Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

We consider a fluid flow in a time dependent domain $\Omega_f(t)=\Omega \setminus \Omega_s(t)\subset {\mathbb R}^3$, surrounding a deformable obstacle $\Omega_s(t)$. We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in $\Omega_f(t)$, $t>0$. We prove that, for any arbitrary exponential decay rate $\omega>0$, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary $\partial \Omega_s(t)$ can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate $\omega>0$.

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2<p<∞ and N<q<∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

Modeling of aortic valve stenosis using fluid-structure interaction method

Background and objectives: Fluid structure interaction (FSI) is defined as interaction of the structures with contacting fluids. The aortic valve experiences the interaction with blood flow in systolic phase. In this study, we have tried to predict the hemodynamics of blood flow through a normal and stenotic aortic valve in two relaxation and exercise conditions using a three-dimensional FSI method. Methods: The aorta valve was modeled as a three-dimensional geometry including a normal model and two others with 25% and 50% stenosis. The geometry of the aortic valve was extracted from CT images and the models were generated by MMIMCS software and then they were implemented in ANSYS software. The pulsatile flow rate was used for all cases and the numerical simulations were conducted based on a time-dependent domain. Results: The obtained results including the velocity, pressure, and shear stress contours in different systolic time sequences were explained and discussed. The maximum blood flow velocity in relaxation phase was obtained 1.62 m/s (normal valve), 3.78 m/s (25% stenosed valve), and 4.73 m/s (50% stenosed valve). In exercise condition, the maximum velocities are 2.86, 4.32, and 5.42 m/s respectively. The maximum blood pressure in relaxation phase was calculated 111.45 mmHg (normal), 148.66 mmHg (25% stenosed), and 164.21 mmHg (50% stenosed). However, the calculated values in exercise situation were 129.57, 163.58, and 191.26 mmHg. The validation of the predicted results was also conducted using existing literature. Conclusions: We believe that such model are useful tools for biomechanical experts. The further studies should be done using experimental data and the data are implemented on the boundary conditions for better comparison of the results.

A reconstructed discontinuous Galerkin method for compressible flows on moving curved grids

A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and invalid elements near moving boundaries.

A stable method for 4D CT-based CFD simulation in the right ventricle of a TGA patient

The paper discusses a stabilization of a finite element method for the equations of fluid motion in a time-dependent domain. After experimental convergence analysis, the method is applied to simulate a blood flow in the right ventricle of a post-surgery patient with the transposition of the great arteries disorder. The flow domain is reconstructed from a sequence of 4D CT images. The corresponding segmentation and triangulation algorithms are also addressed in brief.

A Reconstructed Discontinuous Galerkin Method for Compressible Flows on Moving Curved Grids

A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for compressible inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation on moving and deforming unstructured curved meshes. Taylor basis functions in the rDG method are defined on the time-dependent domain, where the integration and computations are performed. The Geometric Conservation Law (GCL) is satisfied by modifying the grid velocity terms on the right-hand side of the discretized equations at Gauss quadrature points. A radial basis function (RBF) interpolation method is used for propagating the mesh motion of the boundary nodes to the interior of the mesh. A third order Explicit first stage, Single Diagonal coefficient, diagonally Implicit Runge-Kutta scheme (ESDIRK3) is employed for the temporal integration. A number of benchmark test cases are conducted to assess the accuracy and robustness of the rDG-ALE method for moving and deforming boundary problems. The numerical experiments indicate that the developed rDG method can attain the designed spatial and temporal orders of accuracy, and the RBF method is effective and robust to avoid excessive distortion and elements near moving boundaries.