Basset force

Unsteady forces due to acceleration of the relative velocity of a body submerged in a fluid can be divided into two parts: the virtual mass effect and the Basset force.

The Basset force term describes the force due to the lagging boundary layer development with changing relative velocity (acceleration) of bodies moving through a fluid [ C. Crowe et al., Multiphase flows with droplets and particles, CRC Press, 1998, ISBN 0-8493-9469-4, p. 81] .The Basset term accounts for viscous effects and addresses the temporal delay in in boundary layer development as the relative velocity changes with time. It is also known as the "history" term.The Basset force is difficult to implement and is commonly neglected from practical reasons, however, it can be substantially large when the body is accelerated at a high rate [R.W. Johnson, The handbook of fluid dynamics, CRC Press, 1998, ISBN 0-8493-2509-9, p.18-3] .

Acceleration of a flat plate

Consider an infinitely large plate started impulsively with a step change in velocity from $0$ to $u\_0$, in the direction of the plate-fluid interface plane.

The equation of motion for the fluid is

:$frac\{partial\; u\}\{partial\; t\}=\; u\_cfrac\{partial^2u\}\{partial\; y^2\}$,

where $u$ is the velocity of the fluid, at some time $t$, parallel to the plate, at a distance $y$ from the plate, and $u\_c$ is the kinematic viscosity of the fluid (c~continous phase).The solution to this equation is

:$u=u\_0erfleft(frac\{y\}\{2sqrt\{\; u\_ct\; ight)$,

where $erf$ denotes the error function.

Assuming that an acceleration of the plate can be broken up into a series of such step changes in the velocity, it can be shown that the cummulative effect on the shear stress on the plate is

:$au=sqrt\{frac\{\; ho\_cmu\_c\}\{piintlimits\_0^tfrac\{frac\{partial\; u\}\{partial\; t\text{'}\{sqrt\{t-t\text{'}dt\text{'}$,

where $ho\_c$ is the mass density of the fluid, and $mu\_c$ is the viscoisty of the fluid.

Acceleration of a spherical particle

Basset found that the force on a spherical particel is [A.B. Basset, Treatise on hydrodynamics, vol. 2 (original publication 1888), Deighton, Bell and Co., Cambridge, 1961]

:$mathbf\{F\}=frac\{3\}\{2\}D^2sqrt\{pi\; ho\_cmu\_c\}intlimits\_0^tfrac\{frac\{Dmathbf\{u\{Dt\text{'}\}-frac\{partialmathbf\{v\{partial\; t\text{'}\{sqrt\{t-t\text{'}dt\text{'}$,

where $D$ is the particle diameter, $frac\{D\}\{Dt\text{'}\}$ is the material derivative, and $mathbf\{u\}$ and $mathbf\{v\}$ are the fluid and particle velocity vectors, respectively.

References