Drag (physics)

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An object falling through a gas or liquid experiences a force in direction opposite to its motion. Terminal velocity is achieved when the drag force is equal to force of gravity pulling it down.

In fluid dynamics, drag is the force that resists the movement of a solid object through a fluid (a liquid or gas). Drag is made up of friction forces, which act in a direction parallel to the object's surface (primarily along its sides, as friction forces at the front and back cancel themselves out), plus pressure forces, which act in a direction perpendicular to the object's surface (primarily at the front and back, as pressure forces at the sides cancel themselves out).

For a solid object moving through a fluid or gas, the drag is the sum of all the aerodynamic or hydrodynamic forces in the direction of the external fluid flow. (Forces perpendicular to this direction are considered lift). It therefore acts to oppose the motion of the object, and in a powered vehicle it is overcome by thrust.

In astrodynamics, depending on the situation, atmospheric drag can be regarded as inefficiency requiring expense of additional energy during launch of the space object or as a bonus simplifying return from orbit.

1 Details
2 General drag
3 Drag at low velocity; Stokes' Drag
4 Drag at high velocity
5 See also
6 References

[edit] Details

Types of drag are generally divided into three categories: parasitic drag, lift-induced drag and wave drag. Parasitic drag includes form drag, skin friction and interference drag. Lift-induced drag is only relevant when wings or a lifting body are present, and is therefore usually discussed only in the aviation perspective of drag. Beyond these two kinds of drag there is a third kind of drag, called wave drag, that occurs when the solid object is moving through the fluid at or near the speed of sound in that fluid. The overall drag of an object is characterized by a dimensionless number called the drag coefficient, and is calculated using the drag equation. Assuming a constant drag coefficient, drag will vary as the square of velocity. Thus, the resultant power needed to overcome this drag will vary as the cube of velocity. The standard equation for drag is one half the coefficient of drag multiplied by the fluid density, the cross sectional area of your specified green item, and the square of the velocity

Wind resistance is a layman's term used to describe drag. Its use is often vague, and is usually used in a relative sense (e.g. A badminton shuttlecock has more wind resistance than a squash ball).

[edit] General drag

Fluid mechanics is one of the most complicated subjects in physics, and so people have found many approximations to avoid such complexities. The drag equation and viscous resistance equation below are two such approximations. In general, the force of drag experienced by an object moving through a fluid can be expressed by:

$\mathbf{F}_d = - \sum_n a_n v^n \mathbf{\hat v}\$

where

a n

is some constant for every value of n that relates to the properties of the fluid and the object, and

v

is the velocity, and

$\mathbf{\hat v}$ is the unit vector indicating the direction of the velocity (the negative sign indicating the drag is opposite to that of velocity).

Generally, the value of $a n$ decreases as n becomes larger.

[edit] Drag at low velocity; Stokes' Drag

The equation for viscous resistance is appropriate for small objects or particles moving through a fluid at relatively slow speeds. In this case, the force of drag is approximately proportional to velocity, but opposite in direction. [1] The equation for viscous resistance is:

$\mathbf{F}_d = - b \mathbf{v} \,$

where:

b is a constant that depends on the properties of the fluid and the dimensions of the object, and

v is the velocity of the object.

When an object falls from rest, its velocity will be

$v(t) = \frac{mg}{b}(1-e^{-bt/m})$

which asymptotically approaches the terminal velocity $v t = m g / b$ . For a certain b, heavier objects fall faster.

For the special case of small spherical objects moving slowly through a viscous fluid (and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag coefficient,

$b = 6 \pi \eta r\,$

where:

r is the Stokes radius of the particle, and

η is the fluid viscosity.

For example, consider a small sphere with radius r = 1 micrometre moving through water at a velocity v of 10 µm/s. Using 10^-3 as the dynamic viscosity of water in SI units, we find a drag force of 0.2 pN. This is about the drag force that a bacterium experiences as it swims through water.

[edit] Drag at high velocity

The Drag equation approximates the force experienced by an object moving through a fluid at relatively large velocity. The equation is attributed to Lord Rayleigh, who originally used $L^2 \$ in place of $A \$ (L being some length). The force on a moving object due to a fluid is:

$\mathbf{F}_d= - {1 \over 2} \rho v^2 A C_d \mathbf{\hat v}$ ^{see derivation}

where

F_d is the force of drag,

ρ is the density of the fluid (Note that for the Earth's atmosphere, the density can be found using the barometric formula),

v is the velocity of the object relative to the fluid,

A is the reference area,

C_d is the drag coefficient (a dimensionless constant, e.g. 0.25 to 0.45 for a car), and

$\mathbf{\hat v}$ is the unit vector indicating the direction of the velocity (the negative sign indicating the drag is opposite to that of velocity).

The reference area A is related to, but not exactly equal to, the area of the projection of the object on a plane perpendicular to the direction of motion (i.e., cross sectional area). Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given. The reference for a wing would be the plane area rather than the frontal area.

[edit] Discussion

The equation is based on an idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. No real object exactly corresponds to this behavior. C_d is the ratio of drag for any real object to that of the ideal object. In practice a rough unstreamlined body (a bluff body) will have a C_d around 1, more or less. Smoother objects can have much lower values of C_d. The equation is precise, it is the C_d (drag coefficient) that can vary and is found by experiment.

Of particular importance is the v² dependence on velocity, meaning that fluid drag increases with the square of velocity. When velocity is doubled, for example, not only does the fluid strike with twice the velocity, but twice the mass of fluid strikes per second. Therefore the change of momentum per second is multiplied by four. Force is equivalent to the change of momentum divided by time. This is in contrast with solid-on-solid friction, which generally has very little velocity dependence .

[edit] Power

The power required to overcome the aerodynamic drag is given by:

$P_d = \mathbf{F}_d \cdot \mathbf{v} = - {1 \over 2} \rho v^3 A C_d$

Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Exerting four times the force over a fixed distance produces four times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice faster. Since power is the rate of doing work, four times a work in half the time requires eight times the power.

It should be emphasized here that the drag equation is an approximation, and does not necessarily give a close approximation in every instance. Thus one should be careful when making assumptions using these equations.

[edit] Velocity of falling object

Main article: Terminal velocity

The velocity as a function of time for an object falling through a non-dense medium is roughly given by a function involving a hyperbolic tangent:

$v(t) = \sqrt{ \frac{2mg}{\rho A C_d} } \tanh \left(t \sqrt{\frac{g \rho C_d A}{2 m}} \right) \,$

In other words, velocity asymptotically approaches a maximum value called the Terminal velocity:

$v_{t} = \sqrt{ \frac{2mg}{\rho A C_d} } \,$

With all else (gravitational acceleration, density, cross-sectional area, drag constant, etc.) being equal, heavier objects fall faster.

[edit] See also

[edit] References

Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.
Huntley, H. E. (1967). Dimensional Analysis. Dover. LOC 67-17978.

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Categories: Aerodynamics | Equations | Fluid dynamics | Force

Drag (physics)

From Wikipedia, the free encyclopedia

Contents

[edit] Details

[edit] General drag

[edit] Drag at low velocity; Stokes' Drag

[edit] Drag at high velocity

[edit] Discussion

[edit] Power

[edit] Velocity of falling object

[edit] See also

[edit] References

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