**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30174

##### Physical Conserved Quantities for the Axisymmetric Liquid, Free and Wall Jets

**Authors:**
Rehana Naz,
D. P. Mason,
Fazal Mahomed

**Abstract:**

A systematic way to derive the conserved quantities for the axisymmetric liquid jet, free jet and wall jet using conservation laws is presented. The flow in axisymmetric jets is governed by Prandtl-s momentum boundary layer equation and the continuity equation. The multiplier approach is used to construct a basis of conserved vectors for the system of two partial differential equations for the two velocity components. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the axisymmetric liquid and free jet. The multiplier approach applied to the third-order partial differential equation for the stream function yields two local conserved vectors one of which is a non-local conserved vector for the system. One of the conserved vectors gives the conserved quantity for the axisymmetric free jet but the conserved quantity for the wall jet is not obtained from the second conserved vector. The conserved quantity for the axisymmetric wall jet is derived from a non-local conserved vector of the third-order partial differential equation for the stream function. This non-local conserved vector for the third-order partial differential equation for the stream function is obtained by using the stream function as multiplier.

**Keywords:**
Axisymmetric jet,
liquid jet,
free jet,
wall jet,
conservation laws,
conserved quantity.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1331529

**References:**

[1] R. Naz, D. P. Mason and F. M. Mahomed, Conservation laws and physical conserved quantities for laminar two-dimensional and radial jets, Accepted for publication in journal of Nonlinear Analysis: Real World Applications.

[2] S. Goldstein, Modren Developments in Fluid Dynamics, Clarendon Press, Oxford, 1938, 1 pp.148-149.

[3] P. W. Duck and R. J. Bodonyi, The wall jet on an axisymmetric body, Q. J. Mech. Appl. Maths, 39 (1986) 467-483.

[4] D. P. Mason and I. Ruscic, Group invariant solution and conservation law for a steady laminar axisymmetric free jet, Quaestiones Mathematicae, 27 (2004) 171-183.

[5] H. Steudel, Uber die Zuordnung zwischen invarianzeigenschaften und Erhaltungssatzen, Zeit Naturforsch. 17 A (1962) 129-132.

[6] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993, pp. 435-458.

[7] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classification, Euro. J. Appl. Math. 13 (2002) 545-566.

[8] R. Naz, F. M. Mahomed and D. P. Mason, Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics. Applied Mathematics and Computation, Available online 26 June (2008).

[9] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967, pp. 75-79.

[10] R. P. Gillespie, Integration, Oliver and Boyd, Edinburgh, 1967, pp. 113- 116.