Analytical solution in parametric form for the two-dimensional free jet of a power-law fluid. (October 2016) - Magan, A.B.; Mason, D.P.; Mahomed, F.M.
Abstract: The two-dimensional free jet of an incompressible non-Newtonian power-law fluid is investigated. The Reynolds number is defined in terms of the characteristic effective viscosity of the power-law fluid. The boundary layer equations for a power-law fluid are derived in terms of the stream function. The free jet is modelled by making the boundary layer approximation perpendicular to the axis of symmetry. The conservation laws for the partial differential equation for the stream function are investigated using the multiplier method and the conserved quantity for the free jet is obtained by integrating the elementary conservation law across the jet. The Lie point symmetry of the partial differential equation for the stream function which is associated with the elementary conserved vector is derived and it is used to obtain the invariant form of the stream function. An analytical solution for the free jet in parametric form is derived. The solution depends on the exponent n in the power law. For a shear thickening fluid ( n > 1 ) it is found that the jet is bounded in the lateral direction perpendicular to the axis of the jet and the equation of the boundary is derived. For a Newtonian fluid ( n = 1 ) and a shear thinning fluid ( 0 < n < 1 ) the jet is unbounded in the direction perpendicular to the axis. The solution for n = 1 / 2 is a special case and has exponential form. For 1 / 2 < n < ∞ the total flux of mass along the jet and the inflow velocity at the boundary ( 1 < n < ∞ ) and at infinity ( 1 / 2 < n ≤ 1 ) are finite. For 0 < n ≤ 1 / 2 the solution is not physically acceptable because the total flux of mass along the jet and the inflow velocity at infinity are infinite. Abstract : Highlights: Analytical solution in parametric form for a free jet of a power-law fluid. Solution generated by Lie point symmetry associated with elementary conserved vector. The velocity profile is bounded only for a shear thickening fluid. Solutions for n ≤ 1 / 2 have infinite mass flux and not physical. The free jet is an ideal problem to test numerical methods for power-law flow.