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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 615612, 12 pages
http://dx.doi.org/10.1155/2011/615612
Research Article

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

Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan

Received 28 January 2011; Accepted 22 June 2011

Academic Editor: FurongΒ Gao

Copyright Β© 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.

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 functionπœ“π‘¦πœ“π‘₯π‘¦βˆ’πœ“π‘₯πœ“π‘¦π‘¦βˆ’πœˆπœ“π‘¦π‘¦π‘¦=0,(2.1) for an incompressible fluid. The relation between stream function and velocity components is𝑒=πœ“π‘¦,𝑣=βˆ’πœ“π‘₯.(2.2) The Lie point symmetry generator of (2.1) derived by Mason [3] is𝑐𝑋=ξ€Ίξ€·1+𝑐3ξ€Έπ‘₯+𝑐2ξ€»πœ•+ξ€Ίπ‘πœ•π‘₯1ξ€»πœ•π‘¦+π‘˜(π‘₯)+ξ€Ίπ‘πœ•π‘¦3πœ“+𝑐4ξ€»πœ•.πœ•πœ“(2.3)

The boundary conditions and the conserved quantity for two-dimensional free jet in terms of stream function are (see [1–3])𝑦=0:πœ“π‘₯=0,πœ“π‘¦π‘¦=0,(2.4)𝑦=±∞:πœ“π‘¦=0,πœ“π‘¦π‘¦ξ€œ=0,(2.5)𝐽=2𝜌∞0πœ“2𝑦𝑑𝑦.(2.6) The conserved vector𝑇1=πœ“2𝑦,𝑇2=βˆ’πœ“π‘₯πœ“π‘¦βˆ’πœˆπœ“π‘¦π‘¦(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]𝑋[1]𝑇1ξ€Έ+𝑇1π·π‘¦ξ€·πœ‰2ξ€Έβˆ’π‘‡2π·π‘¦ξ€·πœ‰1𝑋=0,(2.8)[1]𝑇2ξ€Έ+𝑇2𝐷π‘₯ξ€·πœ‰1ξ€Έβˆ’π‘‡1𝐷π‘₯ξ€·πœ‰2ξ€Έ=0.(2.9) Equations (2.8) and (2.9) yield𝑇1𝑐3βˆ’12𝑐1ξ‚„=0,𝑇2𝑐3βˆ’12𝑐1ξ‚„=0,(2.10) and thus for conserved vector (2.7), 𝑐3=(1/2)𝑐1. The Lie point symmetry generator associated with conserved vector (2.7) is3𝑋=2𝑐1π‘₯+𝑐2ξ‚„πœ•+ξ€Ίπ‘πœ•π‘₯1ξ€»πœ•π‘¦+π‘˜(π‘₯)+1πœ•π‘¦2𝑐1πœ“+𝑐4ξ‚„πœ•.πœ•πœ“(2.11)

Now, πœ“=πœ™(π‘₯,𝑦) is a group-invariant solution of (2.1) if𝑋(πœ“βˆ’πœ™(π‘₯,𝑦))βˆ£πœ“=πœ™=0,(2.12) which yieldsξ‚΅πœ“=π‘₯+2𝑐23𝑐1ξ‚Ά1/3𝑔(πœ‰)βˆ’2𝑐4𝑐1,𝑦(2.13)πœ‰=ξ€·π‘₯+2𝑐2/3𝑐1ξ€Έ2/3βˆ’πΎ(π‘₯),(2.14) where2𝐾(π‘₯)=3𝑐1ξ€œπ‘₯π‘˜(π‘₯)ξ€·π‘₯+2𝑐2/3𝑐1ξ€Έ5/3𝑑π‘₯.(2.15) The conserved quantity (2.6) is independent of π‘₯ provided 𝐾(π‘₯)=0 which yields π‘˜(π‘₯)=0. Since the stream function is determined up to an arbitrary constant, 𝑐4 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 𝑔(πœ‰):𝑑3𝜈3π‘”π‘‘πœ‰3𝑑+𝑔2π‘”π‘‘πœ‰2+ξ‚΅π‘‘π‘”ξ‚Άπ‘‘πœ‰2=0.(2.16)

Equation (2.16) can be transformed toπ‘“ξ…žξ…žξ…ž+π‘“π‘“ξ…žξ…ž+𝑓′2=0,(2.17) withπœ‚=π΄πœ‰3𝜈,𝐴𝑓=𝑔,(2.18) where 𝐴 is arbitrary constant, and prime denotes differentiation with respect to πœ‚. The boundary conditions and conserved quantity (2.4)–(2.6), in terms of 𝑓(πœ‚), take the following form:𝑓(0)=0,π‘“ξ…žξ…ž(0)=0,π‘“ξ…ž(±∞)=0,π‘“ξ…žξ…ž(±∞)=0,(2.19)𝐽=2𝐴3πœŒξ€œ3𝜈∞0𝑓′2π‘‘πœ‚.(2.20)

