CONSERVATION LAWS AND CONSERVED QUANTITIES FOR LAMINAR RADIAL JETS WITH SWIRL

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1 Mathematical and Computational Applications,Vol. 15, No. 4, pp , 21. c Association for Scientific Research CONSERVATION LAWS AND CONSERVED QUANTITIES FOR LAMINAR RADIAL JETS WITH SWIRL R. Naz 1, I. Naeem 2 and F.M. Mahomed 3 1 Centre for Mathematics and Statistical Sciences Lahore School of Economics, Lahore 532, Pakistan. 2 Department of Mathematics, School of Science and Engineering, LUMS, Lahore Cantt 54792, Pakistan. 3 School of Computational and Applied Mathematics, Centre for Differential Equations, Continuum Mechanics and Applications, Universit of the Witwatersrand, Wits 25, South Africa. rehananaz qau@ahoo.com, inqau@ahoo.com, Fazal.Mahomed@wits.ac.za Abstract- Conserved quantities pla a central role in the solution of jet flow problems. A sstematic wa of deriving conserved quantities for the radial jets with swirl is presented. The multiplier approach is used to derive the conservation laws for the sstem of three boundar laer equations for the velocit components and for the sstem of two partial differential equations for the stream function. When the swirl is zero or at a large distance from the orifice (at infinit), the boundar laer equations for the radial jets with swirl reduce to those of the purel radial jets. The conserved quantities for the radial liquid, free and wall jets with swirl are derived b integrating the conservation laws across the jets. Kewords- Radial jets with swirl, conservation laws, conserved quantit. 1. INTRODUCTION It is of great interest to stud the three dimensional boundar laer flows, for eample the radial jets with swirl due to their etensive range of applications. The concept of two-dimensional boundar laer flow of the purel radial liquid jet was introduced b Watson [1]. The two-dimensional boundar laer flow of the purel radial free jet was first considered b Schwarz [2]. Glauert in [3] studied the two-dimensional boundar laer flow of the purel radial wall jet. Later, Rile in [4] also discussed the two-dimensional boundar laer flow of purel radial liquid, free and wall jets. In all these jet flow problems, the boundar conditions are homogeneous and therefore the unknown eponent in the similarit solution cannot be obtained from the boundar conditions. B integrating Prandtl s momentum boundar laer equation across the jet and using the boundar conditions and the

2 R. Naz, I. Naeem and F. M. Mahomed 743 continuit equation, a condition, known as the conserved quantit, was derived. Rile [5] etended the problem of the purel radial jets to the radial jets with swirl. Later, O Nan and Schwarz [6] also studied the problem of the radial free jet with swirl. The governing equations consist of a continuit equation and two momentum equations in the radial and swirl directions respectivel. This problem consists of homogeneous boundar conditions and thus also requires conserved quantities. The conserved quantities for the radial jets with swirl were established using the same idea as was used in [1, 2, 3, 4] to construct the conserved quantities for the purel radial jets. B integrating the momentum equation in the swirl direction and using the boundar conditions and the continuit equation, one conserved quantit for the radial free jet with swirl and one for the radial wall jet with swirl were derived. The second conserved quantit for the radial free and wall jets with swirl were established b integrating the momentum equation in the radial direction and requiring that swirl is zero at =. The onl conserved quantit for the radial liquid jet with swirl [5] was obtained from a phsical argument based on conservation of mass flu in an incompressible fluid and is the same as for the purel radial liquid jet. The conserved quantit for each problem has in general phsical significance. Rile [5] established the conserved quantities for the radial liquid, free and wall jets with swirl. The onl conserved quantit for the radial liquid jet with swirl is total mass flu per radian which is constant. One of the conserved quantities for the radial free jet with swirl is the radial flu of angular momentum about =. The second conserved quantit for the radial free jet with swirl is the radial flu of momentum and is invariant onl when the swirl is zero or at large distances from the orifice. For a radial wall jet with swirl one conserved quantit is the flu of eterior momentum flu and is constant onl when the swirl is zero or at large distances from the orifice. There is no obvious phsical interpretation of the other conserved quantit for the radial wall jet with swirl. In this paper, we construct the conserved quantities for the radial liquid, free and wall jets with swirl b the method recentl introduced b Naz et al [7]. We use the multiplier approach introduced b Steudel [8] (see also [9, 1]) to derive the conservation laws for the sstem of three partial differential equations for the velocit components as well as for the sstem of two partial differential equations for the stream function. The radial jets with swirl behave as purel radial jets when the swirl is zero or at large distances from the orifice as becomes large the swirl component will tend to zero more rapidl than the radial velocit component. Then the boundar laer equations for radial jets with swirl reduce to that of purel radial jets. The conservation laws for the purel radial jets were derived in [7] with the

3 744 Conservation Laws for Laminar Radial Jets with Swirl help of the multiplier approach. The conserved quantities for the radial, liquid, free and wall jets with swirl are derived with the help of conservation laws. The radial jets (liquid, free and wall) with swirl satisf the same partial differential equations but the boundar conditions for each jet are different. The derivation of conservation laws depends onl on the partial differential equations but the derivation of the phsical conserved quantities depends also on the boundar conditions. The boundar conditions therefore determine which conserved vector is associated with which jet. 2. CONSERVATION LAWS FOR RADIAL LAMINAR JET We consider the radial liquid jet, free jet and wall jet with swirl. The radial liquid jet with swirl is formed when a circular jet of liquid, with an aiall smmetrical swirling component of velocit, strikes a plane boundar normall and spreads over it. The radial free jet with swirl is formed when two circular jets, each with an aiall-smmetrical swirling component of velocit, impinge on one another. When a circular jet, having aiall smmetric swirling component of velocit, strikes a plane surface normall which is surrounded b the same fluid as the jet and spreads out radiall over it then a radial wall jet with swirl is formed. All these jets were described b Rile [5]. The fluid in the jets is viscous and incompressible. In the free jet and wall jet the surrounding fluid consists of the same fluid as in the jet but for the liquid jet the surrounding fluid is a gas. The clindrical polar coordinates (, θ, ) are used with as radial coordinate and = is the ais of smmetr. All fluid variables are independent of θ. For the liquid jet the solid boundar is at = and the free surface is at = ϕ(). The wall jet has a solid boundar at = and the free jet is smmetrical about =. For all three jets an aiall smmetrical swirling component of velocit is considered. The boundar laer equations governing the flow in a radial laminar jet with swirl, in the absence of pressure gradient, are uu + vu w2 = νu, (1) uw + vw + uw = νw, (2) (u) + (v) =, (3) where u(, ) and v(, ) are the velocit components in and directions respectivel, w(, ) is an aiall-smmetrical swirling component of velocit and ν is the

4 R. Naz, I. Naeem and F. M. Mahomed 745 kinematic viscosit of the fluid. Equations (1) and (2) are the momentum equations in the radial and θ directions respectivel and (3) is the continuit equation. The boundar-laer equations (1)-(3) have been derived with the assumption that the swirl velocit w and radial velocit u are of same order of magnitude whereas the transverse velocit v is of lower order (see [5]). The stream function ψ(, ) satisfies [11] u = ψ, v = ψ. (4) The radial free jet with swirl is smmetrical about the plane =, therefore v(, ) =. For the liquid jet and the wall jet there is no suction or blowing of fluid at the solid boundar = and thus v(, ) =. Therefore ψ (, ) = for all three jets and thus ψ(, ) is a constant. We choose the constant as zero and hence ψ(, ) =. (5) Let O be an reference point on the line = in the aial plane θ = θ o, then the stream function at P (, θ o, ) is (see [11]) ψ(, ) = P O [u(, )d v(, )d]. (6) Integrating (6) with respect to from = to = along a straight line with kept fied ields ψ(, ) = u(, )d. (7) Now, ψ(, ) is finite onl if we assume u(, ) sufficientl rapidl as. and Equation (3) is identicall satisfied and equations (1) and (2) ield 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 =, (8) 1 ψ w 1 ψ w ψ w νw =. (9) We will derive the conservation laws for the sstem (1)-(3) and also for the sstem (8)-(9) for the stream function, using the multiplier approach [8, 9].

5 746 Conservation Laws for Laminar Radial Jets with Swirl 2.1. Conservation laws for sstem of equations for velocit components Consider the multipliers of the form Λ 1 = Λ 1 (,, u, v, w), Λ 2 = Λ 2 (,, u, v, w)andλ 3 = Λ 3 (,, u, v, w) for the sstem of equations (1)-(3), such that ( ) ( Λ 1 uu + vu νu w2 + Λ 2 uw + vw + uw ) νw +Λ 3 (u + u + v ) = D T 1 + D T 2, (1) for all functions u(, ), v(, ) and w(, ) and not onl for the solutions of sstem. In (1), D and D are total derivative operators defined b D = + u u + v v + w w + u + v + w u v w +u + v + w +, (11) u v w D = + u u + v v + w w + u + v + w u v w +u + v + w +. (12) u v w The determining equations for the multipliers Λ 1, Λ 2 and Λ 3 are [ E u Λ 1 (uu + vu νu w2 ) + Λ 2(uw + vw + uw νw ) ] +Λ 3 (u + u + v ) =, (13) [ E v Λ 1 (uu + vu νu w2 ) + Λ 2(uw + vw + uw νw ) ] +Λ 3 (u + u + v ) =, (14) [ E w Λ 1 (uu + vu νu w2 ) + Λ 2(uw + vw + uw νw ) ] +Λ 3 (u + u + v ) =, (15) where E u, E v and E w are the standard Euler operators which annihilate divergence epressions: E u = u D D + D 2 + D D + D 2, (16) u u u u u

6 R. Naz, I. Naeem and F. M. Mahomed 747 and E v = v D D + D 2 + D D + D 2, (17) v v v v v E w = w D D + D 2 + D D + D 2. (18) w w w w w The epansion of equations (13)-(15) ields Λ 1u (uu + vu νu w2 ) + Λ 2u(uw + vw + uw νw ) + Λ 3u (u + u + v ) +Λ 1 u + Λ 2 [w + w ] + Λ 3 D [uλ 1 + Λ 3 ] D [vλ 1 ] νd 2 (Λ 1 ) =, (19) Λ 1v (uu + vu νu w2 ) + Λ 2v(uw + vw + uw νw ) + Λ 3v (u + u + v ) +Λ 1 u + Λ 2 w D (Λ 3 ) =, (2) Λ 1w (uu + vu νu w2 ) + Λ 2w(uw + vw + uw νw ) + Λ 3w (u + u + v ) 2w Λ 1 + u Λ 2 D [uλ 2 ] D [vλ 2 ] νd 2 (Λ 2 ) =. (21) Equations (19)-(21) must be satisfied for all functions u(, ), v(, ) and w(, ). After epansion the highest order derivative terms in sstem (19)-(21) are secondorder derivative terms. Therefore, equating the coefficients of u, v and w to zero, we obtain Λ 1u =, Λ 1v =, Λ 1w =, Λ 2u =, Λ 2v =, Λ 2w =, (22) which in turn results in Λ 1 = A(, ), Λ 2 = B(, ), Λ 3 = C(,, u, v, w). (23) Equations (19)-(21), with the help of (23), reduce to v C v +v (A C u )+w (C w B) uc u Bw +ua +C +va +νa =, (24) u C v + u (A C u ) w (C w B) + uc v C =, (25) (u + v )(C w B) + uc w 2w A + u B ub vb νb =. (26) Equating the first-order derivative terms u, u and w in (25) to zero, we obtain C v =, A C u =, C w B =, (27)

7 748 Conservation Laws for Laminar Radial Jets with Swirl which ields C = u A(, ) + w B(, ) + D(, ). (28) Equations (24)-(26) become ( 2u A A ) + va + w ( B 2B ) + D + νa =, (29) ua + wb + D =, (3) ( u B 2B ) + vb + 2w A + νb =. (31) The separation of equations (29)-(31) with respect to u, v and w finall gives Λ 1 =, Λ 2 = c 2 2, Λ 3 = c 1 + c 2 w, (32) where c 1 and c 2 are constants. It is not difficult to construct the conserved vectors b elementar manipulations with the help of the multipliers (32). Equations (1) and (32) give [uu +vu νu w2 ]+c 2 2 [uw +vw + uw νw ]+(c 1 +c 2 w)[u+u +v ] = D [c 1 u + c 2 2 uw] + D [c 1 v + c 2 2 (vw νw )], (33) for arbitrar functions u(, ), v(, ) and w(, ) and therefore when u(, ), v(, ) and w(, ) are the solutions of the sstem of differential equations (1)-(3), then Thus D [c 1 u + c 2 2 uw] + D [c 1 v + c 2 2 (vw νw )] =. (34) T 1 = u, T 2 = v, (35) T 1 = 2 uw, T 2 = 2 (vw νw ), (36) are the conserved vectors for the sstem of differential equations (1)-(3) corresponding to multipliers of the form Λ 1 = Λ 1 (,, u, v, w) and Λ 2 = Λ 2 (,, u, v, w) Conservation Laws for the stream function formulation Consider multipliers of the form Λ 1 = Λ 1 (,, ψ, w) and Λ 2 = Λ 2 (,, ψ, w) in order to derive the conservation laws for the sstem (8)-(9). The determining equations for the multipliers Λ 1 and Λ 2 are E ψ [Λ 1 ( 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 )

8 R. Naz, I. Naeem and F. M. Mahomed 749 +Λ 2 ( 1 ψ w 1 ψ w + 1 ] ψ w νw 2 ) =, (37) and where E w [Λ 1 ( 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ) +Λ 2 ( 1 ψ w 1 ψ w + 1 ] ψ w νw 2 ) =, (38) E ψ = ψ D D + D 2 + D D + D 2, (39) ψ ψ ψ ψ ψ E w = w D D + D 2 + D D + D 2, (4) w w w w w are the standard Euler operators. In (39) and (4), D and D are the total derivative operators and are defined b D = + ψ ψ + w w + ψ + w + ψ + w +, (41) ψ w ψ w D = + ψ ψ + w w + ψ + w + ψ + w +. (42) ψ w ψ w Epansion of equations (37) and (38) ields Λ 1ψ [ 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ] + Λ 2ψ [ 1 ψ w 1 ψ w ψ w νw ] + D [ 1 Λ 1ψ + 1 Λ 2w ] D [Λ 1 ( 1 ψ 2 2 ψ ) +Λ 2 ( w + w 2 )] + D D [ 1 Λ 1ψ ] D 2 [ 1 Λ 1ψ ] + νd 3 [Λ 1 ] =, (43) Λ 1w [ 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ] + Λ 2w [ 1 ψ w 1 ψ w ψ w νw ] 2wΛ 1 + Λ 2 ψ 2 D [ Λ 2 ψ ] + D [ Λ 2 ψ ] νd[λ 2 2 ] =. (44) Equating the coefficients of ψ and w in (44) to zero, we obtain Λ 1w =, Λ 2w =. (45) Now, since Λ 1 and Λ 2 are not functions of w, therefore separating the rest of the terms in (44) (in epanded form) with respect to w ields Λ 1 =. (46)

9 75 Conservation Laws for Laminar Radial Jets with Swirl Equations (43) and (44) reduce to Λ 2ψ [ 1 ψ w 1 ψ w ψ w νw ] 2w Λ w [Λ 2 + ψ Λ 2ψ ] and ( w + w 2 )[Λ 2 + ψ Λ 2ψ ] =, (47) 2ψ Λ 2 2 ψ [Λ 2 + ψ Λ 2ψ ] + ψ [Λ 2 + ψ Λ 2ψ ] ν[λ 2 + ψ Λ 2ψ + ψ Λ 2ψ + ψ Λ 2ψ + ψ 2 Λ 2ψψ ] =. (48) From equation (47), the coefficient of w (or in (48) the coefficient of ψ ) ields Λ 2ψ = and thus Λ 2 = A(, ). (49) Equations (47) and (48) become A w + w [ 2A A 2 ] + w A =, (5) 2 ψ A + ψ [ 2A 2 A ] νa =, (51) which finall ield A = c 3 2 and Λ 1 =, Λ 2 = c 3 2, (52) where c 3 is an arbitrar constant. The multipliers Λ 1 and Λ 2 for the sstem (8)-(9) satisf Λ 1 [ 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ] +Λ 2 [ 1 ψ w 1 ψ w ψ w νw ] (53) = D T 1 + D T 2, for all functions ψ(, ) and w(, ). From equations (52) and (53), we have [ 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ]+c 3 2 [ 1 ψ w 1 ψ w ψ w νw ] = D [c 3 wψ ] + D [c 3 ( wψ ν 2 w )], (54)

10 R. Naz, I. Naeem and F. M. Mahomed 751 for arbitrar ψ(, ) and w(, ). sstem (8)-(9), we have When ψ(, ) and w(, ) are solutions of the D [c 3 wψ ] + D [c 3 ( wψ ν 2 w )] =. (55) Thus T 1 = wψ, T 2 = wψ ν 2 w, (56) are the components of a conserved vector for the sstem (8)-(9). Notice that the conserved vector (56) is equivalent to the conserved vector (36). The multiplier approach gives multipliers onl for the local conserved vectors. The multipliers for the non-local conserved vectors cannot be obtained with the help of the multiplier approach as presented. Notice that if we multipl equation (8) with zero and equations (9) with 2 ψ, we have [ 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ w 2 ]+ 2 ψ[ 1 ψ w 1 ψ w ψ w νw ] ] = D [ψwψ ] + D [ wψψ ν 2 ψw + ν 2 w ψ d, (57) for arbitrar ψ(, ) and w(, ). When ψ(, ) and w(, ) are solutions of the sstem (8)-(9), we have ] D [ψwψ ] + D [ wψψ ν 2 ψw + ν 2 w ψ d =. (58) Thus T 1 = ψwψ, T 2 = wψψ ν 2 ψw + ν 2 w ψ d, (59) are the components of a non-local conserved vector for the sstem (8)-(9). It ma be possible that the non-local conservation law (59) which does not arise from local multipliers can be derived through a potential sstem associated with the sstem (1), (2) and (3) or the sstem (8) and (9) Conservation laws for radial jets at large distance from the orifice The radial jets with swirl at large distance from the orifice, correspond to a radial jet with zero swirl. When, w, then the sstem (1)-(3) reduces to uu + vu = νu, (6) (u) + (v) =, (61)

11 752 Conservation Laws for Laminar Radial Jets with Swirl which is same as the boundar laer equations for purel radial jet flows. The sstem (8)-(9) for the stream function, when, reduces to a single third-order partial differential equation 1 ψ ψ 1 2 ψ2 1 ψ ψ νψ =. (62) The local conserved vectors for sstem (6)-(61) derived in [7] are T 1 = u, T 2 = v, (63) T 1 = u 2, T 2 = uv νu. (64) The conserved vectors for the third-order partial differential equation (62) (see [7]) are and T 1 = 1 ψ2, T 2 = 1 ψ ψ νψ, (65) T 1 = 1 ψψ2, T 2 = 1 ψψ ψ + ν 2 ψ2 νψψ. (66) The conserved vectors given in equations (64) and (65) are equivalent. 3. CONSERVED QUANTITIES FOR RADIAL LIQUID JETS In this section, we derive the conserved quantities for the radial liquid, free and wall jets with swirl b the technique used in [7]. We first consider the conserved vectors (35), (36) and (64) for the sstem of equations for the velocit components and derive the conserved quantities for the radial liquid and the free jets with swirl. Then we consider the stream function formulation to give an alternative derivation of the conserved quantities for the radial free jet with swirl and also derive the conserved quantities for the radial wall jet with swirl. All the conserved vectors (T 1, T 2 ) derived here depend on u(, ), v(, ), w(, ) or ψ(, ) and satisf D T 1 + D T 2 = T 1 + T 2. (67) But for a conserved vector D T 1 + D T 2 = and therefore equation (67) ields T 1 + T 2 =. (68)

