Analytical solutions of singularities moving with an arbitrary path when two fluids are present (original) (raw)

How do singularities move in potential flow?

2011

The equations of motion of point vortices embedded in incompressible flow go back to Kirchhoff. They are a paradigm of reduction of an infinite-dimensional dynamical system, namely the incompressible Euler equation, to a finite-dimensional system, and have been called a "classical applied mathematical playground". The equation of motion for a point vortex can be viewed as the statement that the translational velocity of the point vortex is obtained by removing the leading-order singularity due to the point vortex when computing its velocity. The approaches used to obtain this result are reviewed, along with their history and limitations. A formulation that can be extended to study the motion of higher singularities (e.g. dipoles) is then presented. Extensions to more complex physical situations are also discussed.

Singularities on Free Surfaces of Fluid Flows

Studies in Applied Mathematics, 1998

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120Њ angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120Њ angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120Њ. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130Њ angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.

Experimental and Numerical Investigation of Two-Phase Flow through Enlarging Singularity

2012

Gas-liquid flow is extensively used in industrial systems such as power generation units, cooling and heating systems (i.e. condensers, evaporators and manifolds), safety valves, etc. These systems generally have complex geometries composed by singularities like expansion, contraction, bends and orifices. Thus two-phase flow characteristics through these singularities should be identified in order to be used in designing of the systems.

Two-phase flow characteristics in singularities

This paper aims at presenting the latest scientific progress on two-phase flow in singularities through academic research at INSA Lyon as well as proposing some future possible important issues to be investigated. Flow regimes of third-and fourth-generation refrigerants in horizontal and vertical return bends as well as in a horizontal sudden contraction were experimentally investigated. The dynamical behavior of vapor bubbles or slugs in vertical downward flow return bend was reported. A simplified analysis of the forces acting on the bubble was proposed to better understand the vapour trajectory. Furthermore, void fraction was measured along the sudden contraction using an image analysis technique, which gives very original results. Such experimental studies also brought to the fore the upstream and downstream flow disturbances caused by such singularities as contractions and return bends and their impact on the hydrodynamic performance (e.g. pressure drop) of refrigerants. Especially, these disturbances can be analysed in terms of perturbation lengths up-and downstream of the singularities. Lastly, large pressure drop databases for R-410A, R-134a and HFO-1234yf were obtained. Experimental values of pressure drops in singularities were compared against different prediction methods from the literature without any satisfactory results. Finally, these databases were used to develop new twophase pressure drop prediction methods for such singularities as return bends and sudden contractions.

Approximate method to singularity behaviour of nonlinear problems in fluid dynamics

The singularity behaviour of nonlinear problems in fluid dynamics is numerically investigated in the present paper. The behaviour of the first Painlev'e transcendent is introduced as a model problem. Then the laminar unsteady flow of a viscous fluid away from a plane stagnation point is studied as second model problem. We have applied approximate method to these problems with the aid of algebraic programming language MAPLE. Analysis based on approximate method suggests that the convergence of the series of shear stress is limited by a pair of singularities. The location and nature of the singularities in the complex plane are presented. The shear stress in real plane is also depicted graphically.

Formation of singularities of solutions of the equations of motion of compressible fluids subjected to external forces in the case of several spatial variables

Journal of Mathematical Sciences, 2007

We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compactly supported perturbation of nontrivial constant state is considered. We generalize the known theorem [1] on initial data resulting in singularities. Finally, we investigate the influence of frictional damping and rotation on the singularity formation.

Analytic Solutions of a Two-Fluid Hydrodynamic Model

Mathematical Modelling and Analysis, 2021

We investigate a one dimensional flow described with the non-compressible coupled Euler and non-compressible Navier-Stokes equations in the Cartesian coordinate system. We couple the two fluids through the continuity equation where different void fractions can be considered. The well-known self-similar Ansatz was applied and analytic solutions were derived for both velocity and pressure field as well.

Is a UCM fluid flow near a stationary point always singular? -Part II

Oftentimes observed divergence of numerical solutions to benchmark flows of the UCM viscoelastic fluid is a known and widely discussed issue. Some authors consider such singularities " invincible ". Following the previous research, the article gives more arguments against this position, for which it considers two typical flows with a stagnation point, often a place of the flows' singularity. For the flow spread over a wall, as previously for the counterflows, numerical and asymptotic analytical solutions are presented. Both kinds of flows turn out regular in the stagnation points, in particular , for high Weissenberg numbers. A good accordance is demonstrated between the analytical and numerical results.