Shear stress is proportional to speed gradient¶
An element of flowing fluid will endure forces from the surrounding fluid (stress forces) that will gradually deform the element over time. In a steady (laminar) flow, these stress forces occur between fluid layers. For an incompressible and isotropic fluid, the shear stress exerted to an element of fluid is proportional to the spatial derivative of the velocity component that is perpendicular to the velocity vector (and as such, parallel to the direction of shear). This law is also called the Newton’s law of viscosity, and the fluids that follow it are said to be Newtonian.
As an example of this law, consider two solid flat plates that contain water in between.
The bottom plate is fixed in place, while the top plate moves parallel to the bottom one with
a small speed u such that the flow of the water is steady. If we were to measure the force that
needed to make the top plate continue to move, we would find that it is proportional to the area
of the plate and the ratio \(\frac{u}{d}\) where \(d\) is the distance between the plates.
Conditions:
The flow is one-dimensional. For a two-dimensional flow, replace the derivative with a sum of partial derivatives with respect to both perpendicular directions.
The fluid is incompressible and isotropic.
The fluid flow is steady (laminar).
Links:
- shear_stress¶
Shear stress. See Pressure from force and area.
- Symbol:
tau- Latex:
\(\tau\)
- dynamic_viscosity¶
Dynamic viscosity of the fluid.
- Symbol:
eta- Latex:
\(\eta\)
- fluid_speed¶
Fluid speed as a function of position perpendicular to the fluid flow, or layer position.
- Symbol:
u(y)
- layer_position¶
Layer position, or position in the direction perpendicular to fluid velocity.
- Symbol:
y
- law¶
tau = eta * Derivative(u(y), y)- Latex:
- \[\tau = \eta \frac{d u}{d y}\]