11-17-08LectPHY2048

11-17-08LectPHY2048 - Fluid Dynamics Types of fluid flow We...

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Fluid Dynamics We can visualize the flow of a fluid by injecting smoke or dye into the moving fluid. There are two distinct regimes of flow: laminar flow and turbulent flow . These are both evident in the figure below. laminar flow turbulent flow Types of fluid flow:
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For laminar flow the velocity of the fluid at each point is a constant , i.e. at each point the speed and the direction of the fluid remains the same over time. For turbulent flow this is not true. For the situation above two photographs taken at different times would show that the streamlines on the left and upper right would be unchanged, while the smoke pattern on the lower right would be quite different. laminar flow turbulent flow
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laminar flow turbulent flow For laminar flow a volume element of the fluid that begins on a streamline remains on that streamline and streamlines do not cross each other. In the following development we assume : 2) The fluid is incompressible 1) The flow is laminar 3) The fluid has very low (negligible) viscosity 4) The flow is irrotational (meaning in essence that the flow does not contain eddies, turning back on itself).
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Equation of continuity Consider the flow of an ideal fluid through a uniform pipe. Δ x If the fluid is flowing with speed v then an element of the fluid goes a distance Δ x in time Δ t so x vx v t t Δ = →Δ=Δ Δ The volume of fluid crossing a point along the pipe, in that time is then Δ V = A Δ x = Av Δ t Cross-sectional area = A (important concept)
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The volume flow rate R V , which is the volume of fluid passing a point along the pipe in given time can then be written, V V RA v t Δ == Δ So the volume flow rate also equals the cross-sectional area of the pipe times the velocity of the fluid going through it. Given the volume flow rate through a hose, R V , we can determine e.g. the time taken to fill a pool of volume V P , since with t unknown we can write, PV VR t = P V V t R =
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Consider now a pipe that has a non-uniform cross-section , necking down as shown below The volume of fluid crossing a point along the tube on the left (in the fatter section) in time Δ t must equal the volume of fluid crossing a point on the right (in the thinner section) in the same time, so Δ V (left) = Δ V(right) A 1 v 1 Δ t = A 2 v 2 Δ t A 1 v 1 = A 2 v 2 v 1 A 1 A 2 v 2
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A 1 v 1 = A 2 v 2 This is the equation of continuity . It says that in order to keep the volume flow rate , of the fluid constant, the velocity of the fluid in the thinner section must be greater (in proportion to the ratio of the cross-sectional areas), i.e. V V RA v t Δ == Δ v 1 A 1 A 2 v 2 1 21 2 A vv A =
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Example A stationary lawn sprinkler consists of a hose feeding a dead ended container with 12 identical holes of 0.13 cm diameter. Water in the 1.9 cm inner diameter hose has a speed of 0.91 m/s. With what speed does the water leave the holes ?
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This note was uploaded on 08/25/2011 for the course PHY 2048 taught by Professor Field during the Fall '08 term at University of Florida.

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11-17-08LectPHY2048 - Fluid Dynamics Types of fluid flow We...

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