Parabolic Velocity Distribution Research Articles

Hydrodynamic stability of Poiseuille flow in a bidisperse porous medium with slip effect

This study focuses on examining the hydrodynamic stability of an incompressible fluid flowing through a bidisperse porous medium. Specifically, the impact of slip boundary conditions on instability is investigated. The study looks at a scenario in which the Darcy theory is used for micropores and the Brinkman theory is used for macropores. An incompressible fluid is located within an unbound channel with a constant pressure gradient along its length in the system under investigation. The fluid flows laminarly along the pressure gradient, resulting in a stable parabolic velocity distribution that does not alter over time. Based on our observations, it appears that increasing the values of the slip parameter, permeability ratio, porous parameter, interaction parameter and Darcy Reynolds number leads to an improvement in the stability of the system. The spectrum behavior of eigenvalues in the Orr–Sommerfeld problem for Poiseuille flow exhibits significant sensitivity and is influenced by multiple factors, encompassing both the mathematical attributes of the problem and the specific numerical techniques utilized for approximation.

Plane-Parallel Laminar Flow of Viscous Fluid in the Transition Zone of the Inlet Section

A study was conducted to analyze how hydrodynamic parameters change in the entrance region of plane-parallel flow under stationary flow conditions, with an initial arbitrary distribution of velocities in the entrance section. This study was based on boundary layer equations, and a boundary problem was formed under the conditions of plane-parallel flow. The boundary conditions were chosen to reflect the pattern of arbitrary velocity distribution in the entrance section. A general solution of the approximating Navier-Stokes equations is provided, which depends on the initial conditions and the Reynolds number. The boundary conditions are established based on the nature of the motion, and the boundary value problem is described. A method for integrating the boundary value problem has been developed, and regularities for the change in velocities along the length of the inlet section have been obtained for both constant and parabolic velocity distributions in the entrance sections. Analytical solutions have been derived to obtain patterns of velocity and pressure changes in any given flow direction. Through computer analysis, velocity change patterns in various sections along the inlet transition area have been constructed, allowing for the determination of fluid velocity at any point on the section and an estimation of the length of the transition area. These proposed solutions provide a framework for accurately constructing individual units of hydromechanical equipment.

Pattern identification of the non-stationary laminar flow of a viscous fluid in the round pipe inlet section

The study of the patterns of change in the hydrodynamic parameters under the conditions of non-stationary flow at the entry of the cylindrical pipe and the initial arbitrary distribution of velocities in the entry section was conducted based on the boundary layer equations. A boundary problem was formed under the axisymmetric change conditions in the flow. The boundary conditions were chosen in accordance with the pattern of an arbitrary distribution of velocities in the entry section. A general solution of the approximating Navier-Stokes equations is presented depending on the initial conditions and the Reynolds number. In accordance with the type of flow, the boundary conditions of the problem are established, and the boundary-value problem is formulated. Regularities for the change in velocities lengthwise in the entrance region have been obtained for a constant and parabolic velocity distribution in the inlet cross-sections. Analytical solutions have been obtained, allowing to obtain patterns of changes in velocities and pressures toward flow at any section and at any time. For the mentioned cases, the composite graphs of velocity changes in different sections along the length of the entrance transition area were constructed by computer analysis, for different time conditions. With the obtained composite graphs, the patterns of change over the entire length of the transition area of the entrance region were constructed, enabling to obtain fluid flow velocity at any point of the section. The length of the transition zone can be estimated based on the condition of reaching a certain percentage (99%) of the maximum velocity of the flow. The proposed solutions create the conditions for correctly constructing separate units of hydromechanical equipment

Method for estimating pulsatile wall shear stress from one‐dimensional velocity waveforms

