In ideal inviscid incompressible flows, the movement of body does not encounter any resistance. This result is known as d'Alembert's Paradox, and this paradox is examined here. Supposed that a two–dimensional diamond–shape body is stationed in a supersonic flow as shown in Figure 12.27.

Fig. 12.27 A simplified diamond shape to illustrate the supersonic d'Alembert's Paradox.

Again, it is assumed that the fluid is inviscid. The net force in flow direction, the drag, is

$D = 2 \left( \dfrac{w }{ 2} \, (P_2 - P_4)\right) = w \, (P_2 - P_4) \label{pm:eq:dragG} \tag{38}$

It can be observed that only the area that "seems'' to be by the flow was used in expressing equation (38). The relation between $$P_2$$ and $$P_4$$ is such that the flow depends on the upstream Mach number, $$M_1$$, and the specific heat, $$k$$. Regardless in the equation of the state of the gas, the pressure at zone 2, $$P_2$$, is larger than the pressure at zone 4, $$P_4$$. Thus, there is always drag when the flow is supersonic which depends on the upstream Mach number, $$M_1$$, specific heat, $$k$$, and the "visible'' area of the object. This drag is known in the literature as (shock) wave drag.

## Contributors

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.