Velocity Derivatives Research Articles

Prodi–Serrin condition for 3D Navier–Stokes equations via one directional derivative of velocity

In this paper, we consider the conditional regularity of weak solution to the 3D Navier–Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂ 3 u , satisfies ∂ 3 u ∈ L p 0 , 1 ( 0 , T ; L q 0 ( R 3 ) ) with 2 p 0 + 3 q 0 = 2 and 3 2 < q 0 < + ∞ , then the weak solution is regular on ( 0 , T ] . The proof is based on the new local energy estimates introduced by Chae-Wolf (2019) [4] and Wang-Wu-Zhang (2020) [21] .

Comparative study of acoustic non-linearity parameter in binary mixtures of N,N-dimethylacetamide with Polyethylene Glycols at different temperatures

The present paper deals with the comparative evaluation of acoustic non-linearity parameter (B/A) for binary liquid mixtures of N, N-dimethylacetamide (DMA) with Polyethylene Glycol 200 (PEG 200) and Polyethylene Glycol 400 (PEG 400) using seven different approaches as Beyer, Hartmann, Ballou, H. Endo, Lu, Tong-Dong and Beyer-Tong Dong at temperatures 308.15, 318.15 and 328.15 K. The excess values of B/A have been computed using all seven approaches and substituted into the Redlich-Kister equation. The temperature and pressure derivatives of ultrasonic velocity using Beyer and Beyer- Tong Dong methods have been presented for both the mixtures. Molecular properties have also been calculated by Beyer-Tong Dong method using Sehgal’s relations. The aim of the present investigation is to check the validity and efficacy of different methods on the dimethylacetamide-Polyethylene Glycol mixtures and to get the better understanding about the nature and strength of the intermolecular interactions in the mixtures.

Recovery of radial-axial velocity in axisymmetric swirling flows of a viscous incompressible fluid in the Lagrangian consideration of vorticity evolution

Swirling laminar axisymmetric flows of viscous incompressible fluids in a potential field of body forces are considered. The study of flows is carried out in a cylindrical coordinate system. In the flows, the regions in which the axial derivative of the circumferential velocity cannot take on zero value in some open neighborhood (essentially swirling flows) and the regions in which this derivative is equal to zero (the region with layered swirl) are considered separately. It is shown that a well-known method (the method of viscous vortex domains) developed for non-swirling flows can be used for regions with layered swirling. For substantially swirling flows, a formula is obtained for calculating the radial-axial velocity of an imaginary fluid through the circumferential vorticity component, the circumferential circulation of a real fluid, and the partial derivatives of these functions. The particles of this imaginary fluid “transfer” vortex tubes of the radial-axial vorticity component while maintaining the intensity of these tubes, and also “transfer” the circumferential circulation and the product of the circular vorticity component by some function of the distance to the axis of symmetry. A non-integral method for reconstructing the velocity field from the vorticity field is proposed. It is reduced to solving a system of linear algebraic equations in two variables. The obtained result is proposed to be used to extend the method of viscous vortex domains to swirling axisymmetric flows.

Solenoidal linear forcing for compressible, statistically steady, homogeneous isotropic turbulence with reduced turbulent Mach number oscillation

This study investigates a solenoidal linear forcing scheme with reduced oscillation of a turbulent Mach number MT for direct numerical simulations (DNS) of statistically steady, homogeneous isotropic turbulence. A conventional linear forcing scheme results in a large temporal oscillation of MT, where the maximum MT reaches about 1.1 times the time-averaged MT. Therefore, strong shocklets are generated when MT becomes large although such strong shocklets hardly appear when MT is close to the time-averaged value. DNS with the proposed forcing scheme confirms that the temporal oscillation of MT is effectively reduced by adjusting a forcing coefficient with a ratio between velocity variance and its steady state value prescribed as a parameter. The time-dependent forcing coefficient results in the variation of the power input to kinetic energy. Therefore, the temporal oscillation of the Reynolds number for this forcing scheme is as large as that for the conventional linear forcing. The ratio between the solenoidal and dilatational kinetic energy dissipation rates increases with MT, and the MT dependence is consistent between the present solenoidal linear forcing and the low-wavenumber solenoidal forcing in wavenumber space. The skewness and flatness of the velocity derivative become large compared with incompressible turbulence when MT exceeds 0.6. Both average and root-mean-squared fluctuation of the shock Mach number of shocklets increase with MT. The most typical thickness of shocklets decreases with MT and asymptotically approaches about 1.5 times the Kolmogorov scale. The shocklet thickness normalized by the Kolmogorov scale hardly depends on the Reynolds number.

