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Before to discuss the? dalembert's Paradox? , I would like to present the life and the work of this extraordinary man. His work in various fields of Science, makes of him one of the man who contributed the most to our modern Mathematics, Aerodynamics thats why I would like to present the global life of this great scientist. The following information are issued from the encyclopaedia Britannica.
Alembert, Jean Le Rond d His work in Mathematics: In 1739, he read his first paper to the Academy of Sciences, of which he became a member in 1741. in 1743, at the age of 26, he published his important Trait? de dynamique, a fundamental treatise on dynamics containing the famous dalembert's Principle, which states that Newtons third law of motion (for every action there is an equal and opposite reaction) is true for bodies that are free to move as well as for bodies rigidly fixed. Other mathematical works followed very rapidly; in 1744, he applied his principle to the theory of equilibrium and motion of fluids, in his trait?
de l? qui libre et du mouvement des fluids. This discovery was followed by the development of partial derivative equations, a branch of the theory of calculus, the first papers on which were published in his R? flexions sur la cause g?
n? rale des vents (1747). It won him a prize at the Berlin Academy, to which he was elected the same year. In 1747 he applied his new calculus to the problem of vibrating strings, in his Recherches sur les cordes vibrates; in 1749 he furnished a method of applying his principles to the motion of any body of a given shape; and in 1749 he found an explanation of the precession of the equinoxes (a gradual change in the position of the Earths orbit), determined its characteristics, and explained the phenomenon of the nutation (nodding) of the Earths axis, in Recherches sur la pr? cession des? quinones et sur la nutation de like de la terre.
In 1752 he published Essai dune nouvelle th? orie de la r? substance des fluids, an essay containing various original ideas and new observations. In it he considered air as an incompressible elastic fluid composed of small particles and, carrying over from the principles of solid body mechanics the view that resistance is related to loss of momentum on impact of moving bodies, he produced the surprising result that the resistance of the particles was zero (the thesis that will be discussed in this research paper); the conclusion was known as the dalembert's Paradox and is not accepted by modern physicists. In the Memoirs of the Berlin Academy he published findings of his research on integral calculus which devises relationships of variables by means of rates of change of their numerical value branch of mathematical science that is greatly indebted to him. In his Recherches sur diff?
rents points important's du syst? me du monde (1754 - 1756) he perfected the solution of the problem of the perturbations (variation of orbit) of the planets that he had presented to the academy some years before. From 1761 to 1780 he published eight volumes of his Opuscules math? magiques.
Meanwhile, it is to note that d Alembert had a big implication in The Encyclop? die with some of his friends philosophers (Voltaire, Rousseau, Montesquieu) So the various works of d Alembert, made of him a great man of our modern science. But among all this brilliant work, there is one theory that let him perplex as he said himself. This concerns his famous paradox, for he found that: there is no drag on a solid in movement in an ir rotational and incompressible flow.
During the following pages, we will try to see how the mathematical tools explained this incongruity. Dalembert's Paradox. After a long study of the mathematical theory of fluids, he nonetheless got a negative result. And he ended with the following conclusion: I do not see then, I admit, how one can explain the resistance of fluids by the theory in a satisfactory manner. It seems to me on the contrary that this theory, dealt with and studied with profound attention, gives, at least in most cases, resistance absolutely zero; a singular paradox which I leave to geometricians to explain. (extract from Aerodynamics of T.
Von Kerman) In order to get through the development of his theory and to explain where the error could have been. We first have to set the hypothesis with which we will work throughout the following derivation: the flow is incompressible the flow is non-viscous and then ir rotational. We will consider the flow around a cylindrical body. We consider a cylinder of radius R, with an imposed flow of velocity Ux far from the cylinder (see fig 1) fig 1 We know that the fluid cannot penetrate the surface of the cylinder, so that the normal component of the velocity at the surface of the cylinder must be zero.
In radial (polar) coordinates this is expressed as: nr (R, q We are now looking for the equation of n. Since we have seen that the flow was ir rotational, we have with? called the del operator and expressed as: ? = /x I + /y j + /z k 1. 3 and from vector calculus, we know that? ? n = 0 (known as curl n) implies that n may be written as the gradient of a scalar function j: n = ?
j 1. 4 If we consider that the fluid is also incompressible (as it is the case here), then: ? . n = 0 (what is called the DIV of n) 1. 5 from 1. 4 and 1. 5 we can have: ? . (-? j) = ? It is important to note that j is a potential function solution of the Laplaces equation. The expression of the Laplaces equation is: ?
j / x ? +? j / y ? +? j / zand under its polar or radial form, for j (r, q): ? j / r ? + 1 /r? ? j / q ? + 1 /r j / it has also been expressed as: 1 /r /r (r. j / r ) + 1 /r? ?
j / q which means that j / r = 0 but why? so in terms of j the boundary condition at infinity is j? Ux; in radial coordinates, x = r. cost, so this becomes: j? -U. r.
cost as r? ? 1. 10 At surface of the cylinder we have: (j (r, q) / r) r = R = 0 1. 11 The strategy is that we will try to solve Laplaces equation for j, and then we will calculate the velocity field from the potential. To do this we first need to realize that in order to satisfy the boundary condition at infinity, j? cost, therefore, we set: j (r, q) = f (r) cos q 1. 12 Substituting this into Laplaces equation, eq 1. 9, becomes a differential equation for f: 1 /r d / dr (r. df / dr ) f / r ? = 0. 1. 13 To solve this equation, we will guess that the solution f (r) = r? , and see if we can find an appropriate value of? . Substituting, we find? = +/- 1, so that our solution is: j (r, q) = (Ar + B/r) cos q. 1. 14 using the boundary condition at infinity, (eq. 1. 10) we have: A = -U. 1. 15 And applying the boundary condition at the surface of the cylinder (eq. 1. 11) we have: B = -U. R? 1. 16 So our final solution of j is: j (r, q) = -U. (r + R? /r) cost 1. 17 The components of the velocity are: nr (r, q) = -j / r = U (1 - R? /r? ) cost 1. 18 nq (r, q) = - 1 /r j / q = -U (1 + R? /r? ) site 1. 19 In this case, the streamlines are symmetric on the upstream and downstream sides of the cylinder (as shown in the figure 1) So the point is now to determine how does the pressure vary around the cylinder?
First the square of the velocity on the surface of the cylinder is: n? (R, q) = nr? (R, q) + nq? (R, q) 1. 20 As for r = R: nr (R, q) = 0 n? (R, q) = nq? (R, q) n? (R, q) = (-U. 2. site)? n? (R, q) = 4 U? sin? q 1. 21 We can now use Bernouillis equation; since the flow is ir rotational, the total pressure as constant is the same everywhere in the fluid.
In particular, at infinity, the total pressure is: ? rU? + po = cst 1. 22 where po is the ambient pressure (which can be choose to be zero without any loss of generality). The pressure on the surface of the cylinder is then: for? rU? + po = ? n? + p (R, q) 1. 23? rU? + po = ? (4 U?
sin? q) + p (R, q) 1. 24 so: p (R, q) = ? rU? (1 - 4 sin? q) 1. 25 From this pressure, we can calculate the net force acting on the cylinder. First, we have to recall that the pressure at the surface of the cylinder is the force per unit area acting on the cylinder; therefore the force dF acting upon an element of the cylinder of surface area dS is: dF = p. n.
dS 1. 26 where n is a unit vector normal to the surface, and the minus sign indicates that this force acts inward. Next, we integrate this over the entire surface of the cylinder to find the net force: F = &# 8747; s p. n. dS 1. 27 For the cylinder, (in polar coordinates) dS = Re. dz 1. 28 with z along the axis of the cylinder. The unit vector depends upon the angle q as n = cost I+ site j 1. 29 then the x and y axis components of the force are: Fx = Lx R &# 8747; p (q) cos q dq. 1. 30 Which is the equation of the drag.
Fy = Lx. R. &# 8747; p (q) site dq. 1. 31 Which is the equation of the lift. Note that the integrals are from 0 to 2 p The value Lx is the length of the cylinder in the z-direction. If we now substitute our expression for the pressure distribution on the surface of the cylinder (eq. 1. 25) into our expression for the drag and for the lift, we have: Fx = -Lx.
R &# 8747; ? rU? (1 - 4 sin? q). cost dq 1. 32 Fx = -? . Lx. R.
rU? . &# 8747; (1 - 4 sin? q). cost. dq But &# 8747; (1 - 4 sin? q). cost.
dq = &# 8747; cost. dq &# 8747; 4 sin? q. cost. dq &# 8747; (1 - 4 sin? q).
cost. dq = [site] (0? 2 p) [ 4 / 3. sin^ 3 ] (0? 2 p) &# 8747; (1 - 4 sin? q). cost. dq = 0 1. 33 so from 1. 32, we have: Fx = 0 So we are arrived at the paradox of Dalembert's: a body in an ir rotational, non-viscous and incompressible flow produces no drag.
The results found here are applicable to any shapes; not only cylinder. CONCLUSION Physically, the fact that the drag is zero is due to the symmetry of the pressure field about the cylinder. The fluid is pushing as hard on the upstream side of the cylinder as on the downstream side, so there are no unbalanced forces which would lead to drag. Whether or not this is paradoxical is a matter of opinion. If there were a drag on the cylinder how would the energy be dissipated in a non viscous fluid?
to really understand drag, we need to go beyond the non viscous approximation and treat real fluids. So the paradox of D Alembert comes from the fact that he neglected the friction forces and then considered it as non-viscous. If we now introduce the viscosity of the fluid, the equation 1. 2 becomes for the flow is no more ir rotational; and then equation 1. 4 also changes for: n? ? j so the motion of the flow of fig 1 can now be described as: and the whole derivation that led to our paradox is changed. We have here the perfect example that illustrates the fact that starting from wrong hypothesis, we can arrive to absurd results.
And this is also true for such brilliant scientist as Mr D Alembert. Drag and Dalembert's paradox, web Britannica. com, Alembert, Jean Le Rond d, web Von Karma Theodore, Aerodynamics. London: McGraw-Hill Paperback Edition, 1963.
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