0.7 Laminar flows with dependence on one dimension

• Couette flow
  • Planar Couette flow
  • Cylindrical Couette flow
  • Planer rotational Couette flow
• Hele-Shaw flow
• Poiseuille flow
  • Friction factor and Reynolds number
  • Non-Newtonian fluids
• Steady film flow down inclined plane
• Unsteady viscous flow
  • Suddenly accelerated plate
  • Developing Couette flow

Reading assignment

One-dimensional (1-D) flow fields are flow fields that vary in only one spatial dimension in Cartesian coordinates. This excludes turbulent flows because it cannot be one-dimensional. Acoustic waves are an example of 1-D compressible flow. We will concern ourselves here with incompressible 1-D flow fields that result from axial or planar symmetry. Cartesian, 1-D incompressible flows do not have a velocity component (other than possibly a uniform translation) in the direction of the spatial dependence because of the condition of zero divergence. Thus the nonlinear convective derivative disappears from the equations of motion in Cartesian coordinates. They may not disappear with curvilinear coordinates.

We can demonstrate that this relation may not apply in curvilinear coordinates by considering an example with cylindrical polar coordinates. Suppose that the only nonzero component of velocity is in the $\theta$ direction and the only spatial dependence is on the $r$ coordinate. The radial component of the convective derivative is non-zero due to centrifugal forces.

The flows can be classified as either forced flow resulting from the gradient of the pressure or the potential of the body force or induced flow resulting from motion of one of the bounding surfaces.

Some flow fields that result in 1-D flow are listed below and illustrated in the following figure (Churchill, 1988)

1. Forced flow through a round tube
2. Forced flow between parallel plates
3. Forced flow through the annulus between concentric round tubes of different diameters
4. Gravitational flow of a liquid film down an inclined or vertical plane
5. Gravitational flow of a liquid film down the inner or outer surface of a round vertical tube
6. Gravitational flow of a liquid through an inclined half-full round tube
7. Flow induced by the movement of one of a pair of parallel planes
8. Flow induced in a concentric annulus between round tubes by the axial movement of either the outer or the inner tube
9. Flow induced in a concentric annulus between round tubes by the axial rotation of either the outer or the inner tube
10. Flow induced in the cylindrical layer of fluid between a rotating circular disk and a parallel plane
11. Flow induced by the rotation of a central circular cylinder whose axis is perpendicular to parallel circular disks enclosing a thin cylindrical layer of fluid
12. Combined forced and induced flow between parallel plates
13. Combined forced and induced longitudinal flow in the annulus between concentric round tubes
14. Combined forced and rotationally induced flow in the annulus between concentric round tubes