Bernoulli Pressure Research Articles

Curved-pipe flow with boundary conditions involving Bernoulli pressure

In this article, we study the steady-state flow of the incompressible viscous fluid through a thin distorted pipe with an arbitrary central curve. We prescribe the inflow and outflow boundary conditions involving the Bernoulli pressure with a given pressure drop. Using the multiscale expansion technique with respect to the pipe's thickness, we construct the higher-order asymptotic approximation of the flow given by the explicit formulae for the velocity and pressure. We also perform a detailed error analysis justifying the usage of the proposed solution and indicating its order of accuracy. For more information see https://ejde.math.txstate.edu/Volumes/2024/63/abstr.html

Homogenization of the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in a perforated pipe

The steady motion of a viscous incompressible fluid in a pipe (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems, without any restriction on the magnitude of the data, as the size of the perforations goes to zero and show that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler equations.

A new class of regularity criteria for the MHD and Navier–Stokes equations

In Skalak (0000) we studied a new class of the regularity criteria for the Navier–Stokes equations based on a remarkable idea that controlling the derivatives of some fundamental quantities like the pressure and the velocity along the streamlines yields the regularity of the weak solutions. We show in the present paper that the results from Skalak (0000) are extendable to the MHD equations and in the framework of the Besov spaces. For example, controlling the directional derivative of the magnetic Bernoulli pressure P along w+/|w+|, where w+=u+b and u and b denote the velocity field and the magnetic field, respectively, yields the regularity. We compare our results with the similar criteria from the literature.

Navier–Stokes equations: regularity criteria in terms of the derivatives of several fundamental quantities along the streamlines—the case of a bounded domain

In this paper we deal with the conditional regularity of the weak solutions of the Navier–Stokes equations on a bounded domain endowed with Navier boundary conditions, Navier-type boundary conditions or Dirichlet boundary conditions. We prove the regularity criteria which are based on the directional derivatives of several fundamental quantities along the streamlines, namely the velocity magnitude, the kinetic energy, the pressure, the velocity field and the Bernoulli pressure. In striking contrast to the known criteria in which the mentioned quantities were differentiated along a fixed vector, our criteria are mostly optimal for the whole range of parameters and have a clear physical meaning.

Improved regularity criteria for the MHD equations in terms of pressure using an Orlicz norm

We present new regularity criteria in terms of the negative part of the pressure p or the positive part of the extended Bernoulli pressure B≔p+12|u|2+12|b|2, where u is the velocity, and b is the magnetic field. The criteria extend the previously known results, and the extension is enabled by the use of an appropriate Orlicz norm.

Analysis and computation of a pressure-robust method for the rotation form of the incompressible Navier–Stokes equations with high-order finite elements

In this work, we develop a high-order pressure-robust method for the rotation form of the incompressible Navier–Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H(div)-conforming velocity reconstruction operator. In order to match the rotation form and ease error analysis, a skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proved to achieve the pressure-independent velocity errors. Optimal convergence rates of H1, L2-error for the velocity and L2-error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and remarkable performance of the proposed method.

Steady-State Navier–Stokes Equations in Thin Tube Structure with the Bernoulli Pressure Inflow Boundary Conditions: Asymptotic Analysis

Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.

Error analysis for a vorticity/Bernoulli pressure formulation for the Oseen equations

Abstract A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L2-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Furthermore, an a posteriori error estimator is designed and its robustness and efficiency are studied using weighted norms. Finally, a set of numerical examples in 2D and 3D is given, where the error indicator serves to guide adaptive mesh refinement. These tests illustrate the behaviour of the new formulation in typical flow conditions, and also confirm the theoretical findings.

