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Resume of the development of the equations

We have now obtained a sufficient number of equations to match the number of unknown quantities in the flow of a fluid. This does not mean that we can solve them nor even that the solution will exists, but it certainly a necessary beginning. It will be well to review the principles that have been used and the assumptions that have been made.

The foundation of the study of fluid motion lies in kinematics , the analysis of motion and deformations without reference to the forces that are brought into play. To this we added the concept of mass and the principle of the conservation of mass, which leads to the equation of continuity,

D ρ D t + ρ ( v ) = ρ t + ( ρ v ) = 0

An analysis of the nature of stress allows us to set up a stress tensor, which together with the principle of conservation of linear momentum gives the equations of motion

ρ D v D t = ρ f + T .

If the conservation of moment of momentum is assumed, it follows that the stress tensor is symmetric, but it is equally permissible to hypothesize the symmetry of the stress tensor and deduce the conservation moment of momentum. For a certain class of fluids however (hereafter called polar fluids) the stress tensor is not symmetric and there may be an internal angular momentum as well as the external moment of momentum.

As yet nothing has been said as to the constitution of the fluid and certain assumptions have to be made as to its behavior. In particular we have noticed that the hypothesis of Stokes that leads to the constitutive equation of a Stokesian fluid (not-elastic) and the linear Stokesian fluid which is the Newtonian fluid.

T i j = ( - p + α ) δ i j + β e i j + γ e i k e k j , Stokesian fluid T i j = ( - p + λ Θ ) δ i j + 2 μ e i j , Newtonian fluid .

The coefficients in these equations are functions only of the invariants of the rate of deformation tensor and of the thermodynamic state. The latter may be specified by two thermodynamic variables and the nature of the fluid is involved in the equation of state, of which one form is

ρ = f ( p , T ) .

If we substitute the constitutive equation of a Newtonian fluid into the equations of motion, we have the Navier-Stokes equation.

ρ D v D t = ρ f - p + ( λ + μ ) ( v ) + μ 2 v

Finally, the principle of the conservation of energy is used to give an energy equation. In this, certain assumptions have to be made as to the energy transfer and we have only considered the conduction of heat, giving

ρ D U D t = ( k T ) - p ( v ) + Y

These equations are both too general and too special. They are too general in the sense that they have to be simplified still further before any large body of results can emerge. They are too special in the sense that we have made some rather restrictive assumptions on the way, excluding for example elastic and electromagnetic effects.

Special cases of the equations

The full equations may be specialized is several ways, of which we shall consider the following:

  1. restrictions on the type of motions,
  2. specializations on the equations of motion,
  3. specializations of the constitutive equation or equation of state.

This classification is not the only one and the classes will be seen to overlap. We shall give a selection of examples and of the resulting equations, but the list is by no means exhaustive.

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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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