An asymptotic analysis to the Cartesian Graetz problem with third-kind boundary conditions is carried out, the temperature profile is calculated for small values of the Biot number. The analysis is performed using asymptotic matching and shows that there are two scales for the axial coordinate, which allows us to find an analytical expression for the temperature field along the channel. This expression can be easily applied in thermal engineering applications.
GraetzVL. Über die wärmeleitungsfähigkeit von flüssigkeiten. Ann Phys1885; 261: 337–357.
2.
SellarsJRTribusMKleinJS. Heat transfer to laminar flow in a round tube or flat conduit—The Graetz problem extended. Trans ASME1956; 78: 441–447.
3.
ArpaciVS. Conduction heat transfer. Boston, MA: Addison-Wesley, 1966, p. 203.
4.
ShahRKLondonAL. Laminar flow forced convection in ducts, advances in heat transfer. New York: Academic Press, 1978.
5.
MansourAR. Two-Dimensional heat or mass transfer in laminar flow between parallel plates: Closed-form solution. J Heat Transfer1989; 111: 566–568.
6.
NickolayMMartinH. Improved approximation for the Nusselt number for hydrodynamically developed laminar flow between parallel plates. Int J Heat Mass Transfer2000; 45: 3263–3266.
7.
RosalesMAFrederickRL. Semi analytic solution to the Cartesian Graetz problem: Results for the entrance region. Int Commun Heat Mass Transfer2004; 31: 733–740.
8.
PrinsJAMulderJSchenkJ. Heat transfer in laminary flow between parallel plates. Appl Sci Res1951; 2: 431–438.
9.
De ByeJVDDSchenkJ. Heat transfer in laminary flow between parallel plates. Appl Sci Res1952; 3: 308–316.
10.
ColtonCKSmithKAStroeveP,et al.Laminar flow mass transfer in a flat duct with permeable walls. AIChE J1971; 17: 773–780.
11.
ÖzişikMNSadeghipourMS. Analytic solution for the eigenvalues and coefficients of the Graetz problem with third kind boundary condition. Int J Heat Mass Transfer1982; 25: 736–739.
12.
VickBÖzişikMNUllrichDF. Effects of axial conduction in laminar tube flow with convective boundaries. J Franklin Inst1983; 316: 159–173.
13.
WijeysunderaNE. Laminar forced convection in circular and flat ducts with wall axial conduction and external convection. Int J Heat Mass Transfer1986; 29: 797–807.
14.
PozziALupoM. The coupling of conduction with forced convection in Graetz problems. J Heat Transfer1990; 112: 797–806.
15.
AriciMMaciaYMCampoA. Finite-difference analysis of the generalized Graetz problem with heat convection boundary condition. Heat Transf Res2020; 51: 323–328.
16.
CessRDShafferEC. Heat transfer to laminar flow between parallel plates with a prescribed wall heat flux. Appl Sci Res1959; 8: 339–344.
17.
BenderCMOrszagSA. Advanced mathematical methods for scientists and engineers. Tokyo: McGraw-Hill, 1978.
18.
Rosales-VeraM. Numerical and asymptotic analysis to the Cartesian Graetz problem with viscous dissipation. Results Appl Math2021; 10: 100144.
19.
AcrivosA. The extended Graetz problem at low Peclet numbers. Appl Sci Res1980; 36: 35–40.
20.
Rosales-VeraM. A Note on the Cartesian Graetz problem with viscous dissipation. Case Studies Therm Eng2021; 28: 101391.
21.
HudsonJLBankoffSG. Asymptotic solutions for the unsteady Graetz problem. Int J Heat Mass Transfer1964; 7: 1303–1307.
22.
SleicherCANotterRHCrippenMD. A solution to the turbulent Graetz problem by matched asymptotic expansions-I The case of uniform wall temperature. Chem Eng Sci1970; 25: 845–857.
23.
WeigandBBogenfeldR. A solution to the turbulent Graetz problem by matched asymptotic expansions for an axially rotating pipe subjected to external convection. Int J Heat Mass Transfer2014; 78: 901–907.
24.
Rosales-VeraM. A note on Leveque's solution to the Cartesian Graetz problem with free convection. Int J Thermofluids2022; 16: 100217.
25.
HickmanHJ. An asymptotic study of the Nusselt-Graetz problem—part I: large x behavior. ASME J Heat Transfer1974; 96: 354–358.
26.
LahjomriJOubarraAAlemanyA. Heat transfer by laminar Hartmann flow in thermal entrance region with a step change in wall temperatures: The Graetz problem extended. Int J Heat Mass Transfer2002; 45: 1127–1148.
27.
AwadMM. Heat transfer for laminar thermally developing flow in parallel-plates using the asymptotic method. In 2010 3rd international conference on thermal issues in emerging technologies theory and applications, 2010, pp. 371–387.
28.
Rosales-VeraM. Cartesian Graetz problem with boundary condition of the third kind: A semi-analytical solution. Int J Thermofluids2022; 14: 100146.
