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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 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: Furong Gao

Copyright © 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 function𝜓𝑦𝜓𝑥𝑦𝜓𝑥𝜓𝑦𝑦𝜈𝜓𝑦𝑦𝑦=0,(2.1) for an incompressible fluid. The relation between stream function and velocity components is𝑢=𝜓𝑦,𝑣=𝜓𝑥.(2.2) The Lie point symmetry generator of (2.1) derived by Mason [3] is𝑐𝑋=1+𝑐3𝑥+𝑐2𝜕+𝑐𝜕𝑥1𝜕𝑦+𝑘(𝑥)+𝑐𝜕𝑦3𝜓+𝑐4𝜕.𝜕𝜓(2.3)

The boundary conditions and the conserved quantity for two-dimensional free jet in terms of stream function are (see [13])𝑦=0:𝜓𝑥=0,𝜓𝑦𝑦=0,(2.4)𝑦=±:𝜓𝑦=0,𝜓𝑦𝑦=0,(2.5)𝐽=2𝜌0𝜓2𝑦𝑑𝑦.(2.6) The conserved vector𝑇1=𝜓2𝑦,𝑇2=𝜓𝑥𝜓𝑦𝜈𝜓𝑦𝑦(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]𝑇1+𝑇1𝐷𝑦𝜉2𝑇2𝐷𝑦𝜉1𝑋=0,(2.8)[1]𝑇2+𝑇2𝐷𝑥𝜉1𝑇1𝐷𝑥𝜉2=0.(2.9) Equations (2.8) and (2.9) yield𝑇1𝑐312𝑐1=0,𝑇2𝑐312𝑐1=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) is3𝑋=2𝑐1𝑥+𝑐2𝜕+𝑐𝜕𝑥1𝜕𝑦+𝑘(𝑥)+1𝜕𝑦2𝑐1𝜓+𝑐4𝜕.𝜕𝜓(2.11)

Now, 𝜓=𝜙(𝑥,𝑦) is a group-invariant solution of (2.1) if𝑋(𝜓𝜙(𝑥,𝑦))𝜓=𝜙=0,(2.12) which yields𝜓=𝑥+2𝑐23𝑐11/3𝑔(𝜉)2𝑐4𝑐1,𝑦(2.13)𝜉=𝑥+2𝑐2/3𝑐12/3𝐾(𝑥),(2.14) where2𝐾(𝑥)=3𝑐1𝑥𝑘(𝑥)𝑥+2𝑐2/3𝑐15/3𝑑𝑥.(2.15) The conserved quantity (2.6) is independent of 𝑥 provided 𝐾(𝑥)=0 which yields 𝑘(𝑥)=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(𝜉):𝑑3𝜈3𝑔𝑑𝜉3𝑑+𝑔2𝑔𝑑𝜉2+𝑑𝑔𝑑𝜉2=0.(2.16)

Equation (2.16) can be transformed to𝑓+𝑓𝑓+𝑓2=0,(2.17) with𝜂=𝐴𝜉3𝜈,𝐴𝑓=𝑔,(2.18) where 𝐴 is arbitrary constant, and prime denotes differentiation with respect to 𝜂. The boundary conditions and conserved quantity (2.4)–(2.6), in terms of 𝑓(𝜂), take the following form:𝑓(0)=0,𝑓(0)=0,𝑓(±)=0,𝑓(±)=0,(2.19)𝐽=2𝐴3𝜌3𝜈0𝑓2𝑑𝜂.(2.20)

The solution of (2.17) subject to (2.19) and condition 𝑓()=1 is (see [2, 11, 12])𝜂𝑓(𝜂)=tanh2,(2.21) and value of 𝐴 in terms of 𝐽 is𝐴=9𝜈𝐽2𝜌1/3.(2.22)

