Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 615612 | https://doi.org/10.1155/2011/615612

R. Naz, "Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice", Mathematical Problems in Engineering, vol. 2011, Article ID 615612, 12 pages, 2011. https://doi.org/10.1155/2011/615612

Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

Academic Editor: Furong Gao
Received28 Jan 2011
Accepted22 Jun 2011
Published18 Aug 2011

Abstract

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

1. Introduction

The governing equations for two-dimensional jet flows are expressed either as the system of two PDEs for the velocity components or by a single nonlinear third-order PDE for the stream function. In [1, 2] the similarity solution and in [3] the group-invariant solution were constructed for the nonlinear third-order PDE for the stream function for two-dimensional free jet with infinite fluid velocity at the orifice. The group-invariant solution for system of equations for the velocity components for the same problem was constructed by Naz et al. [4]. Glauert [5] derived the similarity solution for radial and two-dimensional wall jets having infinite fluid velocity at the orifice.

The general form of similarity solution for the flows having finite velocity at the orifice was suggested by Watson [6], and the similarity solutions for system of equations for velocity components for the radial and two-dimensional liquid jets were derived. The similarity solution for radial and two-dimensional wall jets having finite velocity at orifice was studied by Riley [7], and so our solution has some significance even near axis. The subject of this paper is to find the group-invariant solution for the nonlinear third-order PDE for stream function governing flow in two-dimensional (free, wall, liquid) jets having finite velocity at the orifice which is not considered yet.

The detailed outline of this paper is as follows: In Section 2 the group-invariant solution for two-dimensional free jet is derived. The symmetry associated with the conserved vector which is used to establish the conserved quantity for each jet generates the group-invariant solution for the nonlinear third-order PDE for the stream function. The group-invariant solution for two-dimensional wall and liquid jets is studied in Sections 3 and 4. In Section 5 the comparison between the results for two-dimensional jets, having finite and infinite fluid velocity at orifice, is constructed. The general form of group-invariant solution for two-dimensional free, wall, and liquid jets is given explicitly in Section 6. Finally the conclusions are summarized in Section 7.

2. Group-invariant Solution for Two-Dimensional Free Jet

The flow in two-dimensional free jet is governed by nonlinear third-order PDE for stream function๐œ“๐‘ฆ๐œ“๐‘ฅ๐‘ฆโˆ’๐œ“๐‘ฅ๐œ“๐‘ฆ๐‘ฆโˆ’๐œˆ๐œ“๐‘ฆ๐‘ฆ๐‘ฆ=0,(2.1) for an incompressible fluid. The relation between stream function and velocity components is๐‘ข=๐œ“๐‘ฆ,๐‘ฃ=โˆ’๐œ“๐‘ฅ.(2.2) The Lie point symmetry generator of (2.1) derived by Mason [3] is๐‘๐‘‹=๎€บ๎€ท1+๐‘3๎€ธ๐‘ฅ+๐‘2๎€ป๐œ•+๎€บ๐‘๐œ•๐‘ฅ1๎€ป๐œ•๐‘ฆ+๐‘˜(๐‘ฅ)+๎€บ๐‘๐œ•๐‘ฆ3๐œ“+๐‘4๎€ป๐œ•.๐œ•๐œ“(2.3)

The boundary conditions and the conserved quantity for two-dimensional free jet in terms of stream function are (see [1โ€“3])๐‘ฆ=0:๐œ“๐‘ฅ=0,๐œ“๐‘ฆ๐‘ฆ=0,(2.4)๐‘ฆ=ยฑโˆž:๐œ“๐‘ฆ=0,๐œ“๐‘ฆ๐‘ฆ๎€œ=0,(2.5)๐ฝ=2๐œŒโˆž0๐œ“2๐‘ฆ๐‘‘๐‘ฆ.(2.6) The conserved vector๐‘‡1=๐œ“2๐‘ฆ,๐‘‡2=โˆ’๐œ“๐‘ฅ๐œ“๐‘ฆโˆ’๐œˆ๐œ“๐‘ฆ๐‘ฆ(2.7) gave the conserved quantity (2.6) for two-dimensional free jet (see [8]). The symmetry associated with the conserved vector which is used to establish the conserved quantity for each jet generates the group-invariant solution for nonlinear third-order PDE [4, 9].

The symmetries associated with a known conserved vector can be determined by [10]๐‘‹[1]๎€ท๐‘‡1๎€ธ+๐‘‡1๐ท๐‘ฆ๎€ท๐œ‰2๎€ธโˆ’๐‘‡2๐ท๐‘ฆ๎€ท๐œ‰1๎€ธ๐‘‹=0,(2.8)[1]๎€ท๐‘‡2๎€ธ+๐‘‡2๐ท๐‘ฅ๎€ท๐œ‰1๎€ธโˆ’๐‘‡1๐ท๐‘ฅ๎€ท๐œ‰2๎€ธ=0.(2.9) Equations (2.8) and (2.9) yield๐‘‡1๎‚ƒ๐‘3โˆ’12๐‘1๎‚„=0,๐‘‡2๎‚ƒ๐‘3โˆ’12๐‘1๎‚„=0,(2.10) and thus for conserved vector (2.7), ๐‘3=(1/2)๐‘1. The Lie point symmetry generator associated with conserved vector (2.7) is๎‚ƒ3๐‘‹=2๐‘1๐‘ฅ+๐‘2๎‚„๐œ•+๎€บ๐‘๐œ•๐‘ฅ1๎€ป๐œ•๐‘ฆ+๐‘˜(๐‘ฅ)+๎‚ƒ1๐œ•๐‘ฆ2๐‘1๐œ“+๐‘4๎‚„๐œ•.๐œ•๐œ“(2.11)

Now, ๐œ“=๐œ™(๐‘ฅ,๐‘ฆ) is a group-invariant solution of (2.1) if๐‘‹(๐œ“โˆ’๐œ™(๐‘ฅ,๐‘ฆ))โˆฃ๐œ“=๐œ™=0,(2.12) which yields๎‚ต๐œ“=๐‘ฅ+2๐‘23๐‘1๎‚ถ1/3๐‘”(๐œ‰)โˆ’2๐‘4๐‘1,๐‘ฆ(2.13)๐œ‰=๎€ท๐‘ฅ+2๐‘2/3๐‘1๎€ธ2/3โˆ’๐พ(๐‘ฅ),(2.14) where2๐พ(๐‘ฅ)=3๐‘1๎€œ๐‘ฅ๐‘˜(๐‘ฅ)๎€ท๐‘ฅ+2๐‘2/3๐‘1๎€ธ5/3๐‘‘๐‘ฅ.(2.15) The conserved quantity (2.6) is independent of ๐‘ฅ provided ๐พ(๐‘ฅ)=0 which yields ๐‘˜(๐‘ฅ)=0. Since the stream function is determined up to an arbitrary constant, ๐‘4 can be chosen to be zero. The insertion of (2.13) into (2.1) results in a nonlinear third-order ordinary differential equation (ODE) for ๐‘”(๐œ‰):๐‘‘3๐œˆ3๐‘”๐‘‘๐œ‰3๐‘‘+๐‘”2๐‘”๐‘‘๐œ‰2+๎‚ต๐‘‘๐‘”๎‚ถ๐‘‘๐œ‰2=0.(2.16)

Equation (2.16) can be transformed to๐‘“๎…ž๎…ž๎…ž+๐‘“๐‘“๎…ž๎…ž+๐‘“โ€ฒ2=0,(2.17) with๐œ‚=๐ด๐œ‰3๐œˆ,๐ด๐‘“=๐‘”,(2.18) where ๐ด is arbitrary constant, and prime denotes differentiation with respect to ๐œ‚. The boundary conditions and conserved quantity (2.4)โ€“(2.6), in terms of ๐‘“(๐œ‚), take the following form:๐‘“(0)=0,๐‘“๎…ž๎…ž(0)=0,๐‘“๎…ž(ยฑโˆž)=0,๐‘“๎…ž๎…ž(ยฑโˆž)=0,(2.19)๐ฝ=2๐ด3๐œŒ๎€œ3๐œˆโˆž0๐‘“โ€ฒ2๐‘‘๐œ‚.(2.20)

The solution of (2.17) subject to (2.19) and condition ๐‘“(โˆž)=1 is (see [2, 11, 12])๎‚€๐œ‚๐‘“(๐œ‚)=tanh2๎‚,(2.21) and value of ๐ด in terms of ๐ฝ is๎‚ต๐ด=9๐œˆ๐ฝ๎‚ถ2๐œŒ1/3.(2.22)

The final form of group-invariant solution is๎‚ธ๐œ“=9๐œˆ๐ฝ๎‚ต2๐œŒ๐‘ฅ+2๐‘23๐‘1๎‚ถ๎‚น1/3๎ƒฌ๐‘“(๐œ‚),๐‘ข(๐‘ฅ,๐‘ฆ)=3๐ฝ24๐œŒ2๐œˆ๎€ท๐‘ฅ+2๐‘2/3๐‘1๎€ธ๎ƒญ1/3๐‘“๎…ž๎ƒฌ๐ฝ(๐œ‚),๐œ‚=6๐œŒ๐œˆ2๎€ท๐‘ฅ+2๐‘2/3๐‘1๎€ธ2๎ƒญ1/3๎‚ต3๐‘ฆ,๐‘‹=2๐‘๐‘ฅ+2๐‘1๎‚ถ๐œ•๐œ•๐œ•๐‘ฅ+๐‘ฆ+1๐œ•๐‘ฆ2๐œ“๐œ•,๐œ•๐œ“(2.23) is the symmetry that generated the group-invariant solution. Now๎ƒฌ๐‘ข(๐‘ฅ,0)=3๐ฝ232๐œŒ2๐œˆ๎€ท๐‘ฅ+2๐‘2/3๐‘1๎€ธ๎ƒญ1/3(2.24) is finite at ๐‘ฅ=0 and so our solution may have some significance even near the axis. By taking ๐‘2=0, the results [1โ€“4] for infinite velocity at orifice can be rediscovered.

3. Group-invariant Solution for Two-Dimensional Wall Jet

The flow in two-dimensional wall jet is also governed by (2.1). The boundary conditions for the two-dimensional wall jet are [5]๐‘ฆ=0:๐œ“๐‘ฅ=0,๐œ“๐‘ฆ=0,(3.1)๐‘ฆ=โˆž:๐œ“๐‘ฆ=0,(3.2) and the conserved quantity is๎€œ๐น=โˆž0๐œ“๐‘ฆ๎‚ต๎€œโˆž๐‘ฆ๐œ“2๐‘ฆโˆ—๐‘‘๐‘ฆโˆ—๎‚ถ๐‘‘๐‘ฆ.(3.3) The conserved vector๐‘‡1=๐œ“๐œ“2๐‘ฆ,๐‘‡2=โˆ’๐œ“๐œ“๐‘ฅ๐œ“๐‘ฆ+๐œˆ2๐œ“2๐‘ฆโˆ’๐œˆ๐œ“๐œ“๐‘ฆ๐‘ฆ(3.4) gave the conserved quantity for two-dimensional wall jet [8], and the symmetry associated with this conserved vector is๎‚ƒ4๐‘‹=3๐‘1๐‘ฅ+๐‘2๎‚„๐œ•+๎€บ๐‘๐œ•๐‘ฅ1๎€ป๐œ•๐‘ฆ+๐‘˜(๐‘ฅ)+1๐œ•๐‘ฆ3๐‘1๐œ“๐œ•.๐œ•๐œ“(3.5)

The group-invariant solution of (2.1) for two-dimensional wall jet case is๎‚ต๐œ“=๐‘ฅ+3๐‘24๐‘1๎‚ถ1/4๐‘ฆ๐‘”(๐œ‰),(3.6)๐œ‰=๎€ท๐‘ฅ+3๐‘2/4๐‘1๎€ธ3/4โˆ’๐พ(๐‘ฅ),(3.7) where3๐พ(๐‘ฅ)=4๐‘1๎€œ๐‘ฅ๐‘˜(๐‘ฅ)๎€ท๐‘ฅ+3๐‘2/4๐‘1๎€ธ7/4๐‘‘๐‘ฅ.(3.8) The conserved quantity (3.3) is independent of ๐‘ฅ provided ๐พ(๐‘ฅ)=0 which yields ๐‘˜(๐‘ฅ)=0. The substitution of (3.6) into (2.1) gives rise to a nonlinear third-order ODE for ๐‘”(๐œ‰):๐œˆ๐‘‘3๐‘”๐‘‘๐œ‰3+14๐‘”๐‘‘2๐‘”๐‘‘๐œ‰2+12๎‚ต๐‘‘๐‘”๎‚ถ๐‘‘๐œ‰2=0.(3.9) Define ๐œ‚=(๐ด/4๐œˆ)๐œ‰ and ๐‘”=๐ด๐‘“ Equation (3.9) transforms to๐‘“๎…ž๎…ž๎…ž+๐‘“๐‘“๎…ž๎…ž+2๐‘“โ€ฒ2=0.(3.10) Boundary conditions (3.1) and (3.2) and conserved quantity (3.3) take the following form:๐‘“(0)=0,๐‘“๎…ž(0)=0,๐‘“๎…ž๐ด(โˆž)=0,๐น=4๎€œ4๐œˆโˆž0๐‘“๎…ž๎‚ต๎€œโˆž๐œ‚๐‘“๎…ž2๐‘‘๐œ‚โˆ—๎‚ถ๐‘‘๐œ‚(3.11) Glauert [5] selected a solution of (3.10) with ๐‘“(โˆž)=1, and after integrating (3.10) twice, the following equation was obtained:๐‘‘โ„Ž=1๐‘‘๐œ‚3๎€ท1โˆ’โ„Ž3๎€ธ,whereโ„Ž2=๐‘“,0โ‰คโ„Žโ‰ค1.(3.12) Equation (3.12) yieldsโˆš๐œ‚=log1+โ„Ž+โ„Ž2+โˆš1โˆ’โ„Ž3tanโˆ’1โˆš3โ„Ž.2+โ„Ž(3.13) The conserved quantity gave the unknown constant ๐ด as๐ด=(40๐œˆ๐น)1/4.(3.14)

Thus we finally obtain๎‚ธ๎‚ต๐œ“=40๐น๐œˆ๐‘ฅ+3๐‘24๐‘1๎‚ถ๎‚น1/4๎ƒฌ๐‘“(๐œ‚),๐‘ข(๐‘ฅ,๐‘ฆ)=5๐น๎€ท2๐œˆ๐‘ฅ+3๐‘2/4๐‘1๎€ธ๎ƒญ1/2๐‘“๎…ž๎ƒฌ(๐œ‚),๐œ‚=5๐น32๐œˆ3๎€ท๐‘ฅ+3๐‘2/4๐‘1๎€ธ3๎ƒญ1/4๎‚ต4๐‘ฆ,๐‘‹=3๐‘๐‘ฅ+2๐‘1๎‚ถ๐œ•๐œ•๐œ•๐‘ฅ+๐‘ฆ+1๐œ•๐‘ฆ3๐œ“๐œ•.๐œ•๐œ“(3.15) The results obtained here for 3๐‘2/4๐‘1=๐‘™ agree with Riley [7], and ๐‘™ can be determined from [13]. By taking ๐‘2=0, the results for infinite velocity at orifice obtained by Glauert [5] can be rediscovered.

4. Group-Invariant Solution for Two-Dimensional Liquid Jet

The governing equation for two-dimensional liquid jet in terms of stream function is (2.1). The boundary conditions and conserved quantity for two-dimensional liquid jet are [6]๐‘ฆ=0:๐œ“๐‘ฅ=0,๐œ“๐‘ฆ=0,(4.1)๐‘ฆ=๐œ™(๐‘ฅ):๐œ“๐‘ฆ๐‘ฆ๎€œ=0,(4.2)๐‘€=0๐œ™(๐‘ฅ)๐œ“๐‘ฆ๐‘‘๐‘ฆ.(4.3) The conserved vector๐‘‡1=๐œ“๐‘ฆ,๐‘‡2=โˆ’๐œ“๐‘ฅ(4.4) gave conserved quantity for two-dimensional liquid jet [8].

Equations (2.8) and (2.9) yield the following Lie point symmetry generator associated with the conserved vector (4.4):๎€บ๐‘๐‘‹=1๐‘ฅ+๐‘2๎€ป๐œ•+๎€บ๐‘๐œ•๐‘ฅ1๎€ป๐œ•๐‘ฆ+๐‘˜(๐‘ฅ)๐œ•๐‘ฆ+๐‘4๐œ•.๐œ•๐œ“(4.5) The group-invariant solution for two-dimensional liquid jet is๎‚ต๐‘๐œ“=๐‘”(๐œ‰)+ln๐‘ฅ+2๐‘1๎‚ถ๐‘4/๐‘1๐‘ฆ,๐œ‰=๐‘ฅ+๐‘2/๐‘1โˆ’๐พ(๐‘ฅ),(4.6) where1๐พ(๐‘ฅ)=๐‘1๎€œ๐‘ฅ๐‘˜(๐‘ฅ)๎€ท๐‘ฅ+๐‘2/๐‘1๎€ธ2๐‘‘๐‘ฅ.(4.7) The conserved quantity is independent of ๐‘ฅ only if ๐พ(๐‘ฅ)=0 which gives ๐‘˜(๐‘ฅ)=0. The stream function contains an additive constant so we may choose ๐‘4=0 without loss of generality.

The substitution of (4.6), with ๐‘˜(๐‘ฅ)=0, ๐‘4=0, into (2.1) yields a nonlinear third-order ordinary differential equation for ๐‘”(๐œ‰):๐œˆ๐‘‘3๐‘”๐‘‘๐œ‰3+๎‚ต๐‘‘๐‘”๎‚ถ๐‘‘๐œ‰2=0.(4.8) Equation (4.8) takes the following form:๐‘“๎…ž๎…ž๎…ž+3๐‘“โ€ฒ2=0,(4.9) where ๐œ‚=๐ด/3๐œˆ๐œ‰ and ๐‘”=๐ด๐‘“. Boundary conditions (4.1) and (4.2) are๐‘“(0)=0,๐‘“๎…ž(0)=0,๐‘“๎…ž๎…ž(1)=0,(4.10) where the free surface is chosen to be ๐œ‚=1. The conserved quantity (4.3) becomes๎€œ๐‘€=10๐ด๐‘“โ€ฒ๐‘‘๐œ‚.(4.11)

Equation (4.9) yields (see [11, 12, 14])๐‘‘๐‘ก=๎€บ2๎€ท๐‘‘๐œ‚1โˆ’๐‘ก3๎€ธ๎€ป1/2,๐‘ก=๐‘“โ€ฒ.(4.12) The final form of solution of (4.9) in parametric form is (see [14])2๐œ‚=3โˆš2๎‚ƒ2๐น1๎‚ƒ12,23,32๎‚„โˆ’๎€ท,11โˆ’๐‘ก3๎€ธ1/2ร—2๐น1๎‚ƒ12,23,32,1โˆ’๐‘ก3,๎‚„๎‚„(4.13) where 2๐น1 is the hypergeometric function of first kind. We can tabulate the values of ๐œ‚ for given values of parameter ๐‘ก=๐‘“โ€ฒ from (4.13), and conserved quantity ๐‘€ yields the constant ๐ด.

Thus finally we obtain3โˆš๐œ“(๐‘ฅ,๐‘ฆ)=3๐‘€๐œ‹๐‘“(๐œ‚),๐‘ข(๐‘ฅ,๐‘ฆ)=9๐‘€2๐œˆ๐œ‹2๎€ท๐‘ฅ+๐‘2/๐‘1๎€ธ๐‘“๎…žโˆš(๐œ‚),๐œ‚=3๐‘€๎€ท๐œˆ๐œ‹๐‘ฅ+๐‘2/๐‘1๎€ธ๎‚ต๐‘๐‘ฆ,๐‘‹=๐‘ฅ+2๐‘1๎‚ถ๐œ•๐œ•๐œ•๐‘ฅ+๐‘ฆ.๐œ•๐‘ฆ(4.14) Now ๐‘ข(๐‘ฅ,0) is finite at ๐‘ฅ=0. The results obtained here for ๐‘2/๐‘1=๐‘™ agree with those concluded by Watson [6] by the similarity solution method, and the procedure to obtain ๐‘™ is discussed there.

5. Comparison between Two-Dimensional Jets with Finite Velocity at Orifice and Infinite Velocity at Orifice

The comparison between two-dimensional jets with finite fluid velocity at orifice and infinite velocity at orifice is constructed in Table 1. Table 1 shows that by formally taking ๐‘2=0, the stream function ๐œ“, variable ๐œ‚, and symmetry that generates group-invariant solution for infinite velocity case can be deduced from those of the finite velocity case.


Finite velocity at orificeInfinite velocity at orifice

2-D free jet ๎‚ธ ๐œ“ = 9 ๐œˆ ๐ฝ ๎‚ต 2 ๐œŒ ๐‘ฅ + 2 ๐‘ 2 3 ๐‘ 1 ๎‚ถ ๎‚น 1 / 3 ๐‘“ ( ๐œ‚ ) ๎‚ธ ๐œ“ = 9 ๐œˆ ๐ฝ ๐‘ฅ ๎‚น 2 ๐œŒ 1 / 3 ๐‘“ ( ๐œ‚ )
๎‚ธ ๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 3 ๐ฝ 2 4 ๐œŒ 2 ๐œˆ ( ๐‘ฅ + 2 ๐‘ 2 / 3 ๐‘ 1 ) ๎‚น 1 / 3 ๐‘“ โ€ฒ ( ๐œ‚ ) ๎‚ธ ๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 3 ๐ฝ 2 4 ๐œŒ 2 ๐œˆ ( ๐‘ฅ + 2 ๐‘ 2 / 3 ๐‘ 1 ) ๎‚น 1 / 3 ๐‘“ ๎…ž ( ๐œ‚ )
๎ƒฌ ๐ฝ ๐œ‚ = 6 ๐œŒ ๐œˆ 2 ( ๐‘ฅ + 2 ๐‘ 2 / 3 ๐‘ 1 ) 2 ๎ƒญ 1 / 3 ๐‘ฆ ๎‚ธ ๐ฝ ๐œ‚ = 6 ๐œŒ ๐œˆ 2 ๎‚น 1 / 3 ๐‘ฆ ๐‘ฅ 2 / 3
๐‘“ ๎…ž ๎…ž ๎…ž + ๐‘“ ๐‘“ ๎…ž ๎…ž + ๐‘“ โ€ฒ 2 = 0 ๐‘“ ๎…ž ๎…ž ๎…ž + ๐‘“ ๐‘“ ๎…ž ๎…ž + ๐‘“ โ€ฒ 2 = 0
๎‚ต 3 ๐‘‹ = 2 ๐‘ ๐‘ฅ + 2 ๐‘ 1 ๎‚ถ ๐œ• ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ + 1 ๐œ• ๐‘ฆ 2 ๐œ“ ๐œ• ๐œ• ๐œ“ 3 ๐‘‹ = 2 ๐‘ฅ ๐œ• ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ + 1 ๐œ• ๐‘ฆ 2 ๐œ“ ๐œ• ๐œ• ๐œ“
2-D wall jet ๎‚ธ ๎‚ต ๐œ“ = 4 0 ๐น ๐œˆ ๐‘ฅ + 3 ๐‘ 2 4 ๐‘ 1 ๎‚ถ ๎‚น 1 / 4 ๐‘“ ( ๐œ‚ ) ๐œ“ = [ 4 0 ๐น ๐œˆ ๐‘ฅ ] 1 / 4 ๐‘“ ( ๐œ‚ )
๎ƒฌ ๐œ‚ = 5 ๐น 3 2 ๐œˆ 3 ( ๐‘ฅ + 3 ๐‘ 2 / 4 ๐‘ 1 ) 3 ๎ƒญ 1 / 4 ๐‘ฆ ๎‚ƒ ๐œ‚ = 5 ๐น 3 2 ๐œˆ 3 ๎‚„ 1 / 4 ๐‘ฆ ๐‘ฅ 3 / 4
๎‚ธ ๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 5 ๐น 2 ๐œˆ ( ๐‘ฅ + 3 ๐‘ 2 / 4 ๐‘ 1 ) ๎‚น 1 / 2 ๐‘“ ๎…ž ( ๐œ‚ ) ๎‚ƒ ๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 5 ๐น ๎‚„ 2 ๐œˆ ๐‘ฅ 1 / 2 ๐‘“ ๎…ž ( ๐œ‚ )
๐‘“ ๎…ž ๎…ž ๎…ž + ๐‘“ ๐‘“ ๎…ž ๎…ž + 2 ๐‘“ โ€ฒ 2 = 0 ๐‘“ ๎…ž ๎…ž ๎…ž + ๐‘“ ๐‘“ ๎…ž ๎…ž + 2 ๐‘“ โ€ฒ 2 = 0
๎‚ต 4 ๐‘‹ = 3 ๐‘ ๐‘ฅ + 2 ๐‘ 1 ๎‚ถ ๐œ• ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ + 1 ๐œ• ๐‘ฆ 3 ๐œ“ ๐œ• ๐œ• ๐œ“ 4 ๐‘‹ = 3 ๐‘ฅ ๐œ• ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ + 1 ๐œ• ๐‘ฆ 3 ๐œ“ ๐œ• ๐œ• ๐œ“
2-D liquid jet 3 โˆš ๐œ“ ( ๐‘ฅ , ๐‘ฆ ) = 3 ๐‘€ ๐œ‹ ๐‘“ ( ๐œ‚ ) 3 โˆš ๐œ“ ( ๐‘ฅ , ๐‘ฆ ) = 3 ๐‘€ ๐œ‹ ๐‘“ ( ๐œ‚ )
โˆš ๐œ‚ = 3 ๐‘€ ๐œˆ ๐œ‹ ( ๐‘ฅ + ๐‘ 2 / ๐‘ 1 ) ๐‘ฆ โˆš ๐œ‚ = 3 ๐‘€ ๐‘ฆ ๐œˆ ๐œ‹ ๐‘ฅ
๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 9 ๐‘€ 2 ๐œˆ ๐œ‹ 2 ( ๐‘ฅ + ๐‘ 2 / ๐‘ 1 ) ๐‘“ ๎…ž ( ๐œ‚ ) ๐‘ข ( ๐‘ฅ , ๐‘ฆ ) = 9 ๐‘€ 2 ๐œˆ ๐œ‹ 2 ๐‘ฅ ๐‘“ ๎…ž ( ๐œ‚ )
๐‘“ ๎…ž ๎…ž ๎…ž + 3 ๐‘“ โ€ฒ 2 = 0 ๐‘“ ๎…ž ๎…ž ๎…ž + 3 ๐‘“ โ€ฒ 2 = 0
๎‚ต ๐‘ ๐‘‹ = ๐‘ฅ + 2 ๐‘ 1 ๎‚ถ ๐œ• ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ ๐œ• ๐‘ฆ ๐œ• ๐‘‹ = ๐‘ฅ ๐œ• ๐œ• ๐‘ฅ + ๐‘ฆ ๐œ• ๐‘ฆ

6. General Form of Group-invariant Solutions for Two-Dimensional Jets

The flow in two-dimensional free, wall, and liquid jets is governed by nonlinear third-order PDE (2.1). The symmetry generator (2.3) associated with the conserved vector that gives conserved quantity for jet flow gives the following conditions on constant: ๐‘3=๎‚€1๐›ผ๎‚๐‘โˆ’11,(6.1) where โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ2๐›ผ=33fortwo-dimensionalfreejet,4fortwo-dimensionalwalljet,1fortwo-dimensionalliquidjet.(6.2) We choose ๐‘4=0 because the stream function is determined up to an arbitrary constant. The expression for group-invariant solutions for two-dimensional jet flows is๎‚ต๐‘๐œ“=๐‘ฅ+๐›ผ2๐‘1๎‚ถ1โˆ’๐›ผ๐‘ฆ๐‘”(๐œ‰),๐œ‰=๎€ท๎€ท๐‘๐‘ฅ+๐›ผ2/๐‘1๎€ธ๎€ธ๐›ผโˆ’๐พ(๐‘ฅ),(6.3) where๐›ผ๐พ(๐‘ฅ)=๐‘1๎€œ๐‘ฅ๐‘˜(๐‘ฅ)๎€ท๎€ท๐‘๐‘ฅ+๐›ผ2/๐‘1๎€ธ๎€ธ๐›ผ+1๐‘‘๐‘ฅ.(6.4) The condition where conserved quantity is independent of ๐‘ฅ yields ๐‘˜(๐‘ฅ)=0 in each of free, wall, and liquid jets. Using (6.3)โ€“(6.4), (2.1) yields๐œˆ๐‘‘3๐‘”๐‘‘๐œ‰3๐‘‘+(1โˆ’๐›ผ)๐‘”2๐‘”๐‘‘๐œ‰2๎‚ต+(2๐›ผโˆ’1)๐‘‘๐‘”๎‚ถ๐‘‘๐œ‰2=0.(6.5) Define the transformationsโŽงโŽชโŽจโŽชโŽฉ๐ด๐œ‚=(1โˆ’๐›ผ)๐œˆ๐ด๐œ‰fortwo-dimensionalfreeandwalljets,3๐œˆ๐œ‰fortwo-dimensionalliquidjet,๐‘”=๐ด๐‘“,(6.6) where ๐ด is a constant. The final form of a group-invariant solution is๎‚ต๐‘๐œ“=๐‘ฅ+๐›ผ2๐‘1๎‚ถ1โˆ’๐›ผโŽงโŽชโŽจโŽชโŽฉ๐ด๐‘“(๐œ‚),๐œ‚=(1โˆ’๐›ผ)๐ด๐‘ฆ๐œˆ๎€ท๎€ท๐‘๐‘ฅ+๐›ผ2/๐‘1๎€ธ๎€ธ๐›ผfortwo-dimensionalfreeandwalljets,๐ด๐‘ฆ๎€ท3๐œˆ๐‘ฅ+๐‘2/๐‘1๎€ธfortwo-dimensionalliquidjet.(6.7) For two-dimensional free and wall jets, (6.5) yields๐‘“๎…ž๎…ž๎…ž+๐‘“๐‘“๎…ž๎…ž+2๐›ผโˆ’11โˆ’๐›ผ๐‘“โ€ฒ2=0,(6.8) and for two-dimensional liquid jet, we have๐‘“๎…ž๎…ž๎…ž+3๐‘“โ€ฒ2=0.(6.9) The symmetry๎‚ต๐‘๐‘‹=๐‘ฅ+๐›ผ2๐‘1๎‚ถ๐œ•๐œ•๐œ•๐‘ฅ+๐›ผ๐‘ฆ๐œ•๐œ•๐‘ฆ+(1โˆ’๐›ผ)๐œ“๐œ•๐œ“(6.10) yielded the group-invariant solution.

7. Conclusions

The group-invariant solutions for two-dimensional free, wall, and liquid jets were derived for finite velocity at orifice. For two-dimensional free jet, a Lie point symmetry was associated with the conserved vector that generated the conserved quantity for two-dimensional free jet. This symmetry generated the group-invariant solution for the nonlinear third-order PDE for stream function subject to certain boundary conditions. The nonlinear third-order PDE was transformed to the nonlinear third-order ODE. Using certain transformations we deduced the same nonlinear third-order ODE as was obtained for two-dimensional free jet having infinite fluid velocity at orifice. The analogue analysis was done for the two-dimensional wall and liquid jets. A detailed comparison of results for finite and infinite velocity at orifice was constructed. The general form of group-invariant solution for the two-dimensional jets was derived.

Acknowledgment

R. Naz is thankful to The Lahore School of Economics for providing funding to complete this research work.

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Copyright ยฉ 2011 R. Naz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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