Lie group analysis of gravity currents

Lie symmetry solutions for two-phase mass flows

“…Importantly, with the Lie symmetry transformations, we show that the similarity variable for the reduced equations associated with our two-phase mass flow model takes a form of a product of two exponents in time and space, and that the similarity form (in terms of the new similarity variable) is expressed as a product of an exponent function (in time or space) and the original dynamical variables. Thus, our solutions generalize several similarity variables and similarity forms obtained in the literature by using Lie symmetry group transformations [11,[25][26][27][33][34][35][36][37]. It appears that the choice of group parameters determines the similarity forms and the dynamics of the reduced system of equations.…”

“…Gravity mass flows and gravity currents, including avalanches, debris flows, debris floods, floods, and water waves, play important role both in environment, geophysics and engineering [2][3][4][5][6][7][8][9][10][11][12]. Debris flows are extremely destructive and dangerous natural hazards, so there is a need for reliable methods for predicting the dynamics, runout distances, and possible areas hit by such events.…”

“…The Lie group theory has been extensively used to obtain solutions of differential equations arising from wide range of physical problems, including the reduction of differential equations, order reduction of ordinary differential equations, construction of invariant solutions, and mapping between the solutions in mechanics, applied mathematics and mathematical physics, and applied and theoretical physics [11,[19][20][21][22][23][24][25][26][27]. Lie algebra and symmetry group theory is an important and effective method for solving a system of non-linear partial differential equations via similarity forms and similarity solutions [19,23,[27][28][29][30][31][32].…”

“…Particularly related to our interest, Lie theory has been successively applied to construct and investigate self-similar solutions to gravity currents, and relatively simple shallow water or two-layer shallow water flows, impulse modified shallow water equations, dispersive shallow-water flows, Benney system, generalized Burgers equation, etc. [11,[25][26][27][33][34][35][36][37].…”

Lie Group Analysis on Brownian Motion and Thermophoresis Effect on Free Convective Boundary-Layer Flow on a Vertical Cylinder Embedded in a Nanofluid-Saturated Porous Medium

“…Bhuvaneswari et al [13] presented similarity solutions of natural convection heat and mass transfer in an inclined surface with chemical reaction via Lie group analysis. Using symmetry analysis, Sahin et al [14] studied the self-similarity solutions of the one-layer shallow-water equations representing gravity currents.…”