
In
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, d'Alembert's paradox (or the hydrodynamic paradox) is a paradox discovered in 1752 by French mathematician
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopé ...
. D'Alembert proved that – for
incompressible
Incompressible may refer to:
* Incompressible flow, in fluid mechanics
* incompressible vector field, in mathematics
* Incompressible surface, in mathematics
* Incompressible string, in computing
{{Disambig ...
and
inviscid potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
– the
drag force is zero on a body moving with constant
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
relative to (and simultaneously through) the
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
.
[Grimberg, Pauls & Frisch (2008).] Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to and at the same time through a fluid, such as air and water; especially at high velocities corresponding with high
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
s. It is a particular example of the
reversibility paradox.
D’Alembert, working on a 1749 Prize Problem of the
Berlin Academy on flow drag, concluded:
A
physical paradox
A physical paradox is an apparent contradiction in physical descriptions of the universe. While multiple physical paradoxes have accepted resolutions, others defy resolution and may indicate flaws in theory. In physics as in all of science, c ...
indicates flaws in the theory.
Fluid mechanics was thus discredited by engineers from the start, which resulted in an unfortunate split – between the field of
hydraulics
Hydraulics () is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counterpart of pneumatics, which concer ...
, observing phenomena which could not be explained, and theoretical
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
explaining phenomena which could not be observed – in the words of the Chemistry Nobel Laureate
Sir Cyril Hinshelwood.
According to
scientific consensus
Scientific consensus is the generally held judgment, position, and opinion of the majority or the supermajority of scientists in a particular field of study at any particular time.
Consensus is achieved through scholarly communication at confer ...
, the occurrence of the paradox is due to the neglected effects of
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. In conjunction with scientific experiments, there were huge advances in the theory of viscous fluid friction during the 19th century. With respect to the paradox, this culminated in the discovery and description of thin
boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
s by
Ludwig Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German Fluid mechanics, fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlyin ...
in 1904. Even at very high Reynolds numbers, the thin boundary layers remain as a result of viscous forces. These viscous forces cause
friction drag on streamlined objects, and for
bluff bodies the additional result is
flow separation
In fluid dynamics, flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake.
A boundary layer exists whenever there is relative movement between a fluid and a solid surface with viscous fo ...
and a low-pressure
wake behind the object, leading to
form drag
Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is defined as the combination of '' form drag'' and ''skin friction drag''.
It is named as ...
.
[Landau & Lifshitz (1987), p. 15.][ Batchelor (2000), pp. 264–265, 303, 337.][
, pp. XIX–XXIII.][
]
The general view in the fluid mechanics community is that, from a practical point of view, the paradox is solved along the lines suggested by Prandtl.
[Stewartson (1981).] A formal mathematical proof is lacking, and difficult to provide, as in so many other fluid-flow problems involving the Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
(which are used to describe viscous flow).
Viscous friction: Saint-Venant, Navier and Stokes
First steps towards solving the paradox were made by Adhémar Barré de Saint-Venant, who modelled viscous
Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
fluid friction. Saint-Venant states in 1847:
Soon after, in 1851, George Gabriel Stokes
Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish mathematician and physicist. Born in County Sligo, Ireland, Stokes spent his entire career at the University of Cambridge, where he served as the Lucasi ...
calculated the drag on a sphere in Stokes flow
Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advection, advec ...
, known as Stokes' law
In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the S ...
. Stokes flow is the low Reynolds-number limit of the Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
describing the motion of a viscous liquid.
However, when the flow problem is put into a non-dimensional form, the viscous Navier–Stokes equations converge for increasing Reynolds numbers towards the inviscid Euler equations
In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
, suggesting that the flow should converge towards the inviscid solutions of potential flow
In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
theory – having the zero drag of the d'Alembert paradox. Of this, there is no evidence found in experimental measurements of drag and flow visualisations.[Batchelor (2000), pp. 337–343 & plates.] In the second half of the 19th century this again raised questions concerning the applicability of fluid mechanics.
Inviscid separated flow: Kirchhoff and Rayleigh
In the second half of the 19th century, focus shifted again towards using inviscid flow
In fluid dynamics, inviscid flow is the flow of an ''inviscid fluid'' which is a fluid with zero viscosity.
The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as the ...
theory for the description of fluid drag—assuming that viscosity becomes less important at high Reynolds numbers. The model proposed by Kirchhoff
and Rayleigh Rayleigh may refer to:
Science
*Rayleigh scattering
*Rayleigh–Jeans law
*Rayleigh waves
*Rayleigh (unit), a unit of photon flux named after the 4th Baron Rayleigh
*Rayl, rayl or Rayleigh, two units of specific acoustic impedance and characte ...
[. Reprinted in: ''Scientific Papers'' 1:287–296.
]
was based on the free-streamline theory of Helmholtz and consists of a steady wake behind the body. Assumptions applied to the wake region include: flow velocities equal to the body velocity, and a constant pressure. This wake region is separated from the potential flow outside the body and wake by vortex
In fluid dynamics, a vortex (: vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
sheets with discontinuous jumps in the tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
ial velocity across the interface.
In order to have a non-zero drag on the body, the wake region must extend to infinity. This condition is indeed fulfilled for the Kirchhoff flow perpendicular to a plate. The theory correctly states the drag force to be proportional to the square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of the velocity.
In first instance, the theory could only be applied to flows separating at sharp edges. Later, in 1907, it was extended by Levi-Civita to flows separating from a smooth curved boundary.
It was readily known that such steady flows are not stable, since the vortex sheets develop so-called Kelvin–Helmholtz instabilities.[ But this steady-flow model was studied further in the hope it still could give a reasonable estimate of drag. Rayleigh asks ''"... whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself."][
However, fundamental objections arose against this approach: ]Kelvin
The kelvin (symbol: K) is the base unit for temperature in the International System of Units (SI). The Kelvin scale is an absolute temperature scale that starts at the lowest possible temperature (absolute zero), taken to be 0 K. By de ...
observed that if a plate is moving with constant velocity through the fluid (at rest far from the plate, except for the wake) the velocity in the wake is equal to that of the plate. The infinite extent of the wake—widening with the distance from the plate, as obtained from the theory—results in an infinite kinetic energy in the wake, which must be rejected on physical grounds.[
Moreover, the observed pressure differences between front and back of the plate, and resulting drag forces, are much larger than predicted: for a flat plate perpendicular to the flow the predicted drag coefficient is ''C''D=0.88, while in experiments ''C''D=2.0 is found. This is mainly due to suction at the wake side of the plate, induced by the unsteady flow in the real wake (as opposed to the theory which assumes a constant flow velocity equal to the plate's velocity).][Batchelor (2000), p. 500.]
So, this theory is found to be unsatisfactory as an explanation of drag on a body moving through a fluid. Although it can be applied to so-called cavity flows where, instead of a wake filled with fluid, a vacuum cavity is assumed to exist behind the body.[Batchelor (2000), pp. 493–494.]
Thin boundary layers: Prandtl
The German physicist Ludwig Prandtl
Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German Fluid mechanics, fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlyin ...
suggested in 1904 that the effects of a thin viscous boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
possibly could be the source of substantial drag.[Prandtl (1904).] Prandtl put forward the idea that, at high velocities and high Reynolds numbers, a no-slip boundary condition causes a strong variation of the flow speeds over a thin layer near the wall of the body. This leads to the generation of vorticity
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point an ...
and viscous dissipation
In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...
of kinetic energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
in the boundary layer. The energy dissipation, which is lacking in the inviscid theories, results for bluff bodies in separation of the flow. The low pressure in the wake region causes form drag
Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is defined as the combination of '' form drag'' and ''skin friction drag''.
It is named as ...
, and this can be larger than the friction drag due to the viscous shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
at the wall.[
Evidence that Prandtl's scenario occurs for bluff bodies in flows of high Reynolds numbers can be seen in impulsively started flows around a cylinder. Initially the flow resembles potential flow, after which the flow separates near the rear ]stagnation point
In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero.Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. The Bernoulli equation shows that the static pressure is hi ...
. Thereafter, the separation points move upstream, resulting in a low-pressure region of separated flow.[
Prandtl made the hypothesis that the viscous effects are important in thin layers – called boundary layers – adjacent to solid boundaries, and that ]viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
has no role of importance outside. The boundary-layer thickness becomes smaller when the viscosity reduces. The full problem of viscous flow, described by the non-linear Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
, is in general not mathematically solvable. However, using his hypothesis (and backed up by experiments) Prandtl was able to derive an approximate model for the flow inside the boundary layer, called ''boundary-layer theory''; while the flow outside the boundary layer could be treated using inviscid flow
In fluid dynamics, inviscid flow is the flow of an ''inviscid fluid'' which is a fluid with zero viscosity.
The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as the ...
theory. Boundary-layer theory is amenable to the method of matched asymptotic expansions
In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or simultaneous equations, system of equations. It is particularly used when solving singular pert ...
for deriving approximate solutions. In the simplest case of a flat plate parallel to the incoming flow, boundary-layer theory results in (friction) drag, whereas all inviscid flow theories will predict zero drag. Importantly for aeronautics
Aeronautics is the science or art involved with the study, design process, design, and manufacturing of air flight-capable machines, and the techniques of operating aircraft and rockets within the atmosphere.
While the term originally referred ...
, Prandtl's theory can be applied directly to streamlined bodies like airfoil
An airfoil (American English) or aerofoil (British English) is a streamlined body that is capable of generating significantly more Lift (force), lift than Drag (physics), drag. Wings, sails and propeller blades are examples of airfoils. Foil (fl ...
s where, in addition to surface-friction drag, there is also form drag. Form drag is due to the effect of the boundary layer and thin wake on the pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
distribution around the airfoil.[Batchelor (2000) pp. 302–314 & 331–337.]
Open questions
To verify, as Prandtl suggested, that a vanishingly small cause (vanishingly small viscosity for increasing Reynolds number) has a large effect – substantial drag —
may be very difficult.
The mathematician Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory.
The mathematician George Birkhoff (1884–1944) was his father.
Life
The son of the mathematician Ge ...
in the opening chapter of his book ''Hydrodynamics'' from 1950, addresses a number of paradoxes of fluid mechanics (including d'Alembert's paradox) and expresses a clear doubt in their official resolutions:
:"''Moreover, I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers.''"
In particular, on d'Alembert's paradox, he considers another possible route to the creation of drag: instability of the potential flow solutions to the Euler equations
In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
. Birkhoff states:
:"''In any case, the preceding paragraphs make it clear that the theory of non-viscous flows is incomplete. Indeed, the reasoning leading to the concept of a "steady flow" is inconclusive; there is no rigorous justification for the elimination of time as an independent variable. Thus though Dirichlet flows (potential solutions) and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable.''"
In his 1951 review of Birkhoff's book, the mathematician James J. Stoker sharply criticizes the first chapter of the book:
:"''The reviewer found it difficult to understand for what class of readers the first chapter was written. For readers that are acquainted with hydrodynamics the majority of the cases cited as paradoxes belong either to the category of mistakes long since rectified, or in the category of discrepancies between theory and experiments the reasons for which are also well understood. On the other hand, the uninitiated would be very likely to get the wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter.''"
In the second and revised edition of Birkhoff's ''Hydrodynamics'' in 1960, the above two statements no longer appear.
