Ideal fluid, volume flux and continuity equation

physics

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2. Lesson Outline ideal fluid dynamics volume flux continuity equation Source: V.van Gogh, Starry Night

3. I can describe the concept of ideal fluid I can determine the volume flux in flowing fluid I can solve the problem of continuity equation in flowing fluid

4. An ideal fluid is defined by the following set of characteristics. The flow of the fluid must be steady The velocity at any point in the flow does not change in time. incompressible The density does not change with pressure. non-viscous There are no sources of internal friction that could remove energy from the flow. irrotational An element (small piece) of the flow traces out a straight path aligned with the fluid velocity, not a helical pattern around it. Ideal fluids

5. Ideal fluid flow is a simple, but useful approximation to actual fluid motion. Arrows here represent the flow velocity in a cross-sectional cutout of the pipe. The velocity is constant across the area, there is no rotation, and no drag (viscosity) near the edges of the pipe.

6. STREAMLINE GEOMETRY

7. Small drag in streamlined position Large drag in unstreamlined position Both Images CC: BY-SA BoH (wimimedia commons) http://creativecommons.org/licenses/by-sa/3.0/

8. Turbulent flow: `vortex shedding’ around an obstacle Steady (also called laminar ) flow to the left of an island becomes unsteady (or turbulent ) after passing by it. The grey scale image above shows clouds in the atmosphere. Patches of the flow behind the obstacle are spun up to create a series of vortices that alternately roll off, or ‘shed’, from the top and bottom of the tube. The behavior downstream of the tube is one of many examples of non-ideal fluid flow. Island (from above) ideal fluid flows in Source: NASA

9. Volume flux (Volume flow Rate) Q = V/t Q = Volume flux of fluid (m 3 /s) V = Volume of flowing fluid (m 3 ) t = times taken by flowing fluid (s)

10. For a fluid of density r flowing through a pipe of cross-sectional area A at speed v , the amount of mass D m passing through a joint in the pipe in a time interval D t is given by For regions of flow in which there are no sources or sinks of fluid, the requirement that mass is neither created nor destroyed implies that the product r Av is the same at all parts of the flow Since an incompressible fluid has the same density at all points in the flow ( r =constant ), then two locations (1 and 2) of a pipe with different cross-sectional area will have the same volume flux. D m = r A v D t A 1 v 1 = A 2 v 2 r 1 A 1 v 1 = r 2 A 2 v 2 Conservation of mass for incompressible fluids