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Viscous fluid bilayer subjected to perpendicular vibration

This section is an extension solution towards parametric excitation of inviscid fluid. In experiments the fluid viscosities cannot be ignored and thus an inviscid theory cannot work very precisely. To develop a more inclusive theory with a viscous system, a linear stability is modelled for Newtonian incompressible viscous fluid layers in order to relate theory with experiments. A single mode excitation has been derived from the Navier-Stokes equations, taking the true nature of the fluid system. The nonlinear equations were linearized with determined base state solution and forced excitation. The equations were solved for neutral stability conditions in order to obtain critical threshold – wavenumber curves. This was achieved by a Fourier-Floquet eigenvalue method and solved numerically by a matrix system. The current model considers an initial stable flat horizontal interface separating two distinct fluids, assuming no stresses for horizontally finite size system (see Fig. 2.4). Wavenumbers were then calculated based on finite cell size used in experiments.
The base state was stable with horizontally flat interface separating two distinct fluids. The stress-free boundary condition was assumed when horizontally finite system conditions were applied to these results, constraining on the allowed wavenumbers by the system. A case study has been presented discussing effects of various parameters like interfacial tension, viscosity, gravity to understand fluid behaviour in different circumstances.

Spatially infinite system results

The linear equations defined above can be solved for all wavenumbers and solutions to produce a set of tongues of instability which is similar to the fins produced by the Mathieu equation in case of inviscid flow but tips are smoothed due to viscous effects and not descending to zero amplitude (Benjamin and Ursell, 1954). The viscosity affects harmonic or higher harmonic solution much more compared to sub-harmonic node.

A case study on effects of parameters

Fluid mixtures show a wide variety of properties when studied over extended ranges of temperature and accelerations. Phase equilibrium is one of the most interesting areas of investigation which can provide information on different types of molecular interaction. Faraday instability like any other phenomenon in physics is an outcome of simple properties of the fluids or constituents active in the experimentation. The properties are basically statistical in nature. The common fluid parameters such as density, viscosity, surface tension are prime for the instability phenomenon as explained earlier, and their effects can help us in comprehending real phenomenon with variables such as gravity, temperature etc. It was thus important to understand that changing these parameters such as temperature will disturb geometric packing fraction of fluid mixture and reduced depth of potential for equal concentration of each liquid component in mixtures. In turn, all fluid properties become a function of temperature. Thus while doing the theoretical analysis for comparison with experiments, corrected values need to be taken into account to understand and validate the phenomenon with theory. Apart from that, a mixture has its own variation in properties because of interaction between molecules of two fluids. Any mixture of two fluids can show affinity (increased intermolecular forces) or aversion between its constituent molecules. For example, current binary fluid set has increased affinity with second fluid with increasing temperature. Variation of properties can also change the volume of fluid or its pressure. In certain physical experiments it was important to keep volume or pressure of the system constant in order to avoid extra uncontrollable presence of variables in the results.
For our binary fluid system, properties of fluids were varying depending on the above cited parameters and are defined based on true measurements of properties with enforced experiment like conditions. With these measurements understanding of binary fluid system has been improved and repeatability of experimental work has been validated.

Mass Transfer in binary fluids: Variation of density

A binary fluid behaviour is a parametrical emphasis of the phase equilibrium of fluid mixture systems. The experimented binary fluid system has a very interesting property of mass transfer when a small variation of temperature was introduced. Binary fluids start moving towards its new equilibrium state which includes partial mixing of one fluid into another. Current binary fluids show an increase in affinity towards each other molecules and change its equilibrium position with increasing temperature. Experiments conducted for current set of liquids (FC-72 & 1 cSt Silicone oil), confirm variation of density due to mixing and have been discussed in this thesis in detail.
To study temperature effects for immiscible fluids (where variation of temperature causes variation of density of top and bottom layers), which was an important aim of this thesis, the variation of density difference becomes an inevitable part of study. Mixing of binary fluids result into decrease in density difference, which in turn is a main parameter of Faraday instability phenomenon. We can simulate this effect by defining a function generating values of and with temperature based on measured and tested values in laboratory settings with controlled temperature conditions (See section 3.3). Decreasing density difference results into a domination of subharmonic waves as can be seen in Figure 2.9 (a)-(c). Another effect of decreasing density gap is the vanishing discretization making the system globally unstable.

Wall proximity conditions: effects of viscosity

In a small rectangular system, where Faraday wavelength is comparable to the container size, viscous damping plays an important role (effect of cell walls is discussed through the quantization of admissible modes. For a cell with one very small dimension (Hele-Shaw cell), walls have two effects: they force a quantization of the wavenumbers and they create local friction at the boundaries, which is not taken into account in these figures and and will be discussed in section 3.4.2) and basically intensifies the effects of viscosity. Viscous force is a stabilizing agent in the Faraday instability, which can be seen with rise in viscosity causing a damping of the tongues. Increasing amplitude threshold and smoothing of edges as well as movement of tongues slightly towards higher wavenumber can be observed. It is remarkable to note that damping effect on the higher harmonic tongues is more severe as compared to lower harmonics. This can be understood by the fact that increased frequency of waves results into increased friction between fluid molecules. Friction in fluid molecules is proportional to velocity gradients, and velocity gradients are inversely proportional to the wavelength. Thus it can be said that viscous stresses tend to dominate all processes as wavelengths get smaller and smaller.

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Effect of gravity: Simulating Faraday instability in microgravity

Space applications like vibrating propellant in rocket engine have variable acceleration. The gravity effects have been studied historically in ISS (International Space Station) and parabolic flight. Decrease in gravity causes tongues to shift rightwards to higher wavenumbers and forces the widening of tongues. Interestingly, the rate of spreading of the first sub-harmonic tongue was much higher in comparison to higher harmonics. This movement causes smaller wavelength and spreads nodes causing higher probability of sub-harmonic waves as well the decrease in mode width. Thus, a reduced gravity solution will have continuity of modes and will behave much more like an infinite boundary system.

Immiscible Instability threshold

Experiments were performed for immiscible instability threshold below consolute temperature for temperature range 25 ⁰C – 42 ⁰C as explained above and are shown in Figure 3.4. The mismatch observed in experimental threshold to theoretical threshold cannot be explained with the theoretical approach presented in this thesis (see Chapter 2). Imposed oscillation frequency and the lowest possible excitation amplitude for which a distinct visible interfacial wave occurred was measured and recorded. These experiments had shown the shifting of modes towards lower frequency as predicted qualitatively by the theory (Benjamin and Ursell, 1954). Figure 3.4 shows the critical threshold amplitude, A, (on y axis) obtained for frequencies ranging between 3 and 8 Hz (on x axis) at five different temperatures, viz., 25 0C, 38 0C, 40 0C, 41.5 0C, and 42 0C. In this temperature range, the interfacial tension coefficient and density contrast drastically diminished as the temperature of the fluids approached the consolute value (42.5 0C). The current confined system at low frequencies was characterized by both harmonic and sub-harmonic modes occurring over discrete bands of frequencies. This was in contrast to a continuum of sub-harmonic modes that were observed in large aspect systems. The various modes have been labelled as A, B, C, etc., and shown in Fig. 3.5. Here, mode B alone was harmonic and it consists of two and half waves. The rest of the modes were sub-harmonic and feature as multiples of the half wave, i.e., A consisted of half wave, C consisted of a full wave, D consisted of one and half waves and so-forth. The different modes interconnect at co-dimension points where the pattern consists of a superposition of both modes, dynamically changing from one state to the other. Within each of the discrete bands, the critical amplitude attained a minimum at the natural frequency of the mode. This phenomenon is equivalent to that of a simple pendulum whose fulcrum is subject to enforced pulsations. The response amplitude of the pendulum bob reaches a peak value when the forcing amplitude of the support crosses a threshold. This critical amplitude primarily depends on the forcing frequency and attains a minimum when the parametric frequency equals the system’s natural frequency. For the present case of Faraday instability, a similar effect produces a minimum of criticality at the natural frequency of each mode and that frequency is called tuned frequency for that mode. Generally, any perturbation of the fluid interface will produce ingress of heavier fluid into the lighter fluid along the crests and the case vice-versa along the troughs. The inertial forces acting on these masses induced by the imposed acceleration provoked the onset of instability and effects such as gravity, interfacial tension and the viscous diffusion acted as restoration forces that bring back the interface to its initial flat condition.
In the case of the present experiments where the increase in temperature led to the decrease in both the density difference (note that the density of each layer changes on account of the solubility and depends on temperature, see figure 3.3a) and the interfacial tension, the earlier discussion in section 2.3, aids in understanding the fluid behaviour. The impact of changing fluid temperature on the onset characteristics has been depicted in Figure 3.4. An important feature that can be observed here is that the frequency band for each mode shrunk with an increase in the fluid temperature. This shrinkage was due to a drift of modes toward lower frequencies of excitation. In other words, the natural frequency of each mode became smaller. This did not come to a surprise as reduction in both the density difference and interfacial tension ought to lead to reduction in the natural frequency of each mode.

Table of contents :

Governing Equations
Linear Stability Analysis
Spatially infinite system results
Results in finite systems
Mass Transfer in binary fluids: density
Proximity in wall conditions: viscosity
Interfacial conditions: interfacial tension
Space application: gravity
Immiscible Instability threshold
Miscible Instability Threshold


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