), including some formal aspects ofthe approach that were not previously dealt with in theavailable publications. This effort made the hybrid ap-proach more visible and the positive response of the heattransfer community was soon provided, as demonstratedin the keynote lecture at the 10th IHTC, UK, in 1994(Cotta, 1994a) and the invited review paper in theIn-ternational Journal of Heat and Mass Transferin thatsame year (Cotta, 1994b), celebrating the contributionof Prof. James P. Hartnett to the heat and mass trans-fer field. Two years later a book on heat conduction waspublished (Cotta and Mikhailov, 1997) that would com-

bine the knowledge on improved lumped system analy-sis, GITT, and symbolic computation. Also within thisphase, a compilation of advanced contributions was or-ganized (Cotta, 1998) in order to complement the 1993book, which remains an important source of advanced ref-erence work on the generalized integral transform tech-nique.

The next phase then initiated in the development ofintegral transforms in heat and fluid flow was character-ized by the optimization of the numerical tasks, proposi-tion of more challenging problems among those classesalready handled, and the more ample application of thisknowledge basis in different areas. For instance, onemay recall the analysis of fluid flow and mass transferwithin petroleum reservoirs (Almeida and Cotta, 1995),three-dimensional Navier-Stokes equations (Quaresmaand Cotta, 1997), Navier-Stokes equations in irregular ge-ometries (Perez Guerrero et al., 2000), forced and naturalconvection with variable fluid properties (Machado andCotta, 1999; Leal et al., 2000), compressible flow andheat transfer in ultracentrifuges (Pereira et al., 2002), sta-bility analysis in natural convection (Alves et al., 2002),three-dimensional natural convection in porous enclo-sures (Luz Neto et al., 2002), eigenvalue and diffusionproblems in multidimensional irregular domains (Sphaierand Cotta, 2000; Sphaier and Cotta, 2002), and contami-nant dispersion in fractured media (Cotta et al., 2003), toname a few. The maturity of the approach was then con-solidated in the edited book (Santos et al., 2001), in theinvited editorial of theHeat Transfer Engineeringjournal(Cotta and Orlande, 2003), and finally in the invited chap-ter for theHandbook of Numerical Heat Transfer(Cottaand Mikhailov, 2006).

In recent years, besides the continuous search for morechallenging problems and different application areas, em-phasis has been placed in unifying and simplifying theuse of the GITT, to reach a larger number of users andoffer an alternative hybrid solution to their problems. Hy-brid methods become even more powerful and applicablewhen symbolic manipulation systems, which were alsowidely disseminated along the last two to three decades,are employed. The effort in integrating the knowledge onthe GITT into a symbolic-numerical algorithm resulted inthe so-called UNIT code (unified integral transforms), ini-tially proposed as a project in 2007 and intended to bridgethe gap between simple problems that allow for a straight-forward analytical solution, and more complex and in-volved situations that almost unavoidably require special-ized software systems. The open-source UNIT code isthus an implementation and development platform for re-



searchers and engineers interested in integral transformsolutions of diffusion and convection–diffusion problems(Sphaier et al., 2011; Knupp et al., 2010, Cotta et al.,2010, 2013).

This work is thus aimed at reviewing the current ver-sion of the multidimensional UNIT code in the Math-ematica platform and introduces its partial transforma-tion scheme option. In this alternative partial transforma-tion mode, the user is allowed to choose one of the di-mensional variables not to be integral transformed. Thisstrategy can be very useful for the solution of transientconvective–diffusive problems in which one chooses toperform the integral transformation only on the direc-tions with predominant diffusion effects, whereas alongthe direction with predominant convection effects theproblem is solved numerically, together with the timevariable, in the resulting transformed system of one-dimensional PDEs. In order to illustrate the present im-plementation, we show some results regarding a non-linear three-dimensional formulation of Burgers’ equa-tion, which are critically compared to the default totaltransformation UNIT code solutions. Also, we illustratethe partial transformation approach for the analysis ofconjugated heat transfer in microheat spreaders made ofmicrochannels molded in a polymeric matrix (Ayres etal., 2011), employing a recently advanced single-domainstrategy for conjugated heat transfer problems developedby Knupp et al. (2012, 2013a,b, 2014a,b).

2. PROBLEM FORMULATION

A general transient convection–diffusion problem ofncoupled potentials is considered, defined in the regionVwith boundary surfaceS:

wk(x)Lk,tTk(x, t) = Gk(x, t,T), x ∈ V, t > 0,

k = 1, 2, ..., n (1a)

where thet operatorLk,t for a parabolic or parabolic–hyperbolic formulation is given by

Lk,t ≡∂

∂t(1b)

and

Gk(x, t,T) = ∇ · (Kk(x)∇Tk(x, t))− dk(x)Tk(x, t)

+ gk(x, t,T) (1c)

with initial and boundary conditions given, respectively,by

Tk(x, 0) = fk(x), x ∈ V (1d)

[αk(x) + βk(x)Kk(x)

∂

∂n

]Tk(x, t) = φk(x, t,T),

x ∈ S, t > 0 (1e)

wheren denotes the outward-drawn normal to the surfaceS, and where the potentials vector is given by

T = {T1, T2, ..., Tk, ..., Tn} (1f)

As mentioned before, Eqs. (1a)–(1f) are quite general,since nonlinear and convection terms may be groupedinto the equations and boundary conditions source terms.It may be highlighted that in the case of decoupled lin-ear source terms, i.e.,g ≡ g(x, t), andφ ≡ φ(x, t),this example is reduced to a class I linear diffusion prob-lem for each potential, according to the classification inMikhailov and Ozisik (1984), and exact analytical solu-tions are readily available via the classical integral trans-form technique. Otherwise, this problem shall not beapriori transformable, except for a few linear coupled sit-uations also illustrated in Mikhailov and Ozisik (1984).However, the formal solution procedure provided by theGITT (Cotta, 1993) may be invoked in order to providehybrid numerical-analytical solutions for the nontrans-formable problems.