Emergent behaviors of relativistic flocks on Riemannian manifolds

We present a relativistic counterpart of the Cucker–Smale (CS) model on Riemannian manifolds (manifold RCS model in short) and study its collective behavior. For Euclidean space, the relativistic Cucker–Smale (RCS) model was introduced in Ha et al. (2020) via the method of a rational reduction from the relativistic gas mixture equations by assuming space-homogeneity, suitable ansatz for entropy and principle of subsystem. In this work, we extend the RCS model on Euclidean space to connected, complete and smooth Riemannian manifolds by replacing usual time derivative of velocity and relative velocity by suitable geometric quantities such as covariant derivative and parallel transport along length-minimizing geodesics. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using LaSalle’s invariance principle. As concrete examples, we further analyze the RCS models on the unit sphere Sd and the hyperbolic space Hd. More precisely, we show that the RCS model on Sd exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in Hd. For the hyperbolic space Hd, we also rigorously justify smooth transition from the RCS model to the CS model in any finite time interval, as speed of light tends to infinity.

Reynolds stress tensor and pressure-related turbulence transport terms measured by time-resolved tomographic-PIV

Turbulence is inherently a three-dimensional and time dependent flow phenomenon (Pope, 2001). Because of the ubiquitous existence of turbulent flows in nature, accurate characterization of turbulent flows, either through experimental measurements or through direct numerical simulations, is of paramount importance for modeling turbulence (Liu and Katz, 2018). Since its inception in 1984 (Adrian, 1984), Particle Image Velocimetry (PIV), among several other conventional techniques used for turbulence measurements, has been a valuable tool for providing reliable experimental data for turbulence research. Several advancements in hardware such as high-speed cameras, together with innovative algorithms and procedures, have extended the scope of PIV to a variety of applications. Westerweel et al. (2013) point out in a recent review article that one of the main advantages of the PIV measurement is its unique ability in measuring quantitatively spatial derivatives of the flow field. With the development of Tomographic PIV introduced by Elsinga et al. (2006), it is now possible to measure simultaneously the distributions of three velocity components in a three-dimensional flow field, thus enabling us to measure all the velocity derivatives of a turbulent flow. However, for a thorough characterization of a turbulent flow, in addition to the velocity gradients, the instantaneous pressure distribution in the 3D flow field also needs to be measured.

Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier–Stokes equations

We present a numerical method for two-phase incompressible Navier–Stokes equation with jump discontinuities in the normal projection of the stress tensor and in the material properties. Although the proposed method is only first-order accurate, it does capture discontinuities sharply, not neglecting nor omitting any component of the jump condition. Discontinuities in velocity gradient and pressure are expressed using a linear combination of singular force and tangential derivatives of velocities to handle jump conditions in a fully implicit manner. The linear system for the divergence of the stress tensor is constructed in the framework of the ghost fluid method, and the resulting saddle-point system is solved via an iterative procedure using recently introduced techniques by Egan and Gibou [9]. Numerical results support the inference that the proposed method converges in L∞ norms even when velocities and pressures are not smooth across the interface and can handle a large density ratio that is likely to appear in a real-world simulation.

Mangrove roots model suggest an optimal porosity to prevent erosion

Mangrove swamps are extremely productive ecosystems providing many ecological services in coastal regions. The hydrodynamic interactions of mangrove roots and water flow have been proposed as a key element to mitigate erosion. Several studies reveal that precise prediction of the morphological evolution of coastal areas, in the face of global warming and the consequent sea-level rise, requires an understanding of interactions between root porosity (the fraction of the volume of void space over the total volume), water flows, and sediment transport. Water flows around the mangrove prop roots create a complex energetic process that mixes up sediments and generates a depositional region posterior to the roots. In this work, we investigated the boundary layer behind permeable arrays of cylinders (patch) that represent the mangrove roots to explore the impact of patch porosity on the onset of sediment transport. The flow measurements were performed in a vertical plane along the water depth downstream of the mangrove root models. A high-resolution Particle Image Velocimetry (PIV) was used in a flume to observe the impact of porosity on the mean flow, velocity derivatives, skin friction coefficient, and production of turbulent kinetic energy for Reynolds number of 2500 (based on patch diameter length-scale). Here, we proposed a predictive model for critical velocity for incipient motion that takes into account the mangrove roots porosity and the near-bed turbulence effect. It is found that the patch with the phi =47% porosity, has the maximum critical velocity over which the sediment transport initiates. We found the optimum porosity has the minimum sediment erosion and creates negative vorticity sources near the bed that increases the critical velocity. This signifies an optimum porosity for the onset of sediment transport consistent with the porosity of mangroves in nature. The phenomenological model is elucidated based on an analysis of the vorticity evolution equation for viscous incompressible flows. For the optimum porous patch, a sink of vorticity was formed which yielded to lower the near-bed turbulence and vorticity. The minimum velocity fluctuations were sufficient to initiate the boundary layer transition, however, the viscous dissipation dominated the turbulence production to obstruct the sediment transport. This work identified the pivotal role of mangrove root porosity in sediment transport in terms of velocity and its derivatives in wall-bounded flows. Our work also provides insight into the sediment transport and erosion processes that govern the evolution of the shapes of shorelines.

Aortic Pulse Wave Velocity Derivation From Arterial Waveforms in Critically Ill Infants and Children

Methods Hemodynamically stable pre-pubertal children with an indwelling arterial catheter were enrolled. Applanation tonometry was used to measure aPWV, which was derived from the transit distance and pulse transit time. The carotid and femoral arteries were used as preferential sites and transit distance was measured relative to the suprasternal notch, an anatomical landmark for the aortic valve. Pulse transit time was derived from the time delay between the two pulse waveforms from the ECG-R wave. Peripheral arterial line waveforms were converted into ascending aortic waveforms by Hemolab software. Forward and backward wave forms were derived from the ascending aortic waveforms. Time delays were calculated between forward and backward wave forms. Effective reflecting distance (EfRD) was calculated from the equation 0.173*age+0.661*BMI+ 34.548. Estimated aPWV was derived from EfRD and the time delays. Measured aPWV and estimated aPWVs were compared. Results Aortic pulse wave velocity and arterial pressure waveforms were obtained in 33 infants and children (50% female). 66% (22) patients had history of congenital heart disease. 54% (18) patients were on mechanical ventilation. 39% (13) patients were on vasoactive medications. 0-1 year aged infants (n=18) had aPWV 0.49-15.91 m/s. aPWV in 1-5-year-old patients (n=5) was 1.82-3.72 m/s, and in 6-10-year-old patients (n=6) was 1.27-4.09 m/s. 10-12-year-old patients (n=4) had aPWV 1.7-7.02 m/s. We identified artifacts from ventilators, nasogastric lines, and pacifiers in some patients that necessitated the use of the brachial artery instead of the carotid artery for the measurement of aPWV. Measured and estimated (from arterial waveforms) aPWVs were similar in 6 patients. Estimated aPWV from arterial waveforms by using Effective reflecting distance (EfRD) correlated well with the patients' age, weight, and length. Conclusions Aortic pulse wave velocity can be feasibly measured in a critical care setting, although patient monitoring and instrumentation may make uniform measurement protocols not feasible. Future derivations of aPWV from arterial waveforms will need to take this into account. Calculating aPWV from arterial waveforms can help us to include a larger sample size of the patient population with arterial waveform data, eliminating the need for tonometry.

1-D Optimum Path Problem and Its Application

The 1-D optimum path problem with two end-points fixed or one end-point fixed, the other end-point variable reduces to vector integral equations of Fredhom / Volterra type and is hard to solve. Translating it to scalar components equations would be an easier way of solving it. Here, the solution of the optimum path problem is recommended by connecting it with the Principle of minimum Energy Release (PMER). A lot of optimum path problems with path function E=cu2, where E is the released energy, u is the velocity, c is constant, can be solved by PMER, e.g., the Great Earthquake, the denotation of a nuclear weapon, the strategy of sports games. The one end-point fixed, the other end-point variable is studied for wing moving. High lights: The pulse-mode of nuclear denotation releasing energy is the same as Earthquake, Yun [1], shows that the derivative of wind velocity with respect to time in proportion to the derivative of temperature with respect to the track. Which conforms with the weather forecast in winter that strong wind companies with low temperature for cold wave coming, and also suits for the motion of mushroom cloud [2].

Wave-kinetic approach to zonal-flow dynamics: Recent advances

The basic physics of drift-wave turbulence and zonal flows has long been studied within the framework of the wave-kinetic theory. Recently, this framework has been reexamined from first principles, which has led to more accurate yet still tractable “improved” wave-kinetic equations. In particular, these equations reveal an important effect of the zonal-flow “curvature” (the second radial derivative of the flow velocity) on the dynamics and stability of drift waves and zonal flows. We overview these recent findings and present a consolidated high-level picture of (mostly quasilinear) zonal-flow physics within reduced models of drift-wave turbulence.

