1.

Segment V6.1: Shear Deformation
(Related to Textbook Section 6.1.3 - Angular Motion and Deformation)

Fluid elements located in a moving fluid move with the fluid and generally undergo a change in shape (angular deformation).

A small rectangular fluid element is located in the space between concentric cylinders. The inner wall is fixed. As the outer wall moves, the fluid element undergoes an angular deformation. The rate at which the corner angles change (rate of angular deformation) is related to the shear stress causing the deformation.

2.

Segment V6.2: Vortex In a Beaker
(Related to Textbook Section 6.5.3 - Vortex)

A flow field in which the streamlines are concentric circles is called a vortex.

A vortex is easily created using a magnetic stirrer. As the stir bar is rotated at the bottom of a beaker containing water, the fluid particles follow concentric circular paths. A relatively high tangential velocity is created near the center which decreases to zero at the beaker wall. This velocity distribution is similar to that of a free vortex, and the observed surface profile can be approximated using the Bernoulli equation which relates velocity, pressure, and elevation.

3.

Segment V6.3: Half-Body
(Related to Textbook Section 6.6.1 - Source In a Uniform Stream -- Half-Body)

Basic velocity potentials and stream functions can be combined to describe potential flow around various body shapes. The combination of a uniform flow and a source can be used to describe the flow around a streamlined body placed in a uniform stream. Streamlines created by injecting dye in steadily flowing water show a uniform flow. Source flow is created by injecting water through a small hole. It is observed that for this combination the streamline passing through the stagnation point could be replaced by a solid boundary which resembles a streamlined body in a uniform flow. The body is open at the downstream end and is thus called a halfbody.

4.

    
Segment V6.4: Potential Flow
(Related to Textbook Section 6.7 - Other Aspects of Potential Flow Analysis)

Flow fields for which an incompressible fluid is assumed to be frictionless and the motion to be irrotational are commonly referred to as potential flows.

Paradoxically, potential flows can be simulated by a slowly moving, viscous flow between closely spaced parallel plates. For such a system, dye injected upstream reveals an approximate potential flow pattern around a streamlined airfoil shape. Similarly, the potential flow pattern around a bluff body is shown. Even at the rear of the bluff body the streamlines closely follow the body shape. Generally, however, the flow would separate at the rear of the body, an important phenomenon not accounted for with potential theory.

5.

Segment V6.5: No-Slip Boundary Condition
(Related to Textbook Section 6.9.1 - Steady Laminar Flow Between Fixed Parallel Plates)

Boundary conditions are needed to solve the differential equations governing fluid motion. One condition is that any viscous fluid sticks to any solid surface that it touches.

Clearly a very viscous fluid sticks to a solid surface as illustrated by pulling a knife out of a jar of honey. The honey can be removed from the jar because it sticks to the knife. This no-slip boundary condition is equally valid for small viscosity fluids. Water flowing past the same knife also sticks to it. This is shown by the fact that the dye on the knife surface remains there as the water flows past the knife.

6.

Segment V6.6: Laminar Flow
(Related to Textbook Section 6.9.3 - Steady, Laminar Flow In Circular Tubes)

The velocity distribution is parabolic for steady, laminar flow in circular tubes. A filament of dye is placed across a circular tube containing a very viscous liquid which is initially at rest. With the opening of a valve at the bottom of the tube the liquid starts to flow, and the parabolic velocity distribution is revealed. Although the flow is actually unsteady, it is quasi-steady since it is only slowly changing. Thus, at any instant in time the velocity distribution corresponds to the characteristic steady-flow parabolic distribution.

7.

Segment V6.7: CFD Example
(Related to Textbook Section 6.10.1 - Numerical Methods)

Complex flows can be analyzed using the finite difference method in which the continuous variables are approximated by discrete values calculated at grid points. The governing partial differential equations are reduced to a set of algebraic equations which is solved by approximate numerical methods.

The flow past a circular cylinder at a Reynolds number of 10,000 was calculated using a finite difference method based on the Navier-Stokes equations. The O-shaped grid contained 100 by 400 grid points. Particle paths clearly show the Karmen vortex street. (Video courtesy of K. Kuwahara, Institute of Space and Astronautical Science, Japan)