A New Analytical Procedure to Solve Two Phase Flow in Tubes

Abstract

Abstract:A new formulation for a proposed solution to the 3D Navier-Stokes Equations in cylindrical co-ordinates coupled to the continuity and level set convection equation is presented in terms of an additive solution of the three principle directions in the radial, azimuthal and z directions of flow and a connection between the level set function and composite velocity vector for the additive solution is shown. For the case of a vertical tube configuration with small inclination angle, results are obtained for the level set function defining the interface in both the radial and azimuthal directions. It is found that the curvature dependent part of the problem alone induces sinusoidal azimuthal interfacial waves whereas when the curvature together with the equation for the composite velocity is considered oscillating radial interfacial waves occur. The implications of two extremes indicate the importance of looking at the full equations including curvature.

A Method of Solving Compressible Navier Stokes Equations in Cylindrical Coordinates Using Geometric Algebra

“…Multiplication of Equation (10) by ρ δ and Equation (12) below by L 1 δ , addition of the resulting equations [9], and using the ordinary product rule of differential multivariable calculus a form as in Equation (13) is obtained whereby I set a = ρ L 1 .…”

“…In the present work I introduce a new procedure to write the compressible unsteady Navier Stokes equations with a general spatial and temporal varying density term in terms of an additive solution of the three principle directions in the radial, azimuthal and z directions of flow. This procedure can be used in physics and engineering to simplify a complex system of PDE to a simpler one such as complex multiphase flows [9] and other areas in physics using Geometric Algebra, such as for example the study of the Maxwell equations. In mathematics, the method leads to the present work which shows the necessary conditions for the full solution of the system of equations for compressible flow to blowup at a certain time for certain types of initial conditions and also implications of using assumptions for necessary and physically meaningful functional forms of density.…”

A Method of Solving Compressible Navier Stokes Equations in Cylindrical Coordinates Using Geometric Algebra

“…(10) by ρ δ and Eq. (12) below by L 1 δ , addition of the resulting equations [7], and using the ordinary product rule of differential multivariable calculus a form as in Eq. (13) is obtained whereby I set a = ρ L 1 .…”