Lars H. Söderholm

Force on a spinning sphere moving in a rarefied gas

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 directio ...

Hybrid Burnett Equations. A New Method of Stabilizing

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 terms Kn2Ma2 can be neglected, the only difference from the conventional Burnett equations is the change of coefficients ϖ2→ϖ3, ϖ3→ϖ3.

On the Kuznetsov equation and higher order nonlinear acoustics equations

Compressible flow of a Newtonian fluid is studied in the fully nonlinear approximation and to lowest order in the dissipation, neglecting cross terms Ma⋅Kn (Ma is the Mach number, Kn the Knudsen number). An initially vorticity free flow is shown to remain vorticity free. An acoustic equation is derived. If Kn=Man, the equation obtained is correct to order Man. For n=1, it reduces to the Kuznetsov equation.

Thermophoresis of axially symmetric bodies

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 to 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 antiparallel to the temperature gradient. It is also shown that the gas excerts a torque on the body, which in magnitude and direction depends on the body geometry. Equations of motion describing the body movement are derived. Asymptotic solutions are studied.

Nonlinear acoustics equations to third order - New stabilization of the Burnett equations

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 ρ, v, T, first in an approximation adequate for third order nonlinear acoustics and then in the general case.

Nonlinear acoustics to second order in Knudsen number without unphysical instabilities

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.

On the Relation between the Hilbert and Chapman-Enskog Expansions

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.

Thermophoretic motion of bodies with axial symmetry

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 w ...

Equilibrium temperature of small body in shearing gas flow

A convex body, with high thermal conductivity, is immersed in a nonuniformly flowing gas. The body is small compared to the mean free path, which in turn is small compared to the macroscopic length scale of the gas. The equilibrium temperature Tw of the body is calculated. For an axially symmetric body in a simply shearing gas of temperature T one obtains the equilibrium temperature TwT=1+βa8 pxyp sin2 θ sin(2φ). (this is for the case that the body is at rest with respect to the gas). θ, φ 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 β depends on the Sonine coefficients. β takes the value 1 if only the lowest order Sonine term is retained. p is the pressure and pxy (the non-vanishing component of) the viscous pressure tensor.

Hilbert Fluid Dynamics Equations Expressed in Chapman-Enskog Pressure Tensor and Heat Current

The most convenient way to calculate heat current and pressure tensor is that of the Chapman-Enskog method, see Grad [4]. In particular, the terms up to second and higher order have been calculated and at least to second order there is an agreed convention of notation for the terms. But in many occasions the Hilbert method is preferred because of the difficulties with the Burnett equations, see Sone [6]. Already in the Handbuch article [4], Grad made clear that the Chapman-Enskog method and Hilbert method are two ways of expressing the same thing. But Grad adds ”to confirm this directly is an intricate task due to the different formalisms”. So, his proof is somewhat abstract. It is the object of the present work to do the explicit confirmation that Grad does not do. The benefit of this is that the fluid dynamics equations of the Hilbert expansion are given explicitly in terms of the pressure tensor and heat current of the ChapmanEnskog method. This means that results that have been derived by the ChapmanEnskog method directly can be used in the Hilbert expansion. In the interesting paper [3] by Chekmarev and Chekmareva multiple scale methods are applied in theThe 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.