Oseen equations
Encyclopedia
In fluid dynamics
, the Oseen equations (or Oseen flow) describe the flow of a viscous
and incompressible fluid
at small Reynolds numbers, as formulated by Carl Wilhelm Oseen
in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, by the partial inclusion of convective acceleration.
Using the Oseen equation, Horace Lamb
was capable to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law
towards somewhat higher Reynolds numbers. As well as for the first time deriving a solution for the viscous flow around a circular cylinder.
The Oseen equations are, in case of an object moving with a steady velocity
U through the fluid — which is at rest far from the object — and in a frame of reference
attached to the object:
where
The boundary conditions for the Oseen flow around a rigid object are:
with r the distance from the object's center, and p∞ the undisturbed pressure far from the object.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the Oseen equations (or Oseen flow) describe the flow of a viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
and incompressible fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
at small Reynolds numbers, as formulated by Carl Wilhelm Oseen
Carl Wilhelm Oseen
Carl Wilhelm Oseen was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm....
in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, by the partial inclusion of convective acceleration.
Using the Oseen equation, Horace Lamb
Horace Lamb
Sir Horace Lamb FRS was a British applied mathematician and author of several influential texts on classical physics, among them Hydrodynamics and Dynamical Theory of Sound...
was capable to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law
Stokes law
Stokes' law can refer to:*Stokes' law for friction force*Stokes' law law describing attenuation of sound in Newtonian liquidsFor integration, see Stokes' theorem....
towards somewhat higher Reynolds numbers. As well as for the first time deriving a solution for the viscous flow around a circular cylinder.
The Oseen equations are, in case of an object moving with a steady velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
U through the fluid — which is at rest far from the object — and in a frame of reference
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
attached to the object:
where
- u is the disturbance velocity induced by the moving object, i.e. the total flow velocity in the frame of reference moving with the object is –U+u,
- p is the pressurePressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
, - ρ is the densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the fluid, - μ is the dynamic viscosity,
- ∇ is the gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
operator, and - ∇2 is the Laplace operatorLaplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
.
The boundary conditions for the Oseen flow around a rigid object are:
with r the distance from the object's center, and p∞ the undisturbed pressure far from the object.