A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers

Abstract

A problem in fluid mechanics which has received some attention recently concerns the emergence of an incompressible Newtonian fluid jet from a uniform tube into an inviscid atmosphere. Both the axisymmetric case of a circular tube and the two-dimensional case of flow from between parallel planes are of interest. When the jet falls vertically under gravity, the motion far downstream is dominated by gravity and the expansion procedures of Clarke (3), and Kaye and Vale (10) give details of the flow in this region. When the flow near the exit is at a high Reynolds number, it is reasonable to expect the flow appropriate to that in an infinite tube to prevail right up to the exit (except, perhaps, near the point of discontinuity of the boundary conditions). With this assumption, Duda and Vrentas(5) use a numerical technique to solve for the flow in the axisymmetric jet beyond the exit, both with and without gravity acting in the axial direction. In the absence of gravity, the jet can be expected to attain a constant width some distance downstream, and at high Reynolds numbers the above assumption is sufficient to allow a mass and momentum balance to determine the contraction ratio of the jet as for the axisymmetric case, and for the two-dimensional case (see Harmon (8)). By treating the dynamics of the jet as those of a boundary layer growing on the free surface, Goren and Wronski (6) and Tillett (18) are able to examine the flow in greater detail.

Geometric and thermal evolution of a surge-type glacier in its quiescent state: Trapridge Glacier, Yukon Territory, Canada, 1969–89

“…The magnitude of the displacement discontinuity (corresponding to the Burgers' vector in dislocation theory) vanishes at the position of the singularity line and increases with distance *In the course of editing this manuscript K . Hutter drew our attention to an important paper by Richardson (1970) that presents an exact solution to the problem addressed by Hutter and Olunloyo.…”

On the Numerical Implementation of Nonlinear Viscoelastic Models in a Finite-Volume Method

“…viscoelastic¯ow calculations (e.g., [14,15]). The entry¯ow problem o ers some advantage in that the inlet conditions are easier to imposeÐ in fact, they consist only of a uniform axial velocity pro® le and zero stressesÐ and another reason for using this same simple geometry is that it has been used in a previous work [1] with the UCM¯uid.…”

Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

“…The creeping planar stick-slip problem was solved analytically by Richardson who used a Wiener-Hopf technique [12] and by Sturges who used the method of matched eigenfunction expansions [13]. Both methods have been used by Trogdon and Joseph [14] to obtain analytical solutions for the round stick-slip problem.…”

“…We discuss brie y the literature with emphasis on works reporting results for the Newtonian case. Tanner and Huang [10] applied the J-integral method for solving the planar stick-slip ow of power-law uids, corrected the numerical estimate of singularity strength in Richardson's Newtonian analysis [12], and calculated the ÿrst singular coe cient for various power-law exponents. Owens and Phillips [23] presented solutions of the planar stick-slip problem obtained with a spectral domain decomposition method (DDM) for an Oldroyd-B uid.…”

“…The SFBIM is developed in Section 3, where four di erent formulations of the method corresponding to di erent techniques of imposing the Dirichlet boundary conditions are presented. The numerical results are given in Section 4, where the fast convergence of the method with respect to the number of singular functions is demonstrated and comparisons are made with the the analytical solution [12] and the results obtained with the boundary integral equation method of Kelmanson [15], the spectral domain decomposition method of Owens and Phillips [24], the high-resolution ÿnite element method of Salamon et al [20], the STGFEM of Ngamaramvaranggul and Webster [21], and other methods. The conclusions are summarized in Section 5.…”