The final form of group-invariant solution is𝜓=9𝜈𝐽2𝜌𝑥+2𝑐23𝑐11/3𝑓(𝜂),𝑢(𝑥,𝑦)=3𝐽24𝜌2𝜈𝑥+2𝑐2/3𝑐11/3𝑓𝐽(𝜂),𝜂=6𝜌𝜈2𝑥+2𝑐2/3𝑐121/33𝑦,𝑋=2𝑐𝑥+2𝑐1𝜕𝜕𝜕𝑥+𝑦+1𝜕𝑦2𝜓𝜕,𝜕𝜓(2.23) is the symmetry that generated the group-invariant solution. Now𝑢(𝑥,0)=3𝐽232𝜌2𝜈𝑥+2𝑐2/3𝑐11/3(2.24) is finite at 𝑥=0 and so our solution may have some significance even near the axis. By taking 𝑐2=0, the results [14] 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:𝜓𝑥=0,𝜓𝑦=0,(3.1)𝑦=:𝜓𝑦=0,(3.2) and the conserved quantity is𝐹=0𝜓𝑦𝑦𝜓2𝑦𝑑𝑦𝑑𝑦.(3.3) The conserved vector𝑇1=𝜓𝜓2𝑦,𝑇2=𝜓𝜓𝑥𝜓𝑦+𝜈2𝜓2𝑦𝜈𝜓𝜓𝑦𝑦(3.4) gave the conserved quantity for two-dimensional wall jet [8], and the symmetry associated with this conserved vector is4𝑋=3𝑐1𝑥+𝑐2𝜕+𝑐𝜕𝑥1𝜕𝑦+𝑘(𝑥)+1𝜕𝑦3𝑐1𝜓𝜕.𝜕𝜓(3.5)

The group-invariant solution of (2.1) for two-dimensional wall jet case is𝜓=𝑥+3𝑐24𝑐11/4𝑦𝑔(𝜉),(3.6)𝜉=𝑥+3𝑐2/4𝑐13/4𝐾(𝑥),(3.7) where3𝐾(𝑥)=4𝑐1𝑥𝑘(𝑥)𝑥+3𝑐2/4𝑐17/4𝑑𝑥.(3.8) The conserved quantity (3.3) is independent of 𝑥 provided 𝐾(𝑥)=0 which yields 𝑘(𝑥)=0. The substitution of (3.6) into (2.1) gives rise to a nonlinear third-order ODE for 𝑔(𝜉):𝜈𝑑3𝑔𝑑𝜉3+14𝑔𝑑2𝑔𝑑𝜉2+12𝑑𝑔𝑑𝜉2=0.(3.9) Define 𝜂=(𝐴/4𝜈)𝜉 and 𝑔=𝐴𝑓 Equation (3.9) transforms to𝑓+𝑓𝑓+2𝑓2=0.(3.10) Boundary conditions (3.1) and (3.2) and conserved quantity (3.3) take the following form:𝑓(0)=0,𝑓(0)=0,𝑓𝐴()=0,𝐹=44𝜈0𝑓𝜂𝑓2𝑑𝜂𝑑𝜂(3.11) Glauert [5] selected a solution of (3.10) with 𝑓()=1, and after integrating (3.10) twice, the following equation was obtained:𝑑=1𝑑𝜂313,where2=𝑓,01.(3.12) Equation (3.12) yields𝜂=log1++2+13tan13.2+(3.13) The conserved quantity gave the unknown constant 𝐴 as𝐴=(40𝜈𝐹)1/4.(3.14)

Thus we finally obtain𝜓=40𝐹𝜈𝑥+3𝑐24𝑐11/4𝑓(𝜂),𝑢(𝑥,𝑦)=5𝐹2𝜈𝑥+3𝑐2/4𝑐11/2𝑓(𝜂),𝜂=5𝐹32𝜈3𝑥+3𝑐2/4𝑐131/44𝑦,𝑋=3𝑐𝑥+2𝑐1𝜕𝜕𝜕𝑥+𝑦+1𝜕𝑦3𝜓𝜕.𝜕𝜓(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:𝜓𝑥=0,𝜓𝑦=0,(4.1)𝑦=𝜙(𝑥):𝜓𝑦𝑦=0,(4.2)𝑀=0𝜙(𝑥)𝜓𝑦𝑑𝑦.(4.3) The conserved vector𝑇1=𝜓𝑦,𝑇2=𝜓𝑥(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=1𝑥+𝑐2𝜕+𝑐𝜕𝑥1𝜕𝑦+𝑘(𝑥)𝜕𝑦+𝑐4𝜕.𝜕𝜓(4.5) The group-invariant solution for two-dimensional liquid jet is𝑐𝜓=𝑔(𝜉)+ln𝑥+2𝑐1𝑐4/𝑐1𝑦,𝜉=𝑥+𝑐2/𝑐1𝐾(𝑥),(4.6) where1𝐾(𝑥)=𝑐1𝑥𝑘(𝑥)𝑥+𝑐2/𝑐12𝑑𝑥.(4.7) The conserved quantity is independent of 𝑥 only if 𝐾(𝑥)=0 which gives 𝑘(𝑥)=0. The stream function contains an additive constant so we may choose 𝑐4=0 without loss of generality.

The substitution of (4.6), with 𝑘(𝑥)=0, 𝑐4=0, into (2.1) yields a nonlinear third-order ordinary differential equation for 𝑔(𝜉):𝜈𝑑3𝑔𝑑𝜉3+𝑑𝑔𝑑𝜉2=0.(4.8) Equation (4.8) takes the following form:𝑓+3𝑓2=0,(4.9) where 𝜂=𝐴/3𝜈𝜉 and 𝑔=𝐴𝑓. Boundary conditions (4.1) and (4.2) are𝑓(0)=0,𝑓(0)=0,𝑓(1)=0,(4.10) where the free surface is chosen to be 𝜂=1. The conserved quantity (4.3) becomes𝑀=10𝐴𝑓𝑑𝜂.(4.11)

Equation (4.9) yields (see [11, 12, 14])𝑑𝑡=2𝑑𝜂1𝑡31/2,𝑡=𝑓.(4.12) The final form of solution of (4.9) in parametric form is (see [14])2𝜂=322𝐹112,23,32,11𝑡31/2×2𝐹112,23,32,1𝑡3,(4.13) where 2𝐹1 is the hypergeometric function of first kind. We can tabulate the values of 𝜂 for given values of parameter 𝑡=𝑓 from (4.13), and conserved quantity 𝑀 yields the constant 𝐴.

Thus finally we obtain3𝜓(𝑥,𝑦)=3𝑀𝜋𝑓(𝜂),𝑢(𝑥,𝑦)=9𝑀2𝜈𝜋2𝑥+𝑐2/𝑐1𝑓(𝜂),𝜂=3𝑀𝜈𝜋𝑥+𝑐2/𝑐1𝑐𝑦,𝑋=𝑥+2𝑐1𝜕𝜕𝜕𝑥+𝑦.𝜕𝑦(4.14) Now 𝑢(𝑥,0) is finite at 𝑥=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 𝜓, variable 𝜂, 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=1𝛼𝑐11,(6.1) where 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 is𝑐𝜓=𝑥+𝛼2𝑐11𝛼𝑦𝑔(𝜉),𝜉=𝑐𝑥+𝛼2/𝑐1𝛼𝐾(𝑥),(6.3) where𝛼𝐾(𝑥)=𝑐1𝑥𝑘(𝑥)𝑐𝑥+𝛼2/𝑐1𝛼+1𝑑𝑥.(6.4) The condition where conserved quantity is independent of 𝑥 yields 𝑘(𝑥)=0 in each of free, wall, and liquid jets. Using (6.3)–(6.4), (2.1) yields𝜈𝑑3𝑔𝑑𝜉3𝑑+(1𝛼)𝑔2𝑔𝑑𝜉2+(2𝛼1)𝑑𝑔𝑑𝜉2=0.(6.5) Define the transformations𝐴𝜂=(1𝛼)𝜈𝐴𝜉fortwo-dimensionalfreeandwalljets,3𝜈𝜉fortwo-dimensionalliquidjet,𝑔=𝐴𝑓,(6.6) where 𝐴 is a constant. The final form of a group-invariant solution is𝑐𝜓=𝑥+𝛼2𝑐11𝛼𝐴𝑓(𝜂),𝜂=(1𝛼)𝐴𝑦𝜈𝑐𝑥+𝛼2/𝑐1𝛼fortwo-dimensionalfreeandwalljets,𝐴𝑦3𝜈𝑥+𝑐2/𝑐1fortwo-dimensionalliquidjet.(6.7) For two-dimensional free and wall jets, (6.5) yields𝑓+𝑓𝑓+2𝛼11𝛼𝑓2=0,(6.8) and for two-dimensional liquid jet, we have𝑓+3𝑓2=0.(6.9) The symmetry𝑐𝑋=𝑥+𝛼2𝑐1𝜕𝜕𝜕𝑥+𝛼𝑦𝜕𝜕𝑦+(1𝛼)𝜓𝜕𝜓(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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