The formal solution regarding the standard procedureof the UNIT code is known as the total transformationscheme, described in Cotta et al. (2013), in which all spa-tial variables are integral transformed. Here we focus onthe partial integral transformation scheme option of theUNIT code, as an alternative solution path to problemswith a strong convective direction, which is not elimi-nated through integral transformation but kept within thetransformed system.

3. SOLUTION METHODOLOGY

3.1 Total Transformation

Following the formal solution procedure for nonlinearconvection–diffusion problems through integral trans-forms, the proposition of eigenfunction expansions forthe associated potentials are first required. The preferredeigenvalue problem choice appears from the direct appli-cation of the separation of variables methodology to thelinear homogeneous purely diffusive version of the pro-posed problem. Thus the recommended set of decoupledauxiliary problems is here given by

∇ · [Kk(x)∇ψki(x)] + [µ2kiwk(x)− dk(x)]ψki(x) = 0,

x ∈ V (2a)



[αk(x)+βk(x)Kk(x)

∂

∂n

]ψki(x)=0, x∈S (2b)

where the eigenvaluesµki and associated eigenfunctionsψki(x) are assumed to be known from exact analytical ex-pressions, for instance, obtained through symbolic com-putation (Wolfram, 2005) or application of the GITT it-self (Naveira-Cotta et al., 2009). One should notice thatEqs. (1a)–(1f) are presented in a form that already reflectsthis choice of eigenvalue problems, given by Eqs. (2a)and (2b), with the adoption of linear coefficients in boththe equations and boundary conditions, and incorporatingthe remaining terms (coupling, nonlinear, and convectiveterms) into the general nonlinear source terms, withoutloss of generality.

By making use of the orthogonality properties of theeigenfunctions, it is then possible to define the followingintegral transform pairs:

Tki(t)=

∫V

wk(x)ψki(x)Tk(x, t)dV transforms (3a)

Tk(x, t) =∞∑i=1

ψki(x)Tk,i(t) inverses (3b)

where the symmetric kernelsψki(x) are given by

ψki(x)=ψki(x)√

Nki

; Nki=

∫V

wk(x)ψ2ki(x)dV (3c,d)

with Nki being the normalization integral.The integral transformation of Eq. (1a) is accom-

plished by applying the operator∫Vψki(x) (·) dV and

making use of the boundary conditions given by Eqs. (1e)and (2b), yielding

dTki(t)

dt+ µ2kiTki(t) = gki(t,T) + bki(t,T),

i = 1, 2, ..., t > 0, k = 1, 2, ..., n (4a)

where the first transformed source termgki(t,T) is due tothe integral transformation of the equation source terms,and the second one,bki(t,T), is due to the contributionof the boundary source terms:

gki(t,T) =

∫V

ψki(x)gk(x, t,T)dV (4b)

bki(t,T) =

∫S

Kk(x)

[ψki(x)

∂Tk(x, t)

∂n− Tk(x, t)

× ∂ψki(x)

∂n

]dS (4c)

The boundary conditions contribution may also be ex-pressed in terms of the boundary source terms, after ma-nipulating Eqs. (1e) and (2b), to yield

bki(t,T)=

∫S

φk(x,t,T)

[ψki(x)−Kk(x)

∂ψki(x)∂n

αk(x) + βk(x)

]dS (4d)

The initial conditions given by Eq. (1d) are transformedthrough the operator

∫Vwk(x)ψki(x) (·) dV to provide

Tki(0) = fki ≡∫V

wk(x)ψki(x)fk(x)dV (4e)

For the solution of the infinite coupled system of non-linear ordinary differential equations given by Eqs. (4a)–(4e), one must make use of numerical algorithms, after thetruncation of the system to a sufficiently large finite or-der. For instance, in the present work, the built-in routineof the Mathematica system (Wolfram, 2005) is employed,NDSolve, which is able to provide reliable solutions un-der automatic absolute and relative error control. After thetransformed potentials have been numerically computed,the Mathematica routine automatically provides an inter-polating function object that approximates thet variablebehavior of the solution in a continuous form. Then theinversion formula can be recalled to yield the potentialfield representation at any desired positionx and timet.

The solution procedure described above provides thebasic straightforward working expressions for the integraltransform method. Nevertheless, for an improved compu-tational performance, it is always recommended to reducethe importance of the equation and boundary source termsso as to enhance the eigenfunction expansion’s conver-gence behavior (Cotta and Mikhailov, 1997). The UNITcode for multidimensional applications allows for user-provided filters, but an automatic progressive linear fil-tering option is also implemented. The interested readeris encouraged to refer to Cotta et al. (2013), where thisfiltering strategy is described in detail.

The constructed multidimensional UNIT code in theMathematica platform (Cotta et al., 2010, 2013) encom-passes all of the symbolic derivations that are required inthe above GITT formal solution, besides the numericalcomputations that are required in the solutions of the cho-sen eigenvalue problem and the transformed differentialsystem. The user essentially needs to specify the problemformulation together with the required problem parame-ters, solve the problem using the provided UNIT algo-rithm, and then choose how to present the results accord-ing to the specific needs. Besides the parameters regard-ing the problem formulation, the user is also asked to set



the truncation order of the eigenfunction expansionsNand to choose the coefficients integration methodology,which can be analytical, through the Integrate routine inthe Mathematica system or user-provided, semianalytical,or by an automatic Gaussian quadrature scheme that ac-counts for the information regarding the eigenfunctions’oscillatory behavior. The Gaussian quadratures and thealternative semianalytical integration procedure are par-ticularly convenient in nonlinear formulations that mightrequire costly numerical integration, once analytical inte-gration is not feasible.

The UNIT code is here illustrated through its version2.2.3, in Mathematica 7.0 or up, and has the followingmain features:

1. A system of linear or nonlinear equations (parabolicproblems, parabolic–hyperbolic problems, or ellipticproblems in pseudo-transient formulation);

2. Multidimensional transient formulations, automati-cally defined by one single parameter with the num-ber of space dimensions;

3. Eigenvalue problem analytically solved via theDSolve routine (Sturm-Liouville problem);

4. Transformed coefficients determined by semianalyt-ical integration (zeroth order), numerical integration(Gaussian quadrature or NIntegrate routine), or an-alytical integration (Integrate routine or user sup-plied);

5. User-defined or automatic progressive linear filter-ing;

6. Reordering by squared eigenvalues criterion or com-bination of transformed initial conditions, trans-formed source term, and squared eigenvalues crite-ria;

7. Nonhomogeneous term via Green’s second formula;

8. Error estimator with adjustable residue order.

3.2 Partial Transformation

An alternative hybrid solution strategy to the above-described full integral transformation is of particular in-terest in the treatment of transient convection–diffusionproblems with a preferential convective direction. In suchcases, the partial integral transformation in all but onespace coordinate may offer an interesting combination ofrelative advantages between the eigenfunction expansion

approach and the selected numerical method for handlingthe coupled system of one-dimensional PDEs that resultsfrom the transformation procedure. To illustrate this par-tial integral transformation procedure, again a transientconvection–diffusion problem ofn coupled potentials isconsidered, but this time separating the preferential di-rection that is not to be integral transformed. The po-sition vectorx now includes the space coordinates thatwill be eliminated through integral transformation, heredenoted byx∗, as well as the space variable to be re-tained in the transformed partial differential system. Thusconsider a general three-dimensional problem withx ={x1, x2, x3}, where only the coordinatesx∗ = {x1, x2}are intended to be eliminated by the integral transforma-tion process, while the remaining space variablex3 shallbe retained in the transformed system to be numericallysolved. The problem to be solved is now written in thefollowing form:

wk(x∗)∂Tk(x, t)

∂t= Gk(x, t,T), x ∈ V, t > 0,

k = 1, 2, ..., n (5a)

with

Gk(x, t,T) = ∇∗ · [Kk(x∗)∇∗Tk(x, t)]− dk(x

∗)

× Tk(x, t) + gk(x, t,T) (5b)

where the operator∇∗ refers only to the coordinates tobe integral transformedx∗, and with initial and boundaryconditions given, respectively, by

Tk(x, 0) = fk(x), x ∈ V (5c)

[λk(x3) + γk(x3)

∂

∂x3

]Tk(x, t) = ϕk(x, t,T),

x3 ∈ S3, t > 0 (5d)

[αk(x

∗) + βk(x∗)Kk(x

∗)∂

∂n∗

]Tk(x, t)=φk(x, t,T),

x∗ ∈ S∗, t > 0 (5e)

wheren∗ denotes the outward-drawn normal to the sur-faceS∗ formed by the coordinatesx∗ andS3 refers to theboundary values of the coordinatex3.

The coefficientswk(x∗), dk(x

∗), Kk(x∗), αk(x

∗),andβk(x

∗) in Eqs. (5a)–(5e) inherently carry the infor-mation on the auxiliary problem that will be considered inthe eigenfunction expansion, and all the remaining terms



from this rearrangement are collected into the sourceterms,gk(x, t,T) andφk(x, t,T), including the exist-ing nonlinear terms and diffusion and/or convection termswith respect to the independent variablex3. Also, the co-efficientsλk(x3) and γk(x3) provide any combinationof first to third kind boundary conditions in the untrans-formed coordinate, while thex3 boundary source terms,ϕk(x, t,T), collect the rearranged information that is notcontained in the right-hand side of Eq. (5d).

Following the solution path previously established, theformal integral transform solution of the posed problemrequires the proposition of eigenfunction expansions forthe associated potentials. The recommended set of uncou-pled auxiliary problems is given by

∇ · [Kk(x∗)∇ψki(x

∗)] +[µ2kiwk(x

∗)− dk(x∗)]

×ψki(x∗) = 0, x∗ ∈ V ∗ (6a)[

αk(x∗) + βk(x

∗)Kk(x∗)

∂

∂n∗

]ψki(x

∗) = 0,

x∗ ∈ S∗ (6b)

The problem indicated by Eqs. (6a) and (6b) allows,through the associated orthogonality property of theeigenfunctions, the definition of the following integraltransform pairs:

Tki(x3, t) =

∫V ∗

wk(x∗)ψki(x

∗)Tk(x, t)dV∗,

transforms (7a)

Tk(x, t) =∞∑i=1

ψki(x∗)Tki(x3, t), inverses (7b)

where the symmetric kernelsψki(x∗) are given by

ψki(x∗) =

ψki(x∗)√

Nki

;

Nki =

∫V ∗

wk(x∗)ψ2

ki(x∗)dV ∗ (7c,d)

with Nki being the normalization integral.The integral transformation of Eq. (5a) is accom-

plished by applying the operator∫V ∗ ψki(x

∗) (·) dV ∗ andmaking use of the boundary conditions given by Eqs. (5e)and (6b), yielding

∂Tki(x3, t)

∂t+ µ2kiTki(x3, t) = gki(x3, t,T)

+ bki(x3, t,T), i = 1, 2, ..., k = 1, 2, ..., n,

x3 ∈ V3, t > 0, (8a)

where the transformed source termgki(x3, t,T) is due tothe integral transformation of the equation source term,and the other,bki(x3, t,T), is due to the contribution ofthe boundary source term at the directions being trans-formed:

gki(x3, t,T) =

∫V ∗ψki(x

∗)gk(x, t,T)dV ∗ (8b)

bki(x3, t,T) =

∫S∗

Kk(x∗)

[ψki(x

∗)∂Tk(x, t)

∂n∗

− Tk(x, t)∂ψki(x

∗)

∂n∗

]dS∗ (8c)

The contribution of the boundary conditions at the direc-tions being transformed may also be expressed in termsof the boundary source terms:

bki(x3, t,T) =

∫S∗φk(x, t,T)

×

[ψki(x

∗)−Kk(x∗)∂ψki(x

∗)∂n∗

αk (x∗) + βk (x∗)

]dS∗ (8d)

The initial conditions given by Eq. (5c) are transformedthrough the operator

∫V ∗ wk(x

∗)ψki(x∗) (·) dV ∗ to pro-

vide

Tki(x3, 0) = f1,ki(x3) ≡∫V ∗

wk(x∗)ψki(x

∗)

× fk(x)dV∗ (8e)

Finally, the boundary conditions with respect to the direc-tion x3 are also transformed through the same operator,yielding[λk(x3)+γk(x3)

∂

∂x3

]Tki(x3, t)=ϕkı(x3, t,T), (8f)

with

ϕki(x3, t,T) ≡∫v∗

wk(x∗)ψki(x

∗)ϕk(x, t,T)dV ∗,

x3 ∈ S3, t > 0 (8g)

Equations (8a)–(8g) form an infinite coupled systemof nonlinear one-dimensional PDEs for the transformedpotentials Tki(x3, t), which is unlikely to be analyti-cally solvable. Nonetheless, reliable algorithms are read-ily available to numerically handle this PDE system, af-ter truncation to a sufficiently large finite order. For in-stance, the Mathematica system provides the built-in rou-tine NDSolve, which can handle this system under auto-matic absolute and relative error control. Once the trans-formed potentials have been numerically computed, the



Mathematica routine automatically provides an interpo-lating function object that approximates thex3 andt vari-ables behavior of the solution in a continuous form. Thenthe inversion formula in Eq. (7b) can be recalled to yieldthe potential field representation at any desired positionxand timet.

A major aspect in the practical implementation of thismethodology is the eventual need for improving the con-vergence behavior of the resulting eigenfunction expan-sions, as pointed out in Cotta and Mikhailov (1997). Theoverall simplest and most effective alternative for conver-gence improvement appears to be the proposition of an-alytical filtering solutions, which present both space andtime dependence within specified ranges of the numericalintegration path. For instance, an appropriate quasi-steadyfilter for the above formulations could be written in gen-eral as

Tk(x, t) = θk(x, t) + Tf,k(x; t) (9a)

Tk(x∗, x3, t) = θk(x

∗, x3, t) + Tf,k(x∗;x3, t) (9b)

where the second term in the right-hand sides representsthe quasi-steady filter solution, which is generally soughtin analytic form. The first term on the right-hand side rep-resents the filtered potentials which are obtained throughintegral transformation. Once the filtering problem for-mulation is chosen, Eqs. (9a) and (9b) are substituted backinto Eqs. (1) or (5), respectively, to obtain the resultingformulation for the filtered potential. It is desirable thatthe filtering solution contains as much information on theoperators of the original problem as possible. This infor-mation may include, for instance, linearized versions ofthe source terms, so as to reduce their influence on con-vergence of the final eigenfunction expansions.

