Body force

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A body force is a force that acts on the volume of a body. 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{d F}{d V}\,


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

In general, any acceleration that a body undergoes will cause a body force given by:


f = \rho a\,


Where ρ 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 m a \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{d F}{d V} = \frac{d m}{d V} a\,


By definition, the derivative of mass with respect to volume is density; from this f = ρa 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 body force induced by a force field may be expressed by the application of the chain rule:


f = \frac{d F}{d V} = \frac{\partial F}{\partial x}\frac{d x}{d V} + \frac{\partial F}{\partial y}\frac{d y}{d V} + \frac{\partial F}{\partial z}\frac{d z}{d V} + \frac{\partial F}{\partial t}\frac{d t}{d V} = \frac{1}{L_x^2} \frac{\partial F}{\partial x} + \frac{1}{L_y^2} \frac{\partial F}{\partial y} + \frac{1}{L_z^2} \frac{\partial F}{\partial z} + \frac{\cfrac{\partial F}{\partial t}}{\cfrac{d V}{d t}} \,


Where Lx, Ly, Lz are the scales of the axes (meters, for example).


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

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