In a body submerged in a fluid, unsteady forces due to acceleration of that body with respect to the 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. The Basset term accounts for viscous effects and addresses the temporal delay 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 for practical reasons; however, it can be substantially large when the body is accelerated at a high rate.
This force in an accelerating Stokes flow has been proposed by Joseph Valentin Boussinesq in 1885 and Alfred Barnard Basset in 1888. Consequently, it is also referred to as the BoussinesqâÂÂBasset force.
Consider an infinitely large plate started impulsively with a step change in velocityâÂÂfrom 0 to u<sub>0</sub>âÂÂin a direction parallel to the plateâÂÂfluid interface plane.
The equation of motion for the fluidâÂÂStokes flow at low Reynolds numberâÂÂis
where u(y,t) is the velocity of the fluid, at some time t, parallel to the plate, at a distance y from the plate, and v<sub>c</sub> is the kinematic viscosity of the fluid (c~continuous phase). The solution to this equation is,
where erf and erfc denote the error function and the complementary error function, respectively.
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 cumulative effect on the shear stress on the plate is
where u<sub>p</sub>(t) is the velocity of the plate, ÃÂ<sub>c</sub> is the mass density of the fluid, and ü<sub>c</sub> is the viscosity of the fluid.
Boussinesq (1885) and Basset (1888) found that the force F on an accelerating spherical particle in a viscous fluid is
where D is the particle diameter, and u and v are the fluid and particle velocity vectors, respectively.