The analytical nature of the inversion formula allowsfor a direct error testing procedure at each specified posi-tion within the medium, and the truncation orderN canalso be controlled to fit the user global error requirementsover the entire solution domain (Cotta, 1993). The tol-erance testing formulas employed in the total and partialtransformation are, respectively,

ε = maxx∈V

∣∣∣∣∣∣∣∣∣N∑

i=N∗ψki(x)Tk,i(t)

Tf,k(x; t) +N∑i=1

ψki(x)Tk,i(t)

∣∣∣∣∣∣∣∣∣ (10a)

ε=maxx∗∈V ∗

∣∣∣∣∣∣∣∣∣N∑

i=N∗ψki(x

∗)Tk,i(x3, t)

Tf,k(x∗;x3, t)+N∑i=1

ψki(x∗)Tk,i(x3, t)

∣∣∣∣∣∣∣∣∣(10b)

The numerator in Eqs. (10a) and (10b) represents thoseterms (from ordersN∗ to N ) that in principle might beabandoned in the evaluation of the inverse formula, with-out disturbing the final result to within the user-requestedaccuracy target, once convergence has been achieved.

4. APPLICATIONS

The UNIT partial transformation algorithm is here illus-trated first with a test case based on the nonlinear three-dimensional formulation of Burgers’ equation, with ho-mogeneous and nonhomogeneous boundary conditions,previously considered by employing the total transforma-tion procedure (Cotta et al., 2013). Then we examine thebehavior of the UNIT PT algorithm for an application in-volving conjugated heat transfer in microchannels, as re-cently proposed via a single-domain formulation that en-compasses both the fluid and solid regions (Knupp et al.,2012).

4.1 Nonlinear Burgers’ Equation

The mathematical formulation of the three-dimensionalnonlinear Burgers’ equation with homogeneous boundaryconditions here considered is given by

∂T (x, y, z, t)

∂t+u(T )

∂T (x, y, z, t)

∂x=ν

[∂2T (x, y, z, t)

∂x2

+∂2T (x, y, z, t)

∂y2+

∂2T (x, y, z, t)

∂z2

],

0 < x < 1, 0 < y < 1, 0 < z < 1, t > 0 (11a)

with initial and homogeneous boundary conditions givenby

T (x, y, z, 0) = 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

0 ≤ z ≤ 1 (11b)

T (0, y, z, t) = 0; T (1, y, z, t) = 0, t > 0 (11c,d)

∂T (x, 0, z, t)

∂y= 0; T (x, 1, z, t) = 0, t > 0 (11e,f)

∂T (x, y, 0, t)

∂z=0; T (x, y, 1, t) = 0, t > 0 (11g,h)

and, for the present application, the nonlinear functionu(T ) is taken as

u(T ) = u0 + cT (11i)



Results are also presented for the case with nonhomoge-neous boundary conditions, in which the boundary condi-tion given by Eq. (11c) is replaced by

T (0, y, z, t) = 1. (12)

4.2 Conjugated Heat Transfer in Microchannels

There has been a marked research interest in improvingthe prediction and design tools for transport phenomenadriven microsystems. In this context, Knupp et al. (2012,2013a,b) have recently introduced an approach for thetreatment of conjugated convection-conduction problems,based on the reformulation of the coupled problem into asingle-domain model, which accounts for the heat trans-fer at both the fluid flow and the channel wall regions. Bymaking use of coefficients represented as space variablefunctions, with abrupt transitions occurring at the fluid–wall interface, the mathematical model is fed with infor-mation concerning the two original domains of the prob-lem. Then the GITT is employed with the integral trans-formation being constructed based upon an eigenvalueproblem with space variable coefficients (Naveira-Cottaet al., 2009), thus incorporating all the information re-garding the transition between the two original domains.

The problem here chosen for illustration, schemati-cally represented in Fig. 1, involves a laminar incompress-ible internal flow of a Newtonian fluid between paral-lel plates, in steady state and undergoing convective heattransfer due to a prescribed temperatureTw, at the ex-ternal face of the microchannel wall. The channel wallis considered to participate on the heat transfer problemthrough both transversal and axial heat conduction. Thefluid flows with a known fully developed velocity profileuf (y), and with a uniform inlet temperatureTin. The flowis assumed to be hydrodynamically developed but ther-

FIG. 1: Schematic representation of the conjugated heattransfer problem in a microchannel

mally developing, with negligible viscous dissipation andtemperature-independent physical properties.

We choose to formulate this problem using the single-domain formulation approach by making use of coeffi-cients represented as space variable functions, with abrupttransitions occurring at the fluid–wall interface. Recallingthe problem symmetry aty = 0, the formulation of theconjugated problem as a single region model is written as(Knupp et al., 2012)

u(y)wf∂T (y, z)

∂z= k(y)

∂2T

∂z2+

∂

∂y

(k(y)

∂T

∂y

),

0 < y < Le, 0 < z < z∞ (13a)

∂T

∂y

∣∣∣∣y=0

= 0, T (Le, z) = Tw, 0 < z < z∞ (13b,c)

T (y, 0) = Tin,∂T (y, z)

∂z

∣∣∣∣z=z∞

= 0,

0 < y < Le (13d,e)

with

u(y) =

{uf (y), if 0 < y < Lf/20, if Lf/2 < y < Le

,

k(y) =

{kf , if 0 < y < Lf/2ks, if Lf/2 < y < Le

(13f,g)

wherewf is the fluid heat capacity,ks is the channel wallthermal conductivity,kf is the fluid thermal conductiv-ity, anduf (y) is the fully developed velocity profile. Thefollowing dimensionless groups are defined:

Z =z/Dh

Re Pr=

z

DhPe; Y =

y

Le; U =

u

uav;

θ =T − Tin

Tw − Tin; K =

k

kf; Re=

uavDh

ν;

Pr=ν

α; Pe= Re Pr=

uavDh

α; α =

kfwf

;

σ =Le

Lf; Yi =

Lf

2Le(14a–k)

where the hydraulic diameter is given byDh = 2Lf .After making use of the groups given by Eqs. (14a)–

(14k), one obtains the dimensionless formulation whichis here rewritten already accounting for a pseudo-transientterm in the formulation:

∂θ(Y, Z, t)

∂t= −U(Y )

∂θ

∂Z+

K(Y )

Pe2∂2θ

∂Z2+

4

σ2∂

∂Y

×(K(Y )

∂θ

∂Y

), 0<Y <1, 0 < Z<Z∞, t>0 (15a)



θ(Y, Z, 0) = θt=0, 0 < Y < 1, 0 < Z < Z∞ (15b)

∂θ

∂Y

∣∣∣∣Y=0

= 0, θ(1, Z, t) = 1,

0 < Z < Z∞, t > 0 (15c,d)

θ(Y, 0, t) = θZ=0 = 0,∂θ

∂Z

∣∣∣∣Z=Z∞

= 0,

0 < Y < 1, t > 0 (15e,f)

with

U(Y ) =

{Uf (Y ), if 0 < Y < Yi

0, if Yi < Y < 1,

K(Y ) =

{1, if 0 < Y < Yi

ks/kf , if Yi < Y < 1(15g,h)

where the initial conditionθt=0 is preferably an estimateof the steady-state solution to accelerate convergence, orjust any reasonable function, since we are only interestedin the steady-state results. Figures 2(a) and 2(b) show thespace variable coefficients with abrupt transition at thefluid–solid interface,U(Y ) andK(Y ), respectively.

5. RESULTS AND DISCUSSION

5.1 Nonlinear Burgers’ Equation

In the analysis of the three-dimensional Burgers’ equa-tion, Eqs. (11) and (12), the following governing pa-rameters values have been adopted (Cotta et al., 2013):u0 = 1, c = 5, andν = 1. The Burgers’ equation for-mulation with homogenous boundary conditions is in-vestigated first. In order to offer a benchmark solution,the UNIT code with total transformation was first em-ployed with user-provided analytical integration of thetransformed initial condition and nonlinear source term,and double checked against a dedicated GITT solution ofthe same problem (Cotta et al., 2013). Then, high trunca-tion orders could be achieved with reduced computationalcost so as to offer a reliable hybrid solution with four fullyconverged significant digits, as presented in (Cotta et al.,2013), where convergence is reached at truncation ordersaroundN = 300 with the traditional squared eigenvaluesreordering criterion. Table 1 illustrates the convergencebehavior of the UNIT code with the partial transformationscheme, hereinafter called UNIT PT, in which the direc-tionx has been chosen not to be integral transformed. Theresults are presented for a different number of terms in

(a)

(b)

FIG. 2: Representation of the space variable coefficientsas functions with abrupt transitions occurring at the fluid–solid wall interface:(a)U(Y ) and(b) K(Y )

the eigenfunction expansion,N = 30, 35, 40, and 45, andfixed number of points in the Gaussian quadrature integra-tion,M = {19,19}. In Table 1 the standard total transfor-mation UNIT solution (as obtained from the last columnin Table 1a of Cotta et al., 2013) is also presented, herecalled UNIT TT, obtained with user-provided analyticalintegration. The fully discrete solution is also presented,obtained by the method of lines from the NDSolve rou-tine of the Mathematica system, under the same precisioncontrol with mesh refinement, as in the options employedin the UNIT PT solution. One should notice that, at theselected positions, the UNIT PT results are converged tothe third significant digit fort = 0.02 and to all four digitsshown fort = 0.1, with the expected improved conver-gence for larger values oft. It is also observed that theUNIT PT results agree with the NDSolve numerical re-sults in three significant digits fort = 0.02 andt = 0.1.



TABLE 1: Convergence of UNIT code solution with the partial transformationscheme for a three-dimensional Burgers’ equation with homogeneous boundaryconditions (GITT withN = 30, 35, 40, or 45 terms andM = {19,19} in theGaussian quadrature integration)

T = 0.02,y = 0.5,z = 0.5

x N = 30 N = 35 N = 40 N = 45 UNIT TTa NDSolveb

0.1 0.2780 0.2803 0.2807 0.2808 0.2798 0.2807

0.3 0.7354 0.7401 0.7407 0.7409 0.7396 0.7406

0.5 0.9291 0.9333 0.9336 0.9337 0.9331 0.9336

0.7 0.9127 0.9155 0.9153 0.9152 0.9156 0.9155

0.9 0.4898 0.4898 0.4892 0.4891 0.4905 0.4896

T = 0.1,y = 0.5,z = 0.5

x N = 30 N = 35 N = 40 N = 45 UNIT TTa NDSolveb

0.1 0.04943 0.04945 0.04946 0.04946 0.04922 0.04947

0.3 0.1494 0.1494 0.1494 0.1494 0.1492 0.1494

0.5 0.2232 0.2232 0.2232 0.2232 0.2231 0.2233

0.7 0.2236 0.2236 0.2236 0.2236 0.2235 0.2236

0.9 0.1031 0.1031 0.1031 0.1031 0.1031 0.1031aN = 300 terms and analytical integration and total transformation, from Table 1(a) of(Cotta et al., 2013);bMathematica 7.

