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Eigenvalues for the Graetz problem in slip-flow
Int. Comm. Heat Mass Transfer, Vol.24, No. 3. pp. 449-451. 1997
Copyright© 1997ElsevierScienceLtd Printedin the USA.Allrightsreserved 0735-1933/97 $17.00+ .00
Pergamon PII S0735-1933(97)00030-4
EIGENVALUES FOR THE GRAETZ PROBLEM IN SLIP-FLOW M. D. Mikhailov and R. M. Cotta Mechanical Engineering Department EE/COPPF_JUF1LI Universidade Federal do Rio de Janeiro, Cidade Universitaria CP 68.503, Rio de Janeiro, ILl, 21945-970, Brasil
(Communicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Exact solution of the problem considered in [ 1] is used to compute the eigenvalues required in the analytical solution of laminar thermally developing forced convection between parallel-plates in slip flow. M a t h e m a t i c a statement is given that permits the straight forward reproduction of these results, at any Knudsen number, Kn Copyright © 1997 Elsevier Science Ltd
Introduction In one of the latest issues of this journal [1 ], a technique developed in 1885 [2] is used to evaluate the eigenvalues for the Graetz problem extended to shp-flow. The first four eigenvalues were found with precision of about 4 digits. The method used appears to be unstable after the fifth root so that only the first four eigenvalues were considered reliable. The authors concluded that an improved method with enhanced calculation speed would be of future interest. In reality more effective and efficient methods for solving Sturm-Liouville problems [3-5 ] are widely available. The problem considered in [1] has also exact solution in terms ofhypergeometric function. This communication is a M a t h e m a t i c a notebook [6] that extends the table of eigenvalues given in [1 ] to a larger number of eigenvalues. The results are obtained with working precision of 16 digits but only 6 digits are shown on table below, since the interested reader may readily repeat the calculations. The results for Kn = 0 are in agreement with the results of Brown [7] and Mikhailov & Cotta [5]. 449
450
M.D. Mikhailovand R.M. Cotta
Vol, 24, No. 3
Results
The eigenvaluos of the problem presented in [1] are the roots of the hypergeometric function obtained through the Mathemaaca system by using the following statement: m u = T a b l e [x / . FindRoot [ HypergeometriclFl [ (2-x (i+4 Kn) )/4, i, x] ~=0, {x,4/Sqrt[l+4 Kn]{i-l,i}} ],{ICaz,O,O.12,0.02},{i,l,6}] ;
The eigenvalues are presented below in table form. The rows correspond to the Knudsen numbers Kn = 0, 0.02, 0.04, 0.06, 0.08, 0.1, and 0.12. The eigenvalues are ordered by columns. T a b l e F o r m [mu ] 2.70436
6.67903
10.6734
14.6711
18.6699
22.6691
2.57829
6.32048
10.0716
13.8213
17.5693
21.3157
2.46818
6.01345
9.56108
13.1049
16.6457
20.1842
2.37096
5.74686
9.12118
12.4906
15.8566
19.2202
2.28432
5.51266
8.73714
11.9564
15.1722
18.3858
2.20647
5.30485
8.3981
11.4862
14.571
17.6539
2.13603
5.11883
8.09593
11.0681
14.0373
17.0048
Conclusion
The eigenvalues for the Graetz problem in slip-flow are obtained accurately and efficiently with Mat~matica by using the exact solution of the Sturm-Liouville problem posed in [1]. For higher order eigenvalues the methods described in [3-5] could be preferable. References
1. R. F. Barron, X. Wang, R. O. Warrington, and T. Ameel, Evaluation of the eigenvalues for the Graetz problem in slip-flow, Int. Comm. Heat Mass Transfer, Vol. 23, No 4, pp. 563-574, (1996). 2. L. Graetz, Uber die Warrneleitungsfahigkeit von Flussigkeiten, Annalen der Physik und Chemie, part 1, vol. 18, pp. 79-94, (1883), part 2, vol. 25, pp. 337-357, (1885).
