Sunčica Čanić

A proof of existence of perturbed steady transonic shocks via a free boundary problem

We prove the existence of a solution of a free boundary problem for the transonic small-disturbance equation. The free boundary is the position of a transonic shock dividing two regions of smooth flow. Assuming inviscid, irrotational flow, as modeled by the transonic small-disturbance equation, the equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. To study the stability of a uniform planar transonic shock, we consider perturbation by a steady C1+ϵ upstream disturbance. If the upstream disturbance is small in a C1 sense, then there is a steady solution in which the downstream flow and the transonic shock are Holder-continuous perturbations of the uniform configuration. This result provides a new use of inviscid perturbation techniques to demonstrate, in two dimensions, the stability of transonic shock waves of the type that appear, for example, over the wing of an airplane, along an airfoil, or as bow shocks in a flow with a supersonic free-stream velocity.

Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics, and Experimental Validation

The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming cylindrical geometry, axially symmetric flow, and viscoelasticity of arterial walls. The aim was to develop a reduced model that would capture certain physical phenomena that have been neglected in the derivation of the standard axially symmetric one-dimensional models, while at the same time keeping the numerical simulations fast and simple, utilizing one-dimensional algorithms. The viscous Navier–Stokes equations were used to describe the flow and the linearly viscoelastic membrane equations to model the mechanical properties of arterial walls. Using asymptotic and homogenization theory, a novel closed, “one-and-a-half dimensional” model was obtained. In contrast with the standard one-dimensional model, the new model captures: (1) the viscous dissipation of the fluid, (2) the viscoelastic nature of the blood flow – vessel wall interaction, (3) the hysteresis loop in the viscoelastic arterial walls dynamics, and (4) two-dimensional flow effects to the leading-order accuracy. A numerical solver based on the 1D-Finite Element Method was developed and the numerical simulations were compared with the ultrasound imaging and Doppler flow loop measurements. Less than 3% of difference in the velocity and less than 1% of difference in the maximum diameter was detected, showing excellent agreement between the model and the experiment.

MODELING VISCOELASTIC BEHAVIOR OF ARTERIAL WALLS AND THEIR INTERACTION WITH PULSATILE BLOOD FLOW

A horizontal rotary table is provided and a mounting arm is supported above the table and generally parallels the latter. The arm extends along a path paralleling an axis of rotation of the table and an upright shaft is journalled from the mounting arm for rotation about an upright axis coinciding with the axis of rotation of the table. Drive structure drivingly connects the table and the upright shaft for rotation of the latter at twice the speed of rotation of the table and a support is mounted on the support arm for guided movement therealong. A scribe tool is carried by the support for engagement with and scribing a workpiece disposed on the table for rotation with the latter and motion converting and drive structure is operatively connected between the upright shaft and the support for effecting oscillation of the support along the arm responsive to and an in timed relation with rotation of the shaft. The motion converting and drive structure includes operational features which function to continuously vary the linear displacement rate of the support along the arm during constant angular velocity of the upright shaft.

Flows on networks: recent results and perspectives

The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books.

Effective Equations Modeling the Flow of a Viscous Incompressible Fluid through a Long Elastic Tube Arising in the Study of Blood Flow through Small Arteries

We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time-dependent pressure drop between the inlet and the outlet boundary. The pressure drop is assumed to be small, thereby introducing creeping flow in the tube. Stokes equations for incompressible viscous fluid are used to model the flow, and the equations of a curved, linearly elastic membrane are used to model the wall. Due to the creeping flow and to small displacements, the interface between the fluid and the lateral wall is linearized and supposed to be the initial configuration of the membrane. We study the dynamics of this coupled fluid-structure system in the limit when the ratio between the characteristic width and the characteristic length tends to zero. Using the asymptotic techniques typically used for the study of shells and plates, we obtain a set of Biot-type visco-elastic equations for the effective pressure and the effective displacements. The approximation is rigorously justified through a weak convergence result and through the error estimates for the solution of the effective equations modified by an outlet boundary layer. Applications of the model problem include blood flow in small arteries. We recover the well- known law of Laplace and obtain new improved models that hold in cases when the shear modulus of the vessel wall is not negligible and the Poisson ratio is arbitrary.

Free Boundary Problems for Nonlinear Wave Systems: Mach Stems for Interacting Shocks

We study a family of two-dimensional Riemann problems for compressible flow modeled by the nonlinear wave system. The initial constant states are separated by two jump discontinuities,

Self-Consistent Effective Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries

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Emerging Trends in Heart Valve Engineering: Part I. Solutions for Future

We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time-dependent pressure head difference. The Navier--Stokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a curved, linearly elastic membrane are used to model the wall. Employing the asymptotic techniques typically used in thin domains, we derive a set of effective equations that hold in medium-to-large compliant vessels for laminar flow regimes. The main novelty is the derivation of the effective equations that do not assume any ad hoc closure, typically assumed in the derivation of one-dimensional models. Using ideas from homogenization theory for porous media flows, we obtain a closed system of effective equations that are of Biot type with memory. Memory accounts for the wave-like phenomena in the problem. Although the equations are two-dimensional, their simple structure enables a design of a numerical algorithm that h...

Mixed hyperbolic-elliptic systems in self-similar flows

As the first section of a multi-part review series, this section provides an overview of the ongoing research and development aimed at fabricating novel heart valve replacements beyond what is currently available for patients. Here we discuss heart valve replacement options that involve a biological component or process for creation, either in vitro or in vivo (tissue-engineered heart valves), and heart valves that are fabricated from polymeric material that are considered permanent inert materials that may suffice for adults where growth is not required. Polymeric materials provide opportunities for cost-effective heart valves that can be more easily manufactured and can be easily integrated with artificial heart and ventricular assist device technologies. Tissue engineered heart valves show promise as a regenerative patient specific model that could be the future of all valve replacement. Because tissue-engineered heart valves depend on cells for their creation, understanding how cells sense and respond to chemical and physical stimuli in their microenvironment is critical and therefore, is also reviewed.

Riemann problems for the two-dimensional unsteady transonic small disturbance equation

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.