106			Week 2: Newton’s Laws: Continued
	or		v2
				> tan θ		(171)
			Rg
	then fs	is positive (down the incline), otherwise it is negative (up the incline). When			v2	= tan θ,
					Rg

fs = 0 and the car would round the curve even on ice (as you determined in a previous homework problem).

See if you can use your knowledge of the algebraic form for fsmax to determine the range of v given µs that will permit the car to round the curve. It’s a bit tricky! You may have to go back a couple of steps and find N max (the N associated with fsmax) and fsmax in terms of that N at the same time, because both N and fs depend, in the end, on v...

2.2: Drag Forces

viscous friction turbulence

Pressure increase

Fd

v

Pressure decrease

Figure 21: A “cartoon” illustrating the di erential force on an object moving through a fluid. The drag force is associated with a di erential pressure where the pressure on the side facing into the ‘wind’ of its passage is higher than the pressure of the trailing/lee side, plus a “dynamic frictional” force that comes from the fluid rubbing on the sides of the object as it passes. In very crude terms, the former is proportional to the cross-sectional area; the latter is proportional to the surface area exposed to the flow. However, the details of even this simple model, alas, are enormously complex.

As we will discuss later in more detail in the week that we cover fluids, when an object is sitting at rest in a fluid at rest with a uniform temperature, pressure and density, the fluid around it presses on it, on average, equally on all sides54.

Basically, the molecules of the fluid on one side of the object hit it, on average, with as much force per unit area area as molecules on the other side and the total cross-sectional area of the object seen from any given direction or the opposite of that direction is the same. By the time one works out all of the vector components and integrates the force component along any line over the whole surface area of the object, the force cancels. This “makes sense” – the whole system is in average static force equilibrium and we don’t expect a tree to bend in the wind when there is no wind!

When the same object is moving with respect to the fluid (or the fluid is moving with respect to the object, i.e. – there is a wind in the case of air) then we empirically observe that a friction-like force is exerted on the object (and back on the fluid) called drag55 .

We can make up at least an heuristic description of this force that permits us to intuitively reason about it. As an object moves through a fluid, one expects that the molecules of the fluid will hit

54We are ignoring variations with bulk fluid density and pressure in e.g. a gravitational field in this idealized statement; later we will see how the field gradient gives rise to buoyancy through Archimedes’ Principle. However, lateral forces perpendicular to the gravitational field and pressure gradient still cancel even then.

55Wikipedia: http://www.wikipedia.org/wiki/Drag (physics). This is a nice summary and well worth at least glancing at to take note of the figure at the top illustrating the progression from laminar flow and skin friction to highly turbulent flow and pure form drag.

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on the side facing the direction of motion harder, on the average, then molecules on the other side. Even though we will delay our formal treatment of fluid pressure until later, we should all be able to understand that these stronger collisions correspond (on average) to a greater pressure on the side of the object moving against the fluid or vice versa, and a lower pressure in the turbulent flow on the far side, where the object is moving away from the “chasing” and disarranged molecules of fluid. This pressure-linked drag force we might expect to be proportional to the cross-sectional area of the object perpendicular to its direction of relative motion through the fluid and is called form drag to indicate its strong dependence on the shape of the object.

However, the fluid that flows over the sides of the object also tends to “stick” to the surface of the object because of molecular interactions that occur during the instant of the molecular collision between the fluid and the surface. These collisions exert transverse “frictional” forces that tend to speed up the recoiling air molecules in the direction of motion of the object and slow the object down. The interactions can be strong enough to actually “freeze” a thin layer called the boundary layer of the fluid right up next to the object so that the frictional forces are transmitted through successive layers of fluid flowing and di erent speeds relative to the object. This sort of flow in layers is often called laminar (layered) flow and the frictional force exerted on the object transmitted through the rubbing of the layers on the sides of the object as it passes through the fluid is called skin friction or laminar drag.

