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

Definition

The 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:

$\tau \propto \dfrac{F}{A}$

Using SI coherent units,

$\tau = \dfrac{F}{A}$

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 m².

Velocity gradient

The difference in velocity between adjacent layers of a moving fluid is called the velocity gradient.

Using SI coherent units,

$\text{velocity gradient} = \dfrac{v}{x}$

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,

$\eta = \dfrac{F/A}{v/x}$

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 m².
^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:

$\dfrac{F}{A} = \eta \ \dfrac{v}{x}$

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:

$F = m \ \dfrac{v}{t}$