scholarly journals On the existence of an exact solution of the Navier-Stokes equation pertaining to certain vortex motion

1972 ◽  
Vol 13 (4) ◽  
pp. 456-460
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Incompressible Fluid ◽  
Stokes Equation ◽  
Viscous Incompressible Fluid ◽  
Fluid Motion ◽  
Vortex Motion ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Cylindrical Polar Coordinates

In 1942, Burgers [1] observed that in cylindrical polar coordinates, the steady Navier-Stokes equation governing viscous incompressible fluid motion can be reduced to a set of ordinary differential equations if the velocity components vr, vo and vz are assumed to have a special form.

scholarly journals Exact Solution for Unsteady Flow of Viscous Incompressible Fluid Over a Suddenly Accelerated Flat Plate (Stokes' First Problem) Using Laplace Transforms

2018 ◽  
Vol 7 (3.6) ◽  
pp. 267
Author(s):  
Spainborlang Kharchandy ◽  
. .
Keyword(s):  
Exact Solution ◽  
Unsteady Flow ◽  
Incompressible Fluid ◽  
Flat Plate ◽  
Stokes Equation ◽  
Viscous Incompressible Fluid ◽  
Laplace Transforms ◽  
Simple Approach ◽  
Navier Stokes ◽  
Navier Stokes Equation

With the Navier-Stokes Equation in Cartesian form (in absence of body forces), Laplace Transforms provides a simple approach towards solving the unsteady flow of a viscous incompressible fluid over a suddenly accelerated flat plate. On comparing the results between  Laplace Transforms and similarity methods, it reveals that Laplace Transforms is simple and effective.

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

1957 ◽  
Vol 2 (3) ◽  
pp. 237-262 ◽  
Author(s):  
Ian Proudman ◽  
J. R. A. Pearson
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Small Reynolds Numbers ◽  
Cylindrical Polar Coordinates ◽  
Flow Past A Sphere

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimensions

1986 ◽  
Vol 163 ◽  
pp. 141-147 ◽  
Author(s):  
J. M. Dorrepaal
Keyword(s):  
Exact Solution ◽  
Stagnation Point ◽  
Stokes Equation ◽  
Similarity Solution ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Angle Of Incidence ◽  
Arbitrary Angle ◽  
Navier Stokes Equation ◽  
Flat Wall

A similarity solution is found which describes the flow impinging on a flat wall at an arbitrary angle of incidence. The technique is similar to a method used by Jeffery (1915) and discussed more recently by Peregrine (1981).

scholarly journals Splitting Decomposition Homotopy Perturbation Method To Solve One -Dimensional Navier -Stokes Equation

2017 ◽  
Vol 13 (2) ◽  
pp. 7123-7134 ◽  
Author(s):  
A. S. J Al-Saif ◽  
Takia Ahmed J Al-Griffi
Keyword(s):  
Exact Solution ◽  
Stokes Equation ◽  
Homotopy Perturbation Method ◽  
Differential Operators ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Adomian Decomposition ◽  
Splitting Time

We have proposed in this  research a new scheme to find analytical  approximating solutions for Navier-Stokes equation  of  one  dimension. The  new  methodology depends on combining  Adomian  decomposition  and Homotopy perturbation methods  with the splitting time scheme for differential operators . The new methodology is applied on two problems of  the test: The first has an exact solution  while  the other one has no  exact solution. The numerical results we  obtained  from solutions of two problems, have good convergent  and high  accuracy   in comparison with the two traditional Adomian  decomposition  and Homotopy  perturbationmethods . 

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