In
fluid dynamics, Couette flow is the flow of a
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
fluid in the space between two surfaces, one of which is moving
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
ially relative to the other. The relative motion of the surfaces imposes a
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
on the fluid and induces flow. Depending on the definition of the term, there may also be an applied
pressure gradient
In atmospheric science, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The p ...
in the flow direction.
The Couette configuration models certain practical problems, like the
Earth's mantle
Earth's mantle is a layer of silicate rock between the crust and the outer core. It has a mass of 4.01 × 1024 kg and thus makes up 67% of the mass of Earth. It has a thickness of making up about 84% of Earth's volume. It is predominantly so ...
and
atmosphere, and flow in lightly loaded
journal bearings. It is also employed in
viscometry
A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a spe ...
and to demonstrate approximations of
reversibility.
It is named after
Maurice Couette, a Professor of Physics at the French
University of Angers
The University of Angers (french: Université d'Angers; UA) is a public university in western France, with campuses in Angers, Cholet, and Saumur.
It forms part of thAngers-Le Mans University Community
History
The University of Angers was init ...
in the late 19th century.
Planar Couette flow
Couette flow is frequently used in undergraduate physics and engineering courses to illustrate
shear-driven fluid motion. A simple configuration corresponds to two infinite, parallel plates separated by a distance
; one plate translates with a constant relative velocity
in its own plane. Neglecting pressure gradients, the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
simplify to
:
where
is the spatial coordinate normal to the plates and
is the velocity field. This equation reflects the assumption that the flow is ''unidirectional'' — that is, only one of the three velocity components
is non-trivial. If the lower plate corresponds to
, the boundary conditions are
and
. The exact solution
:
can be found by integrating twice and solving for the constants using the boundary conditions.
A notable aspect of the flow is that
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
is constant throughout the domain. In particular, the first derivative of the velocity,
, is constant. According to
Newton's Law of Viscosity (Newtonian fluid), the shear stress is the product of this expression and the (constant) fluid
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
.
Startup
In reality, the Couette solution is not reached instantaneously. The "startup problem" describing the approach to steady state is given by
:
subject to the initial condition
: