Hidenori Hasimoto

Nonlinear Modulation of Gravity Waves

Slow modulation of gravity waves on water layer with uniform depth is investigated by using singular perturbation methods. It is found, to the lowest order of perturbation, that the complicated system of equations governing such modulation can be reduced to a simple nonlinear Schrodinger equation. A nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave. The linear stability of this plane wave solution is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whithams criterion. The same equation is found to give a weak cnoidal wave derived from the Korteweg-de Vries equation in the shallow-water limit.

Boundary Layer Growth on a Flat Plate with Suction or Injection

Exact solutions of the Navier-Stokes equations are obtained for the boundary layer growth on an infinite flat plate with uniform suction or injection (with velocity V normal to its plane) which is started at time t =0 (with velocity U along its plane), for the two cases: i) U =arbitrary, V =const and ii) U ∞ t α , V ∞ t -1/2 . i) gives simple relations between the cases of suction and injection and ii) gives similar velocity profiles. Rayleighs problem ( U =const) is investigated in detail, and the resulting solutions show the same qualitative natures as the corresponding steady flow solutions for an semi-infinite flat plate so far obtained.

Motion of a Vortex Filament and its Relation to Elastica

Steady rotation of a very thin plane vortex filament with uniform angular velocity (-varOmega) in the direction opposite to the circulatory rotation around it is discussed on the basis of the localized induction equation. It is shown that the possible form of the filament is that of the plane elastic filament of flexural rigidity B under the action of thrust F applied at its ends provided that (G/varOmega{=}B/F), where G is the coefficient of local induction proportional to the circulation of the filament.

On the Flow of a Viscous Fluid past a thin Screen at Small Reynolds Numbers

The slow steady flow of a viscous incompressible fluid past a thin screen with holes or slits is investigated on the basis of the Stokes equations of motion. It is found that the flow conductance σ of the holes or slits (i.e. the ratio of the total flow Q to the pressure drop P ) is given by begin{aligned} sigma{=}frac{Q}{P}{=}frac{M}{8rhomu}, end{aligned} where ρ is the density of the fluid, µ the viscosity, and M the virtual mass of disks or strips which are congruent with the holes or slits, moving broadside-on in a perfect fluid. The drag D acting on a part of the wall δW of the screen is also given by begin{aligned} D{=}Pq_{p}, end{aligned} where q p , is the total flow of the perfect fluid through δW in this movement with unit velocity. As examples, the cases of a single elliptic hole, a single slit, two parallel equal slits, and a series of parallel equal and equidistant slits are considered. Especially, in the last case, the total drag for an intervening strip is shown to be 4πµ U /log |c...

A Sphere Theorem on the Stokes Equation for Axisymmetric Viscous Flow

A theorem is presented which gives the perturbation stream function (tilde{varPsi}(r,theta)) when a sphere ( r = a ) is introduced into an unlimited viscous fluid in axisymmetric motion obeying the Stokes equation, of which the stream function is (varPsi(r,theta)), where ( r ,θ,ϕ) are the polar coordinates. (tilde{varPsi}(r,theta)) is given by

Swirl of a Conducting Gas due to the Hall Effect

The Hall effect is shown to induce a swirling or transverse motion in the flow of an electrically conducting gas in a magnetic field and surrounded by an insulating medium. Both the two-dimmensional and the axi-symmetric floes are discussed. On the assumption that the magnetic Reynolds number and the magnetic interaction parameter are small, the transverse or helical velocity is shown to be independent of the Mach number to first order approximation. This velocity is explicitly determined for a slender symmetric flow with small Hall parameter in terms of the imposed magnetic field along the axis. A numerical example is given for a magnetic field of a single current loop.

Application of the Thin-Wing-Expansion Method to the Compressible Flow past an Elliptic Cylinder

Imais thin-wing-expansion method is applied to a uniform flow past an elliptic cylinder at zero angle of attack, in order to obtain the analytic expressions correct to the third approximation for the complex velocity potential, and for the velocity distribution over the cylinder. Numerical calculation is made for the case t =0.1 where t is the thickness ratio of the cylinder. The convergence of the result seems to be satisfactory except near the stagnation point, where the anomalous behaviour appears.

On the Asymptotic Behaviour of Three-dimensional Compressible Fluid Flow at a Great Distance from a Body, I. (The Force and Moment on a Solid Body in a Stream of Compressible Fluid)

The steady irrotational isentropic continuous subsonic flow of a perfect gas past an arbitary three-dimensional body with arbitary distribution of sources on its surface is considered. Based on the thing-wingexpansion method, the velocity potential Φ is expanded in the form Φ= U ( x + A 1 r -1 + A 2 r -2 +...), A 1 = a 0 f -1 , A 2 = f -3 [ a 10 cos θ+sin θ( a 11 cos ω+ a 1-1 sin ω)], f =(1- M 2 sin 2 θ) 1/2 , where U is the velocity of the undisturbed flow of Mach number M streaming in the positive direction of x -axis, and x = r cos θ, y = r sin θcos ω, Z = r sin θsin ω. Using this expression, the force ( X , Y , Z ) and moment ( M x , M y , M z ) experienced by the body are found to be: X =ρ ∞ Q ∞ U , Y = Z =0, M x =0, M y , z =± 4π a 1,±1 ρ ∞ U 2 /(1- M 2 ) -1 , where ρ ∞ Q ∞ is the total strength of sources, and ρ ∞ the density at infinity.