Simplified Navier-Stokes Equations Research Articles

Imaging-based method to quantify left ventricular diastolic pressures

Abstract Aim To establish an imaging-based method to quantify left ventricular (LV) diastolic pressures. Methods/results In 115 patients suspected of coronary artery disease, LV pressure was measured by micromanometers and images by echocardiography. LV filling pressure was measured as LV pre-atrial contraction pressure (pre-A PLV). Based on previous observations we hypothesized that pre-A PLV approximates the sum of minimum PLV and maximum transmitral pressure difference. Parameters used for pressure estimates included LV volumes and strain, left atrial strain, mitral flow velocities, systolic arterial cuff pressure and body mass index. Minimum PLV was estimated by predictors identified in a derivative cohort (n=81). Mitral pressure difference was calculated by a simplified Navier-Stokes equation. Accuracy of estimates of minimum PLV, pre-A PLV and end-diastolic PLV were investigated in a testing cohort (n=19). Patient-specific LV diastolic pressure curves were constructed by adjusting a reference curve according to pressure estimates at key diastolic events. There was good agreement between estimated and measured pre-A PLV: Bias 0.0, limits of agreement <3.1 mmHg (±1.96SD). Estimated minimum PLV and end-diastolic PLV also showed good agreement with measured pressures. Furthermore, there was good agreement between measured and estimated LV diastolic pressure curves, quantified as mean LV diastolic pressure: Bias 0.2, limits of agreement <3.2 mmHg. Conclusion The proposed non-invasive method provided estimates of minimum PLV, pre-A PLV and end-diastolic PLV, each reflecting different features of diastolic function. Additionally, it provided an estimate of the LV diastolic pressure curve. Validation in larger populations with different phenotypes is necessary to determine the validity of the method in clinical practice.

A numerical model for calculating velocity distribution in cross-section of an open channel

Flow velocity in open channels is a fundamental hydraulic parameter with wide-ranging applications, including the development of rating curves and the study of sediment transport. While some river engineering projects may only require the calculation of average flow velocity, others, such as the design of hydraulic structures and stable channels, as well as the assessment of boundary shear stress, necessitate a more comprehensive understanding of flow dynamics, including the two-dimensional velocity distribution within open channels. To address this, various mathematical models have been proposed for estimating the two-dimensional distribution of flow velocity across transverse and depth directions. However, these models often come with complexities that hinder their practical application. In this research, we introduce a simplified numerical model that combines simplified Navier–Stokes equations with an eddy viscosity formula. This innovative approach aims to estimate velocity distribution in both rectangular and trapezoidal channels. The accuracy of our developed model hinges on the momentum transfer coefficient used in the eddy viscosity formula. Through the execution of the numerical model and the utilization of observational data, we determined the optimal value of the momentum transfer coefficient to be 0.241. To validate the effectiveness of our numerical model, we compared its predictions with laboratory data encompassing diverse hydraulic conditions. The results demonstrated a high level of accuracy, with the calculated velocity distribution and flow discharge differing by no more than 7.6% and 6.8%, respectively.

Stagnation Bioconvection Flow of Titanium and Aluminium Alloy Nanofluid Containing Gyrotactic Microorganisms over an Exponentially Vertical Sheet

The pivotal aim of this research is to address a natural stagnation bioconvection flow of a hybrid nanofluid containing gyrotactic microorganisms over an exponentially stretching and shrinking vertical sheet. The mathematical formulation of simplified Navier-Stokes equations is made in the presence of a few parameters such as Prandtl number, concentration to thermal buoyancy ratio, microorganism to thermal buoyancy ratio, Lewis number, bioconvection Peclet number, bioconvection Lewis number, microorganisms concentration difference and buoyancy parameter. The two types of nanofluid containing titanium alloy (Ti6Al4V) and aluminium alloy (AA7075) immersed in water are considered for the investigation. In the analysis, the governing partial differential equations (PDEs) are transformed into a set of ordinary differential equations (ODEs) by a similarity transformation. The resulting equations are rewritten in MATLAB software through the Bvp4c method to obtain the solutions. The effects of hybrid nanofluid of titanium alloy (Ti6Al4V) and aluminium alloy (AA7075), microorganisms’ concentration difference parameter, and bioconvection Lewis Number are observed in this mathematical model in the presence of stretching and shrinking sheets. The numerical values are obtained for the skin friction coefficient, local Nusselt number, local Sherwood number, and local density of motile microorganisms for the reporting purpose. In addition, the profiles of the velocity, temperature, concentration, and microorganism are visualized as the main findings of this article.