Attitude Interpolation Algorithm for Industrial Robot Based on Quaternion Method

In order to achieve the smooth transition of the end-tool pose of the industrial robot, after determining the pose of the teaching points of the robot. Firstly, the path of the pose change between the adjacent teaching points is constructed by using the SLERP method, then, the position of the start and end points of the attitude transition curve is determined, and the expression of the attitude trajectory curve is obtained. According to the relation between the derivative of quaternion and Angular Velocity, the conversion between unit quaternion and angular velocity is realized by using Newton-Cortez formula. Finally, the velocity of the fitted industrial robot’s tool’s attitude trajectory is planned, and the whole attitude trajectory is interpolated.

Energy dissipation and enstrophy production/destruction at very low Reynolds numbers in the final stage of the transition period of decay in grid turbulence

Decay characteristics of turbulent kinetic energy and enstrophy in grid turbulence have been investigated in the far downstream region (x/M∼103: x is the downstream distance from the grid, M is the mesh size of the grid) through wind tunnel experiments using hot-wire anemometry, with the lowest turbulent Reynolds number Reλ≈5. The non-dimensional dissipation rate Cε increases rapidly toward the final stage of the transition period of decay and the profile agrees well with previous direct numerical simulation [W. D. McComb et al., “Taylor's (1935) dissipation surrogate reinterpreted,” Phys. Fluids 22, 061704 (2010)] and theoretical estimation [D. Lohse, “Crossover from high to low Reynolds number turbulence,” Phys. Rev. Lett. 73, 3223 (1994)] at very low Reλ in decaying and stationary isotropic turbulence. The present result of Cε is an update on the experimental data in grid turbulence toward a very low Reλ, where measurements have been absent. The energy spectrum in the dissipation range at very low Reλ deviates from a universal form observed at high Reynolds numbers. The decay rate of enstrophy is proportional to S+2G/Reλ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient). It is shown that G and S+2G/Reλ increase rapidly with decreasing Reλ at very low Reλ, indicating that the effect of enstrophy destruction is dominant in the final stage of the transition period of decay. The profiles of S+2G/Reλ against Reλ is well fitted by a power-law function even in the final stage of the transition period of decay.

Analysis of viscoelastic non-Newtonian fluid over a vertical forward-facing step using the Maxwell fractional model

In this study, unsteady of the fluid flow and heat transfer of natural convection of viscoelastic non-Newtonian fluid have been investigated with the Maxwell fractional model on a vertical forward-facing step. Fractional derivatives are used to the fluid properties more accurate calculations. The fractional time derivative of the Caputo method and the fractional derivative of the velocity and temperature of the viscoelastic fluid are calculated using the finite difference numerical method by combining the L-1 algorithm. The effect of various factors such as parameters of velocity and temperature fraction derivative (α and β), buoyancy force, step length, and Prandtl number on velocity profile, the temperature of the viscoelastic fluid, Nusselt number and friction coefficient on a vertical forward-facing step are investigated. Numerical results show that with increasing α, β and step length, the coefficient of friction increases and also decreases with increasing Prandtl. With increasing α, the average Nusselt value decreases, and with increasing β, Prandtl, and step length increases. It is also observed that the thickness of the momentum and thermal boundary layer in viscoelastic fluids is higher than that of the Newtonian fluid boundary layer.

Modeling the process of force load generation at the initial periodic change in pressure (a plane problem)

The article deals with modeling the process of force load generation at an initial periodic change in pressure (a plane problem). The subject of research is a pulsating flow in a flat channel at an initial periodic pressure change. The determination of flow parameters with a periodic change in the inlet pressure; the changes in the structure of the working fluid associated with the release of various particles from the pipe walls, the addition of impurities to prevent leaks, and the high-speed modes, are given in the article considering the law of molecular and molar transfer between layers. Research methods are based on Newton’s rheological law, according to which molecular transfer is described by the law of proportionality of stresses to the derivative of the normal velocity; on the method of accounting for molar transfer by proportionality of stresses to the derivative of normal acceleration; on the method of mathematical modeling and the analytical method for their solutions, based on the provisions of operational calculus. An analytical solution to the problem of pulsating fluid motion in a plane-parallel channel is obtained with allowance for single and group transfer of molecules in the flow. The application of the analytical expressions obtained for the velocities is not limited to the critical Reynolds number, i.e. they are applied for any values of this number. Analytical expressions are obtained for the transverse and longitudinal components of the flow velocity. The resulting solution describes two zones of flow: in the first zone, two types of transfer occur, depending on the flow pattern, either molecular or molar transfer of fluid volumes between the layers prevails. In the second zone, only molecular transfer occurs.

