1855_PDF_C07.PDF

Classical Flu ids

7.1

Viscous Stress Tensor, Stokesian, and Newtonian Fluids

A fundamental characteristic of any ﬂuid — be it a liquid or a gas — is that

the action of shear stresses, no matter how small they may be, will cause the

ﬂuid to deform continuously as long as the stresses act. It follows, therefore,

that a ﬂuid at rest (or in a state of rigid body motion) is incapable of

sustaining any shear stress whatsoever. This implies that the stress vector

on an arbitrary element of surface at any point in a ﬂuid at rest is proportional

to the normal

of that element, but independent of its direction. Thus, we

write

() =

= −

p n

(7.1-1)

where the (positive) proportionality constant

is the

thermostatic pressure

or,

as it is frequently called, the

hydrostatic pressure

. We note from Eq 7.1-1 that

−

(7.1-2)

which indicates that for a ﬂuid at rest the stress is everywhere compressive,

that every direction is a principal stress direction at any point, and that the

hydrostatic pressure is equal to the mean normal stress,

3 σ

=−

(7.1-3)

This pressure is related to the temperature

and density

by an equation

of state having the form

(

) = 0

Fp o ,,

ρθ

(7.1-4)

For a ﬂuid in motion the shear stresses are not usually zero, and in this

case we write

−+

(7.1-5)

where

which is a function of the motion

and vanishes when the ﬂuid is at rest. In this equation, the pressure

is called the

viscous stress tensor,

called the

thermodynamic pressure

and is given by the same functional rela-

tionship with respect to

and

as that for the static pressure

in the

equilibrium state, that is, by

(

) = 0

Fp ,,

ρθ

(7.1-6)

Note from Eq 7.1-5 that, for a ﬂuid in motion,

is not equal to the mean

normal stress, but instead is given by

3 στ

(

)

=−

−

(7.1-7)

so that, for a ﬂuid at rest (

In developing constitutive equations for viscous ﬂuids, we ﬁrst remind

ourselves that this viscous stress tensor must vanish for ﬂuids at rest, and

following the usual practice, we assume that

= 0),

equates to

is a function of the rate of

deformation tensor

. Expressing this symbolically, we write

()

τ ij

(7.1-8)

If the functional relationship in this equation is nonlinear, the ﬂuid is called

Stokesian

ﬂuid. When

deﬁnes

as a linear function of

, the ﬂuid is

known as a

Newtonian

ﬂuid, and we represent it by the equation

τ ij

(7.1-9)

ijpq

in which the coefﬁcients

reﬂect the viscous properties of the ﬂuid.

As may be veriﬁed experimentally, all ﬂuids are isotropic. Therefore,

ijpq

in Eq 7.1-9 is an isotropic tensor; this, along with the symmetry properties

to 2. We conclude

that, for a homogeneous, isotropic Newtonian ﬂuid, the constitutive equation

and

, allow us to reduce the 81 coefﬁcients

ijpq

=−

λ δ

(7.1-10)

where

which denote the viscous properties

of the ﬂuid. From this equation we see that the mean normal stress for a

Newtonian ﬂuid is

* and

* are

viscosity coefﬁcients

(

)

=− +

* D

(7.1-11)

where

* )

is known as the

coefﬁcient of bulk viscosity

. The

condition

* )

(7.1-12

)

or, equivalently,

=−

(7.1-12

)

is known as

and we see from Eq 7.1-11 that this condition

assures us that, for a Newtonian ﬂuid at rest, the mean normal stress equals

the (negative) pressure

Stokes condition,

If we introduce the deviator tensors

S ij =σδ σ

(7.1-13

)

for stress and

=−

(7.1-13

)

for rate of deformation into Eq 7.1-10, we obtain

δσ

= −

δ λ

µβ

(7.1-14)

which may be conveniently split into the pair of constitutive equations

S ij

µβ

(7.1-15

)

(

)

=−

−

(7.1-15

)

The ﬁrst of this pair relates the shear effect of the motion with the stress

deviator, and the second associates the mean normal stress with the thermo-

dynamic pressure and the bulk viscosity.

7.2

Basic Equations of Viscous Flow, Navier-Stokes

Equations

Inasmuch as ﬂuids do not possess a “natural state” to which they return

upon removal of applied forces, and because the viscous forces are related

directly to the velocity ﬁeld, it is customary to employ the Eulerian descrip-

tion in writing the governing equations for boundary value problems in

viscous ﬂuid theory. Thus, for the thermomechanical behavior of a Newtonian

ﬂuid, the following ﬁeld equations must be satisﬁed:

(a)

the continuity equation (Eq 5.3-6)

ρ+=

v i,i

(7.2-1)

(b)

the equations of motion (Eq 5.4-4)

σρρ

ij, j

(7.2-2)

(c)

the constitutive equations (Eq 7.1-10)

=−

λ δ

(7.2-3)

(d)

the energy equation (Eq 5.7-13)

u= D q

ρσ

−+

(7.2-4)

i,i

(e)

the kinetic equation of state (Eq 7.1-6)

(

)

ρθ

(7.2-5)

(f)

the caloric equation of state (Eq 5.8-1)

(

)

ρθ

(7.2-6)

(g)

the heat conduction equation (Eq 5.7-10)

q i

=−κθ ,

(7.2-7)

This system, Eqs 7.2-1 through 7.2-7, together with the deﬁnition of the rate

of deformation tensor,

(

)

(7.2-8)

i j

j i

represents 22 equations in the 22 unknowns,

. If

thermal effects are neglected and a purely mechanical problem is proposed,

we need only Eqs 7.2-1 through 7.2-3 as well as Eq 7.2-8 and a temperature

independent form of Eq 7.2-5, which we state as

, and

()

(7.2-9)

Certain of the above ﬁeld equations may be combined to offer a more

compact formulation of viscous ﬂuid problems. Thus, by substituting

Eq 7.2-3 into Eq 7.2-2 and making use of the deﬁnition Eq 7.2-8, we obtain

This provides a system of 17 equations in the 17 unknowns,

, and

(

)

ρρ

=−++

λµ

(7.2-10)

j ji

i jj

which are known as the

equations for ﬂuids. These equations,

along with Eqs 7.2-4, 7.2-5, and 7.2-6, provide a system of seven equations

for the seven unknowns,

Navier-Stokes

. Notice that even though Eq 7.2-3

is a linear constitutive equation, the Navier-Stokes equations are nonlinear

because in the Eulerian formulation

, and

∂

j,j

If Stokes condition

is assumed, Eq 7.2-10 reduces to the form

=−

ρρ

=−+

( v

)

(7.2-11)

j, ji

i, jj

Also, if the kinetic equation of state has the form of Eq 7.2-9, the Navier-

Stokes equations along with the continuity equation form a complete set of

four equations in the four unknowns,

In all of the various formulations for viscous ﬂuid problems stated above,

the solutions must satisfy the appropriate ﬁeld equations as well as boundary

and initial conditions on both traction and velocity components. The bound-

ary conditions at a ﬁxed surface require not only the normal, but also the

tangential component of velocity to vanish because of the “boundary layer”

effect of viscous ﬂuids. It should also be pointed out that the formulations

and

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