The Laminar Two-Dimensional Free Jet of an Incompressible Pseudoplastic Fluid

Abstract

Boundary-layer equations have been used to obtain an exact solution for the free outflow of a non-Newtonian pseudoplastic fluid from a two-dimensional orifice into a mass of the same fluid. Existence of similarity solutions has been proved. The results obtained by Schlichting for the Newtonian fluid flow is shown to be the special case of the more general analysis given here. Though further effort is necessary to investigate the turbulent jet flow, the analysis presented here should make a definite contribution in this definition.

Analytical solution in parametric form for the two-dimensional free jet of a power-law fluid

“…In the numerical solution of Lemieux and Unny [12] the boundary curve = y y max exists for all < < ∞ n 0 with a different shape for < < n 0 2/3, = n 2/3 and < < ∞ n 2/3…”

“…Kutepov et al [11] considered the same problem, where once again there was no distinction made for values > n 1/2 and < n 1/2 and the interpretation of the results for > n 1 did not predict the boundary for the jet. Lemieux and the Unny [12] considered the numerical solution of a two-dimensional free jet of a power-law fluid and were the first to confirm that a boundary curve for a two-dimensional free jet of a power-law fluid exists. However, these authors found that this boundary exists for all values of n. Mitwally [13] who also solved the ordinary differential equation numerically found that the boundary curve exists only for values > n 1 and not over the entire range < < ∞ n 0 of n. In this paper we derive the analytical solution in parametric form for a two-dimensional free jet of an incompressible powerlaw fluid.…”

Jet flows in non-Newtonian fluids

“…The latter is to be determined from coupling the jet flow with the outer flow as described below (Section 2.3). The function f (η) is obtained numerically from solving a set of ordinary differential equations (Gutfinger & Shinnar [9], Rotem [14], Lemieux & Unny [11], Atkinson [2], Serth [21], Vlachopoulos & Stournaras [27], Mitwally [13]). Results are shown in Figures 2.1 and 2.2 for various values of the power-law index n. Note that in the axisymmetric case self-similar boundary-layer solutions exist only for n > 1/2 (Rotem [14], Serth [21]).…”

Discussion: “On the Laminar Two-Dimensional Free Jet of an Incompressible Pseudoplastic Fluid” (Atkinson, C., 1972, ASME J. Appl. Mech., 39, pp. 1162–1164)

“…A Simple Test for Asymptotic Stability in Partially Dissipative Symmetric Systems 1 P. K. C. Wang. 2 The authors gave a necessary and sufficient condition for the asymptotic stability of equilibrium of the vector equation My + Cy + Ky = 0 in terms of the rank of a certain matrix. This result is basically a controllability condition in linear system theory [l].…”

“…When C, K, and M are symmetric, the foegoing condition is equivalent to the one given by the authors. The sufficiency of (2) for asymptotic stability by feedback control u = -y can be also established by making use of the well-known results on stabilizability and pole assignment [2]. In particular, if (1) and (2) is completely controllable, then there exists a real matrix F such that the spectrum of (A + BF) corresponds to any preassigned set (for complex numbers, they must be given in conjugate pairs).…”