These results, in comparison with those obtained by theUNIT TT, indicate a faster convergence rate of the UNITPT solution, as expected, since three eigenfunction ex-pansions are represented in the UNIT TT solution, whileonly two are represented in the UNIT PT solution. Nev-ertheless, it should be remembered that the UNIT PT so-lution demands additional numerical efforts per equation,since a system of one-dimensional PDEs is being solved,while for the UNIT TT scheme the total integral transfor-mation process leads to a transformed ODE system. Thismay also help explaining the fact that the UNIT PT resultsare closer to the fully discrete ones than to the benchmarkUNIT TT solution. In the UNIT PT, even considerablyreducing the numerical effort when transforming froma three-dimensional partial differential system to a one-dimensional one, the numerical solution of the coupledPDE’s transformed system still yields a considerable ef-fect on the final computed results. Apparently, only undera priori imposed variable meshing or adaptive meshingrefinement, the UNIT PT and the fully discrete approachwould be able to provide final results of closer accuracy tothe hybrid numerical–analytical solution with total trans-formation, here adopted as a benchmark. Figure 3 depicts

FIG. 3: Comparison between the UNIT PT and TT so-lutions and the NDSolve routine solution for the three-dimensional Burgers’ equation problem with homoge-neous boundary conditions



the graphical comparison of the UNIT PT and UNIT TTsolutions, at different times, against those obtained withthe NDSolve routine, where it is observed to be in excel-lent agreement between the three solutions throughout, tothe graph scale.

Proceeding to the Burgers’ equation problem with non-homogeneous boundary conditions, Table 2 illustrates theconvergence behavior of the UNIT solution with partialtransformation, by varying the truncation order,N = 55,60, 65, and 70, with a fixed number of points in the Gaus-sian quadrature integration,M = {19,19}. We point outthat no filtering procedure was performed in this solution,since the direction with nonhomogeneous boundary con-ditionsx is the one chosen not to be integral transformed.In these results, an excellent convergence behavior is ob-served at the selected positions, the results being con-verged to the fourth significant digit throughout. It shouldbe stressed that, based on the truncation order selected bythe user, the UNIT code is able to automatically provide a

safe number of points in the Gaussian numerical integra-tion procedure, in case the user does not wish to manuallyset this option.

Figure 4 brings the comparison between the solutionsobtained with the UNIT PT and the UNIT TT with theuser-provided filtering option, against the fully discreteNDSolve routine solution, along thex coordinate. Inthese results one may again confirm the excellent adher-ence between the UNIT solutions as well as with the ND-Solve solution curves throughout thex variable domain.

5.2 Conjugated Heat Transfer in Microchannels

The dimensionless thermal conductivity has been calcu-lated, motivated by an application with a microchanneletched on a polyester resin substrate (ks = 0.16 W/m◦C)and water as the working fluid (kf = 0.64 W/m◦C), sothat ks/kf = 0.25 (Ayres et al., 2011; Knupp et al.,2013b). The single-domain approach combined with inte-

TABLE 2: Convergence of UNIT code solution with the partial transformationscheme for three-dimensional Burgers’ equation with nonhomogeneous boundaryconditions (GITT withN = 55, 60, 65, or 70 terms andM = {19,19} in the Gaus-sian quadrature integration)

t = 0.04

(x, y, z) N = 55 N = 60 N = 65 N = 70 UNIT TTa NDSolveb

(0.2,0.5,0.5) 0.9407 0.9409 0.9408 0.9408 0.9433 0.9408

(0.5,0.5,0.5) 0.8523 0.8523 0.8523 0.8523 0.8493 0.8523

(0.8,0.5,0.5) 0.5945 0.5945 0.5945 0.5945 0.5904 0.5948

(0.5,0.2,0.5) 0.9119 0.9119 0.9119 0.9119 0.9089 0.9119

(0.5,0.8,0.5) 0.4897 0.4896 0.4896 0.4897 0.4912 0.4896

(0.5,0.5,0.2) 0.9119 0.9119 0.9119 0.9119 0.9089 0.9119

(0.5,0.5,0.8) 0.4897 0.4897 0.4897 0.4897 0.4912 0.4896

t = 0.1

(x, y, z) N = 55 N = 60 N = 65 N = 70 UNIT TTa NDSolveb

(0.2,0.5,0.5) 0.8798 0.8801 0.8799 0.8799 0.8805 0.8798

(0.5,0.5,0.5) 0.6400 0.6400 0.6400 0.6400 0.6340 0.6398

(0.8,0.5,0.5) 0.3365 0.3365 0.3365 0.3365 0.3307 0.3364

(0.5,0.2,0.5) 0.7457 0.7457 0.7457 0.7457 0.7402 0.7456

(0.5,0.8,0.5) 0.3282 0.3282 0.3282 0.3282 0.3279 0.3281

(0.5,0.5,0.2) 0.7457 0.7457 0.7457 0.7457 0.7402 0.7456

(0.5,0.5,0.8) 0.3282 0.3282 0.3282 0.3282 0.3279 0.3281aUNIT code solution with total transformation and user-provided polynomial filter (N = 60,M = {8,8,10});bMathematica 7.



FIG. 4: Comparison between the UNIT PT and UNIT TTsolutions vs the NDSolve routine solution for the Burgers’equation problem with nonhomogeneous boundary condi-tions, alongx

gral transforms for the solution of conjugated heat transferproblems has been introduced, verified, and validated inKnupp et al. (2012), while the same problem consideredhere was previously tackled in Knupp et al. (2013a) withtwo alternative approaches, thus offering some bench-mark results for the comparison of the UNIT solutionshere presented. For the solution via the UNIT PT scheme,the simplest possible auxiliary eigenvalue problem wasadopted, with constant coefficients, so as to lead to anauxiliary problem with analytical solution readily ob-tained from the DSolve routine of the Mathematica sys-

tem, the automatic option of the UNIT code. In this leastinformative choice, the terms with abrupt spatial transi-tion that are responsible for the information on the tran-sition of the two regions (fluid stream and channel walls)are grouped into the source term. In the results below, weillustrate the solution obtained for the case with Pe = 0.5.

Table 3 presents the convergence behavior of the UNITPT scheme with respect to the number of terms in theeigenfunction expansion,N = 10, 15, 20, 25, or 30, withfixed M = 96 points in the Gaussian quadrature integra-tion. One may observe that the results are converged tothe third or even up to the fourth significant digit withrespect to the truncation order, at the selected positions.The comparison with the reference GITT values (Knuppet al., 2013a), which is fully converged to all digits shown,indicates a three-significant-digits agreement with the au-tomatic UNIT PT solution at the selected positions.

Figure 5 depicts the comparison of the UNIT PT so-lution with the reference curves for the transversal tem-perature profiles for some different longitudinal positionsalong the flow,Z = 0.05, 0.1, 0.2, 0.3, 0.5, 0.75, 1.0, 1.5,and 4.5, for both the fluid and the channel wall regions.Despite the adoption of the simplest possible auxiliaryeigenvalue problem, a very good graphical agreement be-tween the automatic UNIT PT solution and the dedicatedGITT solution is observed throughout. In fact, in orderto investigate the behavior of the solution of the abovesingle-domain formulation by a purely numerical method,we have attempted to solve the original partial differentialproblem through the NDSolve routine of the Mathemat-ica package under automatic accuracy control. However,under default specifications, it was not able to accuratelysolve the problem with the intrinsic maximum refinementoptions. We have also tried to impose limits of refinement

TABLE 3: Convergence behavior of the conjugated problem solution withPe = 0.5 obtained with the UNIT code via partial transformation withN =10, 15, 20, 25, or 30 terms andM = 96 points in the Gaussian quadratureintegration

NZ = 0.1 Z = 0.2

Y = 0.3 Y = 0.6 Y = 0.9 Y = 0.3 Y = 0.6 Y = 0.910 0.06511 0.1247 0.4899 0.1328 0.2321 0.692015 0.06672 0.1236 0.4928 0.1330 0.2333 0.692120 0.06667 0.1233 0.4935 0.1328 0.2333 0.692025 0.06670 0.1231 0.4934 0.1330 0.2329 0.692030 0.06634 0.1230 0.4932 0.1329 0.2329 0.6920

GITTa 0.06626 0.1241 0.4935 0.1321 0.2348 0.6923a GITT solution with analytical integration (Knupp et al., 2013a).



FIG. 5: Transversal temperature profiles in conjugatedproblem, calculated using the single-domain formulationvia the UNIT PT scheme, in comparison with benchmarkGITT results.a(Knupp et al., 2013a) for Pe = 0.5

higher than the default options, up to the maximum avail-able computation capability, and satisfactory results werestill not obtained, reconfirming the relative merits of thehybrid solution approach.

6. CONCLUSIONS

This work compared the total and partial transformationschemes featured into the unified integral transform al-gorithm, implemented in the so-called UNIT code, aspart of the ongoing project related to the progressive con-struction of an open-source symbolic-numerical compu-tational code for finding solutions of partial differentialsystems based on the generalized integral transform tech-nique (GITT). The implementations have been comparedfor nonlinear three-dimensional formulations of Burgers’equation, where it has been illustrated that the partialtransformation scheme can be straightforward and accu-rate, leading to a hybrid solution path that may offer ad-vantages inherent to the analytical development in com-bination with the powerful numerical solution procedureavailable. As an application illustration, the solution ofa conjugated conduction–convection heat transfer prob-lem of laminar flow inside parallel plate microchannelshas been demonstrated. This problem has been reformu-lated by means of a single-domain approach, recently in-troduced with the main objective of simplifying the con-

jugated heat transfer analysis, without requiring any sortof iterative procedure. Again, the UNIT code under thepartial transformation scheme was capable of recoveringwith accuracy and mild computational cost, the bench-mark results obtained with a dedicated GITT solutionwith analytical integration.

ACKNOWLEDGMENTS

The authors would like to acknowledge financial supportprovided by the federal and state Brazilian research spon-soring agencies CNPq and FAPERJ. The present paper isthe modified and extended version of a keynote lecturepresented at the 14th Brazilian Congress of Thermal Sci-ences and Engineering, ENCIT 2012, in Rio de Janeiro,Brazil, November 2012, as a tribute to Prof. Mikhail D.Mikhailov. This work is also dedicated to the memory ofTzvetanka Ivantcheva, our dear Zeza, that with her lifechoice of staying always beside Misho Mikhailov, herhusband, she also gave us the chance to get to know, ad-mire, and love her along her last 17 years while living inBrazil. We shall all miss her very much.

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