Jitender Singh

Convective instability in a ferrofluid layer with temperature-modulated rigid boundaries

Under terrestrial as well as gravity-free conditions, a time-periodic modulation in temperatures of two horizontal rigid planes containing an initially quiescent ferrofluid layer induces time-periodic oscillations in the fluid layer at the onset of instability. This results in a series of patterns of time-periodically oscillating magnetoconvective rolls, along the vertical. The onset of instability in the ferrofluid layer is either a harmonic response or a subharmonic response depending upon the modulation. The instability is found to be significantly affected by the application of magnetic field across the ferrofluid layer. Under modulation, subcritical instabilities are found to occur in the form of subharmonic response. Also, the onset of instability in the ferrofluid layer when it is driven solely by the magnetic forces alone is found to heavily depend upon the frequency of modulation, the effect being greatest for the low-frequency modulation and negligible for the case of high-frequency modulation. The gravity-free limit is also evaluated as a function of the magnetic susceptibility, under modulation. To carry out this extensive study, the classical Floquet theory is utilized.

STABILITY OF NONAXISYMMETRIC FERROFLUID FLOW IN ROTATING CYLINDERS WITH MAGNETIC FIELD

Effect of an axially applied magnetic field on the stability of a ferrofluid flow in an annular space between two coaxially rotating cylinders with nonaxisymmetric disturbances has been investigated numerically. The critical value of the ratio Ω∗ of angular speeds of the two cylinders, at the onset of the first nonaxisymmetric mode of disturbance, has been observed to be affected by the applied magnetic field.

Rayleigh-Bénard convection with two-frequency temperature modulation.

The response of Rayleigh-Bénard convection in a horizontal fluid layer to time-periodic heating of its horizontal boundaries with a mixture of two frequencies is analyzed numerically. The ratio of the two forcing frequencies and the mixing angle of the amplitudes of modulation provide a control on the instability of the system. In addition to the existence of well-known harmonic and subharmonic instability responses under modulation, the time-periodic oscillation of the boundary temperatures of the fluid-layer with two frequencies results in more bicritical states in comparison to the single-frequency excitation. The onset of instability depends strongly on the modulation parameters and the Prandtl number of the fluid.

A Noninductive Proof of de Moivre's Formula

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A Short Proof That Lebesgue Outer Measure of an Interval Is Its Length

Proof. Given two real numbers a and b with a < b, it is enough to prove that m∗([a, b]) = b− a. Clearly, m∗([a, b]) ≤ b− a. Now let [a, b] ⊂ ∪k=1Ik for some positive integer n, which is always possible since [a, b] is compact. Without loss of generality, assume that the set [a, b] ∩ Ik is nonempty for each k. Observe that the set ∪k=1Ik is connected. (Otherwise, if (P,Q) is its separation, then for each k, by connectedness of Ik, either Ik ⊂ P or Ik ⊂ Q. Thus each of P and Q is equal to union of sets from the list {I1, . . . , In}. So the pair (P ∩ [a, b],Q ∩ [a, b]) determines a separation of [a, b], which contradicts connectedness of [a, b].) So ∪k=1Ik is an open interval containing [a, b]. Thus, b− a ≤ (∪k=1Ik ) ≤ ∑n k=1 (Ik ), where the last inequality holds since some intervals overlap. Hence, b− a ≤ m∗([a, b]).

Instability in temperature modulated rotating Rayleigh–Bénard convection

The problem of instability in an infinite horizontal thin layer of a Boussinesq fluid, uniformly rotated and heated from below with time periodic oscillations of the wall temperatures is investigated theoretically and numerically. In doing so, modest rotation rate is considered such that the Froude number does not exceed 0.05 which allows neglect of centrifugal effects. The Floquet analysis is used to obtain the critical Rayleigh number as a function of the parameters controlling the system. The instability is found to manifest itself in the form of a flow which oscillates either harmonically or subharmonically depending upon the control parameters. Instability regions in an appropriate parametric space of the dimensionless wave number of disturbance imposed over the flow and the dimensionless amplitude of modulation, at the onset of time periodic fluid flows are obtained numerically. The modulation amplitude, the modulation frequency and the rotation rate, are observed to affect the stability of the flow considerably.