12 R. Naz, I. Naeem and F. M. Mahomed 753 The conserved quantities for the radial liquid, free and wall jets with swirl are obtained b integrating equation (68) for the corresponding conservation law across the jet and imposing the boundar conditions Conserved quantit for the radial liquid jet with swirl The conserved quantit for the radial liquid jet with swirl is derived with the help of conserved vector (35). The derivation for conserved quantit for radial liquid jet with swirl is same as in [7] for the purel radial jet flow. The boundar conditions for the radial liquid jet with swirl are = : u =, v =, w =, (69) = ϕ() : u =, w =. (7) The velocit component v(, ϕ()) is where v(, ϕ()) = D [ϕ()] = u(, ϕ())dϕ() Dt d, (71) D Dt = + u(, ) t + v(, ) (72) is the material time derivative. The conserved quantit for the radial liquid jet with swirl is obtained b integrating (68) with respect to from = to = ϕ() keeping as fied. For the conserved vector (35), we obtain ϕ() [ (u) + (v)]d =. (73) Differentiating (73) under integration sign [12], we have d d ϕ() [u]d u(, ϕ()) dϕ() d + [v(, )]ϕ() =. (74) The boundar condition (69) for v(, ) and epression (71) for v(, ϕ()), reduces (74) to ϕ() ud = constant, independent of, (75) which gives that the total mass flu is constant along the jet. Therefore F = ϕ() ud (76)

13 754 Conservation Laws for Laminar Radial Jets with Swirl is the conserved quantit for the radial liquid jet with swirl [5] Conserved quantities for the radial free jet with swirl The boundar conditions for the radial free jet with swirl are = : v =, u =, w =, (77) = ± : u =, u =, w =, w =. (78) For the conserved vector (36), equation (68) on integration with respect to from = to = with kept as fied during the integration, ields d d [2 (uw)d] + [ ] 2 vw ν 2 w =. (79) We assume that v(, ± ) is finite. The boundar conditions (77) and (78) on (79) gives 2 2 uwd = constant, independent of. (8) The conserved quantit for the radial free jet with swirl is F = 2 2 uwd, (81) where F is a constant independent of. If ρ is the densit of fluid, then the constant ρf is the total radial flu of angular momentum about = (Rile [5]). The second conserved quantit which is valid onl when there is no swirl or at large distance from the orifice can be obtained from the conserved vector (64). Integrate (68) with respect to for the conserved vector (64), we obtain d d [ u 2 d] + [uv νu ] =. (82) The boundar condition (78) and v(, ± ) is finite, ields Thus 2 G = 2 u 2 d = constant, independent of. (83) u 2 d (84) and ρg is the second conserved quantit for the radial free jet with swirl and is valid onl at large distance from orifice.

14 R. Naz, I. Naeem and F. M. Mahomed 755 The conserved quantities for the radial free jet with swirl can also be derived for the stream function formulation. We now outline an alternative derivation of the conserved quantities for the radial free jet with swirl. Also we derive the conserved quantities for the radial wall jet with swirl. In terms of the stream function the boundar conditions (77) and (78) take the following form: = : ψ =, ψ =, w =, (85) = ± : ψ =, ψ =, w =, w =. (86) Integrate equation (68) with respect to form = to = keeping as fied. For the conserved vector (56), we have d d [ wψ d] + [ ] wψ ν 2 w =. (87) Equation (87) with the help of boundar condition (86) and requirement ψ (, ± ) = v(, ± ) is finite, gives 2 wψ d = constant, independent of. (88) Equation (88) is equivalent to (8) and thus we finall obtain the conserved quantit F given in (81). The second conserved quantit which is valid onl when there is no swirl or at large distance from the orifice can be obtained from conserved vector (65). Integration of (68) with respect to with kept as fied for the conserved vector (65) gives d d [ 1 ψ d] 2 + [ 1 ψ ψ νψ ] =. (89) Equation (89) with the help of boundar condition (86) and the requirement ψ (, ± ) = v(, ± ) is finite, ields 2 ψ 2 d = constant, independent of. (9) Equation (9) is equivalent to (83) and hence we obtain the conserved quantit (84) Conserved quantities for the radial wall jet with swirl For the radial wall jet with swirl the boundar conditions are = : u =, v =, w =, (91)

15 756 Conservation Laws for Laminar Radial Jets with Swirl = : u =, u =, w =, w =. (92) In terms of the stream function, we obtain = : ψ =, ψ =, w =, (93) = : ψ =, ψ =, w =, w =. (94) B integrating equation (68) with respect to from = to = keeping as fied and considering the conserved vector (59), we have d d [ ψwψ d] + [ ] wψψ ν 2 ψw + ν 2 w ψ d =. (95) But ψ(, ) = whereas ψ(, ), ψ (, ) and w (, ) are assumed to be finite. Thus boundar conditions (93) and (94) reduce equation (95) to [ d ] ψwψ d + ν 2 w ψ d =. (96) d Integrating equation (96) b parts and using boundar conditions, we deduce ( d ( ) ψ wψ d )d = ν 2 wψ d, (97) d and in terms of the velocit components, we have d ( ) ( 3 u wud )d = ν 3 wu d. (98) d To make the right side of equation (98) zero, Rile [5] assumed that u/w is a function of onl and this condition is satisfied b the similarit solution. Thus in the similarit solution regime, we have ( ) 3 u wud d = constant, independent of. (99) Therefore M = 3 ( ) u wud d, (1) is a conserved quantit. Rile [5] used ρ 2 M as a conserved quantit for the radial wall jet with swirl. There is no obvious phsical significance of this conserved quantit. The second conserved quantit is derived with no swirl or at large distance from the orifice and the conserved vector (66) is used in its derivation. Integrating (68)

16 R. Naz, I. Naeem and F. M. Mahomed 757 with respect to from = to = keeping as fied during the integration, we have d d [ 1 ψψ 2 d] + [ 1 ψψ ψ + ν 2 ψ2 νψψ ] The boundar conditions (93) and (94) ield 1 =. (11) ψψ 2 d = constant, independent of, (12) we have used the requirements ψ(, ) =, ψ(, ) and ψ (, ) are finite. Integrating (12) b parts, we obtain N = 1 or equivalentl N = 2 ( ) ψ ψ 2 d d, (13) ( u u 2 d )d, (14) which is valid onl for no swirl or at large distance from orifice. The constant ρ 2 N is interpreted as the flu of eterior momentum flu Comparison between conservation laws for radial jets with swirl and purel radial jet flows The results for radial jets with swirl and for purel radial jets derived in [7] are displaed in Table 1. The boundar laer equations for the velocit components as well as for the stream function formulation are given in Table 1 for both the radial jets with swirl and for the purel radial jets. The conservation laws for the purel radial jet flows hold for the radial jets with swirl but onl at large distances from the orifice (as ). The multiplier approach on the sstem of equations for the velocit components for purel radial jets ields two local conserved vectors and for the stream function formulation also two local conserved vectors are obtained. The multiplier approach on the sstem of equations for the velocit components for radial jets with swirl ields three conserved vectors, one of these conserved vectors is valid onl when. For the stream function formulation for the radial jets with swirl four conserved vectors are obtained and two of them are valid onl when. The conserved vectors for the radial jets with swirl as and for the purel radial jets are the same. The conserved quantit for the radial liquid jet with swirl and the purel radial liquid jet are the same. The onl conserved quantit for the purel radial free jet

17 758 Conservation Laws for Laminar Radial Jets with Swirl is that the total radial flu of momentum is constant in the jet. The radial free jet with swirl has two conserved quantities. One conserved quantit for the radial free jet with swirl is the total radial flu of angular momentum about = which is constant. The second conserved quantit is the same as for the purel radial free jet but it holds onl as. The conserved quantit for a purel radial wall jet is the flu of eterior momentum flu in the wall jet which is constant and holds for the radial wall jet with swirl at. Another conserved quantit for the radial wall jet with swirl is established with the help of a non-local conserved vector for the sstem of two partial differential equations for the stream function formulation. There is no obvious phsical interpretation of that conserved quantit.

18 R. Naz, I. Naeem and F. M. Mahomed 759 Boundar laer equations Radial jets with swirl Purel radial jets uu + vu w2 = νu uu + vu = νu uw + vw + uw = νw (u) + (v) = (u) + (v) = Conserved vectors T 1 = 2 uw for T 2 = 2 (vw νw ) velocit components T 1 = u T 1 = u T 2 = v T 2 = v As T 1 = u 2 T 1 = u 2 T 2 = (uv νu ) T 2 = (uv νu ) Stream function 1 ψ 1 ψ 2 2 1ψ ψ 1 ψ 1 ψ 2 2 formulation νψ w 2 = 1ψ ψ νψ = 1 ψ w 1ψ w + 1 ψ 2 w νw = Conserved vectors for stream function T 1 = wψ T 2 = wψ ν 2 w T 1 = ψwψ T 2 = wψψ ν 2 ψw + ν2 w ψ As T 1 = 1 ψ2 T 2 = 1ψ ψ νψ T 1 = 1 ψψ2 T 2 = 1ψψ ψ + ν 2 ψ2 νψψ T 1 = 1 ψ2 T 2 = 1ψ ψ νψ T 1 = 1 ψψ2 T 2 = 1ψψ ψ + ν 2 ψ2 νψψ

19 76 Conservation Laws for Laminar Radial Jets with Swirl Conserved quantities For liquid jet For free jet ϕ() ud ϕ() ud 2 2 uwd 2 u 2 d as 2 u 2 d For wall jet ( 3 u wud )d ( 2 ) u u 2 d d as ( 2 u u 2 d )d Table 1: Comparison between radial jets with swirl and purel radial jets 4. CONCLUDING REMARKS The conserved quantities for the radial liquid, free and wall jets with swirl derived b Rile [5] can be constructed with the help of conservation laws. The radial jets with swirl behave as purel radial jets at large distances from the orifice. At large distances from the orifice as, the swirling component w and the boundar laer equations for radial jets with swirl reduce to that of purel radial jets. Therefore, as the conservation laws for radial jets with swirl coincide with those of the purel radial jets. The multiplier approach gave two local conservation laws for the sstem of equations for the velocit components. One of the conserved vectors gave the conserved quantit for the radial liquid jet with swirl and the second conserved vector ielded one conserved quantit for the radial free jet with swirl. Also, another conserved vector for this sstem was obtained as. The second conserved quantit for the radial free jet with swirl was obtained from that conserved vector. The conserved quantities for the wall jet with swirl cannot be obtained from the conserved vectors for the sstem of equations for the velocit components. For the sstem of two partial differential equations for the stream function formulation one local and one non-local conserved vector were obtained. The local conserved vector gave one conserved quantit for the radial free jet with swirl and the non-local conserved vector was used to establish one conserved quantit for the radial wall jet with swirl. Also, two conserved vectors for the stream function for-

20 R. Naz, I. Naeem and F. M. Mahomed 761 mulation were obtained as which were used to derive the second conserved quantities for the radial free and wall jets with swirl. It is of interest to notice that the non-local conserved vector for the sstem of two partial differential equations for the stream function formulation cannot be obtained b the multiplier approach. The reason is that the multiplier approach onl gives multipliers for the local conserved vectors. It ma be possible that the non-local conservation law which does not arise from local multipliers can be derived through a potential sstem associated with the sstem (1)-(3) or the sstem (8)-(9). 5. REFERENCES [1] E. J. Watson, The radial spread of a liquid jet over a horizontal plane, J. Fluid Mech, 2, , [2] W. H. Schwarz, The radial free jet, Chem. Eng, Sci, 18, , [3] M. B. Glauert, The wall jet, J. Fluid Mech,1, , [4] N. Rile, Asmptotic epansions in radial jets, J. Math. Phs, 41, , [5] N. Rile, Radial jets with swirl, Part I. Incompressible Flow, Quart. J. Mech. and Appl. Math, 15, , [6] M. O Nan, W. H. Schwarz, The swirling radial free jet, Appl. Sci. Res. 15, , [7] R. Naz, D. P. Mason and F. M. Mahomed, Conservation laws and conserved quantities for laminar two-dimensional and radial jets, Nonlinear Analsis: Real World Applications, 1, , 29. [8] H. Steudel, Uber die Zuordnung Zwischen Invarianzeigenschaften und Erhaltungssatzen, Zeit Naturforsch, 17, , [9] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, [1] R. Naz, F. M. Mahomed, D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl. Math.Comp, 25, , 28. [11] G. K. Batchelor, An Introduction to Fluid Dnamics, Cambridge Universit Press, 75-79, [12] R. P. Gillespie, Integration, Oliver and Bod, Edinburgh, , 1967.