The solution of (2.17) subject to (2.19) and condition 𝑓(∞)=1 is (see [2, 11, 12])ξ‚€πœ‚π‘“(πœ‚)=tanh2,(2.21) and value of 𝐴 in terms of 𝐽 is𝐴=9πœˆπ½ξ‚Ά2𝜌1/3.(2.22)

The final form of group-invariant solution isξ‚Έπœ“=9πœˆπ½ξ‚΅2𝜌π‘₯+2𝑐23𝑐1ξ‚Άξ‚Ή1/3𝑓(πœ‚),𝑒(π‘₯,𝑦)=3𝐽24𝜌2πœˆξ€·π‘₯+2𝑐2/3𝑐1ξ€Έξƒ­1/3π‘“ξ…žξƒ¬π½(πœ‚),πœ‚=6𝜌𝜈2ξ€·π‘₯+2𝑐2/3𝑐1ξ€Έ2ξƒ­1/3ξ‚΅3𝑦,𝑋=2𝑐π‘₯+2𝑐1ξ‚Άπœ•πœ•πœ•π‘₯+𝑦+1πœ•π‘¦2πœ“πœ•,πœ•πœ“(2.23) is the symmetry that generated the group-invariant solution. Now𝑒(π‘₯,0)=3𝐽232𝜌2πœˆξ€·π‘₯+2𝑐2/3𝑐1ξ€Έξƒ­1/3(2.24) is finite at π‘₯=0 and so our solution may have some significance even near the axis. By taking 𝑐2=0, the results [1–4] 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]𝑦=0:πœ“π‘₯=0,πœ“π‘¦=0,(3.1)𝑦=∞:πœ“π‘¦=0,(3.2) and the conserved quantity isξ€œπΉ=∞0πœ“π‘¦ξ‚΅ξ€œβˆžπ‘¦πœ“2π‘¦βˆ—π‘‘π‘¦βˆ—ξ‚Άπ‘‘π‘¦.(3.3) The conserved vector𝑇1=πœ“πœ“2𝑦,𝑇2=βˆ’πœ“πœ“π‘₯πœ“π‘¦+𝜈2πœ“2π‘¦βˆ’πœˆπœ“πœ“π‘¦π‘¦(3.4) gave the conserved quantity for two-dimensional wall jet [8], and the symmetry associated with this conserved vector is4𝑋=3𝑐1π‘₯+𝑐2ξ‚„πœ•+ξ€Ίπ‘πœ•π‘₯1ξ€»πœ•π‘¦+π‘˜(π‘₯)+1πœ•π‘¦3𝑐1πœ“πœ•.πœ•πœ“(3.5)

The group-invariant solution of (2.1) for two-dimensional wall jet case isξ‚΅πœ“=π‘₯+3𝑐24𝑐1ξ‚Ά1/4𝑦𝑔(πœ‰),(3.6)πœ‰=ξ€·π‘₯+3𝑐2/4𝑐1ξ€Έ3/4βˆ’πΎ(π‘₯),(3.7) where3𝐾(π‘₯)=4𝑐1ξ€œπ‘₯π‘˜(π‘₯)ξ€·π‘₯+3𝑐2/4𝑐1ξ€Έ7/4𝑑π‘₯.(3.8) The conserved quantity (3.3) is independent of π‘₯ provided 𝐾(π‘₯)=0 which yields π‘˜(π‘₯)=0. The substitution of (3.6) into (2.1) gives rise to a nonlinear third-order ODE for 𝑔(πœ‰):πœˆπ‘‘3π‘”π‘‘πœ‰3+14𝑔𝑑2π‘”π‘‘πœ‰2+12ξ‚΅π‘‘π‘”ξ‚Άπ‘‘πœ‰2=0.(3.9) Define πœ‚=(𝐴/4𝜈)πœ‰ and 𝑔=𝐴𝑓 Equation (3.9) transforms toπ‘“ξ…žξ…žξ…ž+π‘“π‘“ξ…žξ…ž+2𝑓′2=0.(3.10) Boundary conditions (3.1) and (3.2) and conserved quantity (3.3) take the following form:𝑓(0)=0,π‘“ξ…ž(0)=0,π‘“ξ…žπ΄(∞)=0,𝐹=4ξ€œ4𝜈∞0π‘“ξ…žξ‚΅ξ€œβˆžπœ‚π‘“ξ…ž2π‘‘πœ‚βˆ—ξ‚Άπ‘‘πœ‚(3.11) Glauert [5] selected a solution of (3.10) with 𝑓(∞)=1, and after integrating (3.10) twice, the following equation was obtained:π‘‘β„Ž=1π‘‘πœ‚3ξ€·1βˆ’β„Ž3ξ€Έ,whereβ„Ž2=𝑓,0β‰€β„Žβ‰€1.(3.12) Equation (3.12) yieldsβˆšπœ‚=log1+β„Ž+β„Ž2+√1βˆ’β„Ž3tanβˆ’1√3β„Ž.2+β„Ž(3.13) The conserved quantity gave the unknown constant 𝐴 as𝐴=(40𝜈𝐹)1/4.(3.14)

