Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids

2019 ◽  
Vol 26 (1/2) ◽  
pp. 167-178 ◽  
Author(s):  
Dongming Wei ◽  
Samer Al-Ashhab
Keyword(s):  
Newtonian Fluids ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Infinite Domain ◽  
Navier Stokes Equation ◽  
Continuity Equations ◽  
Self Similar ◽  
Similar Solutions ◽  
Two Parameters

The reduced problem of the Navier–Stokes and the continuity equations, in two-dimensional Cartesian coordinates with Eulerian description, for incompressible non-Newtonian fluids, is considered. The Ladyzhenskaya model, with a non-linear velocity dependent stress tensor is adopted, and leads to the governing equation of interest. The reduction is based on a self-similar transformation as demonstrated in existing literature, for two spatial variables and one time variable, resulting in an ODE defined on a semi-infinite domain. In our search for classical solutions, existence and uniqueness will be determined depending on the signs of two parameters with physical interpretation in the equation. Illustrations are included to highlight some of the main results.

scholarly journals SELF-SIMILAR ANALYTIC SOLUTION OF THE TWO-DIMENSIONAL NAVIER-STOKES EQUATION WITH A NON-NEWTONIAN TYPE OF VISCOSITY

2016 ◽  
Vol 21 (1) ◽  
pp. 83-94 ◽  
Author(s):  
Imre Ferenc Barna ◽  
Gabriella Bognar ◽  
Krisztian Hriczo
Keyword(s):  
Trial Function ◽  
Analytic Solution ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Newtonian Viscosity ◽  
Navier Stokes Equation ◽  
Continuity Equations ◽  
Eulerian Description ◽  
Self Similar ◽  
Newtonian Type

We investigate Navier-Stokes (NS) and the continuity equations in Cartesian coordinates and Eulerian description for the two dimensional incompressible nonNewtonian fluids. Due to the non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multidimensional generalization of the well-known self-similar Ansatz, which has already been used for non-compressible and compressible viscous flow studies. Geometrical interpretations of the trial function are also discussed. Our recent results are compared to the former Newtonian ones.

scholarly journals GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

2011 ◽  
Vol 21 (03) ◽  
pp. 421-457 ◽  
Author(s):  
RAPHAËL DANCHIN ◽  
MARIUS PAICU
Keyword(s):  
Global Existence ◽  
Turbulent Flows ◽  
Vertical Direction ◽  
Boussinesq System ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Incompressible Euler Equation ◽  
Navier Stokes Equation ◽  
Transport Diffusion ◽  
Anisotropic Viscosity

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

Nonexistence of singular pseudo-self-similar solutions of the Navier–Stokes system

2001 ◽  
Vol 319 (4) ◽  
pp. 809-815 ◽  
Author(s):  
Judith R. Miller ◽  
Mike O'Leary ◽  
Maria Schonbek
Keyword(s):  
Stokes System ◽  
Navier Stokes ◽  
Self Similar ◽  
Similar Solutions

scholarly journals A note on time-fractional Navier–Stokes equation and multi-Laplace transform decomposition method

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hassan Eltayeb ◽  
Imed Bachar ◽  
Yahya T. Abdalla
Keyword(s):  
Numerical Simulation ◽  
Approximate Solution ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Adomian Decomposition ◽  
Stokes Problems

Abstract In this study, the double Laplace Adomian decomposition method and the triple Laplace Adomian decomposition method are employed to solve one- and two-dimensional time-fractional Navier–Stokes problems, respectively. In order to examine the applicability of these methods some examples are provided. The presented results confirm that the proposed methods are very effective in the search of exact and approximate solutions for the problems. Numerical simulation is used to sketch the exact and approximate 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.

scholarly journals Solutions of Navier-Stokes Equation with Coriolis Force

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Sunggeun Lee ◽  
Shin-Kun Ryi ◽  
Hankwon Lim
Keyword(s):  
Diffusion Equation ◽  
Stokes Equation ◽  
Coriolis Force ◽  
Potential Flow ◽  
Convective Diffusion ◽  
Navier Stokes ◽  
Coriolis Effect ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Convective Diffusion Equation

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.

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