van de Aaf Fons Ven

A model for arterial adaptation combining microstructural collagen remodeling and 3D tissue growth

Long-term adaptation of soft tissues is realized through growth and remodeling (G&R). Mathematical models are powerful tools in testing hypotheses on G&R and supporting the design and interpretation of experiments. Most theoretical G&R studies concentrate on description of either growth or remodeling. Our model combines concepts of remodeling of collagen recruitment stretch and orientation suggested by other authors with a novel model of general 3D growth. We translate a growth-induced volume change into a change in shape due to the interaction of the growing tissue with its environment. Our G&R model is implemented in a finite element package in 3D, but applied to two rotationally symmetric cases, i.e., the adaptation towards the homeostatic state of the human aorta and the development of a fusiform aneurysm. Starting from a guessed non-homeostatic state, the model is able to reproduce a homeostatic state of an artery with realistic parameters. We investigate the sensitivity of this state to settings of initial parameters. In addition, we simulate G&R of a fusiform aneurysm, initiated by a localized degradation of the matrix of the healthy artery. The aneurysm stabilizes in size soon after the degradation stops.

Eigencurrent analysis of resonant behavior in finite antenna arrays

Resonant behavior in a finite array that appears as (modulated) impedance or current-amplitude oscillations may limit the array bandwidth substantially. Therefore, simulations should predict such behavior. Recently, a new approach has been developed, called the eigencurrent approach, which can predict resonant behavior in finite arrays. A study of line arrays of either E- or H-plane-oriented strips and rings in free space and in half-spaces confirms our conclusion in earlier research that resonant behavior is caused by the excitation of one of the eigencurrents. The eigenvalue (or characteristic impedance) of this eigencurrent becomes small in comparison to the eigenvalues of the other eigencurrents that can exist on the array geometry. We demonstrate that the excitation of this eigencurrent results in an edge-diffracted wave propagating along the surface of the array, which may turn into a standing wave. In that case, the amplitudes and phases of the element impedances show the same standing-wave pattern as those of the excited eigencurrent. We demonstrate that the phase velocity of this wave is approximately equal to or slightly larger than the free-space velocity of light. Finally, we throw light on the relation between the excitation of eigencurrents with a small eigenvalue and the behavior of super directive arrays

Mathematical model of falling of a viscous jet onto a moving surface

We analyze the stationary flow of a jet of Newtonian fluid that is drawn by gravity onto a moving surface. The situation is modeled by a third-order ODE on a domain of unknown length and with an additional integral condition; by solving part of the equation explicitly we can reformulate the problem as a first-order ODE, again with an integral constraint. We show that there are two flow regimes, and characterize the associated regions in the three-dimensional parameter space in terms of an easily calculable quantity. In a qualitative sense the results from the model are found to correspond with experimental observations.

Transient behaviour and stability points of the Poiseuille flow of a KBKZ-fluid

The Poiseuille flow of a KBKZ-fluid, being a nonlinear viscoelastic model for a polymeric fluid, is studied. The flow starts from rest and especially the transient phase of the flow is considered. It is shown that under certain conditions the steady flow equation has three different equilibrium points. The stability of these points is investigated. It is proved that two points are stable, whereas the remaining one is unstable, leading to several peculiar phenomena such as discontinuities in the velocity gradient near the wall of the pipe (‘spurt’) and hysteresis. Our theoretical results are confirmed by numerical calculationsof the velocity gradient.

A variational principle for magneto-elastic buckling

A variational principle that can serve as the basis for a magneto-elastic stability (or buckling) problem is constructed. For the two cases of soft ferromagnetic media and superconductors, respectively, it is shown how the variational principle directly yields an explicit expression for the buckling value. The formulation starts from a specific choice for a magneto-elastic Lagrangian L (associated with the so-called Maxwell-Minkowski model for magneto-elastic interactions). For the evaluation of the principle the first and second variations of L are calculated both inside and outside the solid magneto-elastic body. Thus, a general buckling criterion, consisting of an expression for the critical field value, together with a set of constraints for the field variables occurring in the right-hand side of this expression, is constructed. Finally, more detailed formulations are given for, successively, soft ferromagnetic bodies and superconductors. Applications to specific structures, yielding explicit numerical values for the magneto-elastic buckling fields, will be given in a forthcoming paper.

