In the present work, we investigate laser heating of nanoscale thin-films irradiated in three dimensions using the dual phase lag (DPL) model. A numerical solution based on mixed-collocation, finite difference method has been employed to solve the DPL heat conduction equation. Direct substitution in the model transforms the differential equation into a linear system of equations in which related system is solved directly without preconditioning. Consistency, stability, and convergence of the proposed method based on a mixed-collocation, finite difference approximation are proved, and numerical results are presented. The general form of matrices and their corresponding eigenvalues are presented.

Issue Section:
Technical Briefs
Keywords:
collocation spectral methods, finite difference methods, heat conduction equation, thin-films, nanoscale
Topics:
Approximation, Eigenvalues, Finite difference methods, Heat conduction, Microscale devices, Nanoscale phenomena, Stability, Thin films, Dimensions, Differential equations, Lasers, Linear systems, Heat

References

1.
Tzou
,
D. Y.
, 1996,
Macro to Micro Heat Transfer
,
Taylor & Francis
,
Washington
.
2.
Li
,
B. Q.
,
Mi
,
C.
,
Liu
,
C. H.
,
Cheng
,
G.
,
Fu
,
M.
, and
Meadow
,
G.
, 2007, “
Nanoparticle Heat Transfer and Its Application to Laser Hyperthermia
,”
Presented at the ASME International Mechanical Engineering Congress and Exposition (IMECE)
, November 10 - 16, Seattle, WA, pp.
1093
–
1100
.
3.
Liu
,
C.
,
Li
,
B. Q.
, and
Mi
,
C.
, 2008, “
Analysis of Dual Phase Lag Heat Conduction in Gold Nanoparticle Based Hyperthermia Treatment
,”
Presented at the ASME International Mechanical Engineering Congress and Exposition (IMECE)
, October 31 -November 6, Boston.
4.
Qui
,
T. Q.
, and
Tien
,
C. L.
, 1992, “
Short-Pulse Laser Heating on Metals
,”
Int. J. Heat Mass Transfer
,
35
, pp.
719
–
726
.
5.
Qui
,
T. Q.
, and
Tien
,
C. L.
, 1993, “
Heat Transfer Mechanisms During Short-Pulse Laser Heating of Metals
,”
ASME J. Heat Transfer
,
115
, pp.
835
–
841
.
6.
Chiffell
,
R. J.
, 1994, “
On the Wave Behavior and Rate Effect of Thermal and Thermo-Mechanical Waves
,” M.Sc. thesis, University of New Mexico, Albuquerque.
7.
Dai
,
W.
, and
Nassar
,
R.
, 1999, “
A Finite Difference Method for Solving the Heat Transport Equation at the Microscale
,”
Numer. Methods Partial Differ. Equ.
,
15
(
6
), pp.
697
–
708
.
Crossref
Search ADS
 
8.
Dai
,
W.
, and
Nassar
,
R.
, 2001, “
A Compact Finite Difference Scheme for Solving a One-Dimensional Heat Transport Equation at the Microscale
,”
J. Comput. Appl. Math.
,
132
(
2
), pp.
431
–
441
.
Crossref
Search ADS
 
9.
Dai
,
W.
, and
Nassar
,
R.
, 2000, “
A Compact Finite Difference Scheme for Solving a Three-Dimensional Heat Transport Equation in a Thin Film
,”
Numer. Methods Partial Differ. Equ.
,
16
(
5
), pp.
441
–
458
.
Crossref
Search ADS
 
10.
Dai
,
W.
, and
Nassar
,
R.
, 2001, “
A Finite Difference Scheme for Solving a Three-Dimensional Heat Transport Equation in a Thin Film With Microscale Thickness
,”
Int. J. Numer. Methods Eng.
,
50
(
7
), pp.
1665
–
1680
.
Crossref
Search ADS
 
11.
Zhang
,
J.
, and
Zhao
,
J. J.
, 2001, “
Iterative Solution and Finite Difference Approximations to 3D Microscale Heat Transport Equation
,”
Math. Comput. Simul.
,
57
(
6
), pp.
387
–
404
.
Crossref
Search ADS
 
12.
Zhang
,
J.
, and
Zhao
,
J. J.
, 2001, “
High Accuracy Stable Numerical Solution of 1D Microscale Heat Transport Equation
,”
Commun. Numer. Methods Eng.
,
17
(
11
), pp.
821
–
832
.
Crossref
Search ADS
 
13.
Zhang
,
J.
, and
Zhao
,
J. J.
, 2001, “
Unconditionally Stable Finite Difference Scheme and Iterative Solution of 2D Microscale Heat Transport Equation
,”
J. Comput. Phys.
,
170
(
1
), pp.
261
–
275
.
Crossref
Search ADS
 
14.
Momeni-Masuleh
,
S. H.
, and
Malek
,
A.
, 2007, “
Hybrid Pseudospectral-Finite Difference Method for Solving a 3D Heat Conduction Equation in a Submicroscale Thin Film
,”
Numer. Methods Partial Differ. Equ.
,
123
, pp.
1139
–
1148
.
15.
Malek
,
A.
, and
Momeni-Masuleh
,
S. H.
, 2008, “
A Mixed Collocation-Finite Difference Method for 3D Microscopic Heat Transport Problems
,”
J. Comput. Appl. Math.
,
217
, pp.
137
–
147
.
Crossref
Search ADS
 
16.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
, 1988,
Spectral Methods in Fluid Dynamics
(Springer Series in Computational Physics),
Springer
,
Berlin
.
17.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
, 2006,
Spectral Methods: Fundamentals in Single Domains
,
Springer
,
Berlin
.
18.
Momeni-Masuleh
,
S. H.
, and
Phillips
,
T. N.
, 2004, “
Viscoelastic Flow in an Undulating Tube Using Spectral Methods
,”
Comput. Fluids
,
33
, pp.
1075
–
1095
.
Crossref
Search ADS
 
19.
Smith
,
G. D.
, 1985,
Numerical Solution of Partial Differential Equations Finite Difference Methods
, 3rd ed.,
Oxford University Press
,
Oxford
.
Copyright © 2012
 by American Society of Mechanical Engineers
You do not currently have access to this content.