The Equation of Continuity

You may have noticed that you can increase the speed of the water emerging from a garden hose by partially closing the hose opening with your thumb. Apparently the speed v of the water depends on the cross – sectional area A through which the water flows.

Here we wish to derive an expression that relates v and A for the steady flow of an ideal fluid through a tube with varying cross section, like that in figure

 

The flow there is toward the right, and the tube segment shown ( part of a longer tube ) has length L. The fluid has speeds v₁ at the left end of the segment and v₂ at the right end. The tube has cross – sectional areas A₁ at the left end A₂ at the right end. Suppose that in a time interval t a volume V of fluid enters the tube segments at its left end. Then, because the fluid is incompressible , an indentical volume V must emerge from the right end of the segment .

We can use this common volume V to relate the speeds and areas . To do so, we first consider figure

Which shows a side view of a tube of uniform cross-sectional area A. In Figure a, a fluid element e is about to pass through the dashed line drawn across the tube width. The element’s speed is v , so during a time interval t , the element moves along the tube a distance x = v t . The volume V of fluid that has passed through the dassed line in that time interval t is

Applying to both the left and right ends of the tube segments , we have

This relation between speed and cross-sectional area is called the equation of continuity for the flow of an ideal fluid. It tells us that the flow speed increases when we decrease the cross-sectional area through which the fluid flows.

Equation A₁v₁ = A₂v₂ applies not only to an actual tube but also to any so-called tube of flow, or imaginary tube whose boundary consists of streamlines. Such a tube acts like a real tube because no fluid element can cross a streamline; thus, all the fluid within a tube of flow must remain within its boundary. Figure

Shows a tube of flow in which the cross- sectional area increases from A₁ to area A₂ along the flow direction. From equation A₁v₁ = A₂v₂ we know that, with the increase in area , the speed must decrease, as is indicated by the greater spacing between streamlines at the right in figure. Similarly, you can see that figure

The speed of the flow is greatest just above and just below the cylinder.

We can rewrite equation A₁v₁ = A₂v as

R_v = Av = a constant

In which R_v is the volume flow rate of the fluid ( volume past given point per unit time ). Its SI unit is the cubic meter per second ( m³ /s ) . If the density rho of the fluid is uniform, we can multiply equation R_v = Av by that density to get the mass flow rate R_m ( mass per unit time ):

The SI unit of mass flow rate is the kilogram per second. The equation says that the mass that flow into tube segement of figure

each second must be equal to the mass that flows out of that segment each second.

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