The Physics
Opus in profectus


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Informally, viscosity is the quantity that describes a fluid's resistance to flow. Fluids resist the relative motion of immersed objects through them as well as to the motion of layers with differing velocities within them.

Formally, viscosity (represented by the symbol η "eta") is the ratio of the shearing stress (F/A) to the velocity gradient (Δvxz or dvx/dz) in a fluid.

η = /A


η = F/A

The more usual form of this relationship, called Newton's equation, states that the resulting shear of a fluid is directly proportional to the force applied and inversely proportional to its viscosity. The similarity to Newton's second law of motion (F = ma) should be apparent.

 = η Δvx
 = m  Δv

or if you prefer calculus symbols (and who doesn't)

F = η dvx
F = m  dv

The SI unit of viscosity is the pascal second [Pa s], which has no special name. Despite its self-proclaimed title as an international system, the International System of Units has had very little international impact on viscosity. The pascal second is rarely used in scientific and technical writing today. The most common unit of viscosity is the dyne second per square centimeter [dyne s/cm2], which is given the name poise [P] after the French physiologist Jean Poiseuille (1799–1869). Ten poise equal one pascal second [Pa s] making the centipoise [cP] and millipascal second [mPa s] identical.

1 Pa s =  10 P
1,000 mPa s =  10 P
1 mPa s =  0.01 P
1 mPa s =  1 cP

There are actually two quantities that are called viscosity. The quantity defined above is sometimes called dynamic viscosity, absolute viscosity, or simple viscosity to distinguish it from the other quantity, but is usually just called viscosity. The other quantity called kinematic viscosity (represented by the Greek letter ν "nu") is the ratio of the viscosity of a fluid to its density.

ν = η

Kinematic viscosity is a measure of the resistive flow of a fluid under the influence of gravity. It is frequently measured using a device called a capillary viscometer — basically a graduated can with a narrow tube at the bottom. When two fluids of equal volume are placed in identical capillary viscometers and allowed to flow under the influence of gravity, a viscous fluid takes longer than a less viscous fluid to flow through the tube. Capillary viscometers will be discussed in more detail later in this section.

The SI unit of kinematic viscosity is the square meter per second [m2/s], which has no special name. This unit is so large that it is rarely used. A more common unit of kinematic viscosity is the square centimeter per second [cm2/s], which is given the name stokes [St] after the Irish mathematician and physicist George Stokes (1819–1903). One square meter per second is equal to ten thousand stokes.

1 cm2/s =  1 St
1 m2/s =  10,000 cm2/s
1 m2/s =  10,000 St

Even this unit is a bit too large, so the most common unit is probably the square millimeter per second [mm2/s] or the centistokes [cSt]. One square meter per second is equal to one million centistokes.

1 mm2/s =  1 cSt
1 m2/s =  1,000,000 mm2/s
1 m2/s =  1,000,000 cSt

The stokes is a rare example of a word in the English language where the singular and plural forms are identical — 1 fish, 2 fish, red fish, blue fish; 1 stokes, 2 stokes, 10 stokes, few stokes.

factors affecting viscosity

Viscosity is first and foremost a function of material. The viscosity of water at 20 ℃ is 1.0020 millipascal seconds (which is conveniently close to one by coincidence alone). Most ordinary liquids have viscosities on the order of 1 to 1,000 mPa s, while gases have viscosities on the order of 1 to 10 μPa s. Pastes, gels, emulsions, and other complex liquids are harder to summarize. Some fats like butter or margarine are so viscous that they seem more like soft solids than like flowing liquids. Molten glass is extremely viscous and approaches infinite viscosity as it solidifies. Since this process is not as well defined as true freezing, some believe (incorrectly) that glass may still flow even after it has completely cooled, but this is not the case. At ordinary temperatures, glasses are as solid as true solids.

From everyday experience, it should be common knowledge that viscosity varies with temperature. Honey and syrups can be made to flow more readily when heated. Engine oil and hydraulic fluids thicken appreciably on cold days and significantly affect the performance of cars and other machinery during the winter months. In general, the viscosity of a simple liquid decreases with increasing temperature (and vice versa). As temperature increases, the average speed of the molecules in a liquid increases and the amount of time they spend "in contact" with their nearest neighbors decreases. Thus, as temperature increases, the average intermolecular forces decrease. The actual manner in which the two quantities vary is nonlinear and changes abruptly when the liquid changes phase.

Viscosity is normally independent of pressure, but liquids under extreme pressure often experience an increase in viscosity. Since liquids are normally incompressible, an increase in pressure doesn't really bring the molecules significantly closer together. Simple models of molecular interactions won't work to explain this behavior and, to my knowledge, there is no generally accepted more complex model that does. The liquid phase is probably the least well understood of all the phases of matter.

