The force acting on a spinning sphere moving in a rarefied gas is calculated. It is found to have three contributions with different directions. The transversal contribution is of opposite direction compared to the so-called Magnus force normally exerted on a sphere by a dense gas. It is given by F=-alpha(tau)xi2/3piR(3)mnomegaxv, where alpha(tau) is the accommodation coefficient of tangential momentum, R is the radius of the sphere, m is the mass of a gas molecule, n is the number density of the surrounding gas, omega is the angular velocity, and v is the velocity of the center of the sphere relative to the gas. The dimensionless factor xi is close to unity, but depends on omega and kappa, the heat conductivity of the body.

The effects of the Magnus force on a spinning sphere in a Keplerian orbit is investigated using perturbation theory. The result is that the plane of the orbit will rotate with the angular velocity - 1/4 alpha(tau)mn/rho(S)omega, where alpha(tau) is the accommodation coefficient of tangential momentum, m and It are the mass and number density of the surrounding gas, and where rho S and omega are the mean density and the angular velocity of the sphere. It is shown that under reasonable assumptions, for a spinning satellite in the Earth's atmosphere, this effect is small.

Thermophoresis of axially symmetric bodies is investigated to first order in the Knudsen number, Kn. The study is made in the limit where the typical length of the immersed body is small compared with the mean free path. It is shown that in this case, in contrast to what is the case for spherical bodies, the arising thermal force on the body is not in general anti-parallel to the temperature gradient. It is also shown that the gas exerts a torque on the body, which in magnitude and direction depends on the body geometry. Equations of motion describing the body movement are derived. Stationary solutions are studied.

For nonlinear acoustics for ultrahigh frequencies it is necessary to go beyond the Navier-Stokes equations. For a gas the next set of equations due to Burnett were shown by Bobylev to have an unphysical instability. The Burnett equations are here stabilized as a set of equations for the fluid dynamic variables rho, v, T, first in an approximation adequate for third order nonlinear acoustics and then in the general case.

Compressible flow of a Newtonian fluid is studied in the fully nonlinear approximation and to lowest order in the dissipation. Nonlinear contributions to dissipation are neglected. It is shown that entropy can then be eliminated. An initially vorticity free flow is shown to remain vorticity free. An acoustic equation is derived. If S = M-n, the equation obtained is correct to order M-n. M is the Mach number (the speed of the fluid divided by the local speed of sound). S is the Stokes number, (the kinematic viscosity divided by the product of the wavelength and the local speed of sound). For n = 1, the equation reduces to the Kuznetsov equation.

7.

Söderholm, Lars H.

KTH, School of Engineering Sciences (SCI), Mechanics.

Consistent third order nonlinear acoustics2008In: NONLINEAR ACOUSTICS FUNDAMENTALS AND APPLICATIONS / [ed] Enflo, BO; Hedberg, CM; Kari, L, 2008, Vol. 1022, p. 69-72Conference paper (Refereed)

Abstract [en]

For very high frequency ultrasound the Navier-Stokes equations are inadequate. The Navier-Stokes equations are first order in the mean free path in a gas. In this paper some sets of equations of second order are introduced and the nonlinear evolution of a sound wave is studied for one of these.

Equilibrium temperature of a convex body in a free molecular shearing flow2002In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 66, no 3Article in journal (Refereed)

Abstract [en]

It is shown that the equilibrium temperature T-w of a high conductivity axially symmetric convex body in a simply shearing gas of temperature T is given by T-w/T=1+(betaa/4)(p(xy)/p) sin(2)theta sin(2Phi), theta,Phi are polar angles of the axis of the body (z is the polar axis). a is a geometric shape factor of the body (which vanishes for a sphere) and beta takes the value 1 if only the lowest order Sonine term is retained. p is the pressure and p(xy) the viscous pressure. The body is assumed small compared to the mean free path, which is small compared to the length scale of the velocity field.

The connection between the Chapman-Enskog and Hilbert expansions is investigated in detail. In particular, the fluid dynamics equations of any order in the Hilbert expansion are given in terms of the pressure tensor and heat current of the Chapman-Enskog expansion.

In the original work by Burnett, the pressure tensor and the heat current contain two time derivates. Those are commonly replaced by spatial derivatives using the equations to zero order in the Knudsen number. The resulting conventional Burnett equations were shown by Bobylev to be linearly unstable. In this paper it is shown that the original equations of Burnett have a singularity. A hybrid of the original and conventional equations is constructed and shown to be linearly stable. It contains two parameters, which have to be larger than or equal to some limit values. For any choice of the parameters, the equations agree with each other and with the Burnett equations to second order in Kn, that is, to the accuracy of the Burnett equations. For the simplest choice of parameters the hybrid equations have no third derivative of the temperature, but the inertia term contains second spatial derivatives. For stationary flow, when the term Kn(2) Ma(2) can be neglected, the only difference,from the conventional Burnett equations is the change of coefficients pi(2) -> pi(3), pi(3) -> pi(3).

The Burnett equations are consistently reformulated as a linearly stable first order system. The equations are then applied to study the nonlinear evolution of a sound wave. The initially sinusoidal wave is nonlinearly distorted and a shock wave develops. The shock is gradually dissolved by dissipation and a sinusoidal wave of smaller and decaying amplitude emerges. The amplitude of this old age solution is compared with the classical results from the Burgers equation of nonlinear acoustics and systematic deviations are found.

The connection between the Chapman-Enskog and Hilbert expansions is investigated in detail. In particular the fluid dynamics equations of any order in the Hilbert expansion are given in terms of the pressure tensor and heat current of the Chapman-Enskog expansion.