Note well: When an object is enlongated and passes through a fluid parallel to its long axis with a comparatively small forward-facing cross section compared to its total area, we say that it is a streamlined object as the fluid tends to pass over it in laminar flow. A streamlined object will often have its total drag dominated by skin friction. A blu object, in contrast has a comparatively large cross-sectional surface facing forward and will usually have the total drag dominated by form drag. Note that a single object, such as an arrow or piece of paper, can often be streamlined moving through the fluid one way and blu another way or be crumpled into a di erent shape with any mix in between. A sphere is considered to be a blu body, dominated by form drag.

Unfortunately, this is only the beginning of an heuristic description of drag. Drag is a very complicated force, especially when the object isn’t smooth or convex but is rather rough and irregularly shaped, or when the fluid through which it moves is not in an “ideal” state to begin with, when the object itself tumbles as it moves through the fluid causing the drag force to constantly change form and magnitude. Flow over di erent parts of a single object can be laminar here, or turbulent there (with portions of the fluid left spinning in whirlpool-like eddies in the wake of the object after it passes).

The full Newtonian description of a moving fluid is given by the Navier-Stokes equation56 which is too hard for us to even look at.

We will therefore need to idealize; learn a few nearly universal heuristic rules that we can use to conceptually understand fluid flow for at least simple, smooth, convex geometries.

It would be nice, perhaps, to be able to skip all of this but we can’t, not even for future physicians as opposed to future engineers, physicists or mathematicians. As it happens, the body contains at least two major systems of fluid flow – the vasculature and the lymphatic system – as well as numerous minor ones (the renal system, various sexual systems, even much of the digestive system is at least partly a fluid transport problem). Drag forces play a critical role in understanding blood pressure, heart disease, and lots of other stu . Sorry, my beloved students, you gotta learn it at least well enough to qualitatively and semi-quantitatively understand it.

56Wikipedia: http://www.wikipedia.org/wiki/Navier-Stokes Equation. A partial di erential way, way beyond the scope of this course. To give you an idea of how di cult the Navier-Stokes equation is to solve (in all but a few relatively simple geometries) simply demonstrating that solutions to it always exist and are smooth is one of the seven most important questions in mathematics and you could win a million dollar prize if you were to demonstrate it (or o er a proven counterexample).

108	Week 2: Newton’s Laws: Continued

Besides, this section is the key to understanding how to at least in principle fall out of an airplane without a parachute and survive. Drag forces significantly modify the idealized trajectory functions we derived in week 1, so much so that anyone relying on them to aim a cannon would almost certainly consistently miss any target they aimed at using the idealized no-drag trajectories.

Drag is an extremely complicated force. It depends on a vast array of things including but not limited to:

•The size of the object.

•The shape of the object.

•The relative velocity of the object through the fluid.

•The state of the fluid (e.g. its velocity field including any internal turbulence).

•The density of the fluid.

•The viscosity of the fluid (we will learn what this is later).

•The properties and chemistry of the surface of the object (smooth versus rough, strong or weak chemical interaction with the fluid at the molecular level).

•The orientation of the object as it moves through the fluid, which may be fixed in time (streamlined versus blu motion) or varying in time (as, for example, an irregularly shaped object tumbles).

To eliminate most of this complexity and end up with “force rules” that will often be quantitatively predictive we will use a number of idealizations. We will only consider smooth, uniform, nonreactive surfaces of convex blu objects (like spheres) or streamlined objects (like rockets or arrows) moving through uniform, stationary fluids where we can ignore or treat separately the other non-drag (e.g. buoyant) forces acting on the object.

There are two dominant contributions to drag for objects of this sort.

The first, as noted above, is form drag – the di erence in pressure times projective area between the front of an object and the rear of an object. It is strongly dependent on both the shape and orientation of an object and requires at least some turbulence in the trailing wake in order to occur.

The second is skin friction, the friction-like force resulting from the fluid rubbing across the skin at right angles in laminar flow.

