## Group invariant solutions and conservation laws for jet flow models of non-Newtownian power-law fluids

##### Date

2014-07-18

##### Authors

Magan, Avnish Bhowan

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

The non-Newtonian incompressible power-law
uid in jet
ow models is investigated.
An important feature of the model is the de nition of a suitable
Reynolds number, and this is achieved using the standard de nition of a
Reynolds number and ascertaining the magnitude of the e ective viscosity.
The jets under examination are the two-dimensional free, liquid and wall
jets. The two-dimensional free and wall jets satisfy a di erent partial di erential
equation to the two-dimensional liquid jet. Further, the jets are reformulated
in terms of a third order partial di erential equation for the stream
function. The boundary conditions for each jet are unique, but more signi -
cantly these boundary conditions are homogeneous. Due to this homogeneity
the conserved quantities are critical in the solution process.
The conserved quantities for the two-dimensional free and liquid jet are
constructed by rst deriving the conservation laws using the multiplier approach.
The conserved quantity for the two-dimensional free jet is also derived
in terms of the stream function. For a Newtonian
uid with n = 1 the twodimensional
wall jet gives a conservation law. However, this is not the case for
the two-dimensional wall jet for a non-Newtonian power-law
uid.
The various approaches that have been applied in an attempt to derive a
conservation law for the two-dimensional wall jet for a power-law
uid with
n 6= 1 are discussed. In conjunction with the attempt at obtaining conservation
laws for the two-dimensional wall jet we present tenable reasons for its failure,
and a feasible way forward.
Similarity solutions for the two-dimensional free jet have been derived for
both the velocity components as well as for the stream function. The associated
Lie point symmetry approach is also presented for the stream function. A
parametric solution has been obtained for shear thinning
uid free jets for
0 < n < 1 and shear thickening
uid free jets for n > 1. It is observed that for
values of n > 1 in the range 1=2 < n < 1, the velocity pro le extends over a
nite range.
For the two-dimensional liquid jet, along with a similarity solution the
complete Lie point symmetries have been obtained. By associating the Lie
point symmetry with the elementary conserved vector an invariant solution
is found. A parametric solution for the two-dimensional liquid jet is derived
for 1=2 < n < 1. The solution does not exist for n = 1=2 and the range 0 < n < 1=2 requires further investigation.