# 3D Velocity Profile for Flow between Two Cylinders

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This Demonstration simulates the solution of the Navier–Stokes equations for radially symmetric steady-state incompressible fluid flow between two long cylinders, with external constant pressure gradient. At the boundary conditions, the outer cylinder has zero velocity while the inner cylinder is moving with a constant velocity.

Contributed by: Peeter Joot (April 2012)

Open content licensed under CC BY-NC-SA

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## Details

The Navier–Stokes equations for radially symmetric steady-state incompressible fluid flow between two long cylinders with external constant pressure gradient has the form

.

This can be solved analytically. The solution depends on the boundary conditions, which require the velocity of the fluid at the fluid and cylinder interface to match (the no-slip condition). Here the profile of the resulting velocity field is illustrated for boundary conditions in which the outer cylinder is fixed, and the inner cylinder is moving with velocity . If and are the inner and outer cylinder radii, respectively, those boundary conditions can be written

,

,

from which this solution can be found:

.

This Demonstration shows how the variation of the shape of this resulting velocity profile depends on both the inner cylinder velocity and the pressure gradient to viscosity () ratio.

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