Course content > Newton's Laws > Kinds of Forces > Resistive forces

Prerequisites

• Formulation of Newton's Laws as foothold principles
• Kinds of forces
• Resistive forces 

When an object moves through a fluid, it feels a resistance to its motion for two reasons:

1. It is dragging the fluid along with it and making the fluid slide along itself, and the fluid has an internal resistance to doing that, the so-called viscous force (viscosity).
2. (In dragging the fluid along, the moving object needs to accelerate at least some volume of fluid in front of it, making that part of the fluid go with the same speed that the object is going.  To do that the object has to exert a force on the molecules of fluid in the direction of motion, so, by Newton's 3rd law, the fluid molecules exert a force back on the object, opposite to the direction of motion.  This force needed to accelerate the fluid is often called inertial fluid force.  We will call this second force simply drag. 

Together these forces generate the resistive force a fluid exerts onto a moving object.   When the object is moving slowly the viscosity dominates and we can ignore the inertial fluid force. When the object is moving rapidly the inertial fluid force or drag dominates.

An object moving in a fluid always feels both viscous forces and inertial fluid forces, but depending on the various parameters of the situation, one force can be much more important than the other.  The ratio of the inertial fluid force to the viscous force is called the Reynold's number and tells which of the two forces you have to pay attention to in a given situation. 

Though motion in a fluid is a very complex process, we can get a pretty good idea of what the drag force looks like with a simple model.  As we often do, let's pick the simplest possible model to calculate: a cylinder moving through a fluid as shown in the figure at the left.  We have a cylinder of radius, R, moving in a direction along its axis with a speed v through a mass of molecules of density, d.

We'll model drag in this situation with the simplest model we can create.  We know the molecules in the fluid are jittering around at a very high speed.  But on the average, as many are going in the opposite direction as in any given direction.  We'll assume that on the average all the molecular jiggling cancels out.  The cylinder will be pushing molecules in front of it.  It therefore has to add speed to those molecules, making them move (on the average) the same speed as the cylinder is going.  So how much force does it have to exert to add speed to those molecules?

In a small amount of time, Δt, the cylinder will move a distance Δx = v Δt.  The front surface of the cylinder will therefore sweep out all the molecules in a thin volume equal to the area of the circle on top of the cylinder times Δx. So it sweeps out a small volume ΔV = πR² Δx.  The mass of molecules in this volume is this volume times the density, or Δm = dΔV.  To add a change in velocity from 0 to v to this small mass, the the cylinder has to exert a force of

F_cyl→fluid Δt = mΔv   (general N2)

= Δm v
(since in our case, the mass is Δm and it changes its velocity by an amount v.)

so putting in our whole chain of what we know about the distance and the mass

By Newton's 3rd law, the force the fluid exerts back on the cylinder is equal and opposite to this:

So we conclude that the drag force that a fluid exerts on an object moving through it

• opposes the motion
• is proportional to the density of the fluid.
• is proportional to the area being pushed through the fluid.
• is proportional to the square of the velocity of the object through the fluid.

Now our model was pretty simplified.  We ignored the fact that the molecules of the fluid were moving, and we ignored the fact that they in fact had to go someplace -- they are being pushed into fluid ahead of it so the fluid has to slide through itself.  The result of all this is summarized by putting in a dimensionless coefficient C_drag -- the drag coefficient -- which is determined by measurement.  Typically, it is not very different from 1. 

This is the equation we will use.

For a "streamlined" object where the shape is optimized to minimize the amount of fluid pushed along with the object, the drag coefficient can be 0.04 or even smaller but this optimization of the shape of the object does NOT alter the scaling relation that the drag is proportional to the square of velocity and proportional to the area being pushed through the fluid.

Follow-ons

• Viscosity
• Reynolds' number

Joe Redish 9/30/11

Wolfgang Losert 9/26/2012