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Newton’s Law of Viscosity

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direction). The corresponding expression for this situation is known as
Ohm’s law (1827) and is given by
iy ¼ _ke
de
dy ð1-6Þ
where ke is the ‘‘electrical conductivity’’ of the medium between the plates.
D. Newton’s Law of Viscosity
Momentum is also a conserved quantity, and we can write an equivalent
expression for the transport of momentum. We must be careful here, however,
because velocity and momentum are vectors, in contrast to mass,
energy, and charge, which are scalars. Hence, even though we may draw
some analogies between the one-dimensional transport of these quantities,
these analogies do not generally hold in multidimensional systems or for
complex geometries. Here we consider the top plate to be subject to a force
in the x direction that causes it to move with a velocity V1, and the lower
plate is stationary ðV0
¼ . Since ‘‘x-momentum’’ at any point where the
local velocity is vx is mvx, the concentration of momentum must be _vx. If
we denote the flux of x-momentum in the y direction by ð_yx
Þ
mf , the transport
equation is
ð_yxÞmf ¼ __
dð_vxÞ
dy ð1-7Þ
where _ is called the kinematic viscosity. It should be evident that ð_yx
Þ
mf is
negative, because the faster fluid (at the top) drags the slower fluid (below)
along with it, so that ‘x-momentum’’ is being transported in the _y direction
by virtue of this drag. Because the density is assumed to be independent
of position, this can also be written
ð_yxÞmf ¼ __
dvx
dy ð1-8Þ
where _ ¼ __ is the viscosity (or sometimes the dynamic viscosity). Equation
(1-8) applies for laminar flow in the x direction and is known as Newton’s
law of viscosity. Newton formulated this law in 1687! It applies directly to a
class of (common) fluids called Newtonian fluids, which we shall discuss in
detail subsequently.
1. Momentum Flux and Shear Stress
Newton’s law of viscosity and the conservation of momentum are also
related to Newton’s second law of motion, which is commonly written
Fx
¼ max
¼ mvx
Þ=dt. For a steady-flow system, this is equivalent to

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direction). The corresponding expression for this situation is known as Ohm’s law (1827) and is given by iy ¼ _ke de dy ð1-6Þ where ke is the ‘‘electrical conductivity’’ of the medium between the plates. D. Newton’s Law of Viscosity Momentum is also a conserved quantity, and we can ...
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