Research Article  Open Access
GroupInvariant Solutions for TwoDimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice
Abstract
The groupinvariant solutions for nonlinear thirdorder partial differential equation (PDE) governing flow in twodimensional 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 thirdorder PDE for the stream function. The comparison between results for twodimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for twodimensional jets is given explicitly.
1. Introduction
The governing equations for twodimensional jet flows are expressed either as the system of two PDEs for the velocity components or by a single nonlinear thirdorder PDE for the stream function. In [1, 2] the similarity solution and in [3] the groupinvariant solution were constructed for the nonlinear thirdorder PDE for the stream function for twodimensional free jet with infinite fluid velocity at the orifice. The groupinvariant 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 twodimensional 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 twodimensional liquid jets were derived. The similarity solution for radial and twodimensional 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 groupinvariant solution for the nonlinear thirdorder PDE for stream function governing flow in twodimensional (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 groupinvariant solution for twodimensional free jet is derived. The symmetry associated with the conserved vector which is used to establish the conserved quantity for each jet generates the groupinvariant solution for the nonlinear thirdorder PDE for the stream function. The groupinvariant solution for twodimensional wall and liquid jets is studied in Sections 3 and 4. In Section 5 the comparison between the results for twodimensional jets, having finite and infinite fluid velocity at orifice, is constructed. The general form of groupinvariant solution for twodimensional free, wall, and liquid jets is given explicitly in Section 6. Finally the conclusions are summarized in Section 7.
2. Groupinvariant Solution for TwoDimensional Free Jet
The flow in twodimensional free jet is governed by nonlinear thirdorder PDE for stream function for an incompressible fluid. The relation between stream function and velocity components is The Lie point symmetry generator of (2.1) derived by Mason [3] is
The boundary conditions and the conserved quantity for twodimensional free jet in terms of stream function are (see [1โ3]) The conserved vector gave the conserved quantity (2.6) for twodimensional free jet (see [8]). The symmetry associated with the conserved vector which is used to establish the conserved quantity for each jet generates the groupinvariant solution for nonlinear thirdorder PDE [4, 9].
The symmetries associated with a known conserved vector can be determined by [10] Equations (2.8) and (2.9) yield and thus for conserved vector (2.7), . The Lie point symmetry generator associated with conserved vector (2.7) is
Now, is a groupinvariant solution of (2.1) if which yields where The conserved quantity (2.6) is independent of provided which yields . Since the stream function is determined up to an arbitrary constant, can be chosen to be zero. The insertion of (2.13) into (2.1) results in a nonlinear thirdorder ordinary differential equation (ODE) for :
Equation (2.16) can be transformed to with where is arbitrary constant, and prime denotes differentiation with respect to . The boundary conditions and conserved quantity (2.4)โ(2.6), in terms of , take the following form:
The solution of (2.17) subject to (2.19) and condition is (see [2, 11, 12]) and value of in terms of is
The final form of groupinvariant solution is is the symmetry that generated the groupinvariant solution. Now is finite at and so our solution may have some significance even near the axis. By taking , the results [1โ4] for infinite velocity at orifice can be rediscovered.
3. Groupinvariant Solution for TwoDimensional Wall Jet
The flow in twodimensional wall jet is also governed by (2.1). The boundary conditions for the twodimensional wall jet are [5] and the conserved quantity is The conserved vector gave the conserved quantity for twodimensional wall jet [8], and the symmetry associated with this conserved vector is
The groupinvariant solution of (2.1) for twodimensional wall jet case is where The conserved quantity (3.3) is independent of provided which yields . The substitution of (3.6) into (2.1) gives rise to a nonlinear thirdorder ODE for : Define and Equation (3.9) transforms to Boundary conditions (3.1) and (3.2) and conserved quantity (3.3) take the following form: Glauert [5] selected a solution of (3.10) with , and after integrating (3.10) twice, the following equation was obtained: Equation (3.12) yields The conserved quantity gave the unknown constant as
Thus we finally obtain The results obtained here for agree with Riley [7], and can be determined from [13]. By taking , the results for infinite velocity at orifice obtained by Glauert [5] can be rediscovered.
4. GroupInvariant Solution for TwoDimensional Liquid Jet
The governing equation for twodimensional liquid jet in terms of stream function is (2.1). The boundary conditions and conserved quantity for twodimensional liquid jet are [6] The conserved vector gave conserved quantity for twodimensional liquid jet [8].
Equations (2.8) and (2.9) yield the following Lie point symmetry generator associated with the conserved vector (4.4): The groupinvariant solution for twodimensional liquid jet is where The conserved quantity is independent of only if which gives . The stream function contains an additive constant so we may choose without loss of generality.
The substitution of (4.6), with , , into (2.1) yields a nonlinear thirdorder ordinary differential equation for : Equation (4.8) takes the following form: where and . Boundary conditions (4.1) and (4.2) are where the free surface is chosen to be . The conserved quantity (4.3) becomes
Equation (4.9) yields (see [11, 12, 14]) The final form of solution of (4.9) in parametric form is (see [14]) where is the hypergeometric function of first kind. We can tabulate the values of for given values of parameter from (4.13), and conserved quantity yields the constant .
Thus finally we obtain Now is finite at . The results obtained here for agree with those concluded by Watson [6] by the similarity solution method, and the procedure to obtain is discussed there.
5. Comparison between TwoDimensional Jets with Finite Velocity at Orifice and Infinite Velocity at Orifice
The comparison between twodimensional jets with finite fluid velocity at orifice and infinite velocity at orifice is constructed in Table 1. Table 1 shows that by formally taking , the stream function , variable , and symmetry that generates groupinvariant solution for infinite velocity case can be deduced from those of the finite velocity case.

6. General Form of Groupinvariant Solutions for TwoDimensional Jets
The flow in twodimensional free, wall, and liquid jets is governed by nonlinear thirdorder 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: where We choose because the stream function is determined up to an arbitrary constant. The expression for groupinvariant solutions for twodimensional jet flows is where The condition where conserved quantity is independent of yields in each of free, wall, and liquid jets. Using (6.3)โ(6.4), (2.1) yields Define the transformations where is a constant. The final form of a groupinvariant solution is For twodimensional free and wall jets, (6.5) yields and for twodimensional liquid jet, we have The symmetry yielded the groupinvariant solution.
7. Conclusions
The groupinvariant solutions for twodimensional free, wall, and liquid jets were derived for finite velocity at orifice. For twodimensional free jet, a Lie point symmetry was associated with the conserved vector that generated the conserved quantity for twodimensional free jet. This symmetry generated the groupinvariant solution for the nonlinear thirdorder PDE for stream function subject to certain boundary conditions. The nonlinear thirdorder PDE was transformed to the nonlinear thirdorder ODE. Using certain transformations we deduced the same nonlinear thirdorder ODE as was obtained for twodimensional free jet having infinite fluid velocity at orifice. The analogue analysis was done for the twodimensional wall and liquid jets. A detailed comparison of results for finite and infinite velocity at orifice was constructed. The general form of groupinvariant solution for the twodimensional 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
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.