A novel solution method for unsteady incompressible Euler flow using the vorticity–Bernoulli-pressure formulation

A novel numerical formulation and time-integration method for the incompressible Euler flow equations is presented using the vorticity–Bernoulli-pressure system. Lagrangian advection of vorticity-carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex-dominated flows and allow larger time steps than explicit Eulerian schemes. Solving only one scalar Poisson equation per time step instead of a vector-valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids. The equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity. The velocity is eliminated as an unknown in the continuity equation, which is efficiently solved for the Bernoulli pressure using the predicted vorticity field obtained by Lagrangian advection. The Bernoulli pressure is efficiently obtained from the Poisson equation in finite volume formulation. An estimated velocity field is calculated from the equation of motion. To effectively counterbalance the discretization errors made by the prediction, in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a Lagrangian framework as well. The velocity field is corrected at the end of the time step. Two-dimensional cases on Cartesian grids are considered. Numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second-order spatial accuracy of the solution. The new scheme produces significantly less artificial vortex distortion than the classical vortex method. The favorable properties of the solutions with respect to accuracy and long-time stability of the overall kinetic energy and enstrophy are also emphasized.

Solvability in a finite pipe of steady-state Navier–Stokes equations with boundary conditions involving Bernoulli pressure

Most methods of numerical simulation require a truncation of an infinite domain to a bounded one, thereby introducing artificial boundaries. We prove existence of weak solutions to the stationary Navier–Stokes equations, simulating the steady flow of a viscous fluid through the pipe. We consider the case when at inflow and outflow boundaries conditions involving the Bernoulli pressure are prescribed and study the problems either with given flow rate of the fluid or the given pressure drop. In both cases we prove existence of a weak solution without any restriction on the data.

Intraglottal Aerodynamics at Vocal Fold Vibration Onset

The most frequently observed type of voice onset in spontaneous speech in normal subjects is the soft onset, and it may be considered as the "physiological" onset. It starts from an immobile narrow glottal slit crossed by a continuous airflow, and then a few oscillations (even a single one in some cases) precede the first glottal closure. It is a transient event, during which the acting forces, lung pressure, intraglottal pressure, myoelastic tension of the vocal fold (VF) oscillator and inertance of the supraglottal vocal tract, interact to progressively reach the steady state of a sustained oscillation. Combined measurements of flow, area, and pressure provide a detailed qualitative and quantitative analysis of the intraglottal mechanical events at the precise moment of starting oscillation in a physiological (soft or soft/breathy) onset. Our in vivo measurements of airflow and glottal area show that the very first oscillation occurs exactly at the time when turbulence appears at the level of the glottal narrowing, ie, when the Reynolds number reaches its critical value. The turbulence may be assumed to trigger an oscillator consisting in the ensemble of the VFs and the air of the vocal tract, which is known to be weakly damped. Turbulence can act here as an aspecific flick, triggering the oscillator, the frequency of oscillation being determined by its mechanical properties. Furthermore, the first noticeable glottal oscillations are sinusoidal: the VFs are neither steeply sucked together by a negative Bernoulli pressure, nor burst apart by the lung pressure. Our measurements show that, at the critical time, the rising positive lung pressure is balanced by the rising negative Bernoulli pressure generated by the transglottal flow.

Analysis and Approximation of a Vorticity\u2013Velocity\u2013Pressure Formulation for the Oseen Equations

We introduce a family of mixed methods and discontinuous Galerkin discretisations designed to numerically solve the Oseen equations written in terms of velocity, vorticity, and Bernoulli pressure. The unique solvability of the continuous problem is addressed by invoking a global inf-sup property in an adequate abstract setting for non-symmetric systems. The proposed finite element schemes, which produce exactly divergence-free discrete velocities, are shown to be well-defined and optimal convergence rates are derived in suitable norms. This mixed finite element method is also pressure-robust. In addition, we establish optimal rates of convergence for a class of discontinuous Galerkin schemes, which employ stabilisation. A set of numerical examples serves to illustrate salient features of these methods.

Limitations in the Assessment of Prosthesis-Patient Mismatch.

Prosthesis-patient mismatch (PPM) after aortic valve replacement (AVR) may affect survival but data are conflicting. It is assessed by relating effective orifice area (EOA) to body surface area (EOAi). EOA is patient-specific as the result of flow-velocity times area at the individual patient's outflow tract levels (LVOTA) divided by trans-prosthetic flow velocity. However, some studies use projected EOAs (i.e., valve size associated EOAs from other patient populations) to assess how PPM affects outcome. We analyzed 76 studies addressing hemodynamic outcome and/or mortality after bioprosthetic AVR. In 48 studies, projected or measured EOA for calculation of EOAi and PPM assessment was used (of which 25 demonstrated an effect on survival). We identified 28 additional studies providing measured EOA values and the corresponding Bernoulli's pressure gradients after AVR. Despite EOA being a patient-specific parameter, 77% of studies assessing a PPM impact on survival used projected EOAs. The 28 studies are providing measured EOA values and the corresponding Bernoulli's pressure gradients in patients after AVR showed a highly significant, linear relationship between EOA and Bernoulli's gradient. Considering this relationship, it is surprising that relating EOA to body surface area (BSA) (EOAi) is standard but relating pressure gradients to BSA is not. We conclude that the majority of studies assessing PPM have used false assumptions because EOA is a patient-specific parameter and cannot be transferred to other patients. In addition, the use of EOAi to assess PPM may not be appropriate and could explain the inconsistent relation between PPM and survival in previous studies.

Modified Einstein and Navier–Stokes Equations

The appearance of inertial rest mass-energy is associated with the kinematic slowing-down of time and with the vortex state of the elementary massive space with zero integral of its kinetic and potential energies. An analog of the Einstein equation is found for moving densities of a non-empty metric space in the concept of the Einstein–Infeld material field. The vector consequences of this tensor equation for a metric medium of overlapping elementary carriers of continuous mass-energies allow us to modify the Navier–Stokes equation under inertial motion of the matter of the nonlocal field in the nonrelativistic limit. The nonlocality of massenergy generates kinematic accelerations of feedback to Newtonian acceleration, which impedes asymptotic divergence of energy fluxes. Stabilization of inertial media by dynamic Bernoulli pressure corresponds to nonlocal self-organization of Einstein–Infeld non-empty space and invalidates Newtonian localization of masses in empty space.

A Contribution to the Theory of Regularity of a Weak Solution to the Navier–Stokes Equations via One Component of Velocity and Other Related Quantities

We deal with a suitable weak solution $$(\mathbf {v},p)$$ to the Navier–Stokes equations in $$\Omega \times (0,T)$$ , where $$\Omega $$ is a domain in $${\mathbb R}^3$$ , $$T>0$$ and $$\mathbf {v}=(v_1,v_2,v_3)$$ . We show that the regularity of $$(\mathbf {v},p)$$ at a point $$(\mathbf {x}_0, t_0)\in \Omega \times (0,T)$$ is essentially determined by the Serrin-type integrability of the positive part of a certain linear combination of $$v_1^2,\, v_2^2,\, v_3^2$$ and p in a backward neighborhood of $$(\mathbf {x}_0,t_0)$$ . An appropriate choice of the coefficients in the linear combination leads to the Serrin-type condition on the positive part of the Bernoulli pressure $$\frac{1}{2}|\mathbf {v}|^2+p$$ or the negative part of p (Theorem 1), or one component of $$\mathbf {v}$$ (Theorem 2), etc.

Study of Siphon Breaker Experiment and Simulation for a Research Reactor.

Under the design conditions of a research reactor, the siphon phenomenon induced by pipe rupture can cause continuous outward flow of water. To prevent this outflow, a control device is required. A siphon breaker is a type of safety device that can be utilized to control the loss of coolant water effectively. To analyze the characteristics of siphon breaking, a real-scale experiment was conducted. From the results of the experiment, it was found that there are several design factors that affect the siphon breaking phenomenon. Therefore, there is a need to develop a theoretical model capable of predicting and analyzing the siphon breaking phenomenon under various design conditions. Using the experimental data, it was possible to formulate a theoretical model that accurately predicts the progress and the result of the siphon breaking phenomenon. The established theoretical model is based on fluid mechanics and incorporates the Chisholm model to analyze two-phase flow. From Bernoulli's equation, the velocity, quantity, undershooting height, water level, pressure, friction coefficient, and factors related to the two-phase flow could be obtained or calculated. Moreover, to utilize the model established in this study, a siphon breaker analysis and design program was developed. The simulation program operates on the theoretical model basis and returns the result as a graph. The user can confirm the possibility of the siphon breaking by checking the shape of the graph. Furthermore, saving the entire simulation result is possible and it can be used as a resource for analyzing the real siphon breaking system. In conclusion, the user can confirm the status of the siphon breaking and design the siphon breaker system using the program developed in this study.

