STUDY OF THE NON-DIMENSIONAL SOLUTION OF DYNAMIC EQUATION OF MOVEMENT ON THE PLANE PLAQUE WITH CONSIDERATION OF TWO-ORDER SLIDING PHENOMENON

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ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006, Tome IV, Fascicole, (ISSN 1584 665) FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, 33118, HUNEDOARA STUDY OF THE NON-DIMENSIONAL SOLUTION OF

HUNEDOARA Abstract: Considering a sliding bondary layer of two orders in the neighborhood of an nlimited plane plaqe we emphasize a new approach for the nmerical soltion of the involved viscos flid

1 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006, Tome IV, Fascicole, (ISSN ) FACULTY OF ENGINEERING HUNEDOARA, 5, REVOLUTIEI, 33118, HUNEDOARA STUDY OF THE NON-DIMENSIONAL SOLUTION OF DYNAMIC EQUATION OF MOVEMENT ON THE PLANE PLAQUE WITH CONSIDERATION OF TWO-ORDER SLIDING PHENOMENON MAKSAY Ştefan, STOICA Diana 1, "POLITEHNICA" UNIVERSITY OF TIMISOARA, THE ENGINEERING FACULTY OF HUNEDOARA Abstract: Considering a sliding bondary layer of two orders in the neighborhood of an nlimited plane plaqe we emphasize a new approach for the nmerical soltion of the involved viscos flid flow. Key words: Sliding bondary layer, polynomial velocity profile AMS: 76D10 1. INTRODUCTION Let s envisage a flid stream past a semi-infinite plane plaqe with the "attack" angle zero. Sppose that the flid is viscos incompressible while the flow is plane (in Oxy ). The plane plaqe is considered located on the real axis Ox, its attack edge being at O. According to the well known bondary layer approximation (Prandtl), the Navier -Stokes eqations p ρv r p. = + μδ, ρv r v. v = + μδv, + = 0, x x (1) lead to ρ + v = μ, x y () v + = 0, (3) x where ρ, p, and v r (, v), are the mass density, the pressre and the plane velocity respectively while μ is the viscosity coefficient. To these eqations one attaches the bondary conditions 87

2 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006 TOME IV. Fascicole (x,0) = L 1 v(x,0) = 0, (x,0) M 1 (x, ) = (x,0),, (4) the first one signifying the fact that the flid slides on the plaqe srface (in stead of the adherence on the plaqe, i.e. of the classical non-slip condition (x,0) = 0 ). In a previos attempt we have given a nmerical method for approaching this problem which is based on a particlar polynomial development of the velocity profile inside the bondary layer. Sch sliding bondary conditions have been considered before [1]. In the papers [], [3], [4] and [5] is stdy the dynamic problem in hypotheses (x,0) L (x,0) =. (5). Reslts By sing the expression of v, which comes from (3), the above relation () leads to x y dy = μ x 0 y ρ. (6) Integrating this integro-differential eqation across the bondary layer, namely, from y = 0 to y = (x) - the pper border of the bondary layer, i.e., y ρ dy ρ dy + ρ dy = τ w, (7) 0 x 0 x 0 x 0 we get the integral relation, where d ρ 1 dy = τw, dx 0 (8) w = μ τ. (9) w In this paper, we solve this eqation by considering a velocity profile of a th polynomial type (within the bondary layer) bt this time of higher ( 6 ) degree which seems to be a more accrate approach. Precisely we sppose that and = 1, where = a0 + a1η + aη + a3η + a4η + a5η + a6η, y ( x). η 1, 0 η 1, η (11) 88

3 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006 TOME IV. Fascicole The coefficients conditions a i can be determined by sing the appropriate where 3 = L M, = 0, 3 = 1, = 0, = 0, L1 L =, (x) 4 = 0, 4 3 = 0, 3 for η = 0, (1) for η = 1, (13) M1 M =. (14) (x) Following the calclations, it reslts for the non-dimensional profile of the horizontal component of the velocity the expression = ( 6L + 10M + 6η 5η + η η ). + 6L + 10M (15) In figre (1) and () is present the profile of the velocity for L = 0, respectively L = 0,5. Figre 1. Profile of the velocity for L=0 Figre. Profile of the velocity for L=0,05 In figre (3) and (4) the inflence of the M parameter on the velocity s profile is presented. 89

4 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006 TOME IV. Fascicole Figre 3. The inflence of the y and M parameters on the velocity s profile (L=0). Figre 4. The inflence of the y and M parameter on the velocity s profile (L=0,05). In figre (5) and (6) is present the contor lines for (eta,0.05,m). (eta,0,m) respectively Figre 5. The contor lines for (eta,0,m) Figre 6. The contor lines for (eta,0.05,m) 90

5 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006 TOME IV. Fascicole The thickness of the implse losses 1 θ = 1 dη, 0 (16) has in this case the following form, where N = + 6L + 10M θ = L + M (17) N 7 1 In figre (7) and (8), in the section x = ct., is present the inflence of the L and M parameters on the velocity in y = 0 respectively y = 0,1, and in figre 9 and 10 is present the contors lines corresponding. Figre 7. The inflence of the L and M parameters on the velocity in y=0 Figre 8. The inflence of the L and M parameters on the velocity in y=0,01 Figre 9. The contor lines for Figre 7 Figre 10. The contor lines for Figre 8 91

6 ANNALS OF THE FACULTY OF ENGINEERING HUNEDOARA 006 TOME IV. Fascicole 3. Conclsions This paper stdy the inflence of two order term from bondary condition corresponding to motion of viscos incompressible flid on the plane plaqe. References [1.] SCHAAF, S.A., SHERMANN, F.S., Mechanika, 1 (9), 1955, pp ; [.] MAKSAY, St., Dynamic bondary layer with sliding on a plane plaqe, Politehnica University of Timisoara, Anal. Fac. Ing. of Hnedoara, Tom III, Fasc.5, 001, pp.39; [3.] MAKSAY, St., Solving the dynamical problem of the flids movement on a plane plaqe considering the sliding phenomenon, Politehnica University of Timisoara, Anal. Fac. Ing. of Hnedoara, Tom I, Fasc. 4, 1999, pp ; [4.] PETRILA, T., MAKSAY, St., New nmerical approach for the dynamic bondary layer sliding on a plate, Bl. St. Univ. Pitesti, Ser. Matem & Inform., 9, 003, pp. 49; [5.] PETRILA, T., Maksay, S., A new nmerical approach of a sliding bondary layer, Compters and Mathematics with applications, 50 (005), ; [6.] HASIMOTO, H., Bondary layer slip soltions for a flat plate, Jornal of the Aeronatical Sciences 5, No.1, 68-69(1958). 9

4 Exact laminar boundary layer solutions

4 Exact laminar boundary layer solutions 4 Eact laminar bondary layer soltions 4.1 Bondary layer on a flat plate (Blasis 1908 In Sec. 3, we derived the bondary layer eqations for 2D incompressible flow of constant viscosity past a weakly crved