Page 291 - Intro to Tensor Calculus

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Figure 2.5-2. Viscosity experiment.
Viscosity

Most fluids are characterized by the fact that they cannot resist shearing stresses. That is, if you put a
shearing stress on the fluid, the fluid gives way and flows. Consider the experiment illustrated in the figure
2.5-2 which illustrates a fluid moving between two parallel plane surfaces. Let S denote the distance between
the two planes. Now keep the lower surface fixed or stationary and move the upper surface parallel to the
~
lower surface with a constant velocity V 0 . If you measure the force F required to maintain the constant
velocity of the upper surface, you discover that the force F varies directly as the area A of the surface and
the ratio V 0 /S. This is expressed in the form
F V 0
= µ ∗ . (2.5.14)
A S
The constant µ is a proportionality constant called the coefficient of viscosity. The viscosity usually depends
∗
upon temperature, but throughout our discussions we will assume the temperature is constant. A dimensional
analysis of the equation (2.5.14) implies that the basic dimension of the viscosity is [µ ]= ML −1 T −1 . For
∗
example, [µ ]= gm/(cm sec) in the cgs system of units. The viscosity is usually measured in units of
∗
centipoise where one centipoise represents one-hundredth of a poise, where the unit of 1 poise= 1 gram
per centimeter per second. The result of the above experiment shows that the stress is proportional to the
change in velocity with change in distance or gradient of the velocity.

Linear Viscous Fluids

The above experiment with viscosity suggest that the viscous stress tensor τ ij is dependent upon both
the gradient of the fluid velocity and the density of the fluid.
In Cartesian coordinates, the simplest model suggested by the above experiment is that the viscous
stress tensor τ ij is proportional to the velocity gradient v i,j and so we write

τ ik = c ikmp v m,p , (2.5.15)

where c ikmp is a proportionality constant which is dependent upon the fluid density.
The viscous stress tensor must be independent of any reference frame, and hence we assume that the
proportionality constants c ikmp can be represented by an isotropic tensor. Recall that an isotropic tensor
has the basic form

∗
∗
∗
c ikmp = λ δ ik δ mp + µ (δ im δ kp + δ ip δ km )+ ν (δ im δ kp − δ ip δ km ) (2.5.16)