In a body submerged in a

fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

, unsteady forces due to

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

of that body with respect to the fluid, can be divided into two parts: the

virtual mass effect 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 ob ...

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 Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...

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

viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

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 Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advection, advec ...

has been proposed by

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 appoin ...

in 1885 and

Alfred Barnard Basset Alfred Barnard Basset FRS (25 July 1854 – 5 December 1930) was a British mathematician working on algebraic geometry, electrodynamics and hydrodynamics. In fluid dynamics, the Basset force—also known as the Boussinesq–Basset force—descr ...

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 a direction parallel to the plate–fluid interface plane. The equation of motion for the fluid—

at low

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

—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 Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

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 In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

and the

complementary error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as: \operatorname z = \frac\int_0^z e^\,\mathrm dt. The integral here is a complex Contour integrat ...

, 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 ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter rho), although the Latin letter ''D'' (or ''d'') ...

of the fluid, and ''μ_c'' is the

viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

of the fluid.

Acceleration of a spherical particle

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

spherical A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

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