While liquids get runnier as they get hotter, gases get thicker. (If one can imagine a "thick" gas.) The viscosity of gases increases as temperature increases and is approximately proportional to the square root of temperature. This is due to the increase in the frequency of intermolecular collisions at higher temperatures. Since most of the time the molecules in a gas are flying freely through the void, anything that increases the number of times one molecule is in contact with another will decrease the ability of the molecules as a whole to engage in the coordinated movement. The more these molecules collide with one another, the more disorganized their motion becomes. Physical models, advanced beyond the scope of this book, have been around for nearly a century that adequately explain the temperature dependence of viscosity in gases. Newer models do a better job than the older models. They also agree with the observation that the viscosity of gases is roughly independent of pressure and density. The gaseous phase is probably the best understood of all the phases of matter.

Since viscosity is so dependent on temperature, it shouldn't never be stated without it.

Viscosities of selected materials (note the different unit prefixes)
simple liquids T (℃) η (mPa s)   gases T (℃) η (μPa s)
alcohol, ethyl (grain) 20 1.1   air 15 17.9
alcohol, isopropyl 20 2.4   hydrogen 0 8.42
alcohol, methyl (wood) 20 0.59   helium (gas) 0 18.6
blood 37 3–4   nitrogen 0 16.7
ethylene glycol 25 16.1   oxygen 0 18.1
ethylene glycol 100 1.98   complex materials T (℃) η (Pa s)
freon 11 (propellant) −25 0.74   caulk 20 1000
freon 11 (propellant) 0 0.54   glass 20 1018–1021
freon 11 (propellant) +25 0.42   glass, strain pt.504 1015.2
freon 12 (refrigerant) -15 ?   glass, annealing pt.546 1012.5
freon 12 (refrigerant) 0 ?   glass, softening pt.724 106.6
freon 12 (refrigerant) +15 0.20   glass, working pt. 103
glycerin 20 1420   glass, melting pt. 101
glycerin 40 280   honey 20 10
helium (liquid) 4 K 0.00333   ketchup 20 50
mercury 15 1.55   lard 20 1000
milk 25 3   molasses 20 5
oil, vegetable, canola 25 57   mustard 25 70
oil, vegetable, canola 40 33   peanut butter 20 150–250
oil, vegetable, corn  20 65   sour cream 25 100
oil, vegetable, corn  40 31   syrup, chocolate 20 10–25
oil, vegetable, olive 20 84   syrup, corn 25 2–3
oil, vegetable, olive 40 ?   syrup, maple 20 2–3
oil, vegetable, soybean 20 69   tar 20 30,000
oil, vegetable, soybean  40 26   vegetable shortening 20 1200
oil, machine, light 20 102        
oil, machine, heavy 20 233        
oil, motor, SAE 20 20 125        
oil, motor, SAE 30 20 200        
oil, motor, SAE 40 20 319        
propylene glycol 25 40.4        
propylene glycol 100 2.75        
water 0 1.79        
water 20 1.00        
water 40 0.65        
water 100 0.28        

motor oil

Motor oil is like every other fluid in that its viscosity varies with temperature and pressure. Since the conditions under which most automobiles will be operated can be anticipated, the behavior of motor oil can be specified in advance. In the United States, the organization that sets the standards for performance of motor oils is the Society of Automotive Engineers (SAE). The SAE numbering scheme describes the behavior of motor oils under low and high temperature conditions — conditions that correspond to starting and operating temperatures. The first number, which is always followed by the letter W, describes the low temperature behavior of the oil at start up while the second number describes the high temperature behavior of the oil after the engine has been running for some time. Lower SAE numbers describe oils that are meant to be used under lower temperatures. Oils with low SAE numbers are generally runnier (less viscous) than oils with high SAE numbers, which tend to be thicker (more viscous).

For example, 10W-40 oil would have a viscosity no greater than 7,000 mPa s in a cold engine crankcase even if its temperature should drop to −25 ℃ on a cold winter night and a viscosity no less than 2.9 mPa s in the high pressure parts of an engine very near the point of overheating (150 ℃).

Viscosity grades for motor oils Source: Society of Automotive Engineers (SAE)
* 0W–40, 5W–40, 10W-40  15W–40, 20W–40, 25W–40
  low temperature specifications
dynamic viscosity
cranking maximum
dynamic viscosity
pumping maximum
00W 06,200 mPa s (−35 ℃) 60,000 mPa s (−40 ℃)
05W 06,600 mPa s (−30 ℃) 60,000 mPa s (−35 ℃)
10W 07,000 mPa s (−25 ℃) 60,000 mPa s (−30 ℃)
15W 07,000 mPa s (−20 ℃) 60,000 mPa s (−25 ℃)
20W 09,500 mPa s (−15 ℃) 60,000 mPa s (−20 ℃)
25W 13,000 mPa s (−10 ℃) 60,000 mPa s (−15 ℃)
  high temperature specifications
kinematic viscosity
low shear rate
dynamic viscosity
high shear rate
20 05.6–9.30 mm2/s (100 ℃) >2.6 mPa s (150 ℃)
30 09.3–12.5 mm2/s (100 ℃) >2.9 mPa s (150 ℃)
*40* 12.5–16.3 mm2/s (100 ℃) >2.9 mPa s (150 ℃)
40 12.5–16.3 mm2/s (100 ℃) >3.7 mPa s (150 ℃)
50 16.3–21.9 mm2/s (100 ℃) >3.7 mPa s (150 ℃)
60 21.9–26.1 mm2/s (100 ℃) >3.7 mPa s (150 ℃)

capillary viscometer

The the mathematical expression describing the flow of fluids in circular tubes was determined by the French physician and physiologist Jean Poiseuille (1799–1869). Since it was also discovered independently by the German hydraulic engineer Gotthilf Hagen (1797–1884), it should be properly known as the Hagen-Poiseuille equation, but it is usually just called Poiseuille's equation. I will not derive it here. (Please don't ask me to.) For non-turbulent, non-pulsatile fluid flow through a uniform straight pipe, the volume flow rate (qm) is…

qm =  πΔPr4

Solve for viscosity if that's what you want to know.

η = πΔPr4

capillary viscometer… keep writing…

falling sphere

The mathematical expression describing the viscous drag force on a sphere was determined by the British physicist George Stokes (1819–1903). I will not derive it here. (Once again, don't ask.)

R = 6πηrv

The formula for the buoyant force on a sphere is accredited to the Greek engineer Archimedes of Syracuse (287–212 BCE), but equations weren't invented back then.

B = ρfluidgVdisplaced

The formula for weight had to be invented by someone, but I don't know who.

W = mg = ρobjectgVobject

Let's combine all these things together for a sphere falling in a fluid. Weight goes down, buoyancy goes up, drag goes up. After a while, the sphere will fall with constant velocity. When it does, all these forces cancel. When a sphere is falling through a fluid it is completely submerged, so there is only one volume to talk about — the volume of a sphere. Let's work through this.

B  +  R  =  W  
ρfluidgV  +  6πηrv  =  ρobjectgV
  6πηrv  =  object − ρfluid)gV
6πηrv  =  Δρg  4  πr3  

And here we are.

η = 2Δρgr2

Drop a sphere into a liquid. If you know the size and density of the sphere and the density of the liquid, you can determine the viscosity of the liquid. If you don't know the density of the liquid you can still determine the kinematic viscosity. If you don't know the density of the sphere, but you know its mass and radius, well then you do know its density. Why are you talking to me? Go back several chapters and get yourself some education.

Should I write more?

non-newtonian fluids

Newton's equation relates shear stress and velocity gradient by means of a quantity called viscosity. A newtonian fluid is one in which the viscosity is just a number. A non-newtonian fluid is one in which the viscosity is a function some mechanical variable like shear stress or time. (Non-newtonian fluids that change over time are said to have a memory.)

Some gels and pastes behave like a fluid when worked or agitated and then settle into a nearly solid state when at rest. Such materials are examples of shear-thinning fluids. House paint is a shear-thinning fluid and it's a good thing, too. Brushing, rolling, or spraying are means of temporarily applying shear stress. This reduces the paint's viscosity to the point where it can now flow out of the applicator and onto the wall or ceiling. Once this shear stress is removed the paint returns to its resting viscosity, which is so large that an appropriately thin layer behaves more like a solid than a liquid and the paint does not run or drip. Think about what it would be like to paint with water or honey for comparison. The former is always too runny and the latter is always too sticky.

Toothpaste is another example of a material whose viscosity decreases under stress. Toothpaste behaves like a solid while it sits at rest inside the tube. It will not flow out spontaneously when the cap is removed, but it will flow out when you put the squeeze on it. Now it ceases to behave like a solid and starts to act like a very thick liquid. when it lands on your toothbrush, the stress is released and the toothpaste returns to a solid (or at least a semisolid) state. You do not have to worry about it flowing off the brush as you raise it to your mouth.

Shear-thinning fluids can be classified into one of three general groups. A material that has a viscosity that decreases under shear stress but stays constant over time is said to be pseudoplastic. A material that has a viscosity that decreases under shear stress and then continues to decrease with time is said to be thixotropic. If the transition from high viscosity (or nearly semisolid) to low viscosity (or essentially liquid) takes place only after the shear stress exceeds some minimum value, the material is said to be a bingham plastic.

Materials that thicken when worked or agitated are called shear-thickening fluids. An example that is often shown in science classrooms is a paste made of cornstarch and water (mixed in the correct proportions). The resulting bizarre goo behaves like a liquid when squeezed slowly and an elastic solid when squeezed rapidly. Ambitious science demonstrators have filled tanks with the stuff and then run across it. As long as they move quickly the surface acts like a block of solid rubber, but the instant they stop moving the paste behaves like a liquid and the demonstrator winds up taking a cornstarch bath. The shear-thickening behavior makes it a difficult bath to get out of. The harder you work to get out, the harder the material pulls back on you. The only way to escape is to move slowly.

Materials that turn nearly solid under stress are more than just a curiosity. They're ideal candidates for body armor and protective sports padding. A bulletproof vest or a kneepad made of of shear-thickening material would be supple and yielding to the mild stresses of ordinary body motions, but would turn rock hard in response to the traumatic stress imposed by a weapon or a fall to the ground.

Shear-thickening fluids are are also divided into two groups: those with a time-dependent viscosity (memory materials) and those with a time-independent viscosity (non-memory materials). If the increase in viscosity increases over time, the material is said to be rheopectic. If the increase is roughly directly proportional to the shear stress and does not change over time, the material is said to be dilatant.

Classes of nonlinear fluids with examples and applications
  shear-thinning shear-thickening
(memory materials)
ketchup, honey, quicksand, snake venom, polymeric thick film inks
cream being whipped
(non-memory materials)
paint, styling gel, whipped cream, cake batter, applesauce, ballpoint pen ink, ceramic/metal inks
starch pastes, silly putty, synovial fluid, chocolate syrup, viscous coupling fluids, liquid armor
with a yield stress bingham plastic
toothpaste, drilling mud, blood, cocoa butter, mayonnaise, yoghurt, tomato puree, nail polish, sewage sludge

With a bit of adjustment, Newton's equation can be written as a power law (the Ostwald-de Waele equation) that handles the pseudoplastics and the dilantants.

F  = k
dvx n
A dz

Where η the viscosity is replaced with k the flow consistency index [Pa sn] and the velocity gradient is raised to some power n called the flow behavior index [dimensionless]. The latter number varies with the class of fluid.

n < 1 n = 1 n > 1
pseudoplastic newtonian dilatant

A different modification to Newton's equation is needed to handle Bingham plastics (the Bingham equation).

F  = σy + ηpl  dvx
A dz

Where σy is the yield stress [Pa] and ηpl is the plastic viscosity [Pa s]. The former number separates Bingham plastics from newtonian fluids.

σy < 0 σy = 0 σy > 0
impossible newtonian bingham plastic

Combining the Ostwald-de Waele power law with the Bingham yield stress gives us the more general Herschel-Bulkley equation. Again, σy is the yield stress [Pa], k is the flow consistency index [Pa sn], and n is the flow behavior index [dimensionless].

F  = σy + k
dvx n
A dz


When a force (F) is applied to an object, one of four things can happen.

  1. It could accelerate as a whole, in which case Newton's second law of motion applies…

    F = ma

    This term is not interesting to us right now. We've already discussed this kind of behavior to death in earlier chapters. Mass (m) is resistance to acceleration (a), which is the second derivative of position (x). Let's move on to something new.
  2. It could flow like a fluid, which could be described by this relationship…

    F = −bv

    This is the simplified model where drag is directly proportional to speed (v), the first derivative of position (x). We used this in terminal velocity problems just because it gave differential equations that were easy to solve. We also used it in the damped harmonic oscillator, again because it gave differential equations that were easy to solve (relatively easy, anyway). The proportionality constant (b) is often called the damping factor.
  3. It could deform like a solid…

    F = −kx

    This is Hooke's law. The proportionality constant (k) is famously called the spring constant. Position (x) is not the part of any derivative nor is it raised to any power.
  4. It could get stuck

    F = −ƒ

    That symbol ƒ makes it look like we're discussing static friction. In fluids (non-newtonian fluids, to be specific) a term like this is associated with yield stress. Position (x) is not involved in any way.

Put everything together and admit that acceleration and velocity are derivatives of position.

F = m  d2x  − b  dx  − kx − ƒ
dt2 dt

This differential equation summarizes the possible behaviors of an object. The interesting thing is that it mixes up the behaviors of fluids and solids. The more interesting thing is that there are occasions when both behaviors will be present in one thing. Materials that both flow like fluids and deform like solids are said to be viscoelastic — an obvious mash-up of viscosity and elasticity. The study of materials with fluid and solid properties is called rheology, which comes from the Greek verb ρέω (reo), to flow.

keep writing…

What old book gave me this idea?