Wall shear stress (WSS)-a key regulator of endothelial function-is commonly estimated in vivo using simplified mathematical models based on Poiseuille's flow, assuming a quasi-steady parabolic velocity distribution, despite evidence that more rapidly time-varying, pulsatile blood flow during each cardiac cycle modulates flow-mediated dilation (FMD) in large arteries of healthy subjects. More exact and accurate models based on the well-established Womersley solution for rapidly changing blood flow have not been adopted clinically, potentially because the Womersley solution relies on the local pressure gradient, which is difficult to measure non-invasively. We have developed an open-source method for automatic reconstruction of unsteady, Womersley-derived velocity profiles, and WSS in conduit arteries. The proposed method (available online at https://doi.org/10.5281/zenodo.7576408) requires only the time-averaged diameter of the vessel and time-varying velocity data available from non-invasive imaging such as Doppler ultrasound. Validation of the method with subject-specific computational fluid dynamics and application to synthetic velocity waveforms in the common carotid, brachial, and femoral arteries reveals that the Poiseuille solution underestimates peak WSS 38.5%-55.1% during the acceleration and deceleration phases of systole and underestimates or neglects retrograde WSS. Following evidence that oscillatory shear significantly augments vasodilator production, it is plausible that mischaracterization of the shear stimulus by assuming parabolic flow leads to systematic underestimates of important biological effects of time-varying blood velocity in conduit arteries.

Experimental and numerical investigations of the jetting regime

In this paper we highlight the first stages of liquid injection through a capillary nozzle prior to the jet formation regime. We emphasize the asymmetry of water-air interface at the beginning of the injection of liquid and the axial symmetry which follows when the jet is fully formed. Numerical simulations are performed in order to estimate the relaxation of the velocity distribution of the liquid as it leaves the capillary nozzle. The plug flow condition for low-viscosity liquid jets is investigated in relation with the parabolic velocity distribution in the nozzle. The numerical data is also used to characterize the flow during the drop formation and pinch-off, both driven by the action of surface tension.

Analytical study of the velocity of the lubricating fluid in the hydrodynamic journal bearing with the effect of centrifugal force for short bearing type

The paper aims to obtain the expression of the velocity of lubricating fluid in the hydrodynamic journal bearings by analytical method. In the classical short-bearing theory, the fluid flow was studied by ignoring the effect of centrifugal force of the lubricating fluid film. However, the self-oscillation of the shaft of high-power motors does not follow the rules in classical hydrodynamic lubrication theory. To explain this phenomenon, a modified form of Reynolds equation, in which the influence of centrifugal force of the lubricant is not ignored, is established. The study aims to establish the modified Reynolds equation by including the effect of centrifugal forces for the case of the short bearing type. Integration of the Navier-Stokes equations, yield the expressions for the components of velocity of the lubricating fluid in the gap. The oil’s pressure in the hydrodynamic journal bearing is obtained by solving the modified Reynolds equation. The numerical results are considered in the case of the stable equilibrium position of the motion of the shaft, i.e. the symmetry axis of the shaft does not move. The plots of the velocity components in the tangential axial direction are displayed too. The theoretical results of a parabolic velocity distribution similar to that of a Newtonian fluid, derived from a Bingham plastic flow model. The flow in the gap, which are obtained by analytical method, are completely consistent with its boundary conditions and its physical properties. Further, with these results, the paper as a lemma to solve the dynamics problem in floating ring bearing with the influence of the centrifugal force of the lubricant

Active control of particle position by boundary slip in inertial microfluidics

Inertial microfluidic is able to focus and separate particles in microchannels based on the characteristic geometry and intrinsic hydrodynamic effect. Yet, the vertical position of suspended particles in the microchannel cannot be manipulated in real time. In this study, we utilize the boundary slip effect to regulate the parabolic velocity distribution of fluid in the microchannel and present a scheme to active control the vertical position of particles in inertial microfluidics. The flow field of a microchannel with a unilateral slip boundary is equivalent to that of the microchannel widened by the relevant slip length, and the particle equilibrium positions in the two microchannels are consistent consequently. Then, we simulate the lateral migrations of three kinds of typical particles, namely circle, ellipse, and rectangle in the microchannel. Unlike the smooth trajectories of circular particles, the motions of the elliptical and rectangular particles are accompanied by regular fluctuations and non-uniform rotations due to their non-circular geometries. The results demonstrate that the unilateral slip boundary can effectively control the vertical equilibrium position of particles. Thus, the present scheme enables to active manipulate the particles positions in vertical direction and can promote more accurate focusing, separating, and transport in inertial microfluidics.

A STUDY ON SUDDEN EXPANSION HYDRODYNAMIC PHENOMENA OCCURRING IN CYLINDRICAL PIPES

Introduction: This paper studies the frequency with which hydrodynamic parameters change in the sudden expansion section of axisymmetric pressure flow, based on the boundary layer equations. Methods: The suggested method reveals the regularity of changes in the hydrodynamic parameters of the flow in the transitional area, making it possible to obtain a velocity profile in any cross-section under common initial and boundary conditions. Based on the general solutions, we studied the hydrodynamic processes occurring in the transitional area of the effective sudden cross-section expansion within the axisymmetric pressure movement, in the following cases: a) when the velocity is constant at any point of the inlet face; b) when the velocity is distributed along the inlet face according to the parabolic law. Our calculations were carried out for different values of the expansion factor. Results: Based on the results of the computer-aided experimental study, we obtained velocity diagrams along the length of the transitional area with constant and parabolic velocity distributions for fluid inflowing into the expanded section. We also determined the patterns of pressure distribution along the length of the relevant section.

Study of dynamic characteristics of flow in plane-parallel and taper clearances of drives of electric power systems

The hydrodynamic parameters of the flow of a viscous working fluid in plane-parallel and conical gaps in drives of electric power systems are investigated. A general expression is found for determining the regularity of the change in the flow rate in the gaps of non-contact seals, and special cases are considered. In the case of frictional movement of one of the walls, the Couette flow is observed in the gap. When the pressure is applied to the liquid in the gap, a parabolic velocity distribution is observed with a maximum in the middle of the channel. Under the joint action of the frictional movement of the wall and the pressure flow in the channel, stratification of fluid movement in opposite directions is observed. Frictional flow in conical slots, when the fluid moves in only one direction, is possible at a taper value of -0,5...1,0. In all other cases, the flow in the slot will have two opposite flow directions. The presence of two opposite flows inside the slot can also be for the case when the direction of movement of the movable wall coincides with the direction of the pressure flow. This is caused by an increase in pressure within the gap. A general solution has been found for the pressure force in the gaps, which tends to push the walls apart, i.e. open the sealing surfaces. The analysis of the pressure force is carried out for various cases of the slotted gap and under various effects of the frictional flow and pressure head.

Effects of Operating and Structural Parameters on Removal of Nitric Oxide by Oxidation in a Ceramic Hollow Fiber Membrane Contactor

A numerical study on the oxidation removal of nitric oxide in a ceramic hollow fiber membrane contactor was performed. To represent the transport and absorption process, the model was created by combining multiphase, species, reaction, and porous models. The numerical results were verified by comparing them with experimental data. The tube and lumen sides both have laminar parabolic velocity distributions. The nitric oxide concentration decreases gradually from the membrane wall to axis at the cross-section except on the inner and outer sides of the membrane tube. The equivalent diffusion length was proved useful for evaluating the entrance effect. At low concentrations, the reduction efficiency was proportional to the absorbent concentration, and at large concentrations, it neared a maximum value. The reduction efficiency was positively affected by elevated operating temperature and pressure. With a gas channel width of 13 mm, the reduction flow rate achieves its maximum. The efficiency of NO reduction per area decreases as the effective membrane length increases. Increasing the operating temperature and membrane length are recommended as design priorities due to high relative enhancements. It is not recommended to improve reduction efficiency by increasing membrane tube diameter and operating pressure in design. Changing the gas flow rate, absorbent concentration and gas channel width are moderate recommended as well.

