We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.Comment: 25 pages, 24 figures, slightly revised and corrected versio

We investigate the dynamics of turbulent flow in a two-dimensional trapped Bose-Einstein condensate by solving the Gross-Pitaevskii equation numerically. The development of the quantum turbulence is activated by the disruption of an initially embedded vortex quadrupole. By calculating the incompressible kinetic-energy spectrum of the superflow, we conclude that this quantum turbulent state is characterized by the Kolmogorov-Saffman scaling law in the wave-number space. Our study predicts the coexistence of two subinertial ranges responsible for the energy cascade and enstrophy cascade in this prototype of two-dimensional quantum turbulence.

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The aim of this study is to evaluate the effect of pulsatile blood flow in thermally significant blood vessels on the thermal lesion region during thermal therapy of tumor. A sinusoidally pulsatile velocity profile for blood flow was employed to simulate the cyclic effect of the heart beat on the blood flow. The evolution of temperature field was governed by the energy transport equation for blood flow together with Pennes' bioheat equation for perfused tissue encircling the blood vessel. The governing equations were numerically solved by a novel multi-block Chebyshev pseudospectral method and the accumulated thermal dose in tissue was computed. Numerical results show that pulsatile velocity profile, with various combinations of pulsatile amplitude and frequency, has little difference in effect on the thermal lesion region of tissue compared with uniform or parabolic velocity profile. However, some minor differences on the thermal lesion region of blood vessel is observed for middle-sized blood vessel. This consequence suggests that, in this kind of problem, we may as well do the simulation simply by a steady uniform velocity profile for blood flow.

We describe an immersed-boundary technique which is adopted from the direct-forcing method. A virtual force based on the rate of momentum changes of a solid body is added to the Navier-Stokes equations. The projection method is used to solve the Navier-Stokes equations. The second-order Adam-Bashford scheme is used for the temporal discretization while the diffusive and the convective terms are discretized using the second-order central difference and upwind schemes, respectively. Some benchmark problems for both stationary and moving solid object have been simulated to demonstrate the capability of the current method in handling fluid-solid interactions.