A research on the converging process in a T-shape tube

A T-shape tube is an important structure in chemical engineering and other fields. This article utilizes the CFD method on python with a simplified Navier-Stokes equation to stimulate the converging process of different fluids and studies the influence of inflow speed, viscosity, density, diffusion coefficient, the diameter of the tube and the length of the tube on the transmission characteristics like pressure and flow field. The results indicate that the ratio of the velocity of the main tube to that of the branch, the diameter of tube AC and tube B is inversely related with mixing efficiency and that the increase of pipe length at outlet tube and the diffusion coefficient of fluid is positive to the well-distributed concentration of outflow. However, the density and viscosity of inflow seem to have little impact on the mixing process.

Motion of Droplets in Lyophilic Axially Varying Geometry-Gradient Tubes

Droplet transport occurs frequently in nature and has a wide range of applications. We studied the droplet motion in a lyophilic axially varying geometry-gradient tube (AVGGT). The motion of the AVGGT in two directions─from the large opening side (L) to the small opening side (S) and from S to L─was theoretically and experimentally analyzed. The droplet dynamic behaviors, such as the self-transport behavior and the droplet stuck behavior, are explored from the view points of mechanics and energy. We found that the surface tension force of a three-phase contact line can be either a driving or an impeding force depending on the various droplet geometries in different AVGGTs. An important contributing factor to the self-transport behavior of a droplet moving from L to S in an AVGGT is the bridge liquid force caused by negative pressure inside the droplet, which is always pointing in the direction of S. As a result of experiments, we investigated the relationship between droplet motion and correlated parameters. The theoretical model based on the simplified Navier-Stokes equation was developed to explain the corresponding mechanism of the droplet motion. Additionally, dimensional analysis was carried out for the droplet stuck behavior of a droplet moving from S to L in an AVGGT to investigate the relationship between the droplet stopping location and the correlated parameters and thus obtain the required geometry for the droplet stopping location.

Analysis and Multi-target Optimisation of the Characteristics with Stepped Cavity of the Hybrid Bearing

In this paper, the peak pressure, stiffness, flow rate, and temperature rise of the oil film in the stepped cavity of the hybrid bearing are investigated. The pressure and flow equations for the central section of the stepped cavity and the axial section of the deep and shallow cavity are derived using the simplified Navier-Stokes equations and the boundary conditions of a liquid bearing. Using the control variable method, the influence of the geometrical parameters of the stepped cavity and the process parameters on the oil film characteristics was simulated, and the correctness of the pressure and flow equations was verified. The results show that: the spindle speed is positively correlated with the peak oil film pressure and flow rate; the oil film thickness is negatively correlated with the peak oil film pressure and positively correlated with the flow rate; the depth and width of the shallow cavity are correlated with the peak oil film pressure, but the correlation with the flow rate is low; the maximum error between the theoretical calculation and the simulation analysis is less than 7.3%. The pressure equation and the flow equation are used to establish the objective function with the maximum oil film stiffness, load-bearing capacity, and minimum temperature rise, and the ideal point evaluation function method is used to do a multi-target optimization design to optimize the ladder cavity and obtain the optimal oil cavity geometric parameters. And an 8.75% increase in peak pressure and a 4.1% reduction in temperature rise for the same process parameters.

Non-Newtonian Fluid Flow Modeling in the Inertial Viscometer with a Computer Vision System

Purpose of research. Development of theoretical premises for the new inertial viscometer, as well as the development of an approximate model of the viscosity fluid flow using convolutional neural networks and laser speckle contrast imaging data.Methods. The study consists of two parts. The first is devoted to a theoretical study of viscosity fluid flow in the toroidal channel of еру new inertial viscometer. The mathematical model of the flow includes the dimensionless equations of Navier-Stokes and convective heat conduction, the analysis of which made it possible to estimate the conditions for the uniformity of pressure and temperature fields. The numerical solution of the simplified Navier-Stokes equation was obtained by the control volume method. The computational experiment made it possible to identify additional operating conditions for the viscometer. The second part of the research is aimed at solving the problem of predicting the values of the shear strain rate on the tour surface and the flow rate. The approximate flow model is based on an ensemble of convolutional neural networks trained on data from laser speckle-contrast visualization of a fluid flow in a transparent tube.Results. The recommendations on the operating parameters of the inertial viscometer for the studied types of liquids in a given viscosity range are obtained. An approximate model has been developed in the form of an ensemble of deep neural networks, which makes it possible to determine the volumetric flow rate and the shear strain rate on the flow surface based on fluid flow images.Conclusion. The approximate Navier-Stokes equation obtained as a result of theoretical analysis for the flow of a viscous fluid in a toroidal channel can be used to numerical determination the kinematic viscosity. So, the necessary flow characteristics, such as volumetric flow rate and shear strain rate on the flow surface, can be found using the developed and pretrained convolutional neural network based on laser speck contrast imaging data. The test fluid can be any non-Newtonian fluid capable of reflecting coherent radiation. In particular, it can be physiological fluids, including blood.

Capturing the onset of thermocapillary convection in the Cattaneo–Christov flow of electrically conducting thin film of Welan gum solution

An investigation of heat transfer characteristics of electrically conducting Welan gum solution over a sheet, influenced by thermocapillary convection (thermal Marangoni convection) and magnetic field is carried out. In order to correctly predict the heat transfer characteristics, the more realistic heat flux model i.e. Cattaneo–Christov model has been utilized whereas the Maxwell-power-law model is used to capture the rheological attribute of Welan gum liquid. The Joule dissipation effect produced by the interaction between magnetic field and fluid is also considered in the energy equation. Simplified Navier–Stokes equation under above-mentioned assumptions/considerations have been obtained and are metamorphosed to a set of ODEs using appropriate Lie-group transforms. Set of ODEs are solved by Runge–Kutta fourth-order method facilitated by the shooting technique. Numerical results are displayed via graphs and tables. One of the key findings of this research work is the evidence of Marangoni convection altering the two boundary layers near the interfacial region and having almost no effect on the flow near the sheet. It has been found that the temperature and velocity of Welan gum solution can be decreased by increasing the thermal relaxation time. It has also been observed that the skin friction coefficient, a representative of shear stress is higher for a more solid-like material.

Marangoni convection in the transient flow of hybrid nanoliquid thin film over a radially stretching disk

Within a magnetohydrodynamic environment, Marangoni convection (Thermocapilarity effect) in an unsteady thin film of hybrid nanoliquid flow over a disk has been discussed. A set of simplified Navier-Stokes equation using boundary layer theory is written in order to model the above mentioned flow situation. The dissipative effects caused by viscosity and magnetic field have been incorporated in temperature-balance equation. A suitable choice of transform variables facilitate a system of ordinary differential equations (ODEs) from original partial differential equations (PDEs) representing the flow phenomena. This system of ODEs are solved by shooting technique in conjunction with Runge-Kutta 4th order numerical scheme. This study reveals that by increasing the surface tension along the liquid-air interface, the velocity of hybrid nanoliquid can be increased. In the context of this research work, the hybrid nanoliquid prepared by dispersing blade shaped [Formula: see text] and Cu nanoparticles, is an ideal liquid as far as liquid coolants are concerned.

Experimental investigation on nozzle diameter of vortex gripper

PurposeVortex grippers use tangential nozzles to form vortex flow and are able to grip a workpiece without any physical contact, thus avoiding any unintentional workpiece damage. This study aims to use experimental and theoretical methods to investigate the effects of nozzle diameter on the performance.Design/methodology/approachFirst, various suction force-distance curves were developed to analyze the effects of nozzle diameter on the maximum suction force. This study determines the tangential velocity distribution on the workpiece surface by substituting the experimental pressure distribution data into simplified Navier-Stokes equations and then used these equations to analyze the effects on the flow field. Subsequent theoretical analysis of the distribution of pressure and circumferential velocity further validated the experimental results. Next, by rearranging these relationships, the study considered the effects of nozzle diameter on the inherent vortex gripper characteristics. In addition, this study developed various suction force-energy consumption curves to analyze the effects of nozzle diameter.FindingsThe results of this study indicated that the vortex gripper’s circumferential velocity and maximum suction force decrease with increasing nozzle diameter. Nozzle diameter did not significantly affect the inherent frequency of the vortex gripper-workpiece inertial system or the corresponding suspension stability of the workpiece. However, an increase in nozzle diameter did effectively increase the vortex gripper’s suspension region. Finally, as the nozzle diameter increased, the energy required to achieve the same maximum suction force decreased.Originality/valueThis study’s findings can enable optimization of nozzle design in emerging vortex gripper designs and facilitate informed selection among existing vortex grippers.

Discrete-vortex analysis of high Reynolds number flow past a rotating cylinder

Flow past a rotating cylinder is investigated using a two-dimensional discrete vortex simulation method in this study. The simplified Navier–Stokes equation is solved based on the relationship between the surface pressure gradient and the generated surface vortex strength. The Reynolds number based on the cylinder diameter and flow velocity is 105. The non-dimensional rotation rate, α (the ratio of the cylinder surface velocity and flow velocity), is varied between 0 and 19, and four different wake formations (vortex shedding, weak vortex shedding, wake, and rotating wake formations) have been derived by the imposed rotation. The relationship between the hydrodynamics and wake formation is illustrated. Under vortex shedding and weak vortex shedding formations, periodical hydrodynamics is induced. Under wake formation, no gap between the positive-vorticity and negative-vorticity layers results in the steady hydrodynamics. The separation of the rotating wake induces the huge fluctuation of hydrodynamics under rotating wake formation. These are significant for a flow control technique and for the design of ocean and civil engineering structures. With the increasing rotation rate, the variation of mean hydrodynamics has been discussed and the maximum mean hydrodynamics is considered to be decided by the rotation rate. According to these wake formations, the vortex shedding, weak vortex shedding, wake, and rotating wake areas are identified. Combining the initial, increasing, and equivalent areas for mean hydrodynamics, two different area-divisions have been conducted for mean hydrodynamics and the relationship between the two area-divisions has been illustrated. Finally, the disappearance of vortex shedding and variation of the Strouhal number have been discussed in detail. The critical value for the disappearance of vortex shedding is α ≈ 3.5, and the Strouhal number remains steady initially and then decreases.

A hybrid Windkessel-Womersley model for blood flow in arteries

A hybrid Windkessel-Womersley (WK-W) coupled mathematical model for the study of pulsatile blood flow in the arterial system is proposed in this article. The model consists of the Windkessel-type proximal and distal compartments connected by a tube to represent the aorta. The blood flow in the aorta is described by the Womersley solution of the simplified Navier-Stokes equations. In addition, we defined a 6-elements Windkessel model (WK6) in which the blood flow in the connecting tube is modeled by the one-dimensional unsteady Bernoulli equation. Both models have been applied and validated using several aortic pressure and flow rate data acquired from different species such as, humans, dogs and pigs. The results have shown that, both models were able to accurately reconstruct arterial input impedance, however, only the WK-W model was able to calculate the radial distribution of the axial velocity in the aorta and consequently the model predicts the time-varying wall shear stress, and frictional pressure drop during the cardiac cycle more accurately. Additionally, the hybrid WK-W model has the capability to predict the pulsed wave velocity, which is also not possible to obtain when using the classical Windkessel models. Moreover, the values of WK-W model parameters have found to fall in the physiologically realistic range of values, therefore it seems that this hybrid model shows a great potential to be used in clinical practice, as well as in the basic cardiovascular mechanics research.

Where has all the power gone? Energy production and loss in vocalization

Human voice production for speech is an inefficient process in terms of energy expended to produce acoustic output. A traditional measure of vocal efficiency relates acoustic power radiated from the mouth to aerodynamic power produced in the trachea. This efficiency ranges between 0.001% and 1.0% in speech-like vocalization. Simplified Navier–Stokes equations for non-steady compressible airflow from trachea to lips were used to calculate steady aerodynamic power, acoustic power, and combined total power at seven strategic locations along the airway. A portion of the airway was allowed to collapse to produce self-sustained oscillation for sound production. A conversion efficiency, defined as acoustic power generated in the glottis to aerodynamic power dissipated, was found to be on the order of 10%, but wall vibration, air viscosity, and kinetic pressure losses consumed almost all of that converted power. Thus, the acoustic power, reflected back and forth in the airway was dissipated at a level on the order of 99.9%, with a small fraction being radiated to the listener.

Development of an aerostatic bearing system for roll-to-roll printed electronics

Roll-to-roll printed electronics is proved to be an effective way to fabricate electrical devices on various substrates. High precision overlay alignment plays a key role to create multi-layer electrical devices. Multiple rollers are adopted to support and transport the substrate web. In order to eliminate the negative effect of the machining error and assembling error of the roller, a whole roll-to-roll system including two aerostatic bearing devices with arrayed restrictors is proposed in this paper. Different to the conventional roller, the aerostatic bearing device can create a layer of air film between the web and the device to realize non-contact support and transport. Based on simplified Navier–Stokes equations, the theoretical model of the air film is established. Moreover, the pressure distribution of the whole flow field and single restrictor in different positions are modeled by conducting numerical simulation with computational fluid dynamics (CFD) software FLUENT. The load capacity curves and stiffness curves are generated to provide guidance for optimizing the structure of the device. A prototype of the aerostatic bearing system is set up and the experiment tests are carried out. For the proposed aerostatic bearing roller with a diameter of 100 mm and length of 200 mm, the experimental results show the aerostatic bearing method can achieve the position accuracy in a range of 1 μm in the vertical direction of the web, which is much better than that using existing methods.

Non-invasive assessment of pulsatile intracranial pressure with phase-contrast magnetic resonance imaging.

Invasive monitoring of pulsatile intracranial pressure can accurately predict shunt response in patients with idiopathic normal pressure hydrocephalus, but may potentially cause complications such as bleeding and infection. We tested how a proposed surrogate parameter for pulsatile intracranial pressure, the phase-contrast magnetic resonance imaging derived pulse pressure gradient, compared with its invasive counterpart. In 22 patients with suspected idiopathic normal pressure hydrocephalus, preceding invasive intracranial pressure monitoring, and any surgical shunt procedure, we calculated the pulse pressure gradient from phase-contrast magnetic resonance imaging derived cerebrospinal fluid flow velocities obtained at the upper cervical spinal canal using a simplified Navier-Stokes equation. Repeated measurements of the pulse pressure gradient were also undertaken in four healthy controls. Of 17 shunted patients, 16 responded, indicating high proportion of “true” normal pressure hydrocephalus in the patient cohort. However, there was no correlation between the magnetic resonance imaging derived pulse pressure gradient and pulsatile intracranial pressure (R = -.18, P = .43). Pulse pressure gradients were also similar in patients and healthy controls (P = .26), and did not differ between individuals with pulsatile intracranial pressure above or below established thresholds for shunt treatment (P = .97). Assessment of pulse pressure gradient at level C2 was therefore not found feasible to replace invasive monitoring of pulsatile intracranial pressure in selection of patients with idiopathic normal pressure hydrocephalus for surgical shunting. Unlike invasive, overnight monitoring, the pulse pressure gradient from magnetic resonance imaging comprises short-term pressure fluctuations only. Moreover, complexity of cervical cerebrospinal fluid flow and -pulsatility at the upper cervical spinal canal may render the pulse pressure gradient a poor surrogate marker for intracranial pressure pulsations.

Theoretical and experimental study of condensation rates on radiant cooling surfaces in humid air

The condensation rate on a radiant cooling surface in humid air is a basic index used to evaluate the risk of condensation and is beneficial to the engineering design of radiant cooling systems. In this paper, condensation rates were studied theoretically using simplified Navier–Stokes equations with Boussinesq approximation, and the heat and mass transfer analogy was also derived. In the experiments, the condensation rates on radiant cooling panels of different lengths at various positions (the floor, the wall, and the ceiling) were measured. The experimental results of the mass transfer coefficients are represented in a correlation of average Sh number vs. Ra number. With a sub-cooled degree of 5 °C, the mass transfer coefficients of the radiant floor, wall and ceiling were 1.22 mm/s, 3.35 mm/s, and 4.15 mm/s, respectively. As a result, the condensation rate on the radiant ceiling was 3.5 times greater than that on the radiant floor and 25% greater than that on the radiant wall.

Free Vibration Analysis of Elastic Plates in Contact with an Inviscid Fluid Medium

In the present study, a finite element formulation is presented to investigate vibration response of elastic plates in contact with a fluid medium. The fluid is assumed to be incompressible and inviscid, and the impermeability condition of the plate is taken into account. The classical plate theory (CPT), first-order shear deformation plate theory (FSDT), and Reddy third-order shear deformation plate theory (RSDT) are considered for the kinematic description of the solid medium and the simplified Navier–Stokes equations are used as the governing equations for the fluid medium. For each plate theory, a coupled set of finite element equations is derived. The effect of the fluid pressure is considered as an added mass and its effect on natural frequencies and mode shapes is investigated through several numerical simulations by varying the boundary conditions.

