Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

One‐dimensional freezing of nonheaving unsaturated soils: Model formulation and similarity solution

Abstract: [1] Freezing of unsaturated soils is associated with the formation of a moving freezing zone and liquid water flow toward the zone. An equilibrium thermodynamic formulation of coupled flow and heat transport in variably saturated partially frozen porous media is developed and a self-similar solution is derived for the case of a semi-infinite horizontal porous media column with a constant freezing temperature on one boundary. Solutions to the self-similar equations are derived using a Runge-Kutta solution procedure. The solution is found to yield two possible modes distinguished by zones composed of different combinations of ice, liquid water, and air. One of the modes contains three zones: a frozen zone (WI) with just ice and liquid water; a transition zone (AWI) with ice, liquid water, and air; and an unsaturated zone (AW) with liquid water and air. The second mode contains only the WI zone and the AW zone. It is found that the WI zone is a quintessential part of the solution. The AWI zone is found to exist when the advancement of the freezing zone is relatively fast, while it is absent when the zone advances slowly. Predictions of ice saturation and liquid water saturation with the self-similar solution are compared to published experimental data. Pore pressure is calculated as a linear combination of ice pressure and liquid water pressure, and the calculated figures are used to provide a condition for model limitation in the case of incipient ice lens formation. The developed similarity solution provides insight into the mechanics of liquid water movement and pore filling with ice and the conditions for incipient heaving.

Capillary instability and breakup of a viscous thread

Abstract: Papageorgiou derived a similarity solution that describes the asymptotic behavior of a thinning viscous thread suspended in vacuum, near the critical time and around the location of breakup. The motion is driven by surface tension, and the fluid inertia is neglected throughout the evolution. To assess the physical relevance of the similarity solution, the evolution of an infinite thread immersed in an ambient fluid with arbitrary viscosity, subject to periodic axisymmetrtic perturbations is simulated through solution of the equations of Stokes flow by a boundary integral method. The results show that when the thread is suspended in vacuum, the similarity solution accurately describes the process of thinning over an extended length of the thread between the developing bulges, and captures the late stages of breakup for a broad range of initial conditions. But a small amount of ambient fluid viscosity, as small as 0.05 times the viscosity of the thread fluid, drastically alters the nature of the motion by shifting the location of the breakup points toward the bases of developing bulges, and causing the thread to develop locally asymmetric shapes.

A theoretical model of the pressure field arising from asymmetric intraglottal flows applied to a two-mass model of the vocal folds.

Abstract: A theoretical flow solution is presented for predicting the pressure distribution along the vocal fold walls arising from asymmetric flow that forms during the closing phases of speech. The resultant wall jet was analyzed using boundary layer methods in a non-inertial reference frame attached to the moving wall. A solution for the near-wall velocity profiles on the flow wall was developed based on a Falkner-Skan similarity solution and it was demonstrated that the pressure distribution along the flow wall is imposed by the velocity in the inviscid core of the wall jet. The method was validated with experimental velocity data from 7.5 times life-size vocal fold models, acquired for varying flow rates and glottal divergence angles. The solution for the asymmetric pressures was incorporated into a widely used two-mass model of vocal fold oscillation with a coupled acoustical model of sound propagation. Asymmetric pressure loading was found to facilitate glottal closure, which yielded only slightly higher values of maximum flow declination rate and radiated sound, and a small decrease in the slope of the spectral tilt. While the impact on symmetrically tensioned vocal folds was small, results indicate the effect becomes more significant for asymmetrically tensioned vocal folds.

Mixed Convection to Power-Law Type Non-Newtonian Fluids from a Vertical Wall

Abstract: The mixed convection boundary layer flow on a vertical stationary or moving plate to a power-law non-Newtonian fluid is analyzed. An exact similarity solution is derived for the case when the surface temperature is inversely proportional to the distance from the leading edge of the plate. A discussion is provided on the effects of the flow index and buoyancy parameter on the velocity and temperature fields.