URINE FLOW DYNAMICS THROUGH THE URETHRA IN PATIENTS WITH BLADDER OUTLET OBSTRUCTION

Voiding dysfunction is a common disease among the elderly male population. However, few studies have elucidated the flow dynamics of the disease mechanism. We attempted to clarify the fluid dynamics of urine in the prostatic urethra (PU) in patients with bladder outlet obstruction (BOO), a common disease associated with voiding dysfunction. A model of normal PU and three phase models following the BOO progression were prepared using 3D CAD software tracing cystourethroscopic images. In fluid dynamics computation, the normal detrusor pressure was applied to the bladder side in each model. Results of fluid simulation were compared for flow trajectory line, fluid energy dynamics using Bernoulli's principle and pressure loss coefficient. In the BOO model, a large vortex was observed adjacent to the bladder outlet (BO) with increased hydraulic energy loss during the passage through the tract. In contrast, opening of the BO eliminated the vortex with reduction of hydraulic energy loss. These results corresponded with clinical data measured by catheterized pressure sensors inserted into the urethra, and those of urethral diameter measured by urethrography. Fluid dynamics simulation using CAD models can clarify the mechanism of voiding dysfunction in a less invasive and more acceptable way in patients with BOO.

Evaluation of right and left ventricular diastolic filling.

A conceptual fluid-dynamics framework for diastolic filling is developed. The convective deceleration load (CDL) is identified as an important determinant of ventricular inflow during the E wave (A wave) upstroke. Convective deceleration occurs as blood moves from the inflow anulus through larger-area cross-sections toward the expanding walls. Chamber dilatation underlies previously unrecognized alterations in intraventricular flow dynamics. The larger the chamber, the larger becomes the endocardial surface and the CDL. CDL magnitude affects strongly the attainable E wave (A wave) peak. This underlies the concept of diastolic ventriculoannular disproportion. Large vortices, whose strength decreases with chamber dilatation, ensue after the E wave peak and impound inflow kinetic energy, averting an inflow-impeding, convective Bernoulli pressure rise. This reduces the CDL by a variable extent depending on vortical intensity. Accordingly, the filling vortex facilitates filling to varying degrees, depending on chamber volume. The new framework provides stimulus for functional genomics research, aimed at new insights into ventricular remodeling.

Criteria for thermally induced atomization and catastrophic breakup of acoustically levitated droplet

We report thermally induced instability leading to catastrophic breakup in acoustically levitated vaporizing fuel droplets. Change in surface tension and viscosity with increase in droplet temperature causes wide fluctuations in droplet aspect ratio. If the viscous damping of aspect ratio oscillation is not strong enough, the droplet goes through unbounded stretching. If the droplet exceeds a critical Weber number locally, a bag type and capillary wave induced atomization can occur, which leads to catastrophic breakup. A stability criterion has been established based on the inhomogeneity of Bernoulli (acoustic) pressure and surface tension of the droplet in terms of a local Weber number and Ohnesorge number. This instability is thermally induced in a droplet which does not experience instabilities without heating.

Underwater bubble pinch-off: Transient stretching flow

At the point of pinch-off of an underwater air bubble, the speed of water rushing in diverges. Previous studies that assumed radial flow throughout showed that the local axial shape is two smoothly connected, slender cones that transition very slowly (logarithmically) to a cylindrical segment. Our simulations show that even with initially radial flow, a transient vertical flow develops with comparable speeds. Bernoulli pressure draws water into the singularity region while incompressibility forces it away from the neck minimum, generating significant vertical flows that rapidly slenderize and symmetrize the collapse region. This transition is due to a different mechanism, occurring much faster than previously expected. Vertical flows dictate the neck shape evolution.