# Oseen's Approximation

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Oseen's Approximation

In 1910 Carl Wilhelm Oseen proposed Oseen's Approximation to treat problems in which a flow field consists of small disturbance of a constant mean flow. His work was based on the experiments of Navier Stokes. Stokes studied a sphere of radius “a” falling in a fluid of viscosity ( $\mu\,$ ). Oseen developed a correction term, which included inertial factors, for the velocity used in Stokes Calculations to solve the problem. His approximation lead to a solution to Stokes Calculations.

## Importance

The method and formulation for analysis of flow at 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 fluid containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens. The fluid does not even have to be a liquid, and the particles don’t have to be solid. It can be used in a number of applications such as smog formation and atomization of liquids.

## Bio-Engineering Application

Blood flow in small vessels, such as capillaries, is characterized by small Reynolds and Womersley numbers. A vessel of diameter of 10 µm with a flow of 1mm/s, viscosity of 0.02 poise for blood, density of 1 g/cm3 and a heart rate of 2Hz, 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. Understanding the movement of these particles is essential for drug delivery and studying metastasis of cancers.

## Calculations

Oseen considered the sphere to be stationary and the fluid to be flowing with a velocity (U) at an infinite distance from the sphere.Inertial terms were neglected in stokes’ calculations. It is a limiting solution when the Reynolds number tends to zero. When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed. Oseen substituted the following velocity values into the Navier-Stokes' equation.

v1 = U + $v_1^'$, v2 = v2', v3 = v3'

Inserting these into the Navier-Stokes’ equation and neglected the quadratic terms in the primed quantities, lead to the derivation of Oseen’s approximation

U ${\partial v_1'\over\partial x_1}$ = $\frac{-1}{\rho\,}$ ${\partial p\over\partial x_1}$ + v $\triangledown^2$vi' (i =1,2,3)

When Stokes’ solution was solved on the basis of Oseen's approximation it showed that the resultant hydrodynamic force (drag) is given by $F= 6\pi\,\mu\,a U\left(1+{3\over 8} N_R\right)$

Where:

• NR is the Reynolds number
• F is the hydrodynamic force
• U is the flow
• a is the radius of the sphere
• $\mu\,$ is the fluid viscosity

The force from Oseen's equation differs from Strokes’ by a factor of $\left({3\over 8}\right) N_R$.

## Error in Stokes' Solution $\triangledown v' ~ = 0$ $v \triangledown v'$ = $- \triangledown p + v \triangledown ^2 v'$.

but when the velocity ﬁeld is :

vy =U $\cos \theta\,$(1 + ${a^3 \over 2r^3}$ ${3a \over 2r})$

vz = − U $\sin \theta\,$ (1- ${a^3 \over 4r^3}$ ${3a \over 4r})$

In the far ﬁeld ${r\over a}$ >> 1, the viscous stress is dominated by the last term. That is : $\triangledown v'$ = $0 \left({a^3\over r^3}\right)$

The inertia term is dominated by the term :

U ${\partial v'\over\partial z_1}$ = $0 \left({a^2 \over r^2}\right)$

The error is then given by the ratio :

U ${{\partial v'\over \partial z_1} \over {v \triangledown v'}}$ = $0 \left({r \over a} \right)$

this becomes unbounded for ${r \over a}$ >> 1. therefore the inertia cannot be ignored in the far ﬁeld. By taking the curl, Stokes equation gives $\triangledown ^2 \zeta\,= 0$ Since the body is a source of vorticity, $\zeta\,$ would become unbounded logarithmically for large ${r \over a}$. This is certainly unphysical and is known as Stokes’ paradox.

## Modifications 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 Strokes' equation is well justified. Far away from the sphere the flow velocity approaches U and the Oseen’s approximation is more accurate. But Oseen's equation was obtained applying the equation for the entire flow field. This question was answered by Proudman and Pearson in 1957, who solved the Navier Strokes’ equation and gave an improved Strokes’ solution in the neighborhood of the sphere and an improved Oseen’s solution at infinity, and matched the two solutions in a supposed common region of their validity. They obtained $F= 6\pi\,\mu\,a U\left(1+{3\over 8} N_R + {9\over 40}*N_R^2 ln N_R+0( N_R^2)\right)$

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