Stokes, who had studied the falling of a sphere through a viscous fluid.
A fundamental property of Oseen's equation is that the general solution can be split into longitudinal and transversal waves.
is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out.
Velocity is derived from potential theory and pressure is from linearized Bernoulli's equations.
For certain Oseen flows, further splitting of transversal wave into irrotational and rotational component is possible
The fundamental solution due to a singular point force embedded in an Oseen flow is the Oseenlet.
The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian[4] and micropolar[5] fluids.
Using the Oseen equation, Horace Lamb was able to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law towards somewhat higher Reynolds numbers.
is the given vector, which gives the direction of the singular force, then in the absence of boundaries, the velocity and pressure is derived from the fundamental tensor
is arbitrary function of space, the solution for an unbounded domain is
When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed.
Oseen substituted the following flow velocity values into the Navier-Stokes equations.
Inserting these into the Navier-Stokes equations and neglecting the quadratic terms in the primed quantities leads to the derivation of Oseen's approximation:
and finally, when Stokes' solution was solved on the basis of Oseen's approximation, it showed that the resultant drag force is given by
Consider the case of a solid sphere moving in a stationary liquid with a constant velocity.
The liquid is modeled as an incompressible fluid (i.e. with constant density), and being stationary means that its velocity tends towards zero as the distance from the sphere approaches infinity.
For a real body there will be a transient effect due to its acceleration as it begins its motion; however after enough time it will tend towards zero, so that the fluid velocity everywhere will approach the one obtained in the hypothetical case in which the body is already moving for infinite time.
Thus we assume a sphere of radius a moving at a constant velocity
direction and its magnitude is equivalent to the stream function used in two-dimensional problems.
is more similar to its derivation from the stream function in the two-dimensional case (in polar coordinates).
.is the polar angle originated from the opposite side of the front stagnation point (
These p and u satisfy the equation of motion and thus constitute the solution to Oseen's approximation.
One may question, however, whether the correction term was chosen by chance, because in a frame of reference moving with the sphere, the fluid near the sphere is almost at rest, and in that region inertial force is negligible and Stokes' equation is well justified.
[6] Far away from the sphere, the flow velocity approaches u and Oseen's approximation is more accurate.
The method and formulation for analysis of flow at a very low Reynolds number is important.
The slow motion of small particles in a fluid is common in bio-engineering.
Oseen's drag formulation can be used in connection with flow of fluids under various special conditions, such as: containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens.
It can be used in a number of applications, such as smog formation and atomization of liquids.
A vessel of diameter of 10 µm with a flow of 1 millimetre/second, viscosity of 0.02 poise for blood, density of 1 g/cm3 and a heart rate of 2 Hz, will have a Reynolds number of 0.005 and a Womersley number of 0.0126.
At these small Reynolds and Womersley numbers, the viscous effects of the fluid become predominant.