# Reynolds number and inertial force

The Reynolds number is defined as the ratio of ´inertial´ forces to viscous forces.

$$Re = \frac{\text{Inertial Forces}}{\text{Viscous Forces}}$$

Now, viscous forces make sense to me. They are frictional shear forces that come about due to the relative motion of the different layers in a flowing fluid, resulting in different amount of friction, hence, different viscosity values.

However, I am not really sure how to think about the 'inertial force'. This, to me, is somewhat of a dynamic effect since large Re numbers indicate turbulence in most cases, where there is a lot of motion, vortices and eddies. But what exactly is the inertial force and how can it be explained physically?

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Inertial force, as the name implies is the force due to the momentum of the fluid. This is usually expressed in the momentum equation by the term $(\rho v)v$. So, the denser a fluid and is, and the higher its velocity, the more momentum (inertia) it has. As in classical mechanics, a force that can counteract or counterbalance this inertial force is the force of friction (shear stress).In the case of fluid flow, this is represented by Newtons law, $\tau_x = \mu \frac{dv}{dy}$. This is only dependent on the viscosity and gradient of velocity. Then, $Re = \frac{\rho v L}{\mu}$, is a measure of which force dominates for a particular flow.

The inertial forces are what gives rise to the dynamic pressure. Another way to look at the Reynolds Number is by the ratio of dynamic pressure $\rho u^2$ and shearing stress $μ v/ L$ and can be expressed as $$Re =\frac{\rho u^2} {μ v/ L} = \frac{ u L} {\nu}$$

At very high Reynolds numbers, the motion of the fluid causes eddies to form and give rise to the phenomena of turbulence.

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I think your answer can be improved by putting more emphasis on the dynamic pressure to explain it in the context of forces. – Bernhard Oct 8 '13 at 20:40
Thanks for you suggestion. – mcodesmart Oct 8 '13 at 20:54
I'm not sure if I'm correct, but in the last latex equation, shouldn't it be vL/(v kinematic viscosity) rather than uL/v? Or could you explain why its written as rho.v.L/mu before? – midnightBlue Nov 25 '14 at 20:43

The ratio of the inertial force to the viscous forces of the fluid is known as the Reynolds number. Now the viscous forces are the forces due to to the friction between the the layers of any real fluid. In fluid mechanics we take the fluid as in the continuum condition, which means fluid particles are very closely packed so necessarily there is friction between layers of fluid. The inertial forces are the forces which are due to due the particles of fluid resisting any change in momentum.

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Newton’s second law can be written as m x a = f1 + f2 + … + fn. The term m x a having dimensions of force and being proportional to the mass of the body, which is a measure of its inertia, is often named inertia force. So, according to the previous equation, the inertia force is just the resultant force acting on the body under analysis. In Fluid Mechanics it is advantageous to use mass per unit volume of the body (fluid in this case), that is its density, so that Newton’s law (or rather, the Navier-Stokes equation) is written with the terms having dimensions of force per unit volume of fluid. When fluids flow, different types of forces act on the fluid. These are represented in the previous equation by f1, f2, … , fn. Suppose viscous forces are represented by f2. Back to the original question, the Reynolds number (Re) associated with the fluid flow would be, in this case, m x a / f2. So, in fluid flow, Re is a measure of the ratio between the resultant force (or inertia force) and viscous force acting on the fluid. Notice that the viscous force is part of the inertia force. In other words, Re is the ratio between the resultant force acting on the fluid and one of its components. Our intuition regarding effects of inertia forces is quite good because in daily life our muscles overcame inertia of static bodies all the time. Inertia of moving bodies are also easily perceptible when our velocity (for instance in a moving car) changes magnitude or direction, that is, when we are accelerated relative to the ground (notice that a is a factor in the inertia force). Effects of viscous forces are much more subtle and require specific experiments.

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You can use latex commands to improve your answer. – Ali Jun 13 '14 at 3:30