Basic HTML Version
217
First, we will define viscosity in precise terms.
Consider a liquid flowing through a narrow tube.
The liquid flow is caused by an applied force (F),
such as gravity, or forces applied by a pump.
Forces acting in the flow direction are called shear
forces. In fact, laminar flow is an example of shear
deformation.
We consider only laminar flow (as opposed to
turbulent flow), where the liquid can be regarded as
consisting of infinitely thin layers. Between adjacent
layers friction forces act, leading to the
characteristic flow profile with zero flow at the walls
and maximal flow in the centre of the tube. We thus have a velocity gradient
perpendicular to the direction of flow.
To simplify the situation we move from a tube to parallel plates, with liquid
between the plates. The lower plate is stagnant, the upper is pulled by a force
F in the x-direction. In result, liquid also flows in the x-direction. The flow rate
(v) depends on the distance (z) from the lower plate: v = v(z). The velocity
gradient is still perpendicular to the direction of flow. It is commonly called the
shear rate, and is given the symbol
.
γ
.
.
γ
=
dv
dz
The ratio between the force (F) and the area (A) upon which it works is given
the name shear stress, and has the symbol
τ
:
τ
=
A
F
Definition of viscosity (symbol:
η
):
The viscosity (shear viscosity) of a liquid is defined as the ratio between the
shear stress and the shear rate:
η
=
τ
γ
=
F
A
⎛
⎝⎜
⎞
⎠⎟
dv
dz
⎛
⎝⎜
⎞
⎠⎟
z
x