3D Velocity Profile for Flow between Two Cylinders

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.


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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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