We introduce a new method of solution for the convective heat transfer under forced laminar flow that is confined by two parallel plates with a distance of 2a or by a circular tube with a radius of a. The advection–conduction equation is first mapped onto the boundary. The original problem of solving the unknown field $T(x,r,t)$ is reduced to seek the solutions of T at the boundary (r = a or r = 0, r is the distance from the centerline shown in Fig. 1), i.e., the boundary functions $Ta(x,t)≡T(x,r=a,t)$ and/or $T0(x,t)≡T(x,r=0,t)$. In this manner, the original problem is significantly simplified by reducing the problem dimensionality from 3 to 2. The unknown field $T(x,r,t)$ can be eventually solved in terms of these boundary functions. The method is applied to the convective heat transfer with uniform wall temperature boundary condition and with heat exchange between flowing fluids and its surroundings that is relevant to the geothermal applications. Analytical solutions are presented and validated for the steady-state problem using the proposed method.

## References

References
1.
Graetz
,
L.
, 1883, “
Uber Die Warmeleitungsfahigheit Von Flussingkeiten—Part 1
,”
Ann. Phys. Chem.
,
18
, pp.
79
94
.
2.
Graetz
,
L.
, 1885, “
Ueber Die Wärmeleitungsfähigkeit Von Flüssigkeiten—Part 2
,”
Ann. Phys. Chem.
,
25
, pp.
337
357
.
3.
Nusselt
,
Z.
, 1910, “
Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge
,”
Z. Ver. Dtsch. Ing.
,
54
, pp.
1154
1158
.
4.
Hsu
,
C. J.
, 1971, “
An Exact Analysis of Low Peclet Number Thermal Entry Region Heat Transfer in Transversely Nonuniform Velocity Fields
,”
AIChE J.
,
17
(
3
), pp.
732
740
.
5.
Michelse
,
M. L.
, and
,
J.
, 1974, “
Graetz Problem With Axial Heat-Conduction
,”
Int. J. Heat Mass Transfer
,
17
(
11
), pp.
1391
1402
.
6.
Papoutsakis
,
E.
,
Ramkrishna
,
D.
, and
Lim
,
H. C.
, 1980, “
The Extended Graetz Problem With Prescribed Wall Flux
,”
AIChE J.
,
26
(
5
), pp.
779
787
.
7.
Papoutsakis
,
E.
,
Ramkrishna
,
D.
, and
Lim
,
H. C.
, 1980, “
The Extended Graetz Problem With Dirichlet Wall Boundary-Conditions
,”
Appl. Sci. Res.
,
36
(
1
), pp.
13
34
.
8.
,
M. A.
, and
Zhang
,
H. Y.
, 1989, “
An Exact Solution of Extended Graetz Problem With Axial Heat-Conduction
,”
Int. J. Heat Mass Transfer
,
32
(
9
), pp.
1709
1717
.
9.
Hennecke
,
D. K.
, 1968, “
Heat Transfer by Hagen-Poiseuille Flow in the Thermal Development Region With Axial Conduction
,” Wärme- und Stofföbertragung,
1
, pp.
177
184
.
10.
Barron
,
R. F.
,
Wang
,
X. M.
,
Ameel
,
T. A.
, and
Warrington
,
R. O.
, 1997, “
The Graetz Problem Extended to Slip-Flow
,”
Int. J. Heat Mass Transfer
,
40
(
8
), pp.
1817
1823
.
11.
Tunc
,
G.
, and
Bayazitoglu
,
Y.
, 2001, “
Heat Transfer in Microtubes With Viscous Dissipation
,”
Int. J. Heat Mass Transfer
,
44
(
13
), pp.
2395
2403
.
12.
Larrode
,
F. E.
,
,
C.
, and
Drossinos
,
Y.
, 2000, “
Slip-Flow Heat Transfer in Circular Tubes
,”
Int. J. Heat Mass Transfer
,
43
(
15
), pp.
2669
2680
.
13.
Meakin
,
P.
, and
Xu
,
Z. J.
, 2009, “
Dissipative Particle Dynamics and Other Particle Methods for Multiphase Fluid Flow in Fractured and Porous Media
,”
Prog. Comput. Fluid Dyn.
,
9
(
6–7
), pp.
399
408
.
14.
Xu
,
Z.
, and
Meakin
,
P.
, 2009, “
A Phase-Field Approach to No-Slip Boundary Conditions in Dissipative Particle Dynamics and Other Particle Models for Fluid Flow in Geometrically Complex Confined Systems
,”
J. Chem. Phys.
,
130
(
23
), p.
234103
.
15.
Xu
,
Z.
,
Meakin
,
P.
, and
Tartakovsky
,
A. M.
, 2009, “
Diffuse-Interface Model for Smoothed Particle Hydrodynamics
,”
Phys. Rev. E
,
79
(
3
), p.
036702
.
16.
Xu
,
Z.
, and
Meakin
,
P.
, 2008, “
Phase-Field Modeling of Solute Precipitation and Dissolution
,”
J. Chem. Phys.
,
129
(
1
), p.
014705
.
17.
Xu
,
Z.
, and
Meakin
,
P.
, 2011, “
Phase-Field Modeling of Two-Dimensional Solute Precipitation/Dissolution: Solid Fingers and Diffusion-Limited Precipitation
,”
J. Chem. Phys.
,
134
(
4
), p.
044137
.
18.
Xu
,
Z.
,
Rosso
,
K. M.
, and
Bruemmer
,
S. M.
, 2011, “
A Generalized Mathematical Framework for Thermal Oxidation Kinetics
,”
J. Chem. Phys.
,
135
(
2
), p.
024108M
.
19.
Shah
,
R. K.
, and
London
,
A. L.
, 1978,
Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data
,
New York
.
20.
Wylie
,
C. R.
, and
Barrett
,
L. C.
, 1982,