CONSERVATION EQUATIONS

These equations are expressions of the laws of physics, written in forms appropriate for flows.
(i) Mass is neither created nor destroyed : "continuity equation", or "conservation of mass".

(ii) Rate of change of momentum = Net force: "conservation of momentum"

(iii) Energy is conserved, though it may change form: "conservation of energy"
 
 

Integral forms for a control volume


 
 
Mass:

Momentum:

The terms on the rhs are:

I: pressure forces acting normal to the surface, per unit area.

II: body forces per unit mass.

III: shear forces acting parallel to the surface, per unit area.
 
 

Energy:

 

Note: 

Energy per unit volume:  = r(Internal energy per unit mass + kinetic energy per unit mass).

In addition to these specific conservation laws, specific equations relating the state variables are needed to solve problems for given kinds of fluids. To solve very complicated flow problems, the boundary conditions are specified, and all of these equations are solved simultaneoulsy all over the flowfield, for each step in time. This is a task which usually requires fast computers with large memory, because we have to keep track of a large number of variables and perform a large number of calculations. Long before this became "possible" people figured out more restricted ways of solving specific problems needed to build airplanes. These "smart analytical methods" form the subject of this course.

Relations Used to Reduce the Conservation Equations to Differential Form

1. Stokes' Theorem
where  is the vector quantity of interest,  is the vector along the closed contour of integration C,  is the unit vector normal to the area enclosed by C.
2. Divergence Theorem

3. Gradient Theorem

If p is a scalar field, then

4. where  is a scalar and  is a vector.

Substantial Derivative

The Eulerian Frame of Reference is the one fixed to the control volume. The Lagrangian frame of reference is the one fixed to a packet of fluid (a fluid element)
 

The rate of change of any property as seen by the fluid element is:

The substantial derivative is:

. The first term on the rhs is the "local" or "unsteady" term. The second is the "convective" term.

The rate of change D()/Dt is for two reasons:

1. Things are changing at the point through which the element is moving (unsteady, local)

2. The element is moving into regions with different properties.

Using the vector identity (4) above, the conservation equations can be re-written:
Continuity:

or,
 


 

In terms of veclocity components, this can be written as a scalar equation:


 

Momentum Conservation: Differential Form

Knowing the properties of the particular fluid and problem being considered, the body force term and the viscous force term can be expanded. One very useful form is where the viscous stresses are related to the rate of strain of the fluid, through a linear expression.This is valid for "Newtonian Fluids". This is further simplified using the Stokes hypothesis, which permits us to delete the normal-strain terms from the strain terms, leaving only shear-strain terms. The resulting form of the momentum equation is called the Navier-Stokes equation. This is often used as the general starting point to solve problems in fluid mechanics.
Energy Conservation

The Euler equation

If the Reynolds number  = (Inertial Force divided by Viscous Force) >>1 in our flow problem, we can neglect the viscous stress terms. Thus the differential form of the momentum equation reduces to:

Here u,v,w are the Cartesian components along x,y,z of the vector  , and fx, fy and fz are components of the body force vector. From the continuity equation,

, and the substantial derivative, we can reduce th emomentum equation to:

Euler equation.