Solution of the Graetz -- Brinkman problem with the Laplace transform Galerkin methodIntroductionGoverning equationsLaplace transform Galerkin methodEquation in Laplace domainGalerkin methodNumerical Laplace transform inversionResults and comparison with previous workConstant wall temperature (different from ambient)Constant wall flux (different from zero)Viscous dissipation in adiabatic regimeViscous dissipation with natural coolingConclusionsReferencesTechnical NoteSolution of the Graetz–Brinkman problem with theLaplace transform Galerkin methodPeter P. Valko´Department of Petroleum Engineering, A&M University, 501K Richardson Building, 3116 Tamu, College Station, TX 77843 3116, USAReceived 13 June 2003; received in revised form 17 November 2004Available online 20 January 2005AbstractThe present study concentrates on the effects of viscous dissipation in laminar forced convection. A power law fluidrheology model is applied and the effect of heat conduction in the axial direction is considered negligible. The physicalproperties are considered constant. Assuming fully developed velocity profile, the development of the temperature pro-file and its asymptotic behavior are investigated. For the solution of the problem the Laplace transform Galerkin tech-nique is used. The method allows for the most general boundary conditions. A detailed comparison with previouslypublished results provides a verification of the numerical technique. An important feature of the approach is that deriv-atives and integrals with respect to the axial location can be obtained through the operational rules of the Laplace trans-formation and hence no numerical derivation or integration is needed. As an application of the numerical model, wefocus on the natural cooling regime, when the viscous dissipation of energy is counter-balanced by keeping the walltemperature at the ambient value. We derive a correlation for the asymptotic behavior of the Nusselt number in thenatural cooling regime. This correlation reproduces the known value for the Newtonian case and provides a convenientmeans to normalize the Nusselt number for a wide range of flow behavior indices.Ó 2005 Elsevier Ltd. All rights reserved.Keywords: Graetz problem; Brinkman problem; Power law rheology; Forced convection; Viscous dissipation; Natural cooling; Laplacetransform inversion; Galerkin method1. IntroductionFor the processing of polymer solutions and meltsthe following heat transfer problem is of particular inter-est. Fluid at ambient temperature with a well developedlaminar velocity profile enters a circular pipe whose wallmay be maintained at constant temperature, or cooled(heated) with a constant flux. Heat conduction in theaxial direction is negligible in comparison with the heattransport by the over-all fluid motion. Viscous heating isnot negligible and the rheology of the fluid is describedby a power law. The physical properties can be consid-ered constant. We are concerned with the developmentof the temperature profile and its asymptotic behavior.Considering only Newtonian behavior and neglectingthe effect of viscous dissipation, this is the well knownGraetz–Nusselt problem. It has been thoroughly investi-gated for the case when the boundary condition is inDirichlet form (constant wall temperature) and when itis in the Neumann form (constant heat flux). Resultsare summarized for instance in [1] and [2]. Occasionally0017-9310/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2004.11.013E-mail address: [email protected] Journal of Heat and Mass Transfer 48 (2005) 1874–1882www.elsevier.com/locate/ijhmtAuthor's copythe so called third boundary condition (Robin form) isalso included, [3].Brinkman [4] brought attention to the importance ofviscous dissipation and Lyche and Bird [5] were the firstto consider fluids with power law behavior. Parallely, theGraetz series solution was perfected by Brown [6]. Thiswas followed by applying various numerical methodsto the non-Newtonian problem, [7–10] and asymptoticexpansion techniques [11,12]. Recent and highly reliableresults for the power law case (without viscous dissipa-tion) are available in Johnston [13]. Asymptotic behav-ior for power law behavior with viscous dissipationwas considered by Barletta [14]. Other than Newtonianand power law behavior has been also studied, for in-stance Bingham plastic behavior in [15–17] and recentlyPhan–Thien–Tanner (PTT) rheology, [18]. The effect ofslip at the wall was included, for example, in [19,20]. An-other direction of research has been to incorporate axialheat conduction, basically extending the scope of theoriginal analytical approach of Graetz, [21–25] and todetermine when the axial conduction can be neglected[13].This work departs from previous studies in the fol-lowing. It uses the the Galerkin (weighted residual)method in combination of the Laplace transform. Weare aware of the application of the Galerkin method tothe Graetz problem [26], but not in Laplace space. Ourmethod allows for a more general form of the boundarycondition including the special cases when the constantflux is zero (adiabatic) and when the constant wall tem-perature coincides with the ambient temperature (natu-ral cooling). In addition, the so called third type ofboundary condition is also a special case of our generalboundary condition. One important advantage of theapproach is that all differentiation and integration arehandled analytically in the radial direction and throughthe operational rules of Laplace transform in the axialdirection. High accuracy of the final results is made pos-sible by a novel numerical Laplace transform inversiontechnique.2. Governing equationsThe problem under consideration is to find the tem-perature T as a function of axial location X and radialposition R. The fluid has a fully developed laminarvelocity profile, U(R) corresponding to the power lawrheology:s ¼ joUoRmð1ÞThe energy equation includes the heat generated bythe internal friction of the fluid:qcpUðRÞoToX¼ k1RooRRoToR sRXoUðRÞoRð2ÞNomenclatureBr Brinkman numberc1, c2, c3constants in boundary conditioncpspecific heat (J/kg K)Gz Graetz numberh heat transfer coefficient (W/m2K)M number of terms in Gaver–Wynn–Rhomethodn number of terms in Galerkin methodNu Nusselt numberp(r) polynomial in Galerkin methodPe Peclet numberqwwall heat flux (W/m2)r dimensionless radial