Experimental analysis of the hydrodynamic performance of an industrial-scale cross-flow sieve tray

This work reports the hydrodynamic description of a large-scale sieve tray equipped in an air/water column mockup. The test tray accommodated a novel multi-probe flow profiler for simultaneous conductivity measurements. For the first time, 3D liquid holdup distribution, liquid residence time distribution and effective froth height distribution were obtained at very high spatiotemporal resolution for several gas and liquid loadings. The liquid flow and mixing patterns were visualized via tracer-based experiments. The methodologies used for acquiring these distributions are thoroughly discussed in this work. Diligent examination of the processed experimental data revealed the hydrodynamic characteristics of an operational sieve tray for the studied loadings. Uniform effective froth height persisted over the majority of the tray, whereas homogenous liquid holdup distribution was observed at multiple elevations above the deck with the highest holdup occurring near the average effective froth height. Based on tracer experiments, axisymmetric liquid flow and mixing patterns were detected with parabolic velocity distribution near the inlet. Overall, the liquid velocities over the remaining tray deck were nearly uniform for the studied loadings.

An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation

The rightward representation of the Barakat-Clark ADE scheme is extended for the solution of the Multi-dimensional Transport equation. The first-order derivative of the Transport equation is represented by a one-sided multi-level finite-difference. The resulting scheme is an explicit marching type iterative solution. The visibility of using this method for the solution of the Multi-dimensional Transport equations is demonstrated through the solution of each of the Burgers’ equation and the Graetz-Nusselt problem for the thermally developing flow between two-parallel plates at constant temperature (at high and low Peclet numbers) with parabolic velocity distribution. The results are compared with the solutions using other schemes. All of the obtained results are compared with the exact solutions of the analyzed problems. The results show that the proposed scheme is unconditionally stable, better accuracy, faster convergence, and lower storage capacity requirement.

Inferior vena cava revisited - Real-time flow MRI of respiratory maneuvers.

Recent MRI studies of blood flow in the inferior vena cava (IVC) resulted in findings which are inconsistent with earlier observations by invasive procedures - most likely because ECG-gated MRI techniques are unable to resolve dynamic adjustments due to respiration. The purpose of this work was to apply real-time phase-contrast MRI at 50 ms resolution to re-evaluate IVC flow in response to normal and deep breathing as well as breath holding and Valsalva maneuver (11 young healthy subjects). Real-time flow MRI relied on highly undersampled radial gradient-echo sequences and a model-based nonlinear inverse reconstruction. A frequency analysis of the predominant pulsatility classified IVC flow in individual subjects as "cardiac", "respiratory" or "mixed" type. Peak flow velocities during free breathing ranged from 30 to 58 cm s-1 , while flow rates varied from 15 to 37 ml s-1 . The subject-specific IVC flow pattern persists during deep breathing although the enhanced respiratory influence may shift subjects form "cardiac" to "mixed" or from "mixed" to "respiratory" type. Peak velocities increased relative to normal breathing but led to similar flow rates of 16 to 34 ml s-1 . Inspiration during deep breathing elicited brief periods of flow reversal in all subjects with mean peak velocities of -21 cm s-1 . The observation of only mildly flattened parabolic velocity distributions within the IVC indicated mostly laminar flow. Breath holding reduced blood flow velocities and rates by more than 40% on average, while Valsalva maneuvers completely abolished venous return. In conclusion, IVC blood flow is dominated by the acquired respiratory behavior of individual subjects and its pressure-induced alterations relative to cardiac pulsation. The responses to breath holding and Valsalva maneuver are in full agreement with previous invasive observations of reduced or even ceased flow, respectively.

Lateral Migration and Nonuniform Rotation of Square Particle Suspended in Poiseuille Flow

A square particle suspended in a Poiseuille flow is investigated by using the lattice Boltzmann method with the Galilean-invariant momentum exchange method. The lateral migration of Segré-Silberberg effect is observed for the square particle, accompanied by the nonuniform rotation and regular wave. To compare with the circular particle, its circumscribed and inscribed squares are used in the simulations. Because the circumscribed square takes up a greater difference between the upper and lower flow rates, it reaches the equilibrium position earlier than the inscribed one. The trajectories of the latter are much closer to those of circle; this indicates that the circle and its inscribed square have a similar hydrodynamic radius in a Poiseuille flow. The equilibrium positions of the square particles change with Reynolds number and show a shape of saddle, whereas those of the circular particles are virtually not affected by Reynolds number. The regular wave and nonuniform rotation are owing to the interactions of the square shape and the parabolic velocity distribution of Poiseuille flow, and high Reynolds number makes the square rotating faster and decrease its oscillating amplitude. A series of contours illustrate the dynamic flow fields when the square particle has successive postures in a half rotating period. This study is beneficial to understand the motion of anisotropic particles and the dendrite growth in dynamic environment.

Axisymmetric free jet of Re → 0 from a round tube

An axisymmetric free jet of Reynolds number Re → 0 from a round tube is investigated in a theoretical way in order to elucidate pressure and velocity fields in the downstream of the tube exit. The Stokes approximation is performed to the Navier–Stokes equations expressed by the spherical coordinate system. The whole flow field is divided into two regions, a closed region consisting of both the tube exit and the hemisphere with the tube radius, i.e. inner region, and a region outside of the hemisphere, i.e. outer region. The pressure and velocity components in the inner region are expanded by a series of unknown functions of the angle θ, multiplied by powers of the radial distance r, whilst those in the outer region by a series of unknown functions of θ, multiplied by inverse powers of r, both of which satisfy the respective boundary conditions at the tube exit and infinity. The inner-region and outer-region pressures are patched together well at an arbitrary point over the hemisphere common to both regions, and the velocity components are done at the same point likewise. The parabolic velocity distribution at the exit is derived from the velocity in the inner region. The jet-ejection pressure and the pressure gradient coefficient are related by one parameter σ and the role of the parameter in the patching is clarified. Stream lines show a typical flow characteristic of the free jet flow for Re → 0 in the outer region.

Comprehensive investigation of the flow in a narrow gap between co-rotating disks

Abstract The inward flow between two parallel, co-rotating disks undergoes a thorough examination by analytical, experimental and numerical means. The analytical approach utilizes the asymptotical truncated series solution provided by Batista (2011) and extends it by a correction for an arbitrary mean tangential velocity at the rotor inlet. Taylor series expansions of the analytical results provide an estimate for the orders of magnitude of velocity components and the polynomial order of their profile shapes. The common assumption of parabolic velocity distributions is only appropriate in the radial direction. In parallel, a unique test rig provides the experimental counterpart of the velocity profiles inside a rotor gap, that is suitably narrow for turbomachinery applications. The optical flow measurements are based on a novel calibration technique and volumetric particle tracking evaluation. Both laminar and turbulent operating conditions are examined. Finally, numerical studies using commercial CFD software provide insight into the flow field inside the test rig rotor where experimental methods fall short and provide an additional means to investigate the effects of the approximations made in the derivation of the analytical results. The velocity distributions acquired by analytical, numerical and experimental means agree well, the asymptotical nature of the analytical solution by Batista (2011) can be observed. The comparison of experimental and numerical results of a turbulent case suggests that the Shear Stress Transport turbulence model reproduces turbulent flow inside the rotor gap appropriately.

Oscillating dispersed-phase co-flow microfluidic droplet generation: Multi-droplet size effect.

