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Laminar Flow/Navier-Stokes Equations

Example A viscous fluid is contained between two infinitely long, vertical, concentric

cylinders as shown in the figure below. The outer cylinder has a radius b and is fixed. The

inner cylinder of radius a rotates with and angular velocity ω. Use the Navier-Stokes

equations to find the velocity distribution in the gap between the two cylinders.

Assume that the flow in the gap is axisymmetric (neither velocity nor pressure depend on the

angular position, θ, within the gap and that there are no components of velocity other than the

tangential component). The only body force is the weight of the fluid in the gap.

Strategy:

1. Use cylindrical coordinates, (r, θ, z) with the θ-component of the Navier-Stokes

equation to derive the equation of motion of the fluid in the gap.

Simplifications/Assumptions:

2. v_r, v_θ, and v_z are independent of θ since the flow is axisymmetric.

3. g_z = ˗ g

Boundary Conditions:

4. The boundary conditions are v_θ(b) = 0 and v_θ(a) = a ω