On the Asymptotics of the Schrödinger Equation Solutions and the Euler Model of an Ideal Fluid

In this paper we analyze the asymptotics of the Schrödinger equation solutions with respect to a small parameter ~. It is well known, that short- waveasymptoticstosolutionsofthisequationleadstothepairofequations— the Hamilton–Jacobi equation for the phase and the continuity equation. These equations coincide with the ones for the potential flows of an ideal fluid. The wave function is invariant with respect to the complex plane rotations group, and the asymptotics is constructed as a point-dependent action of this group on some function that is found by solving the transfer equation. It is shown in the paper, that if the Heisenberg group is used instead of the rotation group, then the limit of the equations solutions with ~ tending to zero leads to the equations for vortex flows of an ideal fluid in a potential field of forces. If the original Schrödinger equation is nonlinear, then equations for barotropic processes in an ideal fluid are obtained.

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

A complementary covariant approach to gravito-electromagnetism

From a previous paper where we proposed a description of general relativity within the gravito-electromagnetic limit, we propose an alternative modified gravitational theory. As in the former version, we analyze the vector and tensor equations of motion, the gravitational continuity equation, the conservation of the energy, the energy-momentum tensor, the field tensor, and the constraints concerning these fields. The Lagrangian formulation is also exhibited as an unified and simple formulation that will be useful for future investigation.

Joint Deep-Learning-Enabled Impact of Holistic Care on Line Coagulation in Hemodialysis

In order to investigate the impact of holistic care on line coagulation and safety in hemodialysis and to address limitations of the conventional ultrasound flow vector imaging (VFM) technique, which requires proprietary software to acquire raw Doppler and scatter tracking data, a combined deep-learning-enabled holistic care approach to line coagulation in hemodialysis is proposed. First, velocity along the direction of the sound beam, which is provided by the color Doppler echocardiogram, is obtained as the radial velocity component using a velocity scale. Moreover, the left ventricular wall contour is automatically identified using a U-Net network and the left ventricular wall velocity is calculated as the boundary condition of continuity equation by a retrained PWC-Net model. Likewise, the velocity component of each blood mass in the vertical direction of the sound beam is obtained by solving the continuity equation (i.e. tangential velocity component). Finally, the velocity vector map of cardiac flow field was synthesized and visualized in the flow diagram. For this purpose, sixty patients admitted to receive hemodialysis from February 2019 to June 2020 were randomly divided into two groups of 30 patients where the control group implemented conventional care and the study group implemented all-round care on the basis of conventional care. The nursing effects of both groups were compared. Incidence of pipeline coagulation and complications in the study group were lower than those in the control group and the difference was statistically significant ( P < 0.05 ). The nursing detail score, nursing attitude score, nursing professionalism score, and total satisfaction score in the study group were higher than those in the control group and the difference was statistically significant ( P < 0.05 ). Applying all-round nursing in hemodialysis can effectively reduce the incidence of line coagulation complications and improve the safety of hemodialysis, as well as improve patients’ satisfaction with nursing care.

Quantum Hydrodynamics of Spinning Particles in Electromagnetic and Torsion Fields

We develop a many-particle quantum-hydrodynamical model of fermion matter interacting with the external classical electromagnetic and gravitational/inertial and torsion fields. The consistent hydrodynamical formulation is constructed for the many-particle quantum system of Dirac fermions on the basis of the nonrelativistic Pauli-like equation obtained via the Foldy–Wouthuysen transformation. With the help of the Madelung decomposition approach, the explicit relations between the microscopic and macroscopic fluid variables are derived. The closed system of equations of quantum hydrodynamics encompasses the continuity equation, and the dynamical equations of the momentum balance and the spin density evolution. The possible experimental manifestations of the torsion in the dynamics of spin waves is discussed.

A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.

Numerical Simulation of Dyeing Process of Cotton with Natural Dye

Cotton dyeing is a very complex process with many variables in which different phenomena occur simultaneously. This study aimed to describe the cotton dyeing process by natural dye, using a mathematical model that consists of three mass conservation equations that depict dyeing cotton in cones, taking a representative volume element at the micro, meso, and macroscales. The first equation describes the concentration changes of the dye in the solution, taking into account the diffusive, convective, adsorptive, and reactive effects. The second equation describes the changes in dye concentration in cotton fiber, considering the diffusive, adsorptive, and reactive effects within an intermediate scale. The last equation describes changes in the concentration of dye in the solution on the macroscale, based on the characteristics of the equipment and the difference in concentration before and after passing through the fiber. In addition, a fluid continuity equation was incorporated, taking into account Darcy’s law. In the simulation of the dyeing process with synthetic dye with initial concentrations of 0.408 and 2.06 kg/m3, RMSE of 0.00221 and 0.0289 kg/m3 were obtained, respectively. For the simulation of a dyeing process with natural dyeing, a behavior similar to the experimental data was obtained.

Physics in curved spacetime

Electromagnetic field theory, and the physics of continuous media (fluids, solids) in curved spacetime are discussed. Generalized Maxwell’s equations are written down and their justