ABSTRACT
We study the axisymmetric flow governed by the Navier-Stokes equations, in tubes with abrupt changes of diameter. Our first concern is the asymptotic behaviour of the solution near the corners.
The rotational symmetry allows us to restrict the problem to two dimensions using the cylindrical coordinates r, z. In the case of the Stokes flow the stream function ψ then satisfies the equation 1 r ( ∂ 4 ψ ∂ z 4 + 2 ∂ 4 ψ ∂ z 2 ∂ r 2 + ∂ 4 ψ ∂ r 4 ) − 1 r 2 ( ∂ 3 ψ ∂ z 2 ∂ r + ∂ 3 ψ ∂ r 3 ) + 1 r 3 ∂ 2 ψ ∂ r 2 − 3 r 4 ∂ ψ ∂ r = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq249.tif"/>
This equation is different from that for the stream function in desk geometry, where the asymptotic behaviour of the solution near the corners is known (cf., e.g., Kondratiev and Olejnik [ 15 ]). In our paper we transform the equation into the polar coordinates ρ, ϑ. Using the Fourier transform technique and some of the results of Kondratiev and Olejnik [ 15 ], we obtain the asymptotic behaviour of the stream function ψ near the corner. E.g., for the internal angle of 3 2 π https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq250.tif"/> there exists a function φ, independent of the radius ρ, such that ψ ( ρ , ϑ ) = ρ 1.54448374 φ ( ϑ ) + … , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq251.tif"/>
which means that the velocity components v 1, v 2 behave near the corner like v l ( ρ , θ ) = ρ 0.54448374 φ l ( θ ) + … , l = 1 , 2. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203756034/6b58283d-8100-4221-bafc-5f6bf7d0c46b/content/eq252.tif"/>
These asymptotics agree with those obtained for planar Stokes flow.
In the second part of the paper we deal with the numerical solution of flow of the incompressible fluid in a tube with sharp changes of diameter. Our aim is to 42make use of the information on the local behaviour of the solution near the corner point, in order to suggest local meshing in correspondence with the asymptotics. We present a cheap strategy for various families of triangular elements. For linear and quadratic elements we show tables of adequate meshings.
