Oscillatory instability in a weakly non-Boussinesq fluid layer with a deformable surface

The Galerkin method is applied in a new way to problems of stationary and oscillatory convective instability. By retaining the time derivatives in the equations rather than assuming an exponential time-dependence, the exact solution is approximated by the solution to a set of ordinary differential equations in time. Computations are simplified because the stability of this set of equations can be determined without finding the detailed solution. Furthermore, both stationary and oscillatory instability can be studied by means of the same trial functions. Previous studies which have treated only stationary instability by the Galerkin method can now be extended easily to include oscillatory instability. The method is illustrated for convective instability of a rotating fluid layer transferring heat.

Download Full-text

A Linear Stability Analysis of Thermal Convection in a Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions

Mathematical Problems in Engineering ◽

10.1155/2013/236901 ◽

2013 ◽

Vol 2013 ◽

pp. 1-15 ◽

Cited By ~ 2

Author(s):

Ruben Avila ◽

Ares Cabello

Keyword(s):

Magnetic Field ◽

Thermal Convection ◽

Fluid Layer ◽

Complex Problem ◽

Wave Number ◽

Critical Parameters ◽

Onset Of Convection ◽

Convective Thermal ◽

Boussinesq Fluid ◽

Simultaneous Rotation

The onset of thermal convection of a Boussinesq fluid located in an unbounded layer heated from below and subject simultaneously to rotation and magnetic field, whose vectors act in different directions, is presented. To the knowledge of the authors, the convective thermal instability analysis for this complex problem has not been previously reported. In this paper, we use the Tau Chebyshev spectral method to calculate the value of the critical parameters (wave number and Rayleigh number at the onset of convection) as a function of (i) different kinds of boundaries, (ii) angle between the three vectors, and (iii) different values of the Taylor numberT(rate of rotation) and magnetic parameterQ(strength of the magnetic force). For the classical problems previously reported in the literature, we compare our calculations with Chandrasekhar’s variational method results and show that the present method is applicable.

Download Full-text

On the generation of shelves by long nonlinear waves in stratified flows

Journal of Fluid Mechanics ◽

10.1017/s0022112097006435 ◽

1997 ◽

Vol 346 ◽

pp. 345-362 ◽

Cited By ~ 2

Author(s):

DILIP PRASAD ◽

T. R. AKYLAS

Keyword(s):

Fluid Layer ◽

Outer Region ◽

Finite Depth ◽

Finite Amplitude ◽

Stratified Flows ◽

Wave Disturbance ◽

Necessary Condition ◽

Matched Asymptotics ◽

Weakly Nonlinear ◽

Boussinesq Fluid

The phenomenon of shelf generation by long nonlinear internal waves in stratified flows is investigated. The problem of primary interest is the case of a uniformly stratified Boussinesq fluid of finite depth. In analysing the transient evolution of a finite-amplitude long-wave disturbance, the expansion procedure of Grimshaw & Yi (1991) breaks down far downstream, and it proves expedient to follow a matched-asymptotics procedure: the main disturbance is governed by the nonlinear theory of Grimshaw & Yi (1991) in the ‘inner’ region, while the ‘outer’ region comprises multiple small-amplitude fronts, or shelves, that propagate downstream and carry O(1) mass. This picture is consistent with numerical simulations of uniformly stratified flow past an obstacle (Lamb 1994). The case of weakly nonlinear long waves in a fluid layer with general stratification is also examined, where it is found that shelves of fourth order in wave amplitude are generated. Moreover, these shelves may extend both upstream and downstream in general, and could thus lead to an upstream influence of a type that has not been previously considered. In all cases, transience of the main nonlinear wave disturbance is a necessary condition for the formation of shelves.

Download Full-text

Nonlinear motions induced by moving thermal waves

Journal of Fluid Mechanics ◽

10.1017/s0022112072000606 ◽

1972 ◽

Vol 54 (1) ◽

pp. 163-187 ◽

Cited By ~ 13

Author(s):

Richard E. Young ◽

Gerald Schubert ◽

Kenneth E. Torrance

Keyword(s):

Thermal Wave ◽

Fluid Layer ◽

Lower Boundary ◽

Phase Speed ◽

Thermal Source ◽

Nonlinear Interactions ◽

Thermal Waves ◽

Mean Flow ◽

Mean Velocity ◽

Boussinesq Fluid

The motion induced in a layer of Boussinesq fluid by moving periodic thermal waves is obtained by numerically solving the complete nonlinear two-dimensional momentum and temperature equations. Three sets of boundary conditions are treated: rigid upper and lower boundaries with symmetrical heating; free upper boundary and rigid lower boundary with heating only at the top; free upper and lower boundaries with symmetrical heating. The nonlinear streamline patterns show that, when the velocity fluctuations are larger than the phase speed of the thermal wave and the mean flow, the convection cells have shapes governed by fluctuating nonlinear interactions. Significant mean velocities can be created even without the characteristic tilt in the convection cells expected on the basis of linear theory. Nonlinear interactions can lead to a mean shear even in the absence of motion of the thermal source. When the viscous diffusion time across the fluid layer is less than or of the same order as the period of the thermal wave, the order of magnitude of the induced mean velocity does not exceed that of the phase speed of the wave, even for intense thermal forcing.

Download Full-text

Surface tension driven oscillatory instability in a rotating fluid layer

Journal of Fluid Mechanics ◽

10.1017/s0022112069002035 ◽

1969 ◽

Vol 39 (1) ◽

pp. 49-55 ◽

Cited By ~ 25

Author(s):

G. A. McConaghy ◽

B. A. Finlayson

Keyword(s):

Surface Tension ◽

Convective Instability ◽

Marangoni Number ◽

Fluid Layer ◽

Oscillatory Instability ◽

Taylor Number ◽

Rotating Fluid ◽

Asymptotic Equations ◽

Critical Marangoni Number ◽

Prandtl Numbers

Oscillatory convective instability is shown to occur in a rotating fluid layer when convection is caused by surface-tension gradients at a free surface. The asymptotic equations, valid when the Taylor number approaches infinity, are solved analytically, and the critical Marangoni number is evaluated numerically. Fluids with Prandtl numbers above 0·201 will exhibit only stationary instability. Fluids with smaller Prandtl numbers will exhibit oscillatory instability with the critical Marangoni number varying as M0T½ where M0 depends on the Prandtl number and T is the Taylor number.

Download Full-text