Viscous Flows

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Navier Stokes ◽  
Small Reynolds Number ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Core ◽  
Flow Past A Sphere

Here some solutions of Navier–Stokes equations are found.The flow of a fluid along a pipe (Poisseuille flow) and that between two rotating cylinders (Couette flow) are the simplest. In the limit of large viscosity (small Reynolds number) the equations become linear: Stokes equations. Flow past a sphere is solved in detail. It is used to calculate the drag on a sphere, a classic formula of Stokes. An exact solution of the Navier–Stokes equation describing a dissipating vortex is also found. It is seen that viscosity cannot be ignored at the boundary or at the core of vortices.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Analysis of Solving Galerkin Finite Element Methods with Symmetric Pressure Stabilization for the Unsteady Navier-Stokes Equations Using Conforming Equal Order Interpolation

2017 ◽  
Vol 9 (2) ◽  
pp. 362-377
Author(s):  
Gang Chen ◽  
Minfu Feng
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Finite Element ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Galerkin Finite Element ◽  
Galerkin Finite Element Methods ◽  
Pressure Stabilization ◽  
Uniform Error Estimates

AbstractThis paper gives analysis of a semi-discrete scheme using equal order interpolation to solve unsteady Navier-Stokes equations. A unified pressure stabilized term is added to our scheme. We proved the uniform error estimates with respect to the Reynolds number, provided the exact solution is smooth.

Transient inertial hydrodynamic interaction between two identical spheres settling at small Reynolds number

2008 ◽  
Vol 605 ◽  
pp. 263-279 ◽  
Author(s):  
B. U. FELDERHOF
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Relative Motion ◽  
Hydrodynamic Interaction ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Small Reynolds Number ◽  
Navier Stokes Equations ◽  
Small Constant ◽  
Distance Vector

The flow pattern generated by a sphere accelerated from rest by a small constant applied forceshows scaling behaviour at long times, as can be shown from the solution of the linearized Navier–Stokes equations. In the scaling regime the kinetic energy of the flow grows with thesquare root of time. For two distant settling spheres starting from rest the kinetic energy ofthe flow depends on the distance vector between centres; owing to interference of the flowpatterns. It is argued that this leads to relative motion of the two spheres. Thecorresponding interaction energy is calculated explicitly in the scaling regime.

On a theory of laminar flow in channels of a certain class

1973 ◽  
Vol 73 (2) ◽  
pp. 361-390 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Arbitrary Order ◽  
Stokes Equations ◽  
Formal Series ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Plane Flow ◽  
Laminar Separation ◽  
Steady Plane

This paper is concerned with the steady plane flow of a viscous fluid in symmetrical channels with slowly curving walls. The product of local channel half-width and local wall curvature is bounded by a small parameter ∈. We review the essentials of the formal approximation, in powers of ∈, proposed in (4) and (5); resolve a question, left open there, regarding the existence of the approximate series for the stream function for any value of the Reynolds number and to arbitrary order in ∈ and prove that, under certain restrictions on the Reynolds number and the divergence angle of the channel walls, this formal series is in fact a strict asymptotic expansion (for ∈ → 0) of an exact solution of the Navier–Stokes equations. As a result, the traditional picture of laminar separation, due to Prandtl, emerges as part of the steady flow field predicted by an exact solution that is known explicitly to arbitrary asymptotic order.

The Navier–Stokes Equations

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Reynolds Number ◽  
Scale Invariance ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Incompressible Limit ◽  
Small Reynolds Number ◽  
Navier Stokes Equations ◽  
Dimensionless Ratio ◽  
Energy And Momentum

When different layers of a fluid move at different velocities, there is some friction which results in loss of energy and momentum to molecular degrees of freedom. This dissipation is measured by a property of the fluid called viscosity. The Navier–Stokes (NS) equations are the modification of Euler’s equations that include this effect. In the incompressible limit, the NS equations have a residual scale invariance. The flow depends only on a dimensionless ratio (the Reynolds number). In the limit of small Reynolds number, the NS equations become linear, equivalent to the diffusion equation. Ideal flow is the limit of infinite Reynolds number. In general, the larger the Reynolds number, the more nonlinear (complicated, turbulent) the flow.

scholarly journals A Semianalytical Approach to the Solution of Time-Fractional Navier-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Zeeshan Ali ◽  
Shayan Naseri Nia ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Ming Kwang Tan
Keyword(s):  
Exact Solution ◽  
Stokes Equation ◽  
Fractional Derivatives ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Caputo Fractional Derivatives ◽  
Navier Stokes Equation ◽  
Small Perturbations ◽  
Navier Stokes Equations ◽  
Numerical Approaches

In this manuscript, a semianalytical solution of the time-fractional Navier-Stokes equation under Caputo fractional derivatives using Optimal Homotopy Asymptotic Method (OHAM) is proposed. The above-mentioned technique produces an accurate approximation of the desired solutions and hence is known as the semianalytical approach. The main advantage of OHAM is that it does not require any small perturbations, linearization, or discretization and many reductions of the computations. Here, the proposed approach’s reliability and efficiency are demonstrated by two applications of one-dimensional motion of a viscous fluid in a tube governed by the flow field by converting them to time-fractional Navier-Stokes equations in cylindrical coordinates using fractional derivatives in the sense of Caputo. For the first problem, OHAM provides the exact solution, and for the second problem, it performs a highly accurate numerical approximation of the solution compare with the exact solution. The presented simulation results of OHAM comparison with analytical and numerical approaches reveal that the method is an efficient technique to simulate the solution of time-fractional types of Navier-Stokes equation.

scholarly journals Generalized Navier–Stokes Equations and Dynamics of Plane Molecular Media

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 288
Author(s):  
Alexei Kushner ◽  
Valentin Lychagin
Keyword(s):  
Carbon Dioxide ◽  
Internal Structure ◽  
Stokes Equation ◽  
Hydrogen Cyanide ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

The first analysis of media with internal structure were done by the Cosserat brothers. Birkhoff noted that the classical Navier–Stokes equation does not fully describe the motion of water. In this article, we propose an approach to the dynamics of media formed by chiral, planar and rigid molecules and propose some kind of Navier–Stokes equations for their description. Examples of such media are water, ozone, carbon dioxide and hydrogen cyanide.

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