Bifurcation theory and applications by L. Salvadori

By L. Salvadori

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Suppose that a body is executing an oscillatory motion under the action of an external force f. When the conditions discussed in §10 are fulfilled, the fluid surrounding the body moves in a potential flow, and we can use the relations previously obtained to derive the equations of motion of the body. The force f must be equal to the time derivative of the total momentum of the system, and the total momentum is the sum of the momentum Mu of the body (M being the mass of the body) and the momentum P of the fluid: Mdu/di + dP/di = f.

5), in vector form, as p — + (v-grad)v = — grad/? + >yAv + (C + yrç)graddivv. 6) This is called the Navier-Stokes equation. 6) t That is, on taking the sum of the components with i = k. 46 §15 Viscous Fluids is zero. In discussing viscous fluids, we shall almost always regard them as incompressible, and accordingly use the equation of motion in the formt - ^ + (vgrad)v = St p gradp + - A v . 7) The stress tensor in an incompressible fluid takes the simple form -·~'«·+<£+£) We see that the viscosity of an incompressible fluid is determined by only one coefficient.

10 Incompressible fluids 21 Finally, let us consider the conditions under which the fluid may be regarded as incompressible. When the pressure changes adiabatically by Δρ, the density changes by Ap = (dp/dp)sAp. According to Bernoulli's equation, however, Ap is of the order of pv2 in steady flow. We shall show in §64 that the derivative (dp/dp)s is the square of the velocity c of sound in the fluid, so that Ap ~ pv2/c2. The fluid may be regarded as incompressible if Αρ/ρ<ζ 1. 16) v

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