Controllable generation of microdroplets at desired sizes and throughputs is important in many applications. Many biological assays require size-optimized droplets for effective encapsulation of analytes and reagents. To perform size optimization, different-size droplets must be generated from identical sources of samples to prevent potential cross-sample variations or other sources of error. In this paper, we introduce a novel alteration of the co-flow droplet generation technique to achieve multi-size generation of monodispersed droplets. Using a custom-made mechanism, we oscillate the disperse-phase (d-phase) flow nozzle perpendicular to the continuous phase (c-phase) flow in a co-flow channel. Oscillation of the d-phase nozzle introduces an additional lateral drag force to the growing droplets while exposing them to various levels of axial drag owing to the parabolic velocity distribution of the c-phase flow. Superimposing both effects results in simultaneous and repeatable generation of monodispersed droplets with different sizes. The effect of nozzle oscillation frequency (f = 0-15 Hz) on droplet generation at different d-phase (Qd = 0.05, 0.10, and 0.50 ml/min) and c-phase (Qc = 2, 5, and 10 ml/min) flow rates was studied. A wide range of monodispersed droplets (4nl-4 μl) were generated using this method. Droplet sizes were directly proportional to the We number and inversely proportional to the Ca number and oscillation frequency. Our technique is promising for applications such as aqueous two-phase systems, where due to inherently low interfacial tension, the d-phase flow forms a long stable jet which can be broken into droplets using the additional oscillatory drag in our device.

Lateral migration and nonuniform rotation of suspended ellipse in Poiseuille flow

Suspensions of an elliptical particle in Poiseuille flow are investigated by using the multiple-relaxation-time lattice Boltzmann method coupled with the Galilean-invariant momentum exchange method. The elliptical particle shows lateral migration and final equilibrium, which are similar to the classical Segré–Silberberg effect. Because the elliptical shape interacts with the parabolic velocity distribution of the Poiseuille flow, the particulate behaviors remarkably contain regular wave accompanied by nonuniform rotation. When the blockage ratio grows up, the equilibrium position of the particle moves towards the channel centerline and the rotation period becomes long. With the increase of the aspect ratio, the rotation period becomes short; the equilibrium position shows a shape of saddle and reaches its bottom at nearby aspect ratio 0.5. Surprisingly, the numerical results show that the equilibrium position is insensitive to Reynolds number (Re) of Poiseuille flows in the range 3≤Re≤300. This work can make active promotions to the researches of blood circulation of birds, whose red blood cell is oval or elliptical in shape.

Effect of shim configuration on internal die flows for non-Newtonian coating liquids in slot coating process

In this study, a strategy for designing optimal shim configuration inside a slot die is suggested to assure the uniform coating flow distribution of various non-Newtonian shear-thinning liquids at the die exit in a slot coating system. Flow patterns of non-Newtonian liquids inside the slot die, via three-dimensional computations, have been compared using various shim geometries which can adjust the flow region in a slot manifold. The rather non-uniform (parabolic) velocity distributions of shear-thinning liquids at the die exit under the basic shim condition could be effectively flattened by the modification of shim geometry without the change of die manifold structure. Dimensions of hybrid shims for controlling flow features at edge and center regions within slit channel are positively tuned, according to the shear-thinning level of coating liquids.

The method of calculating the velocity field and shear stresses in incomressible fluid

The velocity field calculation method is based on the use of two special cases of the Newtonian fluid motion equations, not including the Navier-Stokes equations. Two shear stress calculation methods are considered. The first method is the differentiation of the velocity field equation, and the second one requires the solution of the first-order differential equation. The second method provides the distribution of shear stresses for any continuous medium, including the Newtonian fluid.Calculation equations for a laminar flow in a round pipe are found. It is shown that a parabolic velocity distribution along the radius is a special case of a more general equation.The factors affecting the shear stresses for the three flow models are found. Stresses are determined by the linear velocity gradients in the laminar flow. In the 3D vortex, they can be found by various equations, which include vorticity. Total stresses for the averaged turbulent flow are calculated by summing the previously found stresses.The equations of the method are incomplete and may be used for the accurate solution of simple problems.