Vol. 24, No. 3
EIGENVALUES FOR THE GRAETZ PROBLEMIN SLIP-FLOW
451
3. P. B. Bailey, M. K. Gordon, and L. F. Shampine, Automatic Solution of the SturmLiouviUe Problem", ACM Transactions on Mathematical Sottware, Vol. 4, No 3, 1978. 4. M. D. Mikhailov and N. L. Vulchanov, A computational procedure for Sturm-Liouville problems, J. Comp. Phys, 50, 323-336, (1983). 5. M. D. Mikhailov and R. M. Cotta, Integral Transform Method for Eigenvalue Problems, Comm. Num. Meth. Eng., 10, 827-835, (1994). 6. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, AddisonWesley, (1991). 7. G. M. Brown, Heat or mass transfer in a fluid in laminar flow in a circular or flat conduit, AIChE J., 6, 179-183, (1960). Received August 6, 1996
Copyright© 1997ElsevierScienceLtd Printedin the USA.Allrightsreserved 0735-1933/97 $17.00+ .00
Pergamon PII S0735-1933(97)00030-4
EIGENVALUES FOR THE GRAETZ PROBLEM IN SLIP-FLOW M. D. Mikhailov and R. M. Cotta Mechanical Engineering Department EE/COPPF_JUF1LI Universidade Federal do Rio de Janeiro, Cidade Universitaria CP 68.503, Rio de Janeiro, ILl, 21945-970, Brasil
(Communicated by J.P. Hartnett and W.J. Minkowycz)
ABSTRACT Exact solution of the problem considered in [ 1] is used to compute the eigenvalues required in the analytical solution of laminar thermally developing forced convection between parallel-plates in slip flow. M a t h e m a t i c a statement is given that permits the straight forward reproduction of these results, at any Knudsen number, Kn Copyright © 1997 Elsevier Science Ltd
Introduction In one of the latest issues of this journal [1 ], a technique developed in 1885 [2] is used to evaluate the eigenvalues for the Graetz problem extended to shp-flow. The first four eigenvalues were found with precision of about 4 digits. The method used appears to be unstable after the fifth root so that only the first four eigenvalues were considered reliable. The authors concluded that an improved method with enhanced calculation speed would be of future interest. In reality more effective and efficient methods for solving Sturm-Liouville problems [3-5 ] are widely available. The problem considered in [1] has also exact solution in terms ofhypergeometric function. This communication is a M a t h e m a t i c a notebook [6] that extends the table of eigenvalues given in [1 ] to a larger number of eigenvalues. The results are obtained with working precision of 16 digits but only 6 digits are shown on table below, since the interested reader may readily repeat the calculations. The results for Kn = 0 are in agreement with the results of Brown [7] and Mikhailov & Cotta [5]. 449
450
M.D. Mikhailovand R.M. Cotta
Vol, 24, No. 3
Results
The eigenvaluos of the problem presented in [1] are the roots of the hypergeometric function obtained through the Mathemaaca system by using the following statement: m u = T a b l e [x / . FindRoot [ HypergeometriclFl [ (2-x (i+4 Kn) )/4, i, x] ~=0, {x,4/Sqrt[l+4 Kn]{i-l,i}} ],{ICaz,O,O.12,0.02},{i,l,6}] ;
The eigenvalues are presented below in table form. The rows correspond to the Knudsen numbers Kn = 0, 0.02, 0.04, 0.06, 0.08, 0.1, and 0.12. The eigenvalues are ordered by columns. T a b l e F o r m [mu ] 2.70436
6.67903
10.6734
14.6711
18.6699
22.6691
2.57829
6.32048
10.0716
13.8213
17.5693
21.3157
2.46818
6.01345
9.56108
13.1049
16.6457
20.1842
2.37096
5.74686
9.12118
12.4906
15.8566
19.2202
2.28432
5.51266
8.73714
11.9564
15.1722
18.3858
2.20647
5.30485
8.3981
11.4862
14.571
17.6539
2.13603
5.11883
8.09593
11.0681
14.0373
17.0048
Conclusion
The eigenvalues for the Graetz problem in slip-flow are obtained accurately and efficiently with Mat~matica by using the exact solution of the Sturm-Liouville problem posed in [1]. For higher order eigenvalues the methods described in [3-5] could be preferable. References
1. R. F. Barron, X. Wang, R. O. Warrington, and T. Ameel, Evaluation of the eigenvalues for the Graetz problem in slip-flow, Int. Comm. Heat Mass Transfer, Vol. 23, No 4, pp. 563-574, (1996). 2. L. Graetz, Uber die Warrneleitungsfahigkeit von Flussigkeiten, Annalen der Physik und Chemie, part 1, vol. 18, pp. 79-94, (1883), part 2, vol. 25, pp. 337-357, (1885).
Vol. 24, No. 3
EIGENVALUES FOR THE GRAETZ PROBLEMIN SLIP-FLOW
451
3. P. B. Bailey, M. K. Gordon, and L. F. Shampine, Automatic Solution of the SturmLiouviUe Problem", ACM Transactions on Mathematical Sottware, Vol. 4, No 3, 1978. 4. M. D. Mikhailov and N. L. Vulchanov, A computational procedure for Sturm-Liouville problems, J. Comp. Phys, 50, 323-336, (1983). 5. M. D. Mikhailov and R. M. Cotta, Integral Transform Method for Eigenvalue Problems, Comm. Num. Meth. Eng., 10, 827-835, (1994). 6. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, AddisonWesley, (1991). 7. G. M. Brown, Heat or mass transfer in a fluid in laminar flow in a circular or flat conduit, AIChE J., 6, 179-183, (1960). Received August 6, 1996