Motion Equation and Solution of Mushroom Cloud

Using model like hot air bloom with zero-weighted membrane wrapped hot air, surrounded by cold air, this paper sets up a partial differential equation (PDE) of motion of mushroom cloud by modifying Navier-Stokes (N-S) equations. The obtained equation is a vector PDE. It states that the derivative of velocity is with respect to time proportions to the gradient of temperature with respect to trace. Its solution is obtained by the method of separating variables for scalar function. These results have been compared with well agreement with literatures. Highlight: The Principle of Minimum Energy Release (PMER) is used to prove the pulse-mode of explosion of nuclear weapon, as great Earthquake, and optimum path problems.

Numerical modeling of fractional viscoelastic non-Newtonian fluids over a backward facing step – Buoyancy driven flow and heat transfer

The unsteady flow and natural convection heat transfer of fractional Maxwell viscoelastic fluid over sudden expansion geometry such as vertical backward facing step is studied. The time–space fractional derivatives are calculated based on Caputo fractional derivative. The nonlinear governing equations are solved using finite difference method mixed with a L-1 algorithm. The step, upstream and downstream walls were set at a constant temperature. The range of step length is 0≤S≤0.5 and dimensionless temperature is 0.25≤θw≤1. The effects of velocity and temperature fractional derivatives (αand β) and various physical parameters, such as step length, dimensionless temperature, on Nusselt number, friction coefficient, profile of velocity and temperature, are investigated numerically. The results reveal that the fractional derivative parameters, the variation of wall temperature and step length have significant effects on the rate of heat transfer, the friction coefficient, the velocity and temperature of backward facing step. The average of Nusselt number variation increases with increasing of β and decreases of α parameters. The results show that the velocity and thermal boundary layer of the non-Newtonian fluid (ordinary Maxwell) is higher than Newtonian fluid. Also, the Nusselt number for the ordinary Maxwell fluid flow is greater than the Newtonian fluid flow. With the increase of the step length of backward facing step, the average Nusselt number, friction coefficient, velocity boundary layer, and temperature boundary layer increases. The obtained results can improve significantly the performance of the systems design that operate with the non-Newtonian fluids.

Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model

The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H5 × H4 × H4 framework are investigated in this paper. We first show that the lower bound of decay rate for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is when the initial data satisfy some low‐frequency assumption. Moreover, we prove that the lower bound of decay rate of k(k ∈ [1, 3]) order spatial derivative for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is . Then, we show that the lower bound of decay rate for the time derivatives of density and velocity is , but the lower bound of decay rate for the time derivative of magnetic field converging to zero in L2‐norm is .

A regularity criterion for the 3D Boussinesq equations

ABSTRACT In this paper, we consider the regularity criteria of 3D incompressible Boussinesq equations. By using the Littlewood–Paley decomposition technique to establishing a regularity criterion in terms of the one-directional derivative of velocity in Besov spaces. Our result improves some previous works (Liu Q. A regularity criterion for the Navier–Stokes equations in terms of one-directional derivative of the velocity. Acta Appl Math. 2015;140(1):1–9; Gala S, Ragusa M A. On the regularity criterion for the Navier–Stokes equations in terms of one-directional derivative. Asian-Eur J Math. 2017;10(1):1750012).

Excess pressure as an analogue of blood flow velocity.

Derivation of blood flow velocity from a blood pressure waveform is a novel technique, which could have potential clinical importance. Excess pressure, calculated from the blood pressure waveform via the reservoir-excess pressure model, is purported to be an analogue of blood flow velocity but this has never been examined in detail, which was the aim of this study. Intra-arterial blood pressure was measured sequentially at the brachial and radial arteries via fluid-filled catheter simultaneously with blood flow velocity waveforms recorded via Doppler ultrasound on the contralateral arm (n = 98, aged 61 ± 10 years, 72% men). Excess pressure was derived from intra-arterial blood pressure waveforms using pressure-only reservoir-excess pressure analysis. Brachial and radial blood flow velocity waveform morphology were closely approximated by excess pressure derived from their respective sites of measurement (median cross-correlation coefficient r = 0.96 and r = 0.95 for brachial and radial comparisons, respectively). In frequency analyses, coherence between blood flow velocity and excess pressure was similar for brachial and radial artery comparisons (brachial and radial median coherence = 0.93 and 0.92, respectively). Brachial and radial blood flow velocity pulse heights were correlated with their respective excess pressure pulse heights (r = 0.53, P < 0.001 and r = 0.43, P < 0.001, respectively). Excess pressure is an analogue of blood flow velocity, thus affording the opportunity to derive potentially important information related to arterial blood flow using only the blood pressure waveform.