Relations Used to Reduce the Conservation Equations to Differential Form

In this section we will first convert the conservation equations to a form  suitable to apply at each point, so that we can track changes from one point to another. This involves converting from the "integral form" over a control volume or control surface, to a "differential form" which deals with small changes from point to point.

The obvious approach is to say: "if the integral, over an arbitrary control volume, of this sum of terms is zero, then this sum of terms must also be zero in the limit as I reduce the size of the control volume down to a point". Thus we can get rid of the integral signs. Unfortunately, we find that we can't yet bring everything under the same integral sign: in each of the conservation equations, there are some integrals over control volumes, and other integrals over control surfaces, and actually we would also like to know about integrals over just a closed contour in a 2-dimensional flowfield. So the first priority is to find relations between integrals over lines, surfaces and volumes.

We have seen, in the conservation equations, and   which are, respectively, integrals over control surfaces and control volumes.

We'll also use , the line integral over a "closed contour", which means, add up all the things we see as we walk along this line which closes on itself like a snake managing to catch its own tail.
 

1. Stokes' Theorem

where  is the vector quantity of interest, dl is the vector along the closed contour of integration c,  is the vector normal to the area enclosed by c.Now we can convert integrals over closed contours to integrals over surfaces, and vice versa.

2. Divergence Theorem

The divergence theorem thus lets us convert between integrals over surfaces and volumes.

3. Gradient Theorem

If p is a scalar field, (only magnitude, no direction, like pressure or density), then

4        Another Vector Identity: Divergence of the Product of a Vector and Scalar

where  is a scalar and  is a vector.

Substantial Derivative

So far, we derived equations for flow in a Control Volume, which is a region of space, rather than a particular clump of fluid. Now, let us see how to describe the changes to a given element of fluid.  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)

This is the substantial derivative, or,

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. 

In the early 1980s, there was a sleet storm in Atlanta. Atlanta's roads have many ups and downs, and in those days most cars had rear-wheel drive. Snow is so rare in Atlanta that the city did not have many sand-trucks then, and people still don't have snow-tires or chains. Instead they look forward to a "snow day" when schools are closed, and employers are usually nice about letting people go home early and take their kids sliding down slopes on improvised sleds. So most offices in downtown had radios tuned to the local stations, listening to the changing weather forecast.
The local radio and TV stations had an attentive audience. They took to their "traffic-copters" and flew out west to the Georgia-Alabama border to watch the storm rolling in. So, Atlanta office workers heard of rapidly changing weather conditions, but also head that this was far out west of Atlanta. They extrapolated and believed the weather prediction that by 6pm, it would be snowing in Atlanta. Wise and considerate supervisors told their teams to go home early, to beat the expected traffic problems in the evening. Atlanta's 100,000 commuters (that was in 1980: now it would be 500,000) pulled out of their offices, and got on the roads. The traffic jam started at 2pm, but this was not a real concern, because people expected to be home by 5 at least. Unfortunately, the rate of change in the weather was not all due to the "convective term", rolling in: the temperature in Atlanta (which is several hundred feet higher than the surrounding plain) dropped, and the sleet started falling at 2pm, and kept on falling. By 3pm, 100,000 cars were on the roads, all sliding around helplessly. Every side road was blocked, because at every hill there were cars sliding across the road, unable to get up the slope. Most people did not reach home before 9pm, often in the cars of kind fellow-sufferers, after abandoning their own cars on the highway. Those who did not spend their time listening to the radio (or had nasty supervisors) just took one look at the road outside, and came back in to huddle in the office overnight.

 Morals of the story: (1) the rate of change is both local (unsteady) and convective.

(2) Don't waste time listening to weather forecasts: look outside instead.
 
 

Imagine that you are inside a TVNEWS helicopter, flying toward a tornado. You are at point (A). If you stood still at point A, you would feel the pressure decrease rapidly [ tornado going to happen there]. This is . As you move at speed, you feel other changes as you move into the pressure field of the tornado.

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

Continuity Equation:

where

is called the substantial derivative.

In terms of velocity components,

Use the Vector Identity

The Continuity equation is:

Momentum Conservation: Differential Form

These equations are quite general (i.e., they apply to most flows that one can imagine, and are too general to get any useful results for specific problems). Exceptions occur. Imagine an immense cloud of gas jetting out of a star: this can also be described using fluid dynamics, except that one might not be able to assume that it’s a "continuum", and the velocities might be so high that they are comparable to the speed of light, so one has to consider conversions between energy and mass. Temperatures may be so high that nuclear reactions might be occurring. Under those conditions, the above equations are not general enough.

On the other hand, we have to make some specializations if we are to apply these equations to any specific problem. The most widely used form is called the "Navier-Stokes equations". These are derived by incorporating the "Stokes hypothesis" to specialize the stress terms on the right hand side to the usual case where we worry mostly about viscous stresses. We'll see more about this when we consider the forces on the right-hand side.

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 Equation

The energy equation, reduced to differential form, is: