The paradox of d'Alembert

Along with the name of the D'Alembert paradox ^[4] in the scientific literature there are the names of the D'Alembert – Euler paradox , the Euler – D'Alembert paradox ^{[5] [6]} and the Euler paradox ^[7] .

Sommerfeld ^[8] with reference to Oseen mentions Spinoza as an early researcher of the paradox. Apparently, we are talking about the work “The Foundations of Descartes’s Philosophy, Geometrically Proven”, in which Spinoza analyzes the conditions under which “the body, for example our hand, could move in any direction with equal movement, not at all opposing other bodies and not meeting countermeasures from other bodies ” ^[9] . In the particular case of a flow around a body symmetrical about the transverse plane inside the channel, resistance vanishing was discovered by D'Alembert in 1744 ^[10] . In a general form (for a body of arbitrary shape), the conversion of the resistance force to zero was established by Euler in 1745 ^[11] . The term " paradox " to describe the conversion of resistance to zero was first used by D'Alembert in 1768 ^[12] .

By virtue of Galileo's principle of relativity, one can speak of the D'Alembert paradox in the case of translational rectilinear motion of a body at a constant speed in an unlimited volume of ideal fluid, which rests at infinity.

In addition, the d'Alembert paradox is valid when a stream flows around a body enclosed in an infinite cylindrical channel.

It is important to note that the formulation of the paradox refers only to the absence of a component of the force acting on the body that is parallel to the flow at infinity (the absence of the resistance force ). The force component that is perpendicular to the flow ( lift ) can be nonzero even if all the conditions of the paradox are fulfilled (for example, this is the case for two-dimensional problems: the lift is calculated using the well-known Zhukovsky formula ).

Let us pay attention to the fact that the moment of forces acting on the body from the flow side can, generally speaking, be nonzero. So, during continuous flow around a plate inclined to the flow, even at zero velocity circulation (and, therefore, at zero lifting force), a torque arises, which tends to rotate the plate across the flow.

In the presence of volume forces (for example, gravity) from the liquid side, the Archimedes force can act on the body, however, it cannot be considered a component of the resistance force, because it does not vanish in the liquid at rest.

As is well known, when a real fluid flows around a body, there is always a nonzero resistance force, the presence of which is explained by the violation of certain conditions included in the formulation of the D'Alembert paradox. In particular,

• if the fluid is not ideal (has a finite viscosity), a drag force may arise, directly or indirectly associated with the action of viscous friction;
• if the body’s motion in the fluid is not stationary, then even in the inviscid fluid model, an inertial drag force arises, due to the fact that when the body moves at a variable speed, the kinetic energy of the surrounding fluid changes with time;
• if the flow is not continuous (for example, there are discontinuity surfaces in the flow), then the flow parameters far ahead and behind the body may not coincide, which leads to nonzero resistance. Examples are
  • a body in a flat stream, generating a chain of concentrated vortices behind itself ( Karman vortex track model);
  • wing of finite magnitude, from which the surface of discontinuity, which goes to infinity, descends of the tangent velocity component (the so-called vortex sheet); the resistance associated with this phenomenon is called inductive;
  • the formation of shock waves during supersonic flow around a body with a gas stream;
• if the fluid does not occupy the entire space around the body, then the d'Alembert paradox may also be violated. Typical examples are
  • the formation behind the body of a cavity that goes to infinity, filled with a resting liquid (Kirchhoff – Helmholtz jet flow scheme simulating a cavitation cavity);
  • the formation of waves on the surface of a liquid ( gravitational waves on water), the creation of which requires energy costs, which leads to the appearance of wave resistance ; resistance is similar in nature due to the appearance of internal waves during body motion in a stratified fluid (say, at the boundary of two fluid layers with different densities);
• if the flow parameters are far in front and behind the body are not aligned, then the resistance force can also be different from zero. In particular, this is the case with the supply of thermal energy to the flow or with the formation of a region (“trace”) behind the body, the parameters in which are different from the parameters in the main flow at infinity.

If we create conditions in which the flow around the body is close enough to the conditions in the formulation of the D'Alembert paradox, for example, to give the body a streamlined (drop-like or ellipsoidal) shape, then it is possible to achieve a significant - tens and hundreds of times - decrease in resistance compared to poorly streamlined (for example , in the form of a cube) by bodies with the same mid-section . The foregoing applies to flows with large Reynolds numbers ; in the opposite case of small Reynolds numbers (the so-called creeping flows ), the resistance of elongated drop-shaped bodies having a large surface area can, on the contrary, be greater than the resistance of "poorly streamlined" bodies.

When particles move in solids , the effect of “superdeep penetration” is known ^[13] . One of the explanations for this effect is qualitatively similar to the D'Alembert paradox: a decrease in resistance is achieved due to the fact that under certain conditions the effect of the particle on its environment is reduced (the channel formed behind the particle collapses ^{[14] [15]} , and there are significant plastic deformations only in a thin wake behind a particle ^[16] ).

• Grimberg G., Pauls W., Frisch U. Genesis of d'Alembert's paradox and analytical elaboration of the drag problem // Physica D. - 2008 .-- T. 237 .

• D'Alembert - Euler paradox - an article from the Great Soviet Encyclopedia .

• Paradox Dubois