Numerical study of geometrical dispersion in self-affine rough fractures

We report a numerical study of passive tracer dispersion in fractures with rough walls modeled as the space between two complementary self-affine surfaces rigidly translated with respect to each other. Geometrical dispersion due to the disorder of the velocity distribution is computed using the lubrication approximation. Using a spectral perturbative scheme to solve the flow problem and a mapping coordinate method to compute dispersion, we perform extensive ensemble averaged simulations to test theoretical predictions on the dispersion dependence on simple geometrical parameters. We observe the expected quadratic dispersion coefficient dependence on both the mean aperture and the relative shift of the crack as of well as the anomalous dispersion dependence on tracer traveling distance. We also characterize the anisotropy of the dispersion front, which progressively wrinkles into a self-affine curve whose exponent is equal to that of the fracture surface

Instanton Approach to Large NN Harish-Chandra-Itzykson-Zuber Integrals

We reconsider the large $N$ asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.Comment: 5 pages, 1 figur

Flow channelling in a single fracture induced by shear displacement

The effect on the transport properties of fractures of a relative shear displacement $\vec u$ of rough walls with complementary self-affine surfaces has been studied experimentally and numerically. The shear displacement $\vec u$ induces an anisotropy of the aperture field with a correlation length scaling as $u$ and significantly larger in the direction perpendicular to $\vec u$. This reflects the appearance of long range channels perpendicular to $\vec u$ resulting in a higher effective permeability for flow in the direction perpendicular to the shear. Miscible displacements fronts in such fractures are observed experimentally to display a self affine geometry of characteristic exponent directly related to that of the rough wall surfaces. A simple model based on the channelization of the aperture field allows to reproduces the front geometry when the mean flow is parallel to the channels created by the shear displacement

A computational model for the design of a nitrous oxide--paraffin wax hybrid rocket engine

A computational model is developed to assist the design of the first hybrid rocket engine of the McGill Rocket Team, using liquid nitrous oxide as an oxidizer and solid paraffin wax as a fuel. The model is developed in three phases: In the first phase, a steady-state model which neglects transient performance decrease of the hybrid engine is considered. The steady-state model considers: a constant oxidizer mass flow rate; a combustion chamber in chemical equilibrium; a one-dimensional isentropic nozzle; and a one-dimensional constant-thrust rocket ascent. The steady-state model is used to conduct parametric studies on engine performance as a function of design parameters such as: oxidizer mass flow rate, fuel grain dimensions, and nozzle dimensions. The engine design point is selected to optimize specific impulse, given physical and structural constraints of the system. In the second phase, an unsteady model incorporating transient performance decrease of the hybrid engine is considered. The transient model considers: a self-pressurizing oxidizer tank in quasi-steady liquid-vapor equilibrium; a combustion chamber in quasi-steady chemical equilibrium; a one-dimensional non-isentropic nozzle; and a one-dimensional rocket ascent model. The transient performance profile of the engine is established, and the unsteady model is used to predict propellant requirements for the launch vehicle, given a target apogee of 3048 m (10,000 ft). In the third stage, the unsteady model is compared to hot fire testing data of the McGill Rocket Team. Semi-empirical parameters used in the formulation are calibrated against testing data, and the validity of the model is assessed