J. Comp. & Math. Sci. Vol.2 (1), 135-138 (2011)
Asymptotic Behaviour of the Velocity Profiles in the Rotation of a Disk B. B. SINGH Department of Mathematics, Dr. Babasaheb Ambedkar Technological University, Lonere – 402 103, Dist. – Raigad (Maharashtra State), India E-mail : brijbhansingh@yahoo.com ABSTRACT This paper deals with the asymptotic behaviours of the solutions, as the similarity variable ζ→∞, of the differential equations F″ - HF′ - ( 1 + F 2 – G 2 ) = 0 …(1) G″ - HG′ - 2 F G = 0 …(2) H′ + 2F = 0 …(3) ζ=0:F=G=H=0 …(4) ζ = ∞ : F = 0, G = 1 which govern the rotation of disk in a fluid (Schlichting 1 ). The method of asymptotic integrations of second order linear differential equation has been used in the study of asymptotic behaviours. It has been found that the behaviours of the velocity profiles F, G and H for ζ→∞ are in agreement with those obtained by Nydahl who used numerical methods to study the flow-field. Key words: asymptotic behaviour ; asymptotic integration; principal solution ; linearly independent solution
INTRODUCTION The system of equations (1) – (4) was first solved by Bodwatt2 in a very laborious way by means of power series expansion at ζ = 0 and an asymptotic expansion for ζ = 0. Recently, the solution was corrected by Nydahl3. The objective of the present paper is to study the asymptotic behaviours of the solutions of (1)-(4) by using the method of asymptotic integrations of second order linear differential equations.
The study of the asymptotic behaviour of the solutions of equations governing problems of physical significance in boundary layer theory is an interesting aspect of discussion in Fluid Mechanics. One of the most important problems in the study of differential equations or differential-difference equations and their applications is that of describing the nature of the solutions for large positive values of the independent variable and this purpose is completely served by the study of the asymptotic behaviours. Thus the asymptotic
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
B. B. Singh, J. Comp. & Math. Sci. Vol.2 (1), 135-138 (2011)
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behaviour pays particular attention towards a desired problem for finding conditions under which a solution approaches zero as the independent variable tends to infinity, or is very small for all independent variables, or is bounded as the independent variable tends to infinity. Because of the above facts, the study of the asymptotic nature of the solutions of Falkner-Skan4 equations governing steady two-dimensional flow of a slightly viscous incompressible fluid past a wedge was initiated by Hartman5. Serrin6 also studied the asymptotic behaviour of velocity profiles in the Prandtl boundary layer equation for the steady twodimensional laminar flow of an incompressible viscous fluid past a rigid wall. Tiryaki and Yaman7 studied the asymptotic behaviour of a class of non-linear functional differential equations of third order. Later, the study of asymptotic behaviours was further carried out by Singh8-10, Singh and Singh11-16, Singh and Kumar17, Singh and Verma18, etc.
(5)
h = 0
(6)
ς
∫ Hds ) 0
to get
F 2 +1− G2 ε+F
(9)
Since q(ζ) > 0 on 0 ≤ ζ < ∞ and possesses a continuous second order derivative, and satisfies the conditions ∞
∫q
1/ 2
( s )ds = ∞ and
∞
5q ′ 2 q ′′ 1/2 ∫ 16q 3 − 4q 2 q (s) ds < ∞,
therefore (8) has a pair of principal and linearly independent solutions satisfying x~c q
−1 / 4
ς 1/ 2 exp ∫ q ( s ) ds , c ≠ 0, (10) 0
and x~c q
−1 / 4
ς 1/ 2 exp ∫ q ( s ) ds , c ≠ 0, (11) 0
ς
where ε ≠ 0 is an arbitrarily chosen constant. To eliminate the middle term from (6), let us put h = x exp ( ½
q(ζ) = - ½ H′ + ¼ H 2 +
Using (9) in (10) and (11), we have
Let us put h=ε+F in (1) to get
G2 −1− F 2 ε+F
(8)
as ζ→∞ (cf. Hartman 19 ).
MATHEMATICAL ANALYSIS
h″ - Hh′ +
x″ - q(ζ)x = 0 where
(7)
2 2 x ~ C 1 exp (- { 1 H − H ' + F + 1 − G } ∫0 2 2H H (ε + F ) ds ), C 1 ≠0, (12)
ς
2 2 x ~ C 1 exp ( { 1 H − H ' + F + 1 − G } ∫0 2 2H H (ε + F ) ds), C 1 ≠0, (13) as ζ→∞.
Using (12) and (13) in (7), one can get ε + F = h ~ C 1 exp ς
( - ∫{ − 0
H ' F 2 +1− G2 + } ds),C1 ≠0, (14) 2H H (ε + F )
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
B. B. Singh, J. Comp. & Math. Sci. Vol.2 (1), 135-138 (2011) ς 2 2 ε + F = h ~ C 1 exp ( {H − H ' + F + 1 − G } ∫
2H
0
H (ε + F )
ds), C 1 ≠0, as ζ→∞.
(15)
As F→0, G→1 as ζ→∞, the solutions (14) and (15) get simplified to ε + F = h ~ C 0 , C0 ≠ 0, (16) ς
∫ Hds ), C
ε + F = h ~ C 0 exp (
0
≠ 0, (17)
0
as ζ→∞. From (16), (17) we get F ~ C0 - ε
(18) ς
F ~ - ε + C 0 exp (
∫ Hds )
(19)
0
as ζ→∞. Similarly, by taking the substitution, ς
G = X exp ( ½
∫ Hds )
(20)
0
in (2) and proceeding in the similar way as equation (1) was handled, one can get the principal solutions of (2) and (4) as (21) G ~ C 0 ′ , C0 ′ ≠ 0, and linearly independent solutions as G ~ C 0 ′ , C0 ′ ≠ 0, (22) as ζ→∞. From (3) and (4), one can get ∞
H (∞) = - 2
∫ Fds ≠ 0
(23)
0
RESULTS AND DISCUSSIONS One of the most important problems in the study of differential difference equations is to study the behaviours of their solutions as the independent solution tends to infinity. This property is fulfilled completely by
137
the study of asymptotic behaviours. Asymptotic behaviour means to study the nature of the solution as the independent variable tends to infinity. If the solution tends to a zero limit, or to an infinitesimally small limit or is bounded, that is said to have asymptotic behaviour. Based on the afore-mentioned criteria, we can study the asymptotic behaviours of the solutions (18), (19) and (21), (22). From (18), it is obvious that F→∞ as ζ→∞, provided that ε = C0 . As ε is an arbitrarily chosen constant, we may take ε = C 0 without any loss of generality. As a matter of fact, (18) will show asymptotic behaviour as ζ→∞. On the otherhand, the linearly independent solution (19) will not show asymptotic behaviour as ζ→∞. This is simply because of the fact that F does not tend to zero as ζ→∞ even if ε = C0 . Therefore, the principal and linearly independent solutions can not exhibit asymptotic behaviour simultaneously. They can do so only when they are similar. This is clear from the asymptotic behaviours of (21) and (22) as ζ→∞. By choosing C′ 0 as 1, one can find that these two solutions will exhibit asymptotic behaviour as ζ→∞ simultaneously. Lastly, the velocity profile H will also show asymptotic behaviour as ζ→∞. This assertion is clear from (23). The results shown by the asymptotic formulae of the velocity profiles F, G and H are highly in agreement with those obtained by Nydahl.
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)
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B. B. Singh, J. Comp. & Math. Sci. Vol.2 (1), 135-138 (2011)
CONCLUDING REMARKS
REFERENCES
The asymptotic integration method to find out the solutions of nonlinear boundary layer equations is the corner-stone of Applied Mathematics. This is a method to find the approximate solutions for velocity profiles for very large values of the independent variables. One of the other corner-stones of Applied Mathematics is scientific computing and it is interesting to note that these two subjects have grown together. However, this is not unexpected given their respective capabilities. By using computer, one is capable of solving problems that are non-linear, non-homogeneous and multidimensional. Moreover, it is possible to achieve very high accuracy. The drawbacks are that the computer solutions do not provide much insight into the physics of the problem, particularly for those who do not have access to the appropriate software or computer, and there is always the question as to whether or not the computed solution is correct. On the other hand, the asymptotic integration methods are also capable of dealing with non-linear, non-homogeneous and multidimensional problems. So, the main objective behind the use of the asymptotic integration method, at least as far as the author is concerned, is to provide reasonably accurate expression for the solution for large values of Îś. By doing this one is able to derive an understanding of the physics of the problem. Also, one can use the result in conjunction with the original problem, to obtain the more efficient numerical procedures for computing the solution.
1. Schlichting H, Boundary Layer Theory (McGraw-Hill Book Company, New York), 227 (1954). 2. Bodwadt U T, ZAMM, 20, 241 (1940). 3. Nydahl J E, Heat Transfer for the Bodwadt Problem, Dissertation, (Colorado State University, Fort Colline, Colorado) (1971). 4. Falkner V M & Skan S W, Phil Mag, 12, 865 (1931). 5. Hartman P, Z angew, Math. Mech. 17(8), 1067 (1986). 6. Serrin J, Proc Royal Soc, 299 (A). 491 (1967). 7. Tiryaki A & Yaman S, Appl Math Letters, 14, 327 (2001). 8. Singh B. B., Ind J Pure Appl Math, 17(6),1067 (1986). 9. Singh B. B., The J Sci Research, 37, 87 (1987). 10. Singh B. B., Ganit : J Bangladesh Math Soc, 15, 51 (1995). 11. Singh B. B. & Singh A K, J Ind Acad Math, 9(1), 37 (1987). 12. Singh B. B. & Singh J, Proc Nat Acad Sci, India, 58(A), 515 (1988). 13. Singh B. B. & Singh J, Indian J Theor Phys, 36(1), 25 (1988). 14. Singh J & Singh B. B., Indian J Theor Phys, 36(2), 127 (1988). 15. Singh B. B. & Singh A K, Indian J Tech, 26, 27 (1988). 16. Singh B. B. & Singh J, Tamkang J Math, 19(3), 35 (1988). 17. Singh B. B. & Kumar A, J Hunan University, 16(1), 12 (1989). 18. Singh B. B. & Verma S. K., Indian J Engg & Material Sci, 6, 346 (1999). 19. Hartman P, Ordinary Differential Equations (Birkhauser Boston), 534 (1982).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)