6
1. Basic Concepts of Fluid Flow
These two equations may be written, in index notation in Cartesian coordinates, as follows:
where Sij is Kronecker symbol (Sij = 1 if i = j and Sij = 0 otherwise).
For incompressible flows, the second term in brackets in Eq. (1.11) is zero
by virtue of the continuity equation. The following notation is often used in
literature to describe the viscous part of the stress tensor:
2
rij = 2pDij - -pSij div v .
(1.13)
3
For non-Newtonian fluids, the relation between the stress tensor and the
velocity is defined by a set of partial differential equations and the total
problem is far more complicated. In fact, different types of non-Newtonian
fluids require different constitutive equations which may, in turn, require
different solution methods. This subject is complex and is just beginning to
be explored. For these reasons, it will not be considered further in this book.
With the body forces (per unit mass) being represented by b, the integral
form of the momentum conservation equation becomes:
A coordinate-free vector form of the momentum conservation equation (1.14)
is readily obtained by applying Gauss' divergence theorem to the convective
and diffusive flux terms:
+ div ( p v v ) = div T + pb
at
The corresponding equation for the ith Cartesian component is:
a ( ~ u i )
a t
+ div ( p u i v ) = div ti + phi .
Since momentum is a vector quantity, the convective and diffusive fluxes
of it through a CV boundary are the scalar products of second rank tensors
( p v v and T) with the surface vector n d S . The integral form of the above
equations is:
1. Basic Concepts of Fluid Flow
These two equations may be written, in index notation in Cartesian coordinates, as follows:
where Sij is Kronecker symbol (Sij = 1 if i = j and Sij = 0 otherwise).
For incompressible flows, the second term in brackets in Eq. (1.11) is zero
by virtue of the continuity equation. The following notation is often used in
literature to describe the viscous part of the stress tensor:
2
rij = 2pDij - -pSij div v .
(1.13)
3
For non-Newtonian fluids, the relation between the stress tensor and the
velocity is defined by a set of partial differential equations and the total
problem is far more complicated. In fact, different types of non-Newtonian
fluids require different constitutive equations which may, in turn, require
different solution methods. This subject is complex and is just beginning to
be explored. For these reasons, it will not be considered further in this book.
With the body forces (per unit mass) being represented by b, the integral
form of the momentum conservation equation becomes:
A coordinate-free vector form of the momentum conservation equation (1.14)
is readily obtained by applying Gauss' divergence theorem to the convective
and diffusive flux terms:
+ div ( p v v ) = div T + pb
at
The corresponding equation for the ith Cartesian component is:
a ( ~ u i )
a t
+ div ( p u i v ) = div ti + phi .
Since momentum is a vector quantity, the convective and diffusive fluxes
of it through a CV boundary are the scalar products of second rank tensors
( p v v and T) with the surface vector n d S . The integral form of the above
equations is: