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183_notes:drag [2014/08/25 12:50] – caballero | 183_notes:drag [2021/02/04 23:39] (current) – [Models of fluid resistance] stumptyl | ||
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===== Drag ===== | ===== Drag ===== | ||
- | In most real world situations, there are some kind of resistive | + | In most real world situations, there is some kind of resistive |
+ | |||
+ | ==== Fluid Resistance ==== | ||
+ | |||
+ | //An object moving in any fluid experiences some form of resistance to its motion due collisions with molecules of the fluid.// Each of these little collisions with the surrounding fluid contribute to the overall resistive force that the fluid exerts on a moving object. | ||
+ | |||
+ | //Unlike friction forces, which are velocity-independent, | ||
==== Models of fluid resistance ==== | ==== Models of fluid resistance ==== | ||
+ | |||
+ | Which model of fluid resistance is most useful (or valid) depends on the properties of the system in question. Specifically, | ||
+ | |||
+ | A discussion of Reynolds number is beyond the scope of this course, but suffice it to say that an small, slow-moving object in a viscous fluid will have a low Reynolds number. A large, fast moving object in a less viscous fluid will have a high Reynolds number. | ||
+ | |||
+ | An excellent, but long video that describes these different kinds of flows is shown below. | ||
+ | |||
+ | {{youtube> | ||
=== Laminar drag === | === Laminar drag === | ||
+ | |||
+ | For a situation where the __Reynolds number is low__ (e.g., a small, slow-moving object in a viscous fluid), the fluid resistance is proportional to the velocity of the object: | ||
$$\vec{F}_{drag} = -b\vec{v}$$ | $$\vec{F}_{drag} = -b\vec{v}$$ | ||
- | $$\vec{F}_{drag} = -6\pi\eta\vec{v}$$ | + | |
+ | where $b$ is a constant factor the depends on different fluid and object parameters. For a spherical object with radius $r$, the fluid resistance takes the form, | ||
+ | |||
+ | $$\vec{F}_{drag} = -6\pi\eta | ||
+ | |||
+ | where** $\eta$** is the **fluid viscosity**. | ||
+ | |||
+ | \\ | ||
=== Turbulent drag === | === Turbulent drag === | ||
+ | |||
+ | For a situation where the __Reynolds number is high__ (e.g., a large, fast-moving object in a less viscous fluid), the fluid resistance is proportional to the speed of the object squared: | ||
$$\vec{F}_{drag} = -cv^2\hat{v}$$ | $$\vec{F}_{drag} = -cv^2\hat{v}$$ | ||
+ | |||
+ | where $c$ is a constant factor the depends on different fluid and object parameters. The constant $c$ can be unpacked f into 3 different parameters: | ||
+ | |||
$$\vec{F}_{drag} = -\dfrac{1}{2} \rho C_d A v^2 \hat{v}$$ | $$\vec{F}_{drag} = -\dfrac{1}{2} \rho C_d A v^2 \hat{v}$$ | ||
+ | |||
+ | where $\rho$ is the// density of the fluid//, $A$ //is the cross-sectional area of the object in the fluid//, and $C_d$ //is the drag coefficient of the object, which is often measured experimentally// | ||
+ | |||
+ | \\ | ||
+ | === What about " | ||
+ | |||
+ | If your system does not exist at either end of the spectrum, where one or the other model dominates, you must use them both at the same time. In these situtations the fluid resistance is given by: | ||
+ | |||
+ | $$\vec{F}_{drag} = -b\vec{v}-cv^2\hat{v}$$ | ||
+ | |||
+ | However, in many cases you can reasonable assume either a low Reynolds number (a small sphere moving in oil) or a high Reynolds number (most macroscopic things moving in air). | ||