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Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

Fluid Dynamics (Chemical Engineering) 47

B. Equation of Continuity which is known variously as Cauchy’s equations of mo-
tion, Cauchy’s first law of motion, the stress equations
If the generic property ψ is identified as the mass density ρ
of motion, or Newton’s second law for continuum fluids.
of a material, then Eq. (2) represents the generic principle
Regardless of the name applied to Eq. (10), Truesdell and
of conservation of mass. The diffusive flux vector j Dρ is
Toupin (1960) identify it and the statement of symmetry of
equal to 0 and also ˙ r ρ equals 0. Thus, the statement of
T as the fundamental equations of continuum mechanics.
conservation of mass, or equation of continuity, is
By using the vector identities relating Eulerian and
∂ρ/∂t =−∇ · ρv (5) Lagrangian frames together with the equation of conti-
nuity, one can convert Eq. (10) to an equivalent form:
in the Eulerian frame or
Dv
ρ = ρg − ∇p − ∇ · τ. (11)
Dρ/Dt =−ρ∇ · v (6) Dt
in the Lagrangian frame. The following are specificex- In this equation the stress tensor T has been partitioned
pressions for Eq. (5) in the three most commonly used into two parts in accordance with
systems:
T =−pδ + P =−pδ − τ, (12)
Cartesian
where −p is the mean normal stress defined by
∂ρ ∂ ∂ ∂
− = (ρv x ) + (ρv y ) + (ρv z ) (7)
1
∂t ∂x ∂y ∂z −p = (T xx + T yy + T zz ) (13)
3
Cylindrical Polar and P is known variously as the viscous stress tensor, the
∂ρ 1 ∂ 1 ∂ ∂ extra stress tensor, the shear stress tensor, or the stress
− = (rρv r ) + (ρv θ ) + (ρv z ) (8) deviator tensor. It contains both shear stresses (the off-
∂t r ∂r r ∂θ ∂z
diagonal elements) and normal stresses (the diagonal ele-
Spherical Polar
ments), both of which are related functionally to velocity
∂ρ 1 ∂ 2 1 ∂ gradient components by means of constitutive relations. In
− = (r ρv r ) + [(sin θ)ρv θ ]
2
∂t r ∂r r sin θ ∂θ purely viscous fluids only the shear stresses are important,
but the normal stresses become important when elasticity
1 ∂
+ (ρv φ ) (9) becomes a characteristic of the fluid. In incompressible
r sin θ ∂φ
liquids the mean normal stress is a dynamic parameter
that replaces the thermodynamic pressure. It is the gradi-
C. Equations of Motion ent of this pressure that is always dealt with in engineering
design problems.
The vector quantity ρv represents both the convective
If one performs the vector operation x × (equations of
mass flux and the concentration of linear momentum. Its
motion), the balance of rotational momentum or moment
vector product x × ρv with a position vector x from some
of momentum about an axis of rotation is obtained. It
axis of rotation represents the concentration of angular
is this equation that forms the basis of design of rotating
momentum about that axis. If g =−∇ is an external
machinery such as centrifugal pumps and turbomachinery.
body or action-at-a-distance force per unit mass, where
Equation (11) is written in the form of Newton’s sec-
is a potential energy field, then the vector ρg represents
ond law and states that the mass times acceleration of a
the volumetric rate of generation or production of linear
fluid particle is equal to the sum of the forces causing
momentum. The vector x × ρg is the volmetric production
that acceleration. In flow problems that are acceleration-
rate of angular momentum.
less(Dv/Dt = 0)itissometimespossibletosolveEq.(11)
Surface tractions or contact forces produce a stress field
for the stress distribution independently of any knowledge
in the fluid element characterized by a stress tensor T. Its
of the velocity field in the system. One special case where
negative is interpreted as the diffusive flux of momentum,
this useful feature of these equations occurs is the case of
and x × (−T ) is the diffusive flux of angular momentum
rectilinear pipe flow. In this special case the solution of
or torque distribution. If stresses and torques are presumed
complex fluid flow problems is greatly simplified because
to be in local equilibrium, the tensor T is easily shown to
the stress distribution can be discovered before the consti-
be symmetric.
tutive relation must be introduced. This means that only a
When all of these quantities are introduced into Eq. (2),
first-order differential equation must be solved rather than
one obtains
a second-order (and often nonlinear) one. The following
∂ are the components of Eq. (11) in rectangular Cartesian,
(ρv) =−∇ · ρvv − ∇ · (−T) + ρg, (10)
∂t cylindrical polar, and spherical polar coordinates: