In a body submerged in a fluid, unsteady forces due to

acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...

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 In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...

development with changing relative velocity (acceleration) of bodies moving through a fluid. The Basset term accounts for

viscous 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 inte ...

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 Boussinesq–Basset force.

Acceleration of a flat plate

Consider an infinitely large plate started impulsively with a step change in velocity—from 0 to ''u₀''—in the direction of the plate–fluid interface plane. The equation of motion for the fluid— Stokes flow at low Reynolds number—is :

\frac=\nu_c\,\frac,

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_c'' is the

kinematic 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 int ...

of the fluid (c~continuous phase). The solution to this equation is, :

u=u_0 - u_0\, \operatorname\left(\frac\right) = u_0 \operatorname\left(\frac\right),

where erf and erfc denote the error function and the

complementary error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementar ...

, 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 :

\tau=\sqrt\int\limits_0^t\frac \, dt',

where ''u_p(t)'' is the velocity of the plate, ''ρ_c'' is the

mass 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. Mathematicall ...

of the fluid, and ''μ_c'' is the

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 inte ...

of the fluid.

Acceleration of a spherical particle

Boussinesq (1885) and Basset (1888) found that the force ''F'' on an accelerating

spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...

particle in a viscous fluid is :

\mathbf=\fracD^2\sqrt\int\limits_0^t\fracdt',

where ''D'' is the particle diameter, and ''u'' and ''v'' are the fluid and particle velocity vectors, respectively.

References

{{DEFAULTSORT:Basset Force Fluid dynamics

Acceleration of a flat plate

Acceleration of a spherical particle

See also

References