Drag (physics)

# Drag (physics)

Overview
Discussion

Encyclopedia
Shape and flow Form
drag
Skin
friction
0% 100%
~10% ~90%
~90% ~10%
100% 0%

In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, drag (sometimes called air resistance or fluid resistance) refers to forces which act on a solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...

object in the direction of the relative fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

flow velocity
Flow velocity
In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...

. Unlike other resistive forces such as dry friction, which is nearly independent of velocity, drag forces depend on velocity.

Drag forces always decrease fluid velocity relative to the solid object in the fluid's path.

## Examples of drag

Examples of drag include the component of the net
Net force
In physics, net force is the total force acting on an object. It is calculated by vector addition of all forces that are actually acting on that object. Net force has the same effect on the translational motion of the object as all actual forces taken together...

aerodynamic or hydrodynamic force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

acting opposite to the direction of the movement of the solid object relative to the Earth as for cars, aircraft and boat hulls; or acting in the same geographical direction of motion as the solid, as for a sails on a down wind sail boat, or in intermediate directions on a sail depending on points of sail. In the case of viscous drag of fluid in a pipe
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...

, drag force on the immobile pipe decreases fluid velocity relative to the pipe.

## Types of drag

Types of drag are generally divided into the following categories:
• parasitic drag
Parasitic drag
Parasitic drag is drag caused by moving a solid object through a fluid medium . Parasitic drag is made up of many components, the most prominent being form drag...

, consisting of
• form drag,
• skin friction,
• interference drag,
• lift-induced drag
Lift-induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars...

, and
• wave drag
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...

(aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...

) or wave resistance (ship hydrodynamics).

The phrase parasitic drag is mainly used in aerodynamics, since for lifting wings drag is in general small compared to lift. For flow around bluff bodies, drag is most often dominating, and then the qualifier "parasitic" is meaningless. Form drag, skin friction and interference drag on bluff bodies are not coined as being elements of parasitic drag, but directly as elements of drag.

Further, lift-induced drag is only relevant when wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...

s or a lifting body
Lifting body
A lifting body is a fixed-wing aircraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage with little or no conventional wing...

are present, and is therefore usually discussed either in the aviation perspective of drag, or in the design of either semi-planing or planing hulls. Wave drag
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...

occurs when a solid object is moving through a fluid at or near the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

in that fluid—or in case there is a freely-moving fluid surface with surface waves
Ocean surface wave
In fluid dynamics, wind waves or, more precisely, wind-generated waves are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They usually result from the wind blowing over a vast enough stretch of fluid surface. Waves in the...

radiating from the object, e.g. from a ship.

For high velocities—or more precisely, at high Reynolds numbers—the overall drag of an object is characterized by a dimensionless number called the drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...

, and is calculated using the drag equation
Drag equation
In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid....

. Assuming a more-or-less constant drag coefficient, drag will vary as the square of velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. 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 mass density, the cross sectional area of the specified 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
Badminton is a racquet sport played by either two opposing players or two opposing pairs , who take positions on opposite halves of a rectangular court that is divided by a net. Players score points by striking a shuttlecock with their racquet so that it passes over the net and lands in their...

shuttlecock
Shuttlecock
A shuttlecock, sometimes called a bird or birdie, is a high-drag projectile used in the sport of badminton. It has an open conical shape: the cone is formed from sixteen or so overlapping feathers, usually goose or duck and from the left wing only, embedded into a rounded cork base...

has more wind resistance than a squash
Squash (sport)
Squash is a high-speed racquet sport played by two players in a four-walled court with a small, hollow rubber ball...

ball).

## Drag at high velocity

The drag equation
Drag equation
In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid....

calculates the force experienced by an object moving through a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

at relatively large velocity (i.e. high Reynolds number, Re > ~1000), also called quadratic drag. The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (L being some length). The force on a moving object due to a fluid is:

see derivation

where is the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

of drag, is the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

of the fluid, is the speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

of the object relative to the fluid, is the drag coefficient
Drag coefficient
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment such as air or water. It is used in the drag equation, where a lower drag coefficient indicates the object will have less aerodynamic or...

