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Continuity equation - Continuity Equation(Conservation of...

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Continuity Equation (Conservation of Mass) There are several physical conservation laws that we use to predict the behavior of a fluid. In Chapter 1 we discussed Newtons's Second Law ( F = ma ) which is really a statement of the conservation of momentum, and the Law of Conservation of Energy. In addition, we use the physical law describing the conservation of mass. This law is typically described for a fluid in terms of an equation called the "continuity" equation. Before we derive the continuity equation, we must first define three terms regarding fluid flow fields. The first is steady state . Steady state means that at any given point, the flow does not fluctuate with time. Imagine looking at a single point in a flow field. If, at that point, the direction and speed (velocity) of the flow remains constant, the flow is steady . The second term is the streamline . Consider a small fluid element moving along a path in steady state. This moving fluid element traces out a fixed path in space. The fixed path is called a streamline. The fluid flow is always parallel to the streamline since that is how the streamline is defined. There can be no flow across a streamline. The third term is the stream tube . A stream tube consists of all the streamlines surrounding a volume of flowing fluid. A hose or a pipe can be thought of as a stream tube. The flow can only go through the stream tube. It cannot go through the walls of the stream tube since the walls are streamlines and there can be no flow across a streamline. Now we can derive the continuity equation. We start the the phyical principle "conservation of mass" that says mass can be neither created nor destroyed. To apply this principle to a flowing gas, let's look at the figure below, and consider an imaginary arbitrary shape drawn perpendicular to the flow direction. The long dotted lines represent a stream tube, thus, the fluid flows within these lines. Flow enters the stream tube on the left with a speed of V 1 through an arbitrarily shaped area A 1 . The flow exits on the right with a speed of V 2 through an arbitrarily shaped area A 2 . (Note: A 2 is in general not the same shape as A 1 . ) If the flow is steady, then the mass that flows through the cross- section at Point 1 is the same as the mass that flow through the cross-section at Point 2. The mass flowing through the stream tube is confined by the streamlines of the boundary, just as the flow in a garden hose is confined by the wall of the hose.
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