Thus we finally obtainξ‚Έξ‚΅πœ“=40𝐹𝜈π‘₯+3𝑐24𝑐1ξ‚Άξ‚Ή1/4𝑓(πœ‚),𝑒(π‘₯,𝑦)=5𝐹2𝜈π‘₯+3𝑐2/4𝑐1ξ€Έξƒ­1/2π‘“ξ…žξƒ¬(πœ‚),πœ‚=5𝐹32𝜈3ξ€·π‘₯+3𝑐2/4𝑐1ξ€Έ3ξƒ­1/4ξ‚΅4𝑦,𝑋=3𝑐π‘₯+2𝑐1ξ‚Άπœ•πœ•πœ•π‘₯+𝑦+1πœ•π‘¦3πœ“πœ•.πœ•πœ“(3.15) The results obtained here for 3𝑐2/4𝑐1=𝑙 agree with Riley [7], and 𝑙 can be determined from [13]. By taking 𝑐2=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]𝑦=0:πœ“π‘₯=0,πœ“π‘¦=0,(4.1)𝑦=πœ™(π‘₯):πœ“π‘¦π‘¦ξ€œ=0,(4.2)𝑀=0πœ™(π‘₯)πœ“π‘¦π‘‘π‘¦.(4.3) The conserved vector𝑇1=πœ“π‘¦,𝑇2=βˆ’πœ“π‘₯(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):𝑐𝑋=1π‘₯+𝑐2ξ€»πœ•+ξ€Ίπ‘πœ•π‘₯1ξ€»πœ•π‘¦+π‘˜(π‘₯)πœ•π‘¦+𝑐4πœ•.πœ•πœ“(4.5) The group-invariant solution for two-dimensional liquid jet isξ‚΅π‘πœ“=𝑔(πœ‰)+lnπ‘₯+2𝑐1𝑐4/𝑐1𝑦,πœ‰=π‘₯+𝑐2/𝑐1βˆ’πΎ(π‘₯),(4.6) where1𝐾(π‘₯)=𝑐1ξ€œπ‘₯π‘˜(π‘₯)ξ€·π‘₯+𝑐2/𝑐1ξ€Έ2𝑑π‘₯.(4.7) The conserved quantity is independent of π‘₯ only if 𝐾(π‘₯)=0 which gives π‘˜(π‘₯)=0. The stream function contains an additive constant so we may choose 𝑐4=0 without loss of generality.

The substitution of (4.6), with π‘˜(π‘₯)=0, 𝑐4=0, into (2.1) yields a nonlinear third-order ordinary differential equation for 𝑔(πœ‰):πœˆπ‘‘3π‘”π‘‘πœ‰3+ξ‚΅π‘‘π‘”ξ‚Άπ‘‘πœ‰2=0.(4.8) Equation (4.8) takes the following form:π‘“ξ…žξ…žξ…ž+3𝑓′2=0,(4.9) where πœ‚=𝐴/3πœˆπœ‰ and 𝑔=𝐴𝑓. Boundary conditions (4.1) and (4.2) are𝑓(0)=0,π‘“ξ…ž(0)=0,π‘“ξ…žξ…ž(1)=0,(4.10) where the free surface is chosen to be πœ‚=1. The conserved quantity (4.3) becomesξ€œπ‘€=10π΄π‘“β€²π‘‘πœ‚.(4.11)

Equation (4.9) yields (see [11, 12, 14])𝑑𝑑=ξ€Ί2ξ€·π‘‘πœ‚1βˆ’π‘‘3ξ€Έξ€»1/2,𝑑=𝑓′.(4.12) The final form of solution of (4.9) in parametric form is (see [14])2πœ‚=3√22𝐹112,23,32ξ‚„βˆ’ξ€·,11βˆ’π‘‘3ξ€Έ1/2Γ—2𝐹112,23,32,1βˆ’π‘‘3,ξ‚„ξ‚„(4.13) where 2𝐹1 is the hypergeometric function of first kind. We can tabulate the values of πœ‚ for given values of parameter 𝑑=𝑓′ from (4.13), and conserved quantity 𝑀 yields the constant 𝐴.

