The Velocity Gradient
When a fluid is moving through a pipe or a solid object is moving through a fluid, the layer of fluid in contact with the sides of the pipe or the surface of the object tends to be in the same state of motion as the object with which it is in contact; that is, the layer of fluid along the side of the pipe is at rest, while that in contact with the moving object is carried along at the same velocity as the object. If the difference in velocity between the fluid at the sides of the pipe and that at the center, or between the moving object and the fluid through which it is moving, is not too great, then the fluid flows in continuous, smooth layers; that is, the flow is laminar.
The difference in velocity between adjacent layers of the fluid is known as a velocity gradient and is given by v/x, where v is the velocity difference and x is the distance between the layers. To keep one layer of fluid moving at a greater velocity than the adjacent layer, a force F is necessary, resulting in a shearing stress F/A, where A is the area of the surface in contact with the layer being moved.
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