Priyanka Shukla

The problem of finding initial conditions that lead to self-similar motion of point vortices is formulated as a linear system. The linearity in the equations is used to check for the existence of similarity solutions with a given shape and, in particular, to numerically find self-similar vortex configurations with or without any prior knowledge of circulations. Algorithms for computing the one-parameter family of collapse and expansion configurations and the finitely many rigid-vortex configurations present in the family are also discussed. Typical families are shown to have vortices parametrized along closed curves, and the conditions for which they are not closed are investigated via several numerical examples.

We revisit the problem of the weakly nonlinear stability analysis of an immiscible two-fluid viscosity-stratified, density-matched, plane Poiseuille flow (PPF) in a rigid channel. A formal amplitude expansion method, in which the flow variables are expanded in terms of a small amplitude function, is employed to examine the nonlinear development of the uniform wave trains. By employing the Chebyshev spectral collocation method, the linear growth rate and the first Landau coefficient, which determine the weakly nonlinear temporal evolution of a finite amplitude disturbance in the vicinity of linear instability, are computed. The focus is on the parameter regime where the long-waves are stable. The present analysis reveals the existence of both subcritical unstable and supercritical stable bifurcations. It is found that similar to the single fluid PPF, the two-fluid flow remains subcritically unstable at the onset of linear instability. There is a transition from subcritical bifurcation at higher wave numbers to supercritical bifurcation at lower wave numbers. The feedback of the mean flow correction onto the wave is responsible for the subcritical bifurcation. The equilibrium amplitude increases (decreases) as a function of the Reynolds number at a fixed wave number, where the bifurcation is supercritical (subcritical). Similar to the single fluid PPF, there is a reduction in the critical Reynolds number due to even extremely weak but finite amplitude disturbances. Moreover, as the disturbance intensity increases, the percentage reduction in the critical Reynolds number increases. The study helps one to have a better understanding and perspective of the bifurcations that occur close to criticality in the two-fluid interface dominated PPF system. In addition, it calls for devoted experiments supplemented with numerical and theoretical predictions on Poiseuille flow of viscosity and/or density stratified systems that would shed light on the nature of transition.

Point vortex systems that include vortices with constant coordinate functions are largely unexplored, even though they have reasonable physical interpretations in the geophysical context. Here, we investigate the dynamical aspects of the restricted three-vortex problem when one of the point vortices is assumed to be fixed at a location in the plane. The motion of the passive tracer is explored from a rotating frame of reference within which the free vortex with non-zero circulation remains stationary. By using basic dynamical system theory, it is shown that the vortex motion is always bounded, and any configuration of the three vortices must go through at least one collinear state. The present analysis reveals that any non-relative equilibrium solution of the vortex system either has periodic inter-vortex distances or it will asymptotically converge to a relative equilibrium configuration. The initial conditions required for different types of motion are explained in detail by exploiting the Hamiltonian structure of the problem. The underlying effects of a fixed vortex on the motion of vortices are also explored.

Grad's method of moments is employed to develop higher-order Grad moment equations-up to the first 26 moments-for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff's law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff's law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff's law. The constitutive relations for the stress and heat flux (i.e. the Navier-Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman-Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

One prototypical instability in granular flows is the shear-banding instability, in which a uniform granular shear flow breaks into alternating bands of dense and dilute clusters of particles having low and high shear (shear stress or shear rate), respectively. In this work, the shear-banding instability in an arbitrarily inelastic granular shear flow is analyzed through the linear stability analysis of granular hydrodynamic equations closed with Navier-Stokes-level constitutive relations. It is shown that the choice of appropriate constitutive relations plays an important role in predicting the shear-banding instability. A parametric study is carried out to study the effect of the restitution coefficient, channel width, and mean density. Two global criteria relating the control parameters are found for the onset of the shear-banding instability.

The hydrodynamic viscous fingering instability can be influenced by a simple viscosity-changing chemical reaction of type A+B→C, when a solution of reactant A is injected into a solution of B and a product C of different viscosity is formed by reaction. We investigate here numerically such reactive viscous fingering in the case of a reaction decreasing the viscosity to define the optimal conditions on the chemical and hydrodynamic parameters for controlling fingering. In particular, we analyze the influence of the Péclet number, Pe, on the efficiency of the chemical control of fingering. We show that the viscosity-decreasing reaction has an increased stabilizing effect when Pe is decreased. On the contrary, fingering is more intense and the system more unstable when Pe is increased. The related reactive fingering patterns cover then respectively a smaller (larger) area than in the nonreactive equivalent. Depending on the value of the Péclet number, a given chemical reaction may thus either enhance or suppress a fingering instability. This stabilization and destabilization at low and high Pe are shown to be related to the Pe-dependent characteristics of a minimum in the viscosity profile that develops around the miscible interface thanks to the effect of the chemical reaction.

A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an A+B→C chemical reaction takes place, where A and B are reactants in solution and C is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether A invades B or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product C has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one.

Linear stability of a plane Poiseuille flow in a channel bounded by anisotropic permeable walls supported by rigid walls is studied. Characteristic instability features due to two-dimensional infinitesimal disturbances of the most unstable wall mode are investigated in detail. A detailed parametric study displays the existence of wall modes, porous modes, and center modes in both the presence and absence of inertial effects. The results reveal that an increase in mean permeability decreases the critical Reynolds number, destabilizing smaller wavenumbers. Although anisotropy has no significant effect on the growth rate at smaller wavenumbers, the impact is substantial at larger wavenumbers, particularly destabilizing short-wave modes and enlarging the bandwidth of unstable wavenumbers. Furthermore, in relation to the configuration with isotropic permeability, the one with larger (smaller) relative wall-normal permeability is more (less) unstable with a large bandwidth of unstable wavenumbers covering short-wave lengths when mean permeability is high and when the fluid channel thickness is the same as the thickness of each of the porous walls. The critical Reynolds number increases with an increase in anisotropic permeability, while the critical wavenumber decreases with an increase in anisotropic permeability. This demonstrates the possibility of enhancing (suppressing) instability by designing the channel walls as one with anisotropic permeability and appropriately tuning the relative wall-normal permeability to be higher (lower). Furthermore, anisotropic permeability can be used to control instabilities for any arbitrary relative thickness of the porous medium beyond a minimum relative thickness that depends on the relative magnitude of wall normal anisotropic permeability.

The linear stability analysis of a plane Poiseuille flow in a channel with anisotropic and inhomogeneous porous layers is performed. The generalized Darcy equation, along with the Navier–Stokes equation, governs the flow in the porous–fluid–porous channel. The Beaver–Joseph interface condition is assumed, which considers the coupling between the flows in the porous and fluid layers. The spectral collocation method is employed to solve the Orr–Sommerfeld type eigenvalue problem for the amplitude of the disturbances of arbitrary wavenumbers. The effect of anisotropy and inhomogeneous permeability on the stability characteristics is addressed in detail. The stability characteristics of the anisotropy parameter (the ratio of permeability in the streamwise and the transverse direction) and the inhomogeneity function are presented in detail.

The dynamics of a constrained three-vortex system, a pair of point vortices of arbitrary non-zero circulations in the velocity field of a fixed point vortex, is investigated. The underlying dynamical system is simplified using a coordinate transformation and categorized into two cases based on the zero and non-zero values of the constant of angular impulse. For each case, dynamical features of the vortex motion are studied analytically in the transformed plane to completely classify the vortex motions and understand the boundedness and periodicity of the inter-vortex distances. The theoretical predictions are also verified numerically and illustrated for various sets of initial conditions and circulations.