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[link] shows how viscosity is measured for a fluid. Two parallel plates have the specific fluid between them. The bottom plate is held fixed, while the top plate is moved to the right, dragging fluid with it. The layer (or lamina) of fluid in contact with either plate does not move relative to the plate, and so the top layer moves at v size 12{v} {} while the bottom layer remains at rest. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from v size 12{v} {} to 0 as shown. Care is taken to insure that the flow is laminar; that is, the layers do not mix. The motion in [link] is like a continuous shearing motion. Fluids have zero shear strength, but the rate at which they are sheared is related to the same geometrical factors A size 12{A} {} and L size 12{L} {} as is shear deformation for solids.

The figure shows the laminar flow of fluid between two rectangular plates each of area A. The bottom plate is shown as fixed. The distance between the plates is L. The top plate is shown to be pushed to right with a force F. The direction of movement of the layer of fluid in contact with the top plate is also toward right with velocity v. The fluid in contact with the plate in the bottom is shown to be in rest with v equals zero. As we see through the layers above the one on the bottom plate, each show a small displacement toward right in increasing order of value with the topmost layer showing the maximum.
The graphic shows laminar flow of fluid between two plates of area A size 12{A} {} . The bottom plate is fixed. When the top plate is pushed to the right, it drags the fluid along with it.

A force F size 12{F} {} is required to keep the top plate in [link] moving at a constant velocity v size 12{v} {} , and experiments have shown that this force depends on four factors. First, F size 12{F} {} is directly proportional to v size 12{v} {} (until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on v size 12{v} {} ). Second, F size 12{F} {} is proportional to the area A size 12{A} {} of the plate. This relationship seems reasonable, since A size 12{A} {} is directly proportional to the amount of fluid being moved. Third, F size 12{F} {} is inversely proportional to the distance between the plates L size 12{L} {} . This relationship is also reasonable; L size 12{L} {} is like a lever arm, and the greater the lever arm, the less force that is needed. Fourth, F size 12{F} {} is directly proportional to the coefficient of viscosity , η size 12{η} {} . The greater the viscosity, the greater the force required. These dependencies are combined into the equation

F = η vA L , size 12{F=η { { ital "vA"} over {L} } } {}

which gives us a working definition of fluid viscosity     η size 12{η} {} . Solving for η size 12{η} {} gives

η = FL vA , size 12{F=η { { ital "FL"} over { ital "vA"} } } {}

which defines viscosity in terms of how it is measured. The SI unit of viscosity is N m/ [ ( m/s ) m 2 ] = ( N/m 2 ) s or Pa s size 12{N cdot "m/" \[ \( "m/s" \) m rSup { size 8{2} } \] = \( "N/m" rSup { size 8{2} } \) "sorPa" cdot s} {} . [link] lists the coefficients of viscosity for various fluids.

Viscosity varies from one fluid to another by several orders of magnitude. As you might expect, the viscosities of gases are much less than those of liquids, and these viscosities are often temperature dependent. The viscosity of blood can be reduced by aspirin consumption, allowing it to flow more easily around the body. (When used over the long term in low doses, aspirin can help prevent heart attacks, and reduce the risk of blood clotting.)

Laminar flow confined to tubes—poiseuille’s law

What causes flow? The answer, not surprisingly, is pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate Q size 12{Q} {} is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as

Q = P 2 P 1 R , size 12{Q= { {P rSub { size 8{2} } - P rSub { size 8{1} } } over {R} } } {}

where P 1 size 12{P rSub { size 8{1} } } {} and P 2 size 12{P rSub { size 8{2} } } {} are the pressures at two points, such as at either end of a tube, and R size 12{R} {} is the resistance to flow. The resistance R size 12{R} {} includes everything, except pressure, that affects flow rate. For example, R size 12{R} {} is greater for a long tube than for a short one. The greater the viscosity of a fluid, the greater the value of R size 12{R} {} . Turbulence greatly increases R size 12{R} {} , whereas increasing the diameter of a tube decreases R size 12{R} {} .

Questions & Answers

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Timothy
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Timothy
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Timothy
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corona charge can verify
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Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Chidalu
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Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
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Chidalu
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Chidalu
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Sean Reply
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Practice Key Terms 5

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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