We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as 'tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems

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An initial deterministic mathematical model for the dynamic motion of a simple pressurised liquid film bearing is derived and utilised to evaluate the possibility of bearing contact for thin film operation. For a very thin film bearing the flow incorporates a Navier slip boundary condition as parametrised by a slip length that in general is subject to significant variability and is difficult to determine with precision. This work considers the formulation of a modified Reynolds equation for the pressurised liquid flow in a highly rotating coned bearing. Coupling of the axial motion of the stator is induced by prescribed axial oscillations of the rotor through the liquid film. The bearing gap is obtained from solving a nonlinear second-order non-autonomous ordinary differential equation, via a mapping solver. Variability in the value of the slip length parameter is addressed by considering it as a random variable with prescribed mean and standard deviation. The method of derived distributions is used to exactly quantify the impact of variability in the slip length with a parametric study investigating the effect of both the deterministic and distribution parameters on the probability of contact. Additionally, as the axial rotor oscillations also have a random aspect due to possible varying excitations of the system, the probability of contact is investigated for both random amplitude of the periodic rotor oscillations and random slip length, resulting in a two parameter random input problem. The probability of contact is examined to obtain exact solutions and evaluate a range of bearing configurations.

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International audiencen this work, we describe a meshless numerical method based on local collocation with RBFs for the solution of the poroelasticity equation. The RBF finite collocation approach forms a series of overlapping nodal stencils, over which an RBF collocation is performed. These local collocation systems enforce the governing PDE operator throughout their interior, with the intersystem communication occurring via the collocation of field variables at the stencil periphery. The method does not rely on a generalised finite differencing approach, whereby the governing partial differential operator is reconstructed at the global level to drive the solution of the PDE. Instead, the PDE governing and boundary operators are enforced directly within the local RBF collocation systems, and the sparse global assembly is formed by reconstructing the value of the field variables at the centrepoint of the local stencils. In this way, the solution of the PDE is driven entirely by the local RBF collocation, and the method more closely resembles the approach of the full-domain RBF collocation method. By formulating the problem in this fashion, high rates of convergence may be attained without the computational cost and numerical ill-conditioning issues that are associated with the full-domain RBF collocation approach. An analytical solution is formulated for a 2D poroelastic fluid injection scenario and is used to verify the proposed implementation of the method. Highly accurate solutions are produced, and convergence rates in excess of sixth order are observed for each field variable (i.e. pressure and displacement) and field-variable derivative (i.e. pressure gradients and stresses). The stress and displacement fields resulting from the solution of the poroelasticity equation are then used to describe the formation and propagation of microfractures and microfissures, which may form in the presence of large shear strain, in terms of a continuous damage variable which modifies the mechanical and hydraulic properties of the porous medium. The formation of such hydromechanical damage, and the resulting increase in hydraulic conductivity, is investigated for a pressurised injection into sandstone. Copyright © 2014 John Wiley & Sons, Lt