Page 290 - Intro to Tensor Calculus

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Analysis of these equations implies that if σ ij is to be isotropic, then σ 21 = σ 21 = −σ 12 = −σ 21

or σ 21 = 0 which implies σ 12 = σ 23 = σ 31 = σ 21 = σ 32 = σ 13 =0. (2.5.10)

The above analysis demonstrates that if the stress tensor σ ij is to be isotropic, it must have the form

σ ij = −pδ ij . (2.5.11)

Use the traction condition (2.3.11), and express the stress vector as

(n)
t = σ ij n i = −pn j . (2.5.12)
j
This equation is interpreted as representing the stress vector at a point on a surface with outward unit
normal n i ,where p is the pressure (hydrostatic pressure) stress magnitude assumed to be positive. The
negative sign in equation (2.5.12) denotes a compressive stress.
Imagine a submerged object in a fluid medium. We further imagine the object to be covered with unit
normal vectors emanating from each point on its surface. The equation (2.5.12) shows that the hydrostatic
pressure always acts on the object in a compressive manner. A force results from the stress vector acting on
the object. The direction of the force is opposite to the direction of the unit outward normal vectors. It is
a compressive force at each point on the surface of the object.
The above considerations were for a fluid at rest (hydrostatics). For a fluid in motion (hydrodynamics)
a different set of assumptions must be made. Hydrodynamical experiments show that the shear stress
components are not zero and so we assume a stress tensor having the form

σ ij = −pδ ij + τ ij , i, j =1, 2, 3, (2.5.13)

where τ ij is called the viscous stress tensor. Note that all real fluids are both viscous and compressible.

Definition: (Viscous/inviscid fluid) If the viscous stress ten-
sor τ ij is zero for all i, j, then the fluid is called an inviscid, non-

viscous, ideal or perfect fluid. The fluid is called viscous when τ ij
is different from zero.

In these notes it is assumed that the equation (2.5.13) represents the basic form for constitutive equations
describing fluid motion.