5.1 Conclusions
In this thesis, we use the stability analyses to study the stability problem of unbounded uniform dense granular shear flows. The procedure in this study is: (1) neglecting the body force term; (2) deriving the two-dimensional linear perturbation system; (3) applying the Kelvin mode to obtain the solution; and (4) conducting asymptotic and transient stability analyses of the spatially uniform, layering and non-layering modes for purely kinetic model.
In the spatially uniform mode, this study indicates that the initial disturbance of transversal velocity will cause the streamwise velocity perturbation to increase with time, which is referred to as algebraic instability. This result finds the same conclusion as the paper (the A&N 1997). In asymptotic stability analysis of layering modes, the marginal stability curve is proven that it does not exist, and the real part of the least-stable eigenvalues are always negative for given entire domain. Namely, the linear perturbation system is stable of layering modes. Despite the unstable phenomenon does not occur in this study, the magnitudes of ω are increased with the coefficient lr of restitution and the solids volume fraction. Layering modes decay faster as the coefficient of restitution and the solids volume fraction are increased. It is due to the higher value of coefficient of restitution can maintain the system in basic state and the larger solids volume fraction possesses less space to develop disturbance waves. At a given coefficient of restitution and the solids volume fraction, layering modes of shorter wavelengths possess initial transient growth rate than those of longer wavelengths.
The transient growth function of layering modes decay asymptotically, and it corresponds with the result of asymptotic stable. Furthermore, the value of Gmax for stable flows is increased with the coefficient of restitution and the solids volume fraction. This study presents four features as the non-zero streamwise wavenumber is reduced by one order: (1) the period of G(t) persists longer; (2) multiple peaks of G(t) appear; and (3) the value of Gmax significantly increases and it is shifted to another peak. Hence, no matter layering modes or non-layering modes, the contribution of transient growth is provided by the disturbance of density at initial period and dominated for all period by the disturbance of kinetic energy. The optimal growth is researched in this study, and as the streamwise wavenumber is reduced to be a small value, (1) the value of kopty decreases; (2) the value of Gopt increases significantly; and (3) topt increases notably. The above three results are consistent with the paper (the A&N 1997). The wave vectors can be rotated by the uniform shear flows (the Kevlin mode) and eventually direct to downward as t→∞. This study illustrates that the band-structure of granular particles will become homogeneous layers aligned with the streamwise direction when time is large, and the amplitudes of disturbances decay to zero asymptotically. The result of adding a quasi-state term is presented in Figure 4-25, but this disorderly result is very unreasonable and cannot be accepted to be a constitutive law for dense granular flow. Therefore, more study is needed to clarify this interesting phenomenon.
5.2 Future works
The mechanism of dense granular flows still lacks a unified theory. In this thesis, we base on the revised constitutive equations of the kinetic theory (Savage 2008). By means of the stability analysis, we can observe the features and properties for purely kinetic model. As the quasi-state contribution is considered, the result is too unreliable to accept. Then the approximate way does not completely describe the rheology behavior of dense granular flow. A complete way is that we not only consider the effect of the quasi-state contribution but also all kinetic terms which cannot be omitted. The results of asymptotic stability analysis for non-layering can also be obtained by conducting the verification (cf. p.276 of the A&N 1997). This verification can prove whether the linear system is stable or not in the non-layering modes. Since the behavior of disturbance waves are described by the Fourier modes, the relation of the Fourier modes can be analyzed in more detailed way in a future study.
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