scholarly journals ANALYTIC SOLUTIONS OF A TWO-FLUID HYDRODYNAMIC MODEL

2021 ◽  
Vol 26 (4) ◽  
pp. 582-590
Author(s):  
Imre Ferenc Barna ◽  
László Mátyás
Keyword(s):  
Stokes Equations ◽  
Analytic Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Two Fluids ◽  
Void Fractions ◽  
Self Similar ◽  
Cartesian Coordinate ◽  
Two Fluid

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.

Two-Fluid Flow Between Rotating Annular Disks

1976 ◽  
Vol 98 (2) ◽  
pp. 214-222 ◽  
Author(s):  
J. E. Zweig ◽  
H. J. Sneck
Keyword(s):  
Annular Flow ◽  
Stokes Equations ◽  
Fluid Flows ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Two Fluids ◽  
Small Clearance ◽  
Annular Disks ◽  
Two Fluid

The general hydrodynamic behavior at small clearance Reynolds numbers of two fluids of different density and viscosity occupying the finite annular space between a rotating and stationary disk is explored using a simplified version of the Navier-Stokes equations which retains only the centrifugal force portion of the inertia terms. A criterion for selecting the annular flow fields that are compatible with physical reservoirs is established and then used to determine the conditions under which two-fluid flows in the annulus might be expected for specific fluid combinations.

Mathematical modeling of steady stratified flows

2003 ◽  
Vol 3 ◽  
pp. 195-207
Author(s):  
A.M. Ilyasov ◽  
V.N. Kireev ◽  
S.F. Urmancheev ◽  
I.Sh. Akhatov
Keyword(s):  
Stokes Equations ◽  
Approximate Model ◽  
Immiscible Liquid ◽  
Stratified Flows ◽  
Navier Stokes ◽  
Flat Channel ◽  
Navier Stokes Equations ◽  
Two Fluids ◽  
Flow Parameters ◽  
Direct Numerical Solution

The work is devoted to the analysis of the flow of immiscible liquid in a flat channel and the creation of calculation schemes for determining the flow parameters. A critical analysis of the well-known Two Fluids Model was carried out and a new scheme for the determination of wall and interfacial friction, called the hydraulic approximation in the theory of stratified flows, was proposed. Verification of the proposed approximate model was carried out on the basis of a direct numerical solution of the Navier–Stokes equations for each fluid by a finite-difference method with phase-boundary tracking by the VOF (Volume of Fluid) method. The graphical dependencies illustrating the change in the interfase boundaries of liquids and the averaged over the occupied area of the phase velocities along the flat channel are presented. The results of comparative calculations for two-fluid models are also given, according to the developed model in the hydraulic approximation and direct modeling. It is shown that the calculations in accordance with the hydraulic approximation are more consistent with the simulation results. Thus, the model of hydraulic approximation is the most preferred method for calculating stratified flows, especially in cases of variable volumetric content of liquids.

GLOBAL EXISTENCE TO THE NAVIER–STOKES EQUATIONS THROUGH A SELF-SIMILAR-LIKE UPPER SOLUTION

2012 ◽  
Vol 14 (05) ◽  
pp. 1250031
Author(s):  
GUY BERNARD
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Velocity ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Navier Stokes Equations ◽  
Upper Solution ◽  
Self Similar

A global existence result is presented for the Navier–Stokes equations filling out all of three-dimensional Euclidean space ℝ3. The initial velocity is required to have a bell-like form. The method of proof is based on symmetry transformations of the Navier–Stokes equations and a specific upper solution to the heat equation in ℝ3× [0, 1]. This upper solution has a self-similar-like form and models the diffusion process of the heat equation. By a symmetry transformation, the problem is transformed into an equivalent one having a very small initial velocity. Using the upper solution, the equivalent problem is then solved in the time interval [0, 1]. This local solution is then extended to the time interval [0, ∞) by an iterative process. At each step, the problem is extended further in time in an interval of time whose length is greater than one, thus producing the global solution. Each extension is transformed, by an appropriate change of variables, into the first local problem in the time interval [0, 1]. These transformations exploit the diffusive and self-similar-like nature of the upper solution.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

Pulsatile Waterhammer Subject to Laminar Friction

1972 ◽  
Vol 94 (2) ◽  
pp. 467-472 ◽  
Author(s):  
D. A. P. Jayasinghe ◽  
H. J. Leutheusser
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Present Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Arbitrary Velocity ◽  
Velocity Input ◽  
Special Case ◽  
Signalling Problem

This paper deals with elastic waves which may be generated in a fluid by the sudden movement of a flow boundary. In particular, an analysis of the classical piston, or signalling problem is presented for the special case of arbitrary velocity input into a stationary fluid contained in a circular, semi-infinite waveguide. The decay of the pulse, as well as the resulting flow development in the inlet region of the pipe are analyzed by means of an asymptotic expansion of the suitably nondimensionalized Navier-Stokes equations for a compressible, nonheat-conducting Newtonian fluid. The results differ significantly from those of the more conventional one-dimensional approach based on the so-called telegrapher’s equation of mathematical physics. The present theory realistically predicts the growth of a boundary layer both in time and position and, hence, it appears to represent the transient fluid motion in a manner which is physically more appealing.

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