# What is the body force in the Navier Stokes Equations?

The incompressible Navier Stokes equations are:

$\rho(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}) = -\frac{\partial p}{\partial x_i} + \mu\frac{\partial^2 u_i}{\partial x_j \partial x_j} + f_i$ for $i =1,2,3$

Reading around I have gathered that the force $f_i$ is a body force which can be that due to gravity, or some other force on fluid due to the presence of a body.

My question is, isn't this force explained by pressure gradients? I.e. if there is a body in the flow (like a wing say) then there will be change in pressure of the flow as it comes close to the wing, which is effectively the influence of the wing on the flow. So why do we include this force term in the equations (assuming we dont care about gravity or any other external forces apart from those due to surfaces in the flow?)

• The body force term is there because of hydrostatics. The pressure gradient has to balance the gravity, otherwise the fluid particle would move. – jjack Aug 16 '15 at 19:37

First, Navier-Stokes governs the fluid in your setup. So, anything apart from the fluid will be an external force in N-S equation.

Body-force means an external force that applies in the bulk of the fluid, like gravity or a magnetic force.

Interaction with a "body", as a wing, which is external to the fluid domain, is done through boundary conditions : the integral of the total stress along the normal to the boundary with the wing will give you the force exerted by the wing on the fluid.

• That is what I initially thought, but when for instance in Acoustics when one derives the inhomogeneous wave equation for a dipole source term, one includes a body force which is the effect of some body acting on the fluid (which is derived from the Navier Stokes equation) see this and look at equation 1.3.7 So is the force of a body then really only a consequence of boundary conditions? – Dipole Apr 16 '14 at 15:31
• From your source: "When these terms are absent the equation describes sound propagation from sources on the boundaries of the ﬂuid, such as the vibrating cone of a loudspeaker." That is the same as Joce states. – Bernhard Apr 18 '14 at 15:42

The distinction between body force and surface force on a fluid's control volume (say a cube) is that in the integral form of the momentum equation, the body force is integrated with respect to the volume of the cube, but the surface force is integrated with respect to the surface area. Pressure is a surface force because it acts normal to the unit surface area of the cube. Gravity is a body force because it acts on the volume of the cube.

Maybe the force $f_i$ can be thought of as being defined by the navier stokes equations:

$f_i = \rho(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}) +\frac{\partial p}{\partial x_i} - \mu\frac{\partial^2 u_i}{\partial x_j \partial x_j}$ for $i =1,2,3$

In this sense a body force will be zero if momentum is balanced by pressure and viscous stresses. I guess it would be like the wing in the question being swept away by the flow instead of being fixed and thus imparting a force on the flow.

• There is no wing in the Navier-Stokes equations. The wing is a boundary condition to fluid described by the equation, described by zero velocity on the surface. – jjack Aug 16 '15 at 19:43