Isao Imai

Discontinuous Potential Flow as the Limiting Form of the Viscous Flow for Vanishing Viscosity

The asymptotic behaviour of the viscous flow past an obstacle for vanishingly small viscosity is discussed on the basis of Prandtls boundary layer theory together with Kirchhoffs dead water theory. If the pressure distribution around the obstacle is such that the separation of the boundary layer on the obstacle takes place, then the flow pattern should involve free stream-lines extending from the separation points. From such considerations it is concluded that the asymptotic flow pattern around a smooth obstacle is a discontinuous flow with dead water region of the Kirchhoff type such that the free stream-lines leave the obstacle with a finite curvature equal to that of the obstacle at the separation point.

On the Definition of Macroscopic Electromagnetic Quantities

A new framework of electromagnetic theory is presented on the basis of conservation principles of energy and momentum. First, Maxwells equations and Lorentz force law in vacuum are derived as theorems from the conservation principles. Next, new kinds of averaging procedure, transversal and longitudinal, are introduced to deal with the macroscopic media. Electromagnetic field ( E , H ) and flux density ( D , B ) are defined as the longitudinal and transversal averages of the corresponding microscopic field. The polarization P and magnetization M defined as P = D -e 0 E , M = B -µ 0 H are given explicit expressions in terms of microscopic electromagnetic structure of the matter. Lorentz transformation of space-time and electromagnetic field can be derived entirely within the framework of electromagnetic theory.

Extension of von Kármán's Transonic Similarity Rule

An attempt is made to extend von Karmans transonic similarity theory so as to cover the whole range of Mach number from subsonic to supersonic. First, the equations for the two-dimensional compressible flow past a slender body are reduced to \begin{aligned} \frac{\partial G}{\partial\bar{\zeta}}{=}B\left(\frac{\partial\phi_{1}}{\partial\xi}\right)^{2}\quad (1)\quad\text{or}\quad\frac{\partial^{2}\phi_{2}}{\partial\eta^{2}}{=}4B\frac{\partial\phi_{2}}{\partial\xi}\frac{\partial^{2}\phi_{2}}{\partial\xi^{2}}\quad (2) \end{aligned} by assuming that the perturbation velocity is small of order e and neglecting small quantities of O ( e 3 ). Here we have written \(\varPhi{=}U(x+\phi)\), \(\varPsi{=}U(y+\Psi)\), ζ=ξ+ i η= x + i µ y , Ψ=µχ, φ+ i χ= A (φ 1 + i χ 1 )= A G , φ 1 =φ 2 +ξ/4 B , B =ν A /4, \(\mu{=}\sqrt{1-M^{2}}\), ν=4 M 2 [1+(γ+1) M 2 /4µ 2 ] where \(\varPhi\) is the stream function. U and M are respectively the free stream velocity and Mach number, γ is the ratio of specific heats, A is a constant o...

On the Asymptotic Behaviour of Compressible Fluid Flow at a Great Distance from a Cylinder in the Absence of Circulation

In this paper the method of thin-wing-expansion which has been developed by the author is applied to the case of a uniform flow of a compressible fluid past an arbitrary cylinder in the case of the absence of sources, sinks and circulation. The asymptotic expressions for the velocity potential and the stream function at a great distance, r , from.the cylinder are determined correctly to the order of 1/ r 4 . Hence the velocity can be found exactly to the order of 1/ r 5 . This should be compared with Grobners result which is correct to the order of 1/r 4 and is restricted to the case of a cylinder symmetrical with respect to the x -axis. Moreover, the present analysis is far simpler than Grobners one in view of the fact that the former requires only some elementary quadratures while the latter the integration of a system of differential equations. Finally, it is shown that the general expression for the velocity potential is in accord with the authors previous result for flow past a circular cylinder, ...

On a Refinement of the Transonic Approximation Theory

In this paper it is proposed that the basic equation for the two-dimensional transonic flow of a compressible fluid should be \begin{aligned} \frac{\partial^{2}\psi}{\partial\tau^{2}}+\frac{\partial^{2}\psi}{\partial\theta^{2}}{=}-\frac{5}{36}\frac{1}{\tau^{2}}\psi \tag{A} \end{aligned} rather than the commonly used one \begin{aligned} \frac{\partial^{2}\varPsi}{\partial\eta^{2}}-\eta\frac{\partial^{2}\varPsi}{\partial\theta^{2}}{=}0,\quad\eta{=}(\gamma+1)^{1/3}\frac{q-q_{*}}{q_{*}} \tag{B} \end{aligned} or its equivalent. Here \(\tau{=}-\int\mu q^{-1}dq\), µ=(1- M 2 ) 1/2 , K =(µρ 0 /ρ) 2 , and \(\varPsi{=}K^{-1/4}\psi\) is the stream function, ( q , θ) are the magnitude and direction of the velocity, q * the critical velocity, M the local Mach number, ρ the density, ρ 0 the stagnation density, and γ the ratio of specific heats. A set of fundamental solutions of (A), which have singularity at the point (τ=τ 1 , θ=0), is obtained in a natural and elementary way in the form \begin{aligned} \psi_{m}{=}\tau^...

Equi-Partition of Outflow of Joule's Heat Generated in a Homogeneous Body of Arbitrary Shape

Joules heat generated in a homogeneous body of arbitrary shape flows out just equally through each of two electrodes, when the body is isolated from the environment both electrically and thermally except at the electrodes, which are attached onto the surface of the body. This “Theorem of equi-partition of Joules heat” holds even when both electrical and thermal conductivity of the material vary with temperature in an arbitrary way.

Application of the M2-Expansion Method to Compressible Flow past Isolated and Lattice Aerofoils

In this paper two-dimensional compressible flow past an arbitrary aerofoil in unlimited fluid or past a straight lattice of aerofoils is considered on the basis of the M 2 -expansion method, which consists in expanding various quantities in powers of M 2 , M being the Mach number. The general expressions for the velocity components at any point in the field of flow are obtained, which are correct to M 4 for an isolated aerofoil and to M 2 for a wing lattice respectively. In particular, the formulae giving the velocity distribution on the surface of the aerofoil are of such a simple form that the numerical calculation for any arbitrary aerofoil or lattice of aerofoils is no longer a prohibitive work as generally believed. As an interesting special case, the flow past a symmetrical aerofoil placed symmetrically in a channel is also considered, which was previously dealt with by Goldstein and Lighthill by means of a different method.