The importance and usefulness of the achievements, made on the subject of the d'Alembert paradox, are reviewed by Keith Stewartson thirty years later. His long 1981 survey article starts with:
:"''Since classical inviscid theory leads to the patently absurd conclusion that the resistance experienced by a rigid body moving through a fluid with uniform velocity is zero, great efforts have been made during the last hundred or so years to propose alternate theories and to explain how a vanishingly small frictional force in the fluid can nevertheless have a significant effect on the flow properties. The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction tends to zero. This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved.''"
For many paradoxes in physics, their resolution often lies in transcending the available theory. In the case of d'Alembert's paradox, the essential mechanism for its resolution was provided by Prandtl through the discovery and modelling of thin viscous boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
s – which are non-vanishing at high Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
s.[
]
Proof of zero drag in steady potential flow
Potential flow
The three main assumptions in the derivation of d'Alembert's paradox is that the steady flow
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motio ...
is incompressible
Incompressible may refer to:
* Incompressible flow, in fluid mechanics
* incompressible vector field, in mathematics
* Incompressible surface, in mathematics
* Incompressible string, in computing
{{Disambig ...
, inviscid and irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not chan ...
.[This article follows the derivation in Section 6.4 of Batchelor (2000).]
An inviscid fluid is described by the Euler equations
In mathematics and physics, many topics are eponym, named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, e ...
, which together with the other two conditions read
:
where ''u'' denotes the flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of the fluid, ''p'' the pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
, ''ρ'' the density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
, and ∇ is the gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator.
We have the second term in the Euler equation as:
:
where the first equality is a vector calculus identity and the second equality uses that the flow is irrotational. Furthermore, for every irrotational flow, there exists a velocity potential
A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.
It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a ca ...
''φ'' such that u = ∇''φ''. Substituting this all in the equation for momentum conservation yields
:
Thus, the quantity between brackets must be constant (any ''t''-dependence can be eliminated by redefining ''φ''). Assuming that the fluid is at rest at infinity and that the pressure is defined to be zero there, this constant is zero, and thus
:
which is the Bernoulli equation for unsteady potential flow.
Zero drag
Now, suppose that a body moves with constant velocity v through the fluid, which is at rest infinitely far away. Then the velocity field of the fluid has to follow the body, so it is of the form , where x is the spatial coordinate vector, and thus:
:
Since u = ∇''φ'', this can be integrated with respect to x:
:
The force F that the fluid exerts on the body is given by the surface integral
:
where ''A'' denotes the body surface and ''n'' the normal vector
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the cu ...
on the body surface. But it follows from (2) that
:
thus
:
with the contribution of ''R''(''t'') to the integral being equal to zero.
At this point, it becomes more convenient to work in the vector component
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scal ...
s. The ''k''th component of this equation reads
:
Let ''V'' be the volume occupied by the fluid. The divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume ...
says that
:
The right-hand side is an integral over an infinite volume, so this needs some justification, which can be provided by appealing to potential theory to show that the velocity ''u'' must fall off as ''r''−3 – corresponding to a dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
* An electric dipole moment, electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple ...
potential field in case of a three-dimensional body of finite extent – where ''r'' is the distance to the centre of the body. The integrand in the volume integral can be rewritten as follows:
:
where first equality (1) and then the incompressibility of the flow are used. Substituting this back into the volume integral and another application of the divergence theorem again. This yields
:
Substituting this in (3), we find that
:
The fluid cannot penetrate the body and thus n · u = n · v on the body surface. So and
:
Finally, the drag is the force in the direction in which the body moves, so
:
Hence the drag vanishes. This is d'Alembert's paradox.
Notes
References
Historical
*
*
*
Further reading
*
*
*
*
*{{citation
, last1=Stewartson , first1=K.
, year=1981
, title=D'Alembert's Paradox
, journal=SIAM Review
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific socie ...
, volume=23 , issue=3 , pages=308–343
, doi=10.1137/1023063
External links
Potential Flow and d'Alembert's Paradox
at MathPages
Fluid dynamics
Physical paradoxes