RiverViscosity (1)

RiverViscosity (1) - previous index next Calculating...

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previous index next Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes Michael Fowler, UVa 6/12/06 Introduction In this lecture, we’ll derive the velocity distribution for two examples of laminar flow. First we’ll consider a wide river, by which we mean wide compared with its depth (which we take to be uniform) and we ignore the more complicated flow pattern near the banks. Our second example is smooth flow down a circular pipe. For the wide river, the water flow can be thought of as being in horizontal “sheets”, so all the water at the same depth is moving at the same velocity. As mentioned in the last lecture, the flow can be pictured as like a pile of printer paper left on a sloping desk: it all slides down, assume the bottom sheet stays stuck to the desk, each other sheet moves downhill a little faster than the sheet immediately beneath it. For flow down a circular pipe, the laminar “sheets” are hollow tubes centered on the line down the middle of the pipe. The fastest flowing fluid is right at that central line. For both river and tube flow, the drag force between adjacent small elements of neighboring sheets is given by force per unit area ( ) dv z F Ad z η = where now the z -direction means perpendicular to the small element of sheet. A Flowing River: Finding the Velocity Profile For a river flowing steadily down a gentle incline under gravity, we’ll assume all the streamlines point in the same direction, the river is wide and of uniform depth, and the depth is much smaller than the width. This means almost all the flow is well away from the edges (the river banks), so we’ll ignore the slowing down there, and just analyze the flow rate per meter of river width , taking it to be uniform across the river. The simplest basic question is: given the slope of the land and the depth of the river, what is the total flow rate? To answer, we need to find the speed of flow v ( z ) as a function of depth (we know the water in contact with the river bed isn’t flowing at all), and then add the flow contributions from the different depths (this will be an integral) to find the total flow. The function v ( z ) is called the “velocity profile”. We’ll prove it looks something like this:
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2 Stream velocity v ( z ) at different depths (For a smoothly flowing river, the downhill ground slope would be imperceptible on this scale.) But how do we begin to calculate v ( z )? Recall that (in an earlier lecture) to find how hydrostatic pressure varied with depth, we mentally separated a cylinder of fluid from its surroundings, and applied Newton’s Laws: it wasn’t moving, so we figured its weight had to be balanced by the sum of the pressure forces it experienced from the rest of the fluid surrounding it. In fact, its weight was balanced by the difference between the pressure underneath and that on top.
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RiverViscosity (1) - previous index next Calculating...

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