In-phase and out-of-phase break-up of two immersed liquid threads under influence of surface tension

The dynamical behaviour of two infinitely long adjacent parallel liquid threads immersed in a fluid is considered under influence of small initial perturbations. Assuming all fluids to behave Newtonian, we used the creeping flow approximation, which resulted in Stokes equations. Applying cylindrical coordinates and separation of variables, and writing the dependence on the azimuthal direction in the form of a Fourier expansion, we obtained general representations of the equations for both the threads and the surrounding fluid. Substitution of these expressions into the boundary conditions leads to an infinite set of linear equations for the unknown coefficients. Its solutions for the lowest two orders of the Fourier expansion, the so-called zero- and first-order solutions, are presented. Much attention is paid to the (in)stability of the configuration, in terms of the so-called growth rate of the disturbance amplitudes. The growth rate of these amplitudes determines the behaviour of the break-up process of the threads. It turns out that this breaking up occurs either in-phase or out-of-phase. This depends on the viscosity ratio of the fluids and on the distance between the threads. These findings agree with experimental observations. The results of the present work also show that the zero-order solution yields the qualitatively correct insight in the break-up process. The extension to a one order higher expansion only leads to relatively small quantitative corrections.

Analytical and experimental characterization of a miniature calorimetric sensor in a pulsatile flow

The behaviour of a miniature calorimetric sensor, which is under consideration for catheter-based coronary-artery-flow assessment, is investigated in both steady and pulsatile tube flows. The sensor is composed of a heating element operated at constant power and two thermopiles that measure flow-induced temperature differences over the sensor surface. An analytical sensor model is developed, which includes axial heat conduction in the fluid and a simple representation of the solid wall, assuming a quasisteady sensor response to the pulsatile flow. To reduce the mathematical problem, described by a two-dimensional advection–diffusion equation, a spectral method is applied. A Fourier transform is then used to solve the resulting set of ordinary differential equations and an analytical expression for the fluid temperature is found. To validate the analytical model, experiments with the sensor mounted in a tube have been performed in steady and pulsatile water flows with various amplitudes and Strouhal numbers. Experimental results are generally in good agreement with theory and show a quasi-steady sensor response in the coronary-flow regime. The model can therefore be used to optimize the sensor design for coronary-flow assessment.

A variational approach to the magneto-elastic buckling problem of an arbitrary number of superconducting beams

Based upon a variational principle and the associated theory derived in three preceding papers, an expression for the magneto-elastic buckling value for a system of an arbitrary number of parallel superconducting beams is given. The total current is supposed to be equal both in magnitude and direction for all beams, and the cross-sections are circular. The expression for the buckling value is formulated more explicitly in terms of the so-called buckling amplitudes, the latter following from an algebraic eigenvalue problem. The pertinent matrix is formulated in terms of complex functions, which are replaced by real potentials. The matrix elements are calculated by a numerical method, solving a set of intergral equations with regular kernels. Apart from the buckling value(s) the buckling modes are also obtained. Finally, our results are compared with the results of a mathematically less complicated theory, i.e. the method of Biot and Savart.

Rayleigh-gravity waves in a heavy elastic medium

SummaryA plane wave travelling through an elastic incompressible medium with a plane boundary is considered. The motion is confined largely to the neighbourhood of the latter. In addition to elastic forces, gravitational ones are accounted for as well. The velocity of propagation is calculated applying a Lagrangian description of the problem.

A variational approach to magneto-elastic buckling problems for systems of ferromagnetic or superconducting beams

Based upon a variational principle derived in a preceding paper, expressions for the magneto-elastic buckling values for ferromagnetic or superconducting systems are given. These relations are evaluated for systems of slender beams. Explicit buckling values are calculated for a single ferromagnetic or superconducting beam of arbitrary cross-section, and for systems of two parallel ferromagnetic or superconducting rods. In the analysis needed for the calculation of the intermediate (i.e., rigid-body) and the perturbed magnetic fields, an intensive use of methods inherent in the theory of complex functions is made. In conclusion our results for a set of two superconducting rods are compared with the results of a mathematically less complicated, but also less rigorous, theory.