Thus finally we obtain3βˆšπœ“(π‘₯,𝑦)=3π‘€πœ‹π‘“(πœ‚),𝑒(π‘₯,𝑦)=9𝑀2πœˆπœ‹2ξ€·π‘₯+𝑐2/𝑐1ξ€Έπ‘“ξ…žβˆš(πœ‚),πœ‚=3π‘€ξ€·πœˆπœ‹π‘₯+𝑐2/𝑐1𝑐𝑦,𝑋=π‘₯+2𝑐1ξ‚Άπœ•πœ•πœ•π‘₯+𝑦.πœ•π‘¦(4.14) Now 𝑒(π‘₯,0) is finite at π‘₯=0. The results obtained here for 𝑐2/𝑐1=𝑙 agree with those concluded by Watson [6] by the similarity solution method, and the procedure to obtain 𝑙 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 𝑐2=0, the stream function πœ“, variable πœ‚, and symmetry that generates group-invariant solution for infinite velocity case can be deduced from those of the finite velocity case.

tab1
Table 1: Comparison between two-dimensional jets with finite velocity at orifice and infinite velocity at orifice.

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: 𝑐3=ξ‚€1π›Όξ‚π‘βˆ’11,(6.1) where ⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩2𝛼=33fortwo-dimensionalfreejet,4fortwo-dimensionalwalljet,1fortwo-dimensionalliquidjet.(6.2) We choose 𝑐4=0 because the stream function is determined up to an arbitrary constant. The expression for group-invariant solutions for two-dimensional jet flows isξ‚΅π‘πœ“=π‘₯+𝛼2𝑐1ξ‚Ά1βˆ’π›Όπ‘¦π‘”(πœ‰),πœ‰=𝑐π‘₯+𝛼2/𝑐1ξ€Έξ€Έπ›Όβˆ’πΎ(π‘₯),(6.3) where𝛼𝐾(π‘₯)=𝑐1ξ€œπ‘₯π‘˜(π‘₯)𝑐π‘₯+𝛼2/𝑐1𝛼+1𝑑π‘₯.(6.4) The condition where conserved quantity is independent of π‘₯ yields π‘˜(π‘₯)=0 in each of free, wall, and liquid jets. Using (6.3)–(6.4), (2.1) yieldsπœˆπ‘‘3π‘”π‘‘πœ‰3𝑑+(1βˆ’π›Ό)𝑔2π‘”π‘‘πœ‰2ξ‚΅+(2π›Όβˆ’1)π‘‘π‘”ξ‚Άπ‘‘πœ‰2=0.(6.5) Define the transformations⎧βŽͺ⎨βŽͺβŽ©π΄πœ‚=(1βˆ’π›Ό)πœˆπ΄πœ‰fortwo-dimensionalfreeandwalljets,3πœˆπœ‰fortwo-dimensionalliquidjet,𝑔=𝐴𝑓,(6.6) where 𝐴 is a constant. The final form of a group-invariant solution isξ‚΅π‘πœ“=π‘₯+𝛼2𝑐1ξ‚Ά1βˆ’π›ΌβŽ§βŽͺ⎨βŽͺβŽ©π΄π‘“(πœ‚),πœ‚=(1βˆ’π›Ό)π΄π‘¦πœˆξ€·ξ€·π‘π‘₯+𝛼2/𝑐1𝛼fortwo-dimensionalfreeandwalljets,𝐴𝑦3𝜈π‘₯+𝑐2/𝑐1ξ€Έfortwo-dimensionalliquidjet.(6.7) For two-dimensional free and wall jets, (6.5) yieldsπ‘“ξ…žξ…žξ…ž+π‘“π‘“ξ…žξ…ž+2π›Όβˆ’11βˆ’π›Όπ‘“β€²2=0,(6.8) and for two-dimensional liquid jet, we haveπ‘“ξ…žξ…žξ…ž+3𝑓′2=0.(6.9) The symmetry𝑐𝑋=π‘₯+𝛼2𝑐1ξ‚Άπœ•πœ•πœ•π‘₯+π›Όπ‘¦πœ•πœ•π‘¦+(1βˆ’π›Ό)πœ“πœ•π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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