fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

, the Basset–Boussinesq–Oseen equation (BBO equation) describes the motion of – and forces on – a small particle in

unsteady flow In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

at low

Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...

s. The equation is named after

Joseph Valentin Boussinesq Joseph Valentin Boussinesq (; 13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. Biography From 1872 to 1886, he was appoi ...

, Alfred Barnard Basset and

Carl Wilhelm Oseen Carl Wilhelm Oseen (17 April 1879 in Lund – 7 November 1944 in Uppsala) was a theoretical physicist in Uppsala and Director of the Nobel Institute for Theoretical Physics in Stockholm. Life Oseen was born in Lund, and took a Fil. Kand. degre ...

Formulation

The BBO equation, in the formulation as given by and , pertains to a small spherical particle of diameter

d_p

having mean

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

\rho_p

whose center is located at

\boldsymbol_p(t)

. The particle moves with Lagrangian velocity

\boldsymbol_p(t)=\text \boldsymbol_p / \textt

in a fluid of density

\rho_f

dynamic 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 inter ...

\mu

and Eulerian velocity field

\boldsymbol_f(\boldsymbol,t)

. The fluid velocity field surrounding the particle consists of the undisturbed, local Eulerian velocity field

\boldsymbol_f

plus a disturbance field – created by the presence of the particle and its motion with respect to the undisturbed field

\boldsymbol_f.

For very small particle diameter the latter is locally a constant whose value is given by the undisturbed Eulerian field evaluated at the location of the particle center,

\boldsymbol_f(t)=\boldsymbol_f(\boldsymbol_p(t),t)

. The small particle size also implies that the disturbed flow can be found in the limit of very small Reynolds number, leading to a drag force given by

Stokes' drag In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by so ...

. Unsteadiness of the flow relative to the particle results in force contributions by

added mass In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the obj ...

and the Basset force. The BBO equation states: :

\begin
  \frac \rho_p d_p^3 \frac
  &= \underbrace_  
  - \underbrace_ 
  + \underbrace_ 
  \\ &
  + \underbrace_ 
  + \underbrace_ .
\end

This is

Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

, in which the

left-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.rate of change of the particle's

linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...

, and the

right-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

s acting on the particle. The terms on the right-hand side are, respectively, the: # Stokes' drag, #

Froude–Krylov force In fluid dynamics, the Froude–Krylov force—sometimes also called the Froude–Kriloff force—is a hydrodynamical force named after William Froude and Alexei Krylov. The Froude–Krylov force is the force introduced by the unsteady pressure ...

due to the

pressure gradient In atmospheric science, the pressure gradient (typically of Earth's atmosphere, 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 particu ...

in the undisturbed flow, with

\boldsymbol

the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...

operator and

p(\boldsymbol,t)

the undisturbed pressure field, # added mass, # Basset force and # other forces acting on the particle, such as

gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...

, etc. The particle Reynolds number

R_e:

R_e = \frac

has to be less than unity,

R_e < 1

, for the BBO equation to give an adequate representation of the forces on the particle. Also suggest to estimate the pressure gradient from 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 ...

: :

-\boldsymbol p 
  = \rho_f \frac 
  - \mu \nabla^2 \boldsymbol_f,

with

\text \boldsymbol_f / \text t

the

material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...

\boldsymbol_f.

Note that in the Navier–Stokes equations

\boldsymbol_f(\boldsymbol,t)

is the fluid velocity field, while, as indicated above, in the BBO equation

\boldsymbol_f

is the velocity of the undisturbed flow as seen by an observer moving with the particle. Thus, even in steady Eulerian flow

\boldsymbol_f

depends on time if the Eulerian field is non-uniform.

Notes

References

* * {{DEFAULTSORT:Basset-Boussinesq-Oseen equation Equations of fluid dynamics