# What really is “inertial force”?

In Fluid Mechanics we often see the term inertial force when discussing Reynolds number. The problem is, I didn't really get what's this inertial force. Basically, the notion of inertia I have is that given by Newton's laws where we think of inertia as the resistance of a body to change its state of motion.

This inertial force, in Fluid Mechanics seems to be associated to the left hand side of Navier-Stokes equation:

$$\rho \left(\dfrac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot \nabla )\mathbf{u}\right) = -\nabla p+(\lambda + \mu)\nabla(\nabla\cdot\mathbf{u}) + \mu \nabla^2\mathbf{u}$$

But I don't really get why is that. So, what really is inertial force in a more general context? And how, in Fluid Mechanics, we associated inertial force with the left hand side of Navier-Stokes equation?

EDIT: there's the following piece of text on Wikipedia's article

A fictitious force, also called a pseudo force,[1] d'Alembert force[2][3] or inertial force,[4][5] is an apparent force that acts on all masses whose motion is described using a non-inertial frame of reference, such as a rotating reference frame..

This makes clearer what is one inertial force in general. But I can't see how this relates to that term in Navier-Stokes equation. I don't get why one would consider that to have any relationship with description of the motion using a non-inertial frame.

## 3 Answers

This is d'Alembert's principle. The basic, very general idea is to take Newton's second law applied to an accelerating mass, and write it as $F-ma=0$. That is, we take the $ma$ term and pretend it's another force balancing the $F$ term. This allows us to think about the dynamic, accelerating mass as if it's a static system. The $ma$ term is what's referred to as an "inertial force".

We can of course do the same thing with the Navier-stokes equations. If we move the left-hand side (as you've written it) over to the right, it looks like a collection of forces-per-unit-volume that sum to zero.

I'm not sure of the historical reasons that caused this to be referred to as an "inertial force" in fluid mechanics much more than in other fields. However, the significance of the Reynolds number must surely have played a role. Since the ratio of the $ma$-derived term to the $F$-related terms is so important, thinking of them as being in some way "the same sort of thing" makes sense.

In this case, by inertial force, they do not refer to the pseudoforce from an non intertial reference frame. Instead, by inertial force, they refer to the force due to the momentum of the fluid. This is usually expressed in the momentum equation by the term (ρv)v. So, the denser a fluid is, and the higher its velocity, the more momentum (inertia) it has.

As a label, a force is sometimes called inertial if it is proportional to the mass of the object the force is applied to. Fictitious forces due to accelerating frames are an example because they are really just the opposite of the acceleration of the frame itself. Gravitational forces are "another" example.

So if you are in a frame moving with a fluid, you expect there to be inertial forces if the frame/fluid itself is accelerating.