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Concept of Viscosity

Arunn Narasimhan 1
Associate Professor,
Department of Mechanical Engineering,
Indian Institute of Technology Madras,
Chennai, Tamil Nadu 600036
Web http: // www. nonoscience. info/

All matter that obeys conservation laws of physics [1 - see endnote], offer
resistance to change of their shape. A fluid is differentiated from a solid by its
mode of resistance to the impressed force that purports to change the shape.
Both liquids and gases are classified as fluids due to the similarity in their
mode of resistance.

When a normal stress [2] σ (i.e. force applied per unit area normal to the
boundary of matter) prevail, both fluids and solids respond (resist) identically,
by undergoing a strain  (deformation), obeying the mathematical statement
σ = E , where E is the modulus of elasticity. The relationship is called
Hooke’s Law [3].

However, when a //shear// stress latexτ (i.e. force applied per unit area tan-
gential to the boundary of matter) acts, a solid responds with a finite angular
(tangential) strain, whereas a fluid strains (deforms) continuously (in time) as
long as the shear stress prevails. The stresses in an elastic solid are propor-
tional to the finite deformation while in the fluid they are proportional to the
rate of deformation.

At the beginning of section 9 of Book 2 of his Principia [4], Sir Isaac Newton
(1642-1727) hypothesized [5] this behaviour of a fluid as ”the resistance arising
from the friction (or lack - or want - of lubricity or slipperiness) of the parts
of a fluid, other things being equal, is proportional to the velocity with which
the parts of the fluid are separated”.

Today, the above hypothesis is restated as the shear stress inside a fluid be-
ing proportional to the velocity gradient, with viscosity being the constant
of proportionality. This physical interpretation of viscosity as a constitutive
1 intended as course notes. Arunn
c Narasimhan
Fig. 1. schematic of a thought experiment to determine dynamic viscosity

Fig. 2. elemental fluid region subject to shear

property of any fluid was due to the works of Navier (1822), Poisson (1831),
St. Venant (1843), and Stokes (1845), almost two centuries after the Principia.

To introduce viscosity more rigorously, let us retrace the widely used example
of a fluid between two parallel surfaces, as shown in Fig. 1.

Now, consider a square fluid element, as shown in Fig. 2, near the bottom wall
of Fig. 1.

When subjected to a constant shear stress on the top surface (which in 2D,
is seen here as a line), the element will deform continuously in time, with (an
assumed) constant strain rate xy equivalent to the angle dθ that AD makes
while straining to AD’, over a certain time dt.
0 0
To put in simple mathematics, for DD (and CC ) being small, from Fig. 2,

ds = dy × dθ ∴ dθ = ds/dy (1)

and hence for time dt, the strain rate can be expressed as

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dθ d ds du
xy = = = (2)
dt dt dt dy

Assuming the shear stress to be linearly related to the strain rate, for our fluid
element we can write

τxy ∼ xy ∴ τxy = µ (3)

where the proportionality constant is µ and is known as the coefficient of

viscosity, introduced by Navier (1822). It is also known as the shear viscosity
or dynamic viscosity.

The above relation can essentially be carried over to Fig. 1, for the entire
fluid experiencing the shear stress between the parallel surfaces. This can be
visually perceived by considering another element similar to ABCD sitting
0 0
on top of ABC D and offering identical strain rate latexd in time dt, for a
identical τxy on its top surface and by repeating the building procedure until
we reach the top wall of Fig. 1. Then, the extended elemental line AD will
match the slanted dashed line in Fig. 1, that denotes the apparent velocity
profile. With this, we essentially have constructed a careful experiment to
determine the viscosity of the fluid subjected to simple shear [6]

The experiment, when performed with different fluids, with the restrictions 1
through 7 shown in the inset of Fig. 1, would result in the applied shear stress
(in Fig. 1, τxy is the shear stress applied in the x direction, and normal to the
y direction, i.e. on the x-z plane of the fluid) being a unique function of xy ,
the resulting strain rate.

That is, for simple fluids such as water, air etc. where the functional relation-
ship is indeed linear, we can write

U 0−U du
τxy ∼ xy (or)τxy = µ =µ (or)τxy = µ (4)
D 0−D dy

According to Eq. (4), a plot between τxy and xy would result in a straight
line passing through the origin, the slope of which indicating the viscosity µ
of the tested fluid.

When the plot yields a curve other than the straight line, the fluid tested is
said to be exhibiting non-Newtonian behaviour. The subject of non-Newtonian
fluid flow is subsumed under a larger endeavor called rheology, the science of
deformation and flow, which includes the study of the mechanical properties

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of a cornucopia of materials from gases and liquids to asphalt and clay. In this
chapter we deal only with fluids exhibiting Newtonian viscosity behaviour.

A good reference point to explore more historical evolution of such fluid me-
chanics concepts is Nemenyi (1962) and Tokaty (1971).


(1) by this we exclude from our present discussion, plasma-like state of matter
such as bio-plasma, the aurorae or corona discharge etc. wherein (instead
of distinct solid, liquid, gas) other states may exist where the conservation
laws may not hold. See Granger (1995) for the origin of this note.
(2) For a solid and fluid at rest, stress, the mechanical pressure and the
thermodynamic pressure are identical while for a fluid in motion, stress
and pressure are different and in principle, all three of them can have
different values. See chapter 2, section 2.4-3, of White (1991), for more
(3) named after the British scientist Robert Hooke (1635 -1703). He was
originally employed by Robert Boyle in 1655 to work on the Boylean air
pump and five years later discovered this law of elasticity. He was also
the first man to state publicly all matter expands when heated and air is
composed of particles separated by large distances.
(4) we consult here, two English translations that covered material from all
of the three editions of the Principia published respectively in 1687, 1713
and 1729 (see the Reference section, for details).
(5) Although Newton probably had a strong reason for putting it as an hy-
pothesis, the concept of viscosity as a fundamental property of any fluid
nevertheless emerges only from that hypothesis. In the subsequent part
of section 9, Newton then uses this hypothesis to explain the motions
generated by an infinitely long cylinder and sphere rotating in a fluid, as
a means to propound his theory of vortices.
(6) more detailed analysis of viscosity, based on Stokes (1845) method, can
be obtained from text books like Prandtl and Teitjens (1934) and White


(1) Granger,R. A., 1995 Fluid Mechanics, Dover Pub., page 41.
(2) Navier, C. L. M. H., 1822 Mmoire sur les Lois du Mouvement des Fluides.
Mm. de l’ Acad. des Sciences, 6, 389.
(3) Nemnyi, P. F., 1962 The Main Concepts and Ideas of Fluid Dynamics in
their Historical Development. Archives for History of Exact Sciences, 2,

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(4) Newton, Sir. I., 1687(1st ed., 1713, 2nd ed. and 1726, 3rd ed.) Philosophiae
Naturalis Principia Mathematica. 1) translated by A. Motte, Prometheus
Books, New York, 1995; originally published as: Newton’s Principia. 1st
American ed., New York, 1848. 2) Cajori, F., Newton’s Principia (Motte’s
translation revised). Uty. of Cal. press, Berkeley, CA, 1946.
(5) Poisson, S. D., 1831 Mmoire sur les quations gnrales de l’quilibre et du
Mouvement des Corps solides lastiques et des Fluides. J. de l’cole Poly.
13, 1.
(6) Prandtl, L. and Tietjens, O. A., 1934 Fundamentals of Hydro- and Aerome-
chanics. Dover Pub. Inc., New York.
(7) Stokes, G. G., 1851 On the Effect of the Internal Friction of Fluids on
the Motion of Pendulums. Cambridge Phil. Trans. 9, 8-106.
(8) St. Venant, 1843 Comptes Rendus, 17, p 1240.
(9) Tokaty, G. A., 1971 A History and Philosophy of Fluid Mechanics. Dover
Pub. Inc., New York.
(10) White, 1991 Viscous Fluid Flow 2nd edition. McGraw Hill Publishers,
New York.

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