High Reynolds Channel Flows: Variable curvature

Two-dimensional laminar flow, at high Reynolds number Re, of an incompressible Newtonian fluid in a curved channel connected to 2 fitting tangent straight channels at its upstream and downstream extremities is considered. The Successive Complementary Expansion Method (SCEM) is adopted. This method leads to an asymptotic reduced model called Global Interactive Boundary Layer (GIBL) which gives a uniformly valid approximate solution of the flow field in the whole domain. To explore the effect of the variable curvature on the flow field, the bend has an elliptical median line. The validity of the proposed GIBL model is confronted to the numerical solution of complete Navier- Stokes equations. This comparison includes the wall shear stress which is a very sensitive measure of the flow field. The GIBL results match very well the complete Navier-stokes results for curvatures $K_{max}$ up to 0.4, curvature variations $\vert K'_{max}\vert$ up to 0.7 and eccentricities $e$ up to $\simeq 0.943$ in the whole geometrical domain. The upstream and downstream effects as well as the impact of the curvature discontinuities and the behaviour in the entire bend are well captured by the GIBL model

High Reynolds Channel Flows: Upstream interaction of various wall deformations

The flow at high Reynolds number in the entrance of distorted channel is considered. We analyse the anticipated fluid responds to a downstream wall distortion, and we find that the non linear upstream length \Delta={\mbox{O}}(R_e^{1/7}), using either a new asymptotic approach called Successive Complementary Expansions Method (SCEM) with generalized asymptotic expansions and a modal analysis of the perturbed flow. Comparisons with Navier-Stokes solutions show that the mathematical model is well founded

Uniformly Valid Asymptotic Flow Analysis in Curved Channels

The laminar incompressible flow in a two-dimensional curved channel having at its upstream and downstream extremities two tangent straight channels is considered. A global interactive boundary layer (GIBL) model is developed using the approach of the successive complementary expansions method (SCEM) which is based on generalized asymptotic expansions leading to a uniformly valid approximation. The GIBL model is valid when the non dimensional number μ = δmath is O(1) and gives predictions in agreement with numerical Navier-Stokes solutions for Reynolds numbers Re ranging from 1 to 10 puissance 4 and for constant curvatures δ = math ranging from 0.1 to 1, where H is the channel width and Rc the curvature radius. The asymptotic analysis shows that μ, which is the ratio between the curvature and the thickness of the boundary layer of any perturbation to the Poiseuille flow, is a key parameter upon which depends the accuracy of the GIBL model. The upstream influence length is found asymptotically and numerically to be O(math)

Curved channel flow with stenoses and aneurysms

Curvature is everywhere, in man or nature made devices. Its implication on physiological flows may be as important as other inertial or viscous effects. Relatively to an otherwise straight vessel, the so called centrifugal forces induce secondary flow that modify the whole flow structure and give rise to Dean vortices. Since the pioneering work of Dean many fundamental and applied investigations were performed. More specifically, in the blood dynamics studies, experimental and numerical simulations were also achieved. On the other hand many studies dealt with stenoses or aneurysms mainly situated in otherwise straight vessels. Thus the coupling between the global curvature effects and local section variations due to stenoses or aneurysms is not enough investigated. In the present work we will quantify how the impact of a stenosis or an aneurysm is modified when it occurs in a curved vessel compared to when it is located in a straight environment

Simple Patient-Based Transmantle Pressure and Shear Estimate From Cine Phase-Contrast MRI in Cerebral Aqueduct

From measurements of the oscillating flux of the cerebrospinal fluid (CSF) in the aqueduct of Sylvius, we elaborate a patient-based methodology for transmantle pressure (TRP) and shear evaluation. High-resolution anatomical magnetic resonance imaging first permits a precise 3-D anatomical digitalized reconstruction of the Sylvius’s aqueduct shape. From this, a very fast approximate numerical flow computation, nevertheless consistent with analytical predictions, is developed. Our approach includes the main contributions of inertial effects coming from the pulsatile flow and curvature effects associated with the aqueduct bending. Integrating the pressure along the aqueduct longitudinal center-line enables the total dynamic hydraulic admittances of the aqueduct to be evaluated, which is the pre-eminent contribution to the CSF pressure difference between the lateral ventricles and the subarachnoidal spaces also called the TRP. The application of the method to 20 healthy human patients validates the hypothesis of the proposed approach and provides a first database for normal aqueduct CSF flow. Finally, the implications of our results for modeling and evaluating intracranial cerebral pressure are discussed

A one-dimensional model of wave propagation within the co-axial viscous fluid filled spinal cavity

One–Dimensional models have been used to simulate pulse waves propagation in the spinal cavity and the interactions between CSF, blood and the spinal cord

Wave propagation into the spinal cavity: a 1d model with coaxial compliant tubes

One–Dimensional models have been used to simulate pulse waves propagation in the spinal cavity and the interactions between CSF, blood and the spinal cord. Some adopted compliant coaxial configurations but neglected the fluid's viscosity [1, 2] while others took into account CSF viscosity but simplified the cavity as one equivalent distensible tube [3]. Previous studies in the inviscid coaxial configuration have shown that the confinement reduces the wave propagation speed of the compliant part by a factor equal to the square root of the area parameter, i.e. the ratio of the tubes cross-sectional areas, when the dura is considered rigid

A coaxial coupled model of cerebral flows: blood and cerebrospinal fluid

International audienceThis study aimed to develop a one dimensional (1D) model to simulate the Cerebrospinal Fluid (CSF) flows in the cerebral sub-arachnoid spaces, and its coupling with the entire cerebral blood flow vascular network. The model consist in a network of coaxial tubes: the interior network represents the cerebral vasculature from the carotid and vertebral arteries to the sinuses and jugular veins (Zagzoule, 1986), and the coaxial exterior tubes the sub arachnoid spaces where the CSF flows. By integrating the mass and momentum flow conservation equations over the tubes cross-sections, we obtain a 1D coupled coaxial model of the blood and CSF flows. Our model takes into account the viscosity of the fluids (Cathalifaud, 2015), and assumes compliant boundary conditions for the coaxial compartment. Given the input pressure signal at the carotid and vertebral arteries, we therefore obtained an induced CSF flow, as shown in Figure 1. Results depends on the confinement of the coaxial compartment and the compliances of the boundary conditions, and well compared to measured CSF flows of the literature (between 2 and 5 cm3/s). We also investigate the coupling effect of the CSF on the blood flows, especially on the cerebral autoregulation characteristic time. We show that it strongly depends on the confinement of the coaxial compartment

Biomechanical modeling and Magnetic Resonance Imaging to investigate CSF flow physiology

Nowadays Phase contrast Magnetic Resonance Imaging (PC-MRI) is the only sensor able to measure in physiological conditions blood and cerebrospinal fluids (CSF) flow dynamics during cardiac cycle at different levels of the craniospinal system [1]. Combining modelling and in Vivo PC-MRI measurments of CSF Flow at the cervical Spinal (Qs) and Aqueduct of Sylvius (Qv) or Sub-Arachnoidal cerebral (Qc) levels makes it possible to compute Intracranial pressure (ICP) as well as get a deep insight into the LCS dynamical system. Inertia has been neglected in most of previous LCS models. Our simple model show that inertia plays a crucial role in particular in the optimal LCS flow amplitudes and phases

A coaxial coupled model of cerebral flows: blood and cerebrospinal fluid

This study aimed to develop a one dimensional (1D) model to simulate the Cerebrospinal Fluid (CSF) flows in the cerebral sub-arachnoid spaces, and its coupling with the entire cerebral blood flow vascular network. The model consist in a network of coaxial tubes: the interior network represents the cerebral vasculature from the carotid and vertebral arteries to the sinuses and jugular veins (Zagzoule, 1986), and the coaxial exterior tubes the sub arachnoid spaces where the CSF flows. By integrating the mass and momentum flow conservation equations over the tubes cross-sections, we obtain a 1D coupled coaxial model of the blood and CSF flows. Our model takes into account the viscosity of the fluids (Cathalifaud, 2015), and assumes compliant boundary conditions for the coaxial compartment. Given the input pressure signal at the carotid and vertebral arteries, we therefore obtained an induced CSF flow, as shown in Figure 1. Results depends on the confinement of the coaxial compartment and the compliances of the boundary conditions, and well compared to measured CSF flows of the literature (between 2 and 5 cm3/s). We also investigate the coupling effect of the CSF on the blood flows, especially on the cerebral autoregulation characteristic time. We show that it strongly depends on the confinement of the coaxial compartment