1.4 Momentum Conservation
5
index appears twice in any term, summation over the range of that index is
implied:
where xi (i=1,2,3) or (x, y , z ) are the Cartesian coordinates and ui or
(ux, u,, u,) are the Cartesian components of the velocity vector v. The conservation equations in Cartesian form are often used and this will be the case
in this work. Differential conservation equations in non-orthogonal coordinates will be presented in Chap. 8.
1.4 Moment um Conservation
There are several ways of deriving the momentum conservation equation. One
approach is to use the control volume method described in Sect. 1.2; in this
method, one uses Eqs. (1.2) and (1.4) and replaces 4 by v , e.g. for a fixed
fluid-containing volume of space:
To express the right hand side in terms of intensive properties, one has to
consider the forces which may act on the fluid in a CV:
0 surface forces (pressure, normal and shear stresses, surface tension etc.);
0 body forces (gravity, centrifugal and Coriolis forces, electromagnetic forces,
etc.).
The surface forces due to pressure and stresses are, from the molecular point
of view, the microscopic momentum fluxes across a surface. If these fluxes
cannot be written in terms of the properties whose conservation the equations govern (density and velocity), the system of equations is not closed;
that is there are fewer equations than dependent variables and solution is
not possible. This possibility can be avoided by making certain assumptions.
The simplest assumption is that the fluid is Newtonian; fortunately, the Newtonian model applies to many actual fluids.
For Newtonian fluids, the stress tensor T, which is the molecular rate of
transport of momentum, can be written:
where , LL is the dynamic viscosity, I is the unit tensor, p is the static pressure
and D is the rate of strain (deformation) tensor:
5
index appears twice in any term, summation over the range of that index is
implied:
where xi (i=1,2,3) or (x, y , z ) are the Cartesian coordinates and ui or
(ux, u,, u,) are the Cartesian components of the velocity vector v. The conservation equations in Cartesian form are often used and this will be the case
in this work. Differential conservation equations in non-orthogonal coordinates will be presented in Chap. 8.
1.4 Moment um Conservation
There are several ways of deriving the momentum conservation equation. One
approach is to use the control volume method described in Sect. 1.2; in this
method, one uses Eqs. (1.2) and (1.4) and replaces 4 by v , e.g. for a fixed
fluid-containing volume of space:
To express the right hand side in terms of intensive properties, one has to
consider the forces which may act on the fluid in a CV:
0 surface forces (pressure, normal and shear stresses, surface tension etc.);
0 body forces (gravity, centrifugal and Coriolis forces, electromagnetic forces,
etc.).
The surface forces due to pressure and stresses are, from the molecular point
of view, the microscopic momentum fluxes across a surface. If these fluxes
cannot be written in terms of the properties whose conservation the equations govern (density and velocity), the system of equations is not closed;
that is there are fewer equations than dependent variables and solution is
not possible. This possibility can be avoided by making certain assumptions.
The simplest assumption is that the fluid is Newtonian; fortunately, the Newtonian model applies to many actual fluids.
For Newtonian fluids, the stress tensor T, which is the molecular rate of
transport of momentum, can be written:
where , LL is the dynamic viscosity, I is the unit tensor, p is the static pressure
and D is the rate of strain (deformation) tensor: