Masaki Takashima

Surface Tension Driven Instability in a Horizontal Liquid Layer with a Deformable Free Surface. I. Stationary Convection

The effect of a free surface deformation on the onset of surface tension driven instability in a horizontal thin liquid layer subjected to a vertical temperature gradient is examined using linear stability theory. Assuming that the neutral state is a stationary one, the conditions under which instability sets in are determined in detail. It is shown that when the upside of the liquid layer is a free surface the free surface deformation is important only for unusually thin layers of very viscous liquids. It is also shown that when the underside of the liquid layer is a free surface the free surface deformation plays an essentially important role and the presence of a vertical temperature gradient can stabilize the layer which, in case of no temperature gradient, is always unstable.

The Stability of Natural Convection in a Vertical Layer of Dielectric Fluid in the Presence of a Horizontal ac Electric Field

Linear stability theory is applied to examine the effect of a horizontal ac electric field on the stability of natural convection which occurs in a dielectric fluid between two parallel vertical plates maintained at different temperatures. The power series method is used to obtain the eigenvalue equation which is then solved numerically. It is shown that when the electrical Rayleigh number L (defined in the text) is less than about 2130 the electric field has no effect on the stability of natural convection, and that when L exceeds this value the electric field and the natural convection flow are coupled strongly and they exhibit a complicated but interesting effect on the stability of the system. A rough criterion is given for applicability of the power series method to the present problem.

The stability of the modified plane couette flow in the presence of a transverse magnetic field

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number Rc, the critical wave number αc, and the critical wave speed cc are obtained for wide ranges of the magnetic Prandtl number Pm and the Hartmann number M. It is found that except for the case when Pm is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as Pm increases.

Electrohydrodynamic Instability in a Viscoelastic Liquid Layer

The problem of the onset of instability in a horizontal layer of viscoelastic dielectric liquid under the simultaneous action of a vertical ac electric field and a vertical temperature gradient is analyzed. Applying linear stability theory, an equation of eighth order is derived. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. It is shown that oscillatory modes of instability exist only when the thickness of the liquid layer is smaller than about 0.5 mm and for such a thin layer the force of electrical origin is much more important than buoyancy force.

Thermal Instability in a Viscoelastic Fluid Layer. I

The stability of a horizontal layer of a viscoelastic fluid (Oldroyd fluid) heated from below is considered. Linear stability theory is used to derive an eigenvalue equation system of sixth order. Consideration is given to the two cases ( a ) both bounding surfaces free and ( b ) both bounding surfaces rigid, the former being solved exactly and the latter appoximately by the Galerkin method. Tables of the various critical values for the onset of instability are provided for various assigned values of parameters. It is shown that the principle of exchange of stabilities does not hold for moderately elastic fluids; and that the critical Rayleigh number is lowered by the presence of a stress relaxation time and raised by the presence of a strain retardation time.

Thermal Instability of Fluid Layer Bounded Below by a Solid Layer of Finite Conductivity

Linear stability theory is applied to the problem of the onset of convective instability in a horizontal layer of fluid heated from below, when the fluid is bounded above by a free surface and below by a solid layer of finite heat conductivity and finite thickness. As the agencies causing instability, both surface tension and buoyancy are taken into account. A Fourier series method is used to obtain the eigenvalue equation, from which various critical values for the onset of a stationary convection are computed numerically for a broad range of thermal boundary conditions. By solving the time-dependent eigen-value equation numerically, it is shown that, as assumed previously, the neutral state is indeed a stationary rather than an oscillatory one.

The Stability of Natural Convection in an Inclined Fluid Layer with Internal Heat Generation

Linear stability theory is applied to the problem of the stability of natural convection that occurs in an inclined fluid layer with uniformly distributed internal heat sources. It is assumed that one bounding plate is a thermally perfect insulator and the other bounding plate is maintained at constant temperature. The power series method is used to obtain the eigenvalue equation which is then solved numerically with the aid of the Muller method. The stability conditions are obtained for Prandtl numbers ranging from 0.001 to 100 and for angles of inclination ranging from -90° to 90°. It is found that the instability sets in as either transverse travelling wave modes or longitudinal stationary modes and that three-dimensional disturbances are not responsible for instability.

The stability of natural convection in a vertical layer of electrically conducting fluid in the presence of a transverse magnetic field

The stability of natural convection of an electrically conducting fluid which is confined between two parallel vertical plates maintained at constant and different temperatures and is permeated by a transverse magnetic field is analyzed using linear stability theory. In deriving the equations governing the stability, a simplification is made using the fact that the magnetic Prandtl number Pm for most electrically conducting fluids is extremely small. The Chebyshev collocation method is used to obtain the eigenvalue equation, which is then solved numerically. The critical Grashof number Gc, the critical wavenumber ac, and the critical wave speed cc are obtained for a wide range of the Prandtl number P and for several selected values of the Hartmann number M. It is found that the magnetic field has a stabilizing effect on the convection flow against both stationary and travelling-wave disturbances. The detailed values of P, Gc, ac, and cc, at the point of transition from stationary to travelling-wave mode are also obtained for several selected values of M.

The Effect of Rotation on Convective Instability in a Horizontal Fluid Layer with Internal Heat Generation

The effect of uniform rotation on the onset of convective instability in a horizontal fluid layer with uniformly distributed internal heat sources is considered using linear stability theory. Under the assumption that the neutral state is a stationary one, the power series method is adopted to obtain the eigenvalue equation which is then computed numerically. The stability conditions influenced by uniform rotation are found for two sets of thermal boundary conditions.

Surface Tension Driven Instabillity in a Horizontal Layer of Binary Liquid Mixture in the Presence of the Soret Effect

The problem of the onset of surface tension driven instability in a horizontal layer of binary liquid mixture subjected to a vertical temperature gradient is examined using linear stability theory. The solute is assumed to be a non-surfactant and the Soret effect is taken into account. Assuming that the neutral state is a stationary one, the conditions under which instability sets in are determined in detail and it is found that the presence of the solute can play a prominent role through the Soret effect.