Real-Time Visual Animation of Explosions

Realistic explosion effects can enhance the realism and immersion in virtual battlefield. The real-time explosion effects simulation of the explosion products and their interactions, such as explosive fireball, explosive smoke and explosive ash, will be exploded into two phases: explosions and fuel modeling. The explosions modeling describe the formation and spreading of the explosive fireball. The fuel modeling depicts rising of the explosive ash accompanied by the explosive smoke. The paper describes an Eulerian approach for animating explosions in the first phase. The divergence-free velocity field is computed on a grid. In order to ensure the quality of conservation, the velocity divergence in the local is joined to describe smoke diffusion. The particles with the respective attributes are updated in both two phases. Simplified Navier-Stokes (NS) equations modified according to the non-conservation of explosion energy is used to determine the fluid motion in whole simulation process. In order to further improve the fidelity and real-time of the simulation effect, small-scale swirl effect of explosion smoke with vorticity confinement is introduced to compensate for the loss of finite difference methods to solve the NS equations. The whole simulation process of explosion effects are realized on Visual Studio 2010 and OpenGL platform, including multiple explosions, explosions near obstacles. The temperature and velocity in the explosion center, explosive diameter and height, are analyzed to verify the high fidelity of this new simulation method. The high fidelity and computational efficiency of this new modified approach are verified with several experimental simulation results.

Stability of Flow Past Alternate Rigid and Porous Panels in Boundary Layer Flow and in Channel Flow

Propagation of two-dimensional small amplitude Tollmien-Schlichting (TS) waves has been investigated over a rigid panel followed by a porous panel in the presence of cross-flow. In the present work boundary layer flow over alternate rigid-porous panels in which suction is applied through the porous panel is investigated.Also investigated is the problem of flow past alternate rigid and porous panels with cross flow. A general space marching solution has been discussed for calculating the mean velocity profile for the above case, in the developing region of mean flow in the porous panel, following the rigid-porous junction. Numerical solutions are obtained using a finite difference method for the suitably simplified Navier Stokes equations, using appropriate boundary conditions.Detailed two-dimensional analyses have been done for the disturbance waves using both the quasi-parallel (QP) approximation, and more accurately, using the non-parallel (NP) approach. The non-parallel approach has been carried out over the developing mean-flow region of the porous panel, following the rigid-porous junction.Numerical solutions have been obtained by finite difference procedures. In some of the cases results have been validated with the available literature. Finally, the jumps in the amplitude of the disturbance waves across the rigid- porous junction were calculated using the theory of Sen et al.[6].The important outcome from this work is in optimizing the length of the porous panel, following the rigid-porous junction. It is seen that, as compared to the length required to approach the asymptotic mean flow state to within 99%, only a very short porous panel length is sufficient to stabilize the disturbances.Hence, it is foreseen that alternate long rigid panels, with in-between short porous panels, could be a very effective way of stabilizing the disturbances, and thus delaying laminar to turbulent transition.

Fabrication and characterization of non-linear parabolic microporous membranes

Large scale fabrication of non-linear microporous membranes is of technological importance in many applications ranging from separation to microfluidics. However, their fabrication using traditional techniques is limited in scope. We report on fabrication and characterization of non-linear parabolic micropores (PMS) in polymer by utilizing flow properties of fluids. The shape of the fabricated PMS corroborated well with simplified Navier–Stokes equation describing parabolic relationship of the form L–t1/2. Here, L is a measure of the diameter of the fabricated micropores during flow time (t). The surface of PMS is smooth due to fluid surface tension at fluid–air interface. We demonstrate fabrication of PMS using curable polydimethylsiloxane (PDMS). The parabolic shape of micropores was a result of interplay between horizontal and vertical fluid movements due to capillary, viscoelastic, and gravitational forces. We also demonstrate fabrication of asymmetric “off-centered PMS” and an array of PMS membranes using this simple fabrication technique. PMS containing membranes with nanoscale dimensions are also possible by controlling the experimental conditions. The present method provides a simple, easy to adopt, and energy efficient way for fabricating non-linear parabolic shape pores at microscale. The prepared parabolic membranes may find applications in many areas including separation, parabolic optics, micro-nozzles/-valves/-pumps, and microfluidic and microelectronic delivery systems.