In this course, we will wrap up all of our ignorance of the shape and cross-sectional area of the object, the density and viscosity of the fluid, and so on into a single number: b. This (dimensioned) number will only be actually computable for certain particularly “nice” shapes, but it allows us to understand drag qualitatively and treat drag semi-quantitatively relatively simply in two important limits.

2.2.1: Stokes, or Laminar Drag

The first is when the object is moving through the fluid relatively slowly and/or is arrow-shaped or rocket-ship-shaped so that streamlined laminar drag (skin friction) is dominant. In this case there is relatively little form drag, and in particular, there is little or no turbulence – eddies of fluid spinning around an axis – in the wake of the object as the presence of turbulence (which we will discuss in more detail later when we consider fluid dynamics) breaks up laminar flow.

This “low-velocity, streamlined” skin friction drag is technically named Stokes’ drag (as Stokes was the first to derive it as a particular limit of the Navier-Stokes equation for a sphere moving

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109

through a fluid) or laminar drag and has the idealized force rule:

~	(172)
F d = −bl~v

This is the simplest sort of drag – a drag force directly proportional to the velocity of relative motion of the object through the fluid and oppositely directed.

Stokes derived the following relation for the dimensioned number bl (the laminar drag coe cient) that appears in this equation for a sphere of radius R:

bl = −6πµR

(173)

where µ is the dynamical viscosity. Di erent objects will have di erent laminar drag coe cients bl, and in general it will be used as a simple given parameter in any problem involving Stokes drag.

Sadly – sadly because Stokes drag is remarkably mathematically tractable compared to e.g. turbulent drag below – spheres experience pure Stokes drag only when they are very small or moving very slowly through the fluid. To given you an idea of how slowly – a sphere moving at 1 meter per second through water would have to be on the order of one micron (a millionth of a meter) in size in order to experience predominantly laminar/Stokes drag. Equivalently, a sphere a meter in diameter would need to be moving at a micron per second. This is a force that is relevant for bacteria or red blood cells moving in water, but not too relevant to baseballs.

It becomes more relevant for streamlined objects – objects whose length along the direction of motion greatly exceeds the characteristic length of the cross-sectional area perpendicular to this direction. We will therefore still find it useful to solve a few problems involving Stokes drag as it will be highly relevant to our eventual studies of harmonic oscillation and is not irrelevant to the flow of blood in blood vessels.

2.2.2: Rayleigh, or Turbulent Drag

On the other hand, if one moves an object through a fluid too fast – where the actual speed depends in detail on the actual size and shape of the object, how blu or streamlined it is – pressure builds up on the leading surface and turbulence57 appears in its trailing wake in the fluid (as illustrated in figure 21 above) when the Reynolds number Re of the relative motion (which is a function of the relative velocity, the kinetic viscosity, and the characteristic length of the object) exceeds a critical threshold. Again, we will learn more about this (and perhaps define the Reynolds number) later – for the moment it su ces to know that most macroscopic objects moving through water or air at reasonable velocities experience turbulent drag, not Stokes drag.

This high velocity, turbulent drag exerts a force described by a quadratic dependence on the relative velocity due to Lord Rayleigh:

~	1
F d = −	2	ρCdA|v|~v = −bt|v|~v	(174)

It is still directed opposite to the relative velocity of the object and the fluid but now is proportional to that velocity squared. In this formula ρ is the density of the fluid through which the object moves (so denser fluids exert more drag as one would expect) and A is the cross-sectional area of the object perpendicular to the direction of motion, also known as the orthographic projection of the object on any plane perpendicular to the motion. For example, for a sphere of radius R, the orthographic projection is a circle of radius R and the area A = πR2.

57Wikipedia: http://www.wikipedia.org/wiki/Turbulence. Turbulence – eddies spun out in the fluid as it moves o of the surface passing throughout it – is arguably the single most complex phenomenon physics attempts to describe, dwarfing even things like quantum field theory in its di culty. We can “see” a great deal of structure in it, but that structure is fundamentally chaotic and hence subject to things like the butterfly e ect. In the end it is very di cult to compute except in certain limiting and idealized cases.