(a dimensionless parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

, e.g. 0.25 to 0.45 for a car) is the reference area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

,

The reference area A is often defined as the area of the orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...

of the object—on a plane perpendicular to the direction of motion—e.g. for objects with a simple shape, such as a sphere, this is the cross sectional
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

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.

In case of a wing, comparison of the drag to the lift force is easiest when the reference areas are the same, since then the ratio of drag to lift force is just the ratio of drag to lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...

. Therefore, the reference for a wing often is the planform
Planform
In aviation, a planform is the shape and layout of a fixed-wing aircraft's fuselage and wing. Of all the myriad planforms used, they can typically be grouped into those used for low-speed flight, found on general aviation aircraft, and those used for high-speed flight, found on many military...

(or wing) area rather than the frontal area.

For an object with a smooth surface, and non-fixed separation point
Flow separation
All solid objects travelling through a fluid acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent...

s—like a sphere or circular cylinder—the drag coefficient may vary with Reynolds number Re, even up to very high values (Re of the order
Order of magnitude
An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount being scaled is 10 and the scale is the exponent being applied to this amount...

107).
For an object with well-defined fixed separation points, like a circular disk with its plane normal to the flow direction, the drag coefficient is constant for Re > 3,500.
Further the drag coefficient Cd is, in general, a function of the orientation of the flow with respect to the object (apart from symmetrical objects like a sphere).

### Power

The power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...

required to overcome the aerodynamic drag is given by:

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 (80.5 km/h) may require only 10 hp to overcome air drag, but that same car at 100 mph (160.9 km/h) requires 80 hp. 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
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, four times the work done in half the time requires eight times the power.

### Velocity of a falling object

The velocity as a function of time for an object falling through a non-dense medium, and released at zero relative-velocity v = 0 at time t = 0, is roughly given by a function involving a hyperbolic tangent (tanh):

The hyperbolic tangent has a limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

value of one, for large time t. In other words, velocity asymptotically approaches a maximum value called the terminal velocity
Terminal velocity
In fluid dynamics an object is moving at its terminal velocity if its speed is constant due to the restraining force exerted by the fluid through which it is moving....

vt:

For a potato-shaped object of average diameter d and of density ρobj, terminal velocity is about

For objects of water-like density (raindrops, hail, live objects—animals, birds, insects, etc.) falling in air near the surface of the Earth at sea level, terminal velocity is roughly equal to

with d in metre and vt in m/s. For example, for a human body ( ~ 0.6 m) ~ 70 m/s, for a small animal like a cat ( ~ 0.2 m) ~ 40 m/s, for a small bird ( ~ 0.05 m) ~ 20 m/s, for an insect ( ~ 0.01 m) ~ 9 m/s, and so on. Terminal velocity for very small objects (pollen, etc.) at low Reynolds numbers is determined by Stokes law.

Terminal velocity is higher for larger creatures, and thus potentially more deadly. A creature such as a mouse falling at its terminal velocity is much more likely to survive impact with the ground than a human falling at its terminal velocity. A small animal such as a cricket
Cricket (insect)
Crickets, family Gryllidae , are insects somewhat related to grasshoppers, and more closely related to katydids or bush crickets . They have somewhat flattened bodies and long antennae. There are about 900 species of crickets...

impacting at its terminal velocity will probably be unharmed. This, combined with the relative ratio of limb cross-sectional area vs. body mass (commonly referred to as the Square-cube law
Square-cube law
The square-cube law is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. It was first demonstrated in 1638 in Galileo's Two New Sciences...

), explains why small animals can fall from a large height and not be harmed.

## Very low Reynolds numbers—Stokes' drag

The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, ). Note that purely laminar flow only exists up to Re = 0.1 under this definition. In this case, the force of drag is approximately proportional to velocity, but opposite in direction. The equation for viscous resistance is:

where: is a constant that depends on the properties of the fluid and the dimensions of the object, and is the velocity of the object

When an object falls from rest, its velocity will be

which asymptotically approaches the terminal velocity . For a given , heavier objects fall faster.

For the special case of small spherical objects moving slowly through a viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

(and thus at small Reynolds number), George Gabriel Stokes
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...

derived an expression for the drag constant,

The Stokes radius, Stokes-Einstein radius, or hydrodynamic radius RH, named after George Gabriel Stokes, is not the effective radius of a hydrated molecule in solution as often mentioned. Rather it is the radius of a hard sphere that diffuses at the same rate as the molecule. The behavior of this...

of the particle, and is the fluid viscosity.

For example, consider a small sphere with radius = 0.5 micrometre (diameter = 1.0 µm) moving through water at a velocity of 10 µm/s. Using 10−3 Pa·s as the dynamic viscosity of water in SI units,
we find a drag force of 0.09 pN. This is about the drag force that a bacterium experiences as it swims through water.

### Lift induced drag

Lift-induced drag (also called induced drag) is drag which occurs as the result of the creation of lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

on a three-dimensional lifting body
Lifting body
A lifting body is a fixed-wing aircraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage with little or no conventional wing...

, such as the wing
Wing
A wing is an appendage with a surface that produces lift for flight or propulsion through the atmosphere, or through another gaseous or liquid fluid...

or fuselage of an airplane. Induced drag consists of two primary components, including drag due to the creation of vortices (vortex drag) and the presence of additional viscous drag (lift-induced viscous drag). The vortices in the flow-field, present in the wake of a lifting body, derive from the turbulent mixing of air of varying pressure on the upper and lower surfaces of the body, which is a necessary condition for the creation of lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

.

With other parameters remaining the same, as the lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

generated by a body increases, so does the lift-induced drag. For an aircraft in flight, this means that as the angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...

, and therefore the lift coefficient
Lift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...

, increases to the point of stall, so does the lift-induced drag. At the onset of stall
Stall (flight)
In fluid dynamics, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded...

, lift is abruptly decreased, as is lift-induced drag, but viscous pressure drag, a component of parasite drag, increases due to the formation of turbulent unattached flow on the surface of the body.

### Parasitic drag

Parasitic drag (also called parasite drag) is drag caused by moving a solid object through a fluid. Parasitic drag is made up of multiple components including viscous pressure drag (form drag), and drag due to surface roughness (skin friction drag). Additionally, the presence of multiple bodies in relative proximity may incur so called interference drag, which is sometimes described as a component of parasitic drag.

In aviation, induced drag tends to be greater at lower speeds because a high angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...

is required to maintain lift, creating more drag. However, as speed increases the induced drag becomes much less, but parasitic drag increases because the fluid is flowing faster around protruding objects increasing friction or drag. At even higher speeds in the transonic
Transonic
Transonic speed is an aeronautics term referring to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.2, i.e. 600–900 mph...

, wave drag
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...

enters the picture. Each of these forms of drag changes in proportion to the others based on speed. The combined overall drag curve therefore shows a minimum at some airspeed - an aircraft flying at this speed will be at or close to its optimal efficiency. Pilots will use this speed to maximize endurance
Endurance (aircraft)
In aviation, Endurance is the maximum length of time that an aircraft can spend in cruising flight. Endurance is sometimes erroneously equated with range. The two concepts are distinctly different: range is a measure of distance flown while endurance is a measure of time spent in the air...

(minimum fuel consumption), or maximise gliding range in the event of an engine failure.

### Power curve in aviation

The interaction of parasitic and induced drag vs. airspeed can be plotted as a characteristic curve, illustrated here. In aviation, this is often referred to as the power curve, and is important to pilots because it shows that, below a certain airspeed, maintaining airspeed counterintuitively requires more thrust as speed decreases, rather than less. The consequences of being "behind the curve" in flight are important and are taught as part of pilot training. At the subsonic airspeeds where the "U" shape of this curve is significant, wave drag has not yet become a factor, and so it is not shown in the curve.

### Wave drag in transonic and supersonic flow

Wave drag (also called compressibility drag) is drag which is created by the presence of a body moving at high speed through a compressible fluid. In aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...

, Wave drag consists of multiple components depending on the speed regime of the flight.

In transonic flight (Mach numbers greater than about 0.8 and less than about 1.4), wave drag is the result of the formation of shockwaves on the body, formed when areas of local supersonic (Mach number greater than 1.0) flow are created. In practice, supersonic flow occurs on bodies traveling well below the speed of sound, as the local speed of air on a body increases when it accelerates over the body, in this case above Mach 1.0. However, full supersonic flow over the vehicle will not develop until well past Mach 1.0. Aircraft flying at transonic speed often incur wave drag through the normal course of operation. In transonic flight, wave drag is commonly referred to as transonic compressibility drag. Transonic compressibility drag increases significantly as the speed of flight increases towards Mach 1.0, dominating other forms of drag at these speeds.

In supersonic flight (Mach numbers greater than 1.0), wave drag is the result of shockwaves present on the body, typically oblique shockwaves formed at the leading and trailing edges of the body. In highly supersonic flows, or in bodies with turning angles sufficiently large, unattached shockwaves, or bow waves will instead form. Additionally, local areas of transonic flow behind the initial shockwave may occur at lower supersonic speeds, and can lead to the development of additional, smaller shockwaves present on the surfaces of other lifting bodies, similar to those found in transonic flows. In supersonic flow regimes, wave drag is commonly separated into two components, supersonic lift-dependent wave drag and supersonic volume-dependent wave drag.

The closed form solution for the minimum wave drag of a body of revolution with a fixed length was found by Sears and Haack, and is known as the Sears-Haack Distribution. Similarly, for a fixed volume, the shape for minimum wave drag is the Von Karman Ogive.

Busemann's Biplane
Busemann's Biplane
Busemann's Biplane is a conceptual airframe design invented by Adolf Busemann which avoids the formation of N-type shock waves and thus does not create a sonic boom....

is not, in principle, subject to wave drag at all when operated at its design speed, but is incapable of generating lift.

In 1752 d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

proved that potential flow
Potential flow
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications...

, the 18th century state-of-the-art inviscid flow
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....

theory amenable to mathematical solutions, resulted in the prediction of zero drag. This was in contradiction with experimental evidence, and became known as d'Alembert's paradox. In the 19th century the Navier–Stokes equations for the description of viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

flow were developed by Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant was a mechanician and mathematician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern...

, Navier
Claude-Louis Navier
Claude-Louis Navier born Claude Louis Marie Henri Navier , was a French engineer and physicist who specialized in mechanics.The Navier–Stokes equations are named after him and George Gabriel Stokes....

and Stokes
George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...

. Stokes derived the drag around a sphere at very low Reynolds numbers, the result of which is called Stokes law
Stokes law
Stokes' law can refer to:*Stokes' law for friction force*Stokes' law law describing attenuation of sound in Newtonian liquidsFor integration, see Stokes' theorem....

.

In the limit of high-Reynolds numbers the Navier–Stokes equations approach the inviscid Euler equations; of which the potential-flow solutions considered by d'Alembert are solutions. However, at high Reynolds numbers all experiments showed there is drag. Attempts to construct inviscid steady flow solutions to the Euler equations, other than the potential flow solutions, did not result in realistic results.

The notion of boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

s—introduced by Prandtl in 1904, founded on both theory and experiments—explained the causes of drag at high Reynolds numbers. The boundary layer is the thin layer of fluid close to the object's boundary, where viscous effects remain important when the viscosity becomes very small (or equivalently the Reynolds number becomes very large).