. It is also
typical for to
vary strongly with temperature of the liquid.Go directly to derivation of several forms for the viscous force.
Liquids hold themselves together. The forces that hold liquid drops together are much weaker than the forces that hold a solid together, yet they are significant. When astronauts play with drops of weightless water, we observe these forces in action. When a board is pulled across the surface of a fluid, these same forces modify the motion of the board.
The liquid that touches the board wets it, and moves at the same speed as the board. The board transfers momentum to the top layer of the liquid. This layer of liquid communicates with the liquid below and transfers momentum downward. Force by the board on the liquid is required to cause this change in the momentum of the liquid. The third law tells us that the liquid in turn exerts a force on the board, as it absorbs momentum. This force by the liquid on the moving board is called the viscous force.
If the liquid is already moving at the speed of the board, it continues at that speed; no momentum is transferred to that liquid. Most commonly, the fluid far below the board is moving slowly, if at all, while the fluid near the board moves at the same speed as the board. The greater the difference between the speed of neighboring liquid layers, the more effectively is momentum transferred from the "fast" to the "slow" layers. In the end, the force on the board is proportional to the spatial derivative of the liquid velocity field; along the direction perpendicular to the board surface.
The force is also proportional to the strength with which neighboring liquid
layers bond to each other. This strength is represented by a coefficient of
viscosity. Typically every different liquid has its own value for the
coefficient of viscosity,
. It is also
typical for to
vary strongly with temperature of the liquid.
Finally the force is proportional to the area,
, of the board
which is in contact with the liquid. The force on the board is directed
opposite to the board velocity across the water. Its magnitude may be written
where the y direction is perpendicular to the surface of the board.
The same equation holds if a stationary board is held to the surface of a flowing fluid. In this case, of course, the direction of the force on the board is the same as the direction of the fluid velocity. If the board is submerged, in the flowing fluid, the force on it increases because the area of contact has increased.
We are interested in adding the viscous force per unit volume, on the fluid to Euler's equation. To do this we imagine a thin slab of fluid which is carefully oriented in the velocity field of the fluid: The velocity field is the same at each point on the bottom surface, and is parallel to the bottom surface. The velocity field is the same at each point on the top surface, and is parallel to the bottom surface. (The velocity fields on the top and bottom need not be equal to each other.) Velocity fields that allow this construction are said to represent "laminar flow," meaning layered flow.
The viscous force may be calculated in the following representative case: The fluid velocity is to the right in the x direction, and the y direction is perpendicular to the surface of the fluid slab. The fluid slab is moving at the average of the two fields above and below it. The result for the net viscous force on the slab, by neighboring fluid on the top and bottom is
Note that if the first derivative of the velocity is a positive constant, the net force is zero. The top of the fluid slab is pulled towards the right by the faster fluid above. The bottom of the fluid slab is pulled to the left by the slower fluid below. The forces add to zero unless the second derivative of the velocity is non zero. If both the first and second derivatives are positive, the upper surface is pulled more strongly to the right.
This result can be generalized to laminar velocity fields with more complex nature. The result is
If this force per unit volume is added to Euler's equation, the name changes to the Navier-Stokes equation:
This equation can be used to show that the velocity distribution in a pipe is parabolic; zero at the walls and maximum at the center. The calculated velocity distribution can be used to determine the rate of flow, with units volume per second, through a pipe of radius R:
The derivative is the pressure gradient along direction of flow. The equation is known as Poiseuille's Law. Its most remarkable feature is the very strong dependence of flow rate on radius. Doubling the diameter of the pipe will increase the flow rate by a factor of 16.
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