Body force

^{also see Surface force}
A body force is a force that acts on the volume of a body and also can be defined as an external force acting throughout the mass of a body. Gravity and magnetism are body forces, the forces associated with a non-inertial reference frame may be viewed as body forces.

The units on a body force are force per volume; compare this to pressure (a surface force), which has units of force per area. Formally, the body force is defined as:

$f={\frac {dF}{dV}}\,$

Where F is force and V is volume. Examples of common body forces include:

• Gravity
• Centrifugal force
• Coriolis force
• Acceleration felt when a vehicle changes velocity (as in braking).
• A pressure gradient (but not pressure alone).
• Magnetic fields exert a body force on magnetohydrodynamic fluids.[[1]]

Fluid Body Force A body force for a moving Fluid at constant Velocity is given as such

$f=\rho v^{2}a\,$

This is one of two components of force the other being the surface force Both forces together equal the total force from a moving fluid
Force_tot = f_s + f_b in Force units such as Newtons or Lbsf

Where $\rho$ is the density of the substance at a given point and (lowercase) f is the body force. This result may derived by applying Newton's law to a small volume and then taking the appropriate limit:

$\Delta F=\Delta ma\quad \Rightarrow \quad {\frac {\Delta F}{\Delta V}}={\frac {\Delta m}{\Delta V}}a\quad \Rightarrow \quad \lim _{\Delta V\to 0}\left({\frac {\Delta F}{\Delta V}}\right)=\lim _{\Delta V\to 0}\left({\frac {\Delta m}{\Delta V}}a\right)\quad \Rightarrow \quad {\frac {dF}{dV}}={\frac {dm}{dV}}a\,$

By definition, the derivative of mass with respect to volume is density; from this $f=\rho a$ follows. A body force doesn't necessarily involve acceleration. A prime example of such a body force is the action of gravity on an immobile object, other examples include various stresses in solids and pressure gradient of a steady fluid flow.

The Navier-Stokes equations are written in terms of body forces.