Group Invariant Solution and Conservation Law for a Free Laminar Two-Dimensional Jet

Transcription

1 Journal of Nonlinear Mathematical Phsics Volume 9, Supplement 00), Birthda Issue Group Invariant Solution and Conservation Law for a Free Laminar Two-Dimensional Jet DPMASON Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, Universit of the Witwatersrand, Private Bag 3, Wits 050, Johannesburg, South Africa dpmason@cam.wits.ac.za Received Ma, 00 Abstract A group invariant solution for a stead two-dimensional jet is derived b considering a linear combination of the Lie point smmetries of Prandtl s boundar laer equations for the jet. Onl two Lie point smmetries contribute to the solution and the ratio of the constants in the linear combination is determined from conservation of total momentum fluxin the downstream direction. A conservation law for the differential equation for the stream function is derived and it is shown that the Lie point smmetr associated with the conservation law is the same as that which generates the group invariant solution. This establishes a connection between the conservation law and conservation of total momentum flux. 1 Introduction The theor of laminar jets has man applications in science and engineering. In this paper we will consider the two-dimensional stead laminar flow of a thin jet from a long narrow orifice into a fluid at rest. Since the jet is thin, the velocit in the direction of the jet varies more rapidl across the jet than along the jet. Prandtl s boundar laer theor therefore applies. There is no bounding wall. It is a free boundar laer and it is an example of a flow without an outer flow. Since there is no solid boundar present and the pressure gradient in the direction of the jet vanishes, the total fluxof momentum in the downstream direction is constant and independent of the distance from the orifice. This conserved quantit plas an important part in the solution of the problem. Schlichting [7] was the first to appl laminar boundar laer theor to the stead flow produced b a free two-dimensional jet emerging into a fluid at rest. He solved the resulting ordinar differential equation numericall. Later, Bickle [1] solved the differential equation analticall. The application of the boundar laer approximation to laminar jets is discussed full in standard texts on boundar laer theor such as b Schlichting [8], Schlichting and Gersten [9] and Rosenhead [6]. The standard procedure is to obtain a similarit solution b assuming a certain form for the stream function. Copright c 00 b D P Mason

2 Free Laminar Two-dimensional Jet 93 We will derive a group invariant solution for the two-dimensional stead laminar jet b considering a linear combination of the Lie point smmetries of the partial differential equation for the stream function. It is not necessar to assume a specific form for the stream function. This method was introduced b Momoniat et. al. [5] who considered the axismmetric spreading of a thin liquid drop. It is found that the similarit solution of Schlichting [7] and Bickle [1] is the group invariant solution. There is a close connection between conservation laws for a differential equation and the Lie point smmetries of the differential equation. B using a result of Kara and Mahomed [3, 4] connecting conservation laws and Lie point smmetries, we will establish a connection between a simple conservation law for the partial differential equation for the stream function and the condition that the total fluxof momentum in the direction of the jet is independent of the distance from the orifice. Mathematical formulation Consider a stead two-dimensional thin jet which emerges from a long narrow orifice in a wall into a fluid which is at rest. The surrounding fluid consists of the same fluid as the jet itself and is viscous and incompressible. Choose the x-axis along the jet with x = 0 at the wall and the -axis perpendicular to the jet with = 0 at the orifice. Since the jet is thin, v x x, ) varies much more rapidl with than with x. The boundar laer approximation therefore applies to the two-dimensional flow produced b the jet. Since the fluid velocit vanishes outside the jet, it follows from Euler s equation that p/x vanishes outside the jet. Since in the boundar laer approximation, p = px), it follows that dp/ dx vanishes in the jet. Hence, Prandtl s two-dimensional boundar laer equations in the jet are v x v x x + v v x = ν v x,.1) v x x + v =0,.) where ν is the kinematic viscosit of the fluid. The boundar conditions are =0: v =0, v x =0,.3) = ± : v x =0..4) B integrating.1) with respect to from = to = andusing.3)itcanbe verified that J is a constant independent of x where J = ρ v xx, )d =ρ 0 v xx, )d..5) J is the total fluxin the x-direction of the x-component of momentum and it is constant because there is no solid boundar present and dp/ dx is neglected. A stream function ψx, ) is introduced which is defined b v x x, ) = ψ, v x, ) = ψ x..6)

3 94 D P Mason Equation.) is identicall satisfied. Equation.1) becomes ψ ψ x ψ x ψ = ν 3 ψ 3..7) The problem can be stated as follows. Solve the partial differential equation.7) for ψx, ) subject to the boundar conditions =0: = ± : ψ x =0, ψ ψ =0,.8) =0..9) and subject to the condition that J is a given constant independent of x where ) ψ J =ρ x, ) d..10) 0 The approach followed b Schlichting [7, 8] was to assume a similarit solution for the stream function of the form ) ψx, ) =x p f x q..11) The exponents p and q were determined b imposing the condition that.7) reduce to an ordinar differential equation for f and that J is independent of x. We will look for a group invariant solution and it will not be necessar to assume a specific form for the stream function ψx, ). 3 Lie point smmetr generators Equation.7) can be written as F ψ x,ψ,ψ x,ψ,ψ )=0, 3.1) where F = ψ ψ x ψ x ψ νψ 3.) and a subscript denotes partial differentiation. The Lie point smmetr generators X = ξ 1 x,, ψ) x + ξ x,, ψ) + ηx,, ψ) ψ 3.3) of the partial differential equation 3.1) are obtained b solving the determining equation [] X [3] F =0, 3.4) F =0

4 Free Laminar Two-dimensional Jet 95 where and X [3] = X + ζ 1 ψ x + ζ ψ + ζ 11 ψ xx + ζ 1 ψ x + ζ ψ + ζ 111 ψ xxx + ζ 11 ψ xx + ζ 1 ψ x + ζ ψ 3.5) ζ i = D i η) ψ s D i ξ s ), 3.6) ζ ij = D j η i ) ψ is D j ξ s ), 3.7) ζ ijk = D k η ij ) ψ ijs D k ξ s ), 3.8) with summation over repeated indices. In 3.6) to 3.8), D 1 and D are the operators of total differentiation with respect to x and respectivel: D 1 = x + ψ x ψ + ψ xx + ψ x +, 3.9) ψ x ψ D = + ψ ψ + ψ x + ψ ) ψ x ψ Since F depends onl on ψ x, ψ, ψ x, ψ and ψ, the coefficients ζ 11, ζ 111, ζ 11 and ζ 1 do not need to be calculated. The coefficient ζ depends on ψ which is eliminated from 3.4) using the partial differential equation 3.1). Equation 3.4) is separated according to the derivatives of ψ. It is found that X = c 1 X 1 + c X + c 3 X 3 + c 4 X 4 + X g, 3.11) where c 1, c, c 3 and c 4 are constants and X 1 = x x +, X = x x + ψ ψ, X 3 = x, X 4 = ψ, X g = gx), 3.1) where gx) is an arbitrar function. The Lie point smmetr generators of the partial differential equation.7) are given b 3.1). 4 Group invariant solution In order to derive a group invariant solution of.7) we consider the linear combination 3.11) of the Lie point smmetr generators. Since X is determined up to an arbitrar multiplicative constant we divide 3.11) b c 1 which we can expect to be non-zero since X 1 is a non-trivial smmetr generator. We incorporate c 1 into the remaining constants and into gx) which is equivalent to taking c 1 =1. Now, ψ =Φx, ) is a group invariant solution of.7) provided [ 1 + c )x + c 3 ) x + + gx) ) ψ +c ψ + c 4 ) ψ ] ψ ) Φx, ) =0, 4.1) ψ=φ

5 96 D P Mason which ma be rewritten as )Φ 1 + c )x + c 3 x + + gx) ) Φ = c Φ+c 4. 4.) Equation 4.) is a quasi-linear first order partial differential equation for Φx, ). Two independent solutions of the differential equations of the characteristic curves are ) 1 Gx) =a 1, 4.3) x + c 3 where Φ+ c 4 c ) = a c, 4.4) x + c 3 Gx) = ) 1/) x gx)dx [ 1 + c )x + c 3 ] +c 4.5) and a 1 and a are constants. Hence, since ψ =Φx, ), the group invariant solution of.8) is of the form ψx, ) = x + c ) c 1+c 3 c 4 fξ), 4.6) c where fξ) is an arbitrar function of ξ and ξ = ) 1 Gx). 4.7) x + c 3 We now substitute 4.6) into.7). This ields an ordinar differential equation for fξ): ν d3 f dξ 3 + c d fξ) df ) 1 + c ) ) df =0. 4.8) 1 + c ) dξ dξ dξ Consider next the condition that J, defined b.10), is constant independent of x. B making the change of variable from to ξ at an given position x,.10) becomes J =ρ x + c ) c 1 c +1 Thus J is independent of x provided 0 ) df dξ. 4.9) dξ c = ) When c = 1, 4.9) becomes ) df J =ρ dξ 4.11) dξ 0

6 Free Laminar Two-dimensional Jet 97 and 4.6) and 4.7) reduce to ψx, ) = x + 3 c 3) 1/3 fξ) c 4, 4.1) ξ = x + 3 c ) /3 Gx). 4.13) The differential equation 4.8) becomes d 3 f dξ ν d fξ) df ) = ) dξ dξ We see that the result c =1/ puts the differential equation 4.8) in a form that can be integrated analticall. Finall, consider the boundar conditions on fξ). The function gx) in 4.5) is arbitrar. To make ξ = 0 correspond to =0wechoosegx) = 0 and therefore Gx) =0. Since ψ x = 1 3 x + 3 c ) /3 fξ) ξ df ), 4.15) dξ 3 ψ = 1 df x + 3 c ) 1/3 dξ, 3 ψ = 1 x + 3 c ) d f 3 dξ, 4.16) the boundar conditions.8) and.9) on ψx, ) ield the following boundar conditions on fξ): f0) = 0, f 0) = 0, f ± ) = ) For completeness we outline briefl the solution of 4.14) subject to the boundar conditions 4.17). Integration of 4.14) once with respect to ξ gives d f dξ + 1 df fξ) 3ν dξ = k, 4.18) where k is a constant. The boundar conditions 4.17) give k =0provided f0) df0) dξ = ) We will take k = 0 and check that the solution obtained satisfies 4.19). Equation 4.18) becomes d f dξ + 1 6ν d f ξ) ) = 0 4.0) dξ and integration with respect to ξ gives df dξ + 1 6ν f ξ) = α 6ν, 4.1)

7 98 D P Mason where α is a constant. Since v x is positive, it follows from.6) and 4.16) that df/dξ is positive. Hence the left hand side of 4.1) is positive and we therefore wrote α instead of α on the right hand side. Equation 4.1) is a variables separable differential equation. Its solution subject to the boundar condition f0) = 0 is α ) fξ) =α tanh 6ν ξ. 4.) It is readil verified that 4.19) is satisfied b 4.). The constant α is obtained in terms of the given fluxof momentum, J, b substituting 4.) into 4.11): α = ) 9νJ 1/3. 4.3) ρ The boundar condition f ± ) = 0 was not used. It is satisfied b the solution 4.). It remains to determine the constants c 3 and c 4. The long narrow orifice in the wall is assumed to be infinitel small. In order to have a finite volume of flow and a finite momentum it is necessar to assume an infinite fluid velocit at the orifice [8]. Now v x x, 0) = ψ x, 0) = α 6ν x + 3 c ) 1/3. 4.4) 3 We therefore take c 3 = 0 to ensure that v x x, 0) = at x = 0. Also, a stream function ψx, ) is determined up to an arbitrar additive constant. Hence from 4.1) we choose c 4 =0. From 4.1), 4.13) and 4.) we obtain the group invariant solution α ) ψx, ) =αx 1/3 tanh 6ν ξ, 4.5) where α is given b 4.3) and ξ = x /3. 4.6) This result agrees with the solution obtained b the combined work of Schlichting [7] and Bickle [1] and given in standard texts [6, 8]. The solution derived b Schlichting and Bickle is therefore the group invariant solution. Since c 1 =1,c =1/, c 3 =0,c 4 = 0 and gx) = 0, the Lie point smmetr which generates the group invariant solution is X = 3 x x ψ ψ. 4.7) 5 Conservation law We will establish a connection between a conservation law for the differential equation.7), the Lie point smmetr 4.7) which generated the group invariant solution of the differential equation and conservation of the total fluxof momentum in the direction of the jet.

8 Free Laminar Two-dimensional Jet 99 The equation D 1 T 1 + D T =0 5.1) is a conservation law for the differential equation.7) if 5.1) is satisfied for all solutions ψx, ) of.7) []. In 5.1) D 1 and D are the operators of total differentiation defined b 3.9) and 3.10). The quantities T i x,, ψ, ψ x,ψ,...), where i =1and,are the components of the conserved vector T = T 1,T ). It is readil verified that D 1 ψ ) + D ψ x ψ νψ ) = 0 5.) for all solutions ψx, ) of the partial differential equation.7). Equation 5.) is a conservation law for.7) and the components of the conserved vector are T 1 = ψ, T = ψ x ψ νψ. 5.3) Now, from a result due to Kara and Mahomed [3, 4], if X = ξ 1 x + ξ + η ψ 5.4) is a Lie point smmetr of.7) and T 1 and T are the components of the conserved vector of.7) associated with X then X [l] T i + T i D k ξ k T k D k ξ i =0, i =1,, 5.5) where X [l] is the l th prolongation of X and the repeated indexis summed form k =1to k =. We will use this result to determine the Lie point smmetr associated with the conserved vector 5.3). For the conserved vector 5.3), l =. When expanded, 5.5) consists of the two equations i =1: X [] T 1 + T 1 D ξ T D ξ 1 =0, 5.6) i =: X [] T + T D 1 ξ 1 T 1 D 1 ξ =0. 5.7) Consider the Lie point smmetr 3.11) with c 1 = 1. Using 3.6) and 3.7) it follows that X [] = [ ] 1 + c )x + c 3 x + + gx) ) +c ψ + c 4 ) ψ + ψ x dg ) dx ψ ψ x +c 1)ψ + ζ 11 + ζ 1 +c )ψ. 5.8) ψ ψ xx ψ x ψ B using 5.3) for T 1 and T, equations 5.6) and 5.7) become c 1)T 1 =0, c 1)T =0. 5.9) Thus 5.6) and 5.7) are satisfied provided c =1/. Hence X given b 3.11) is the Lie point smmetr generator associated with the conserved vector 5.3) provided c =1/. This is the same condition on c as obtained

9 100 D P Mason from 4.9) b insisting that J is independent of x. Hence c as determined b 5.5) has the same value as in the Lie point smmetr 4.7) which generates the group invariant solution. This establishes a connection between the conserved vector T and the total momentum flux J through the Lie point smmetr generator. No condition is placed on c 3, c 4 or gx) b 5.5). The constants c 3 and c 4 and the arbitrar function gx) are determined b the choice of origin of coordinates and the choice of the arbitrar additive constant in the stream function. The connection between the conserved vector T and the total momentum flux J is that the are related through the identit T 1 = 1 J ρ. 5.10) 6 Conclusions The solution for the stream function of a free two-dimensional jet derived b the combined work of Schlichting [7] and Bickle [1] is the group invariant solution. Onl two of the five Lie point smmetries of.7) contribute to the generation of the group invariant solution. The group invariant solution, given b 4.5) and 4.6), has the form.11) assumed b Schlichting [7, 8]. To obtain the group invariant solution, the form.11) was not assumed, but was derived. The Lie group analsis also shows that there are no other forms for a similarit solution of.7) besides.11). The method used a conserved quantit to determine the constants in the linear combination of Lie point smmetries. In the problem considered here the conserved quantit was the total momentum fluxin the direction of the jet. In the original problem considered b Momoniat et. al. [5] the conserved quantit was the total volume of the liquid drop. The method will be applicable to other problems for which conserved quantities can be derived. We established a connection between a simple conservation law for the differential equation for the stream function and conservation of the total momentum flux. The Lie point smmetr associated with the conservation law is the same as that which generates the group invariant solution which was derived b imposing the condition that the total momentum fluxis conserved. Acknowledgements The author thanks the National Research Foundation, Pretoria, South Africa, for financial support. References [1] Bickle W G,. The Plane Jet, Phil. Mag ), [] Ibragimov N H and Anderson R L, Apparatus of Group Analsis, in Smmetries, Exact Solutions and Conservation Laws, CRC Handbook of Lie Group Analsis of Differential Equations, Vol. 1, Editor: Ibragimov N H., CRC Press, Boca Raton, 1994, 1 14,

10 Free Laminar Two-dimensional Jet 101 [3] Kara A H and Mahomed F M, Action of Lie Bäckland Smmetries on Conservation Laws, in Developments in Theor and Computation, Proceedings of the International Conference at the Sophus Lie Conference Centre, Nordfjordheid, Norwa, 30 June 5 Jul 1997, Editors: Ibragimov N H, Razi Naqvi K, Straume E, Mars Publishers, Trondheim, 1999, [4] Kara A H and Mahomed F M, Relationship between Smmetries and Conservation Laws, Internat. J. Theoret. Phs ), [5] Momoniat M, Mason D P and Mahomed F M, Non-Linear Diffusion of an Axismmetric Thin Liquid Drop: Group-Invariant Solution and Conservation Law, Inter. J. Non-Linear Mech ), [6] Rosenhead L, Lanunar Boundar Laers, Clorendon Press, Oxford, 1963, [7] Schlichting H, Laminare Strahlausbreitung, Z. Angew. Math. Mech ), [8] Schlichting H, Boudar-Laer Theor, Sixth Edition, McGraw-Hill, New York, [9] Schlichting H and Gersten K, Boundar Laer Theor, Springer-Verlag, Berlin, 000, Ch. 7.