pascal second
SI coherent derived unit whose name and symbol includes an SI coherent derived unit with a special name and symbol
Name | Symbol | Quantity | Expressed in terms of SI base units | |
pascal second | Pa s | dynamic viscosity | kg m^{−1} s^{−1} | |
DefinitionThe pascal second, symbol Pa s, is the SI coherent derived unit of dynamic viscosity. |
Viscosity
Viscosity is the quantity that measures a fluid’s resistance to flow. It is the inverse of fluidity, the quantity that measures ease of flow.
Treacle has a higher viscosity than water, and is described as being more viscous. Water in turn is more viscous than air.
The extent to which a fluid resists the relative motion of an object moving through it is dependent on the fluid’s viscosity. The viscosity of a fluid also determines the extent of its resistance to the motion of layers within it that are moving at different relative velocities.
In a moving fluid, for one layer of fluid to maintain a greater velocity than an adjacent layer, a shear force is required, resulting in a shear stress.
Shear stress
In contrast to conventional stress, which arises from a force perpendicular to the material cross section on which it acts, a shear stress arises from a force parallel to the material cross section. The force that produces the shear stress is called a shear force.
Shear stress is proportional to the shear force, and inversely proportional to the cross-sectional area of material to which the shear force is applied:
Using SI coherent units,
where:
- τ is the shear stress in pascals, symbol Pa,
- F is the shear force in newtons, symbol N,
- A is the cross-sectional area of material (parallel to the applied force) in square metres, symbol m^{2}.
Velocity gradient
The difference in velocity between adjacent layers of a moving fluid is called the velocity gradient.
Using SI coherent units,
where:
- v is the velocity difference in metres per second, symbol m s^{-1}.
- x is the distance between the layers in metres, symbol m,
Dynamic viscosity
Dynamic viscosity is the ratio of the shear stress to the velocity gradient between adjacent layers in a fluid.
Using SI coherent units,
where:
- η is the dynamic viscosity in pascal seconds, symbol Pa s,
- F is the shear force in newtons, symbol N,
- A is the cross-sectional area of material (parallel to the applied force) in square metres, symbol m^{2}.
- ^{F}⁄_{A} is the shear stress in pascals, symbol Pa,
- v is the velocity difference in metres per second, symbol m s^{-1},
- x is the distance between the layers in metres, symbol m,
- ^{v}⁄_{x} is the velocity gradient in reciprocal seconds, symbol s^{-1}.
Newton’s equation
The equation defining dynamic viscosity can be re-arranged to form Newton’s equation, which states that the resulting shear of a fluid is directly proportional to the force applied and inversely proportional to its viscosity:
This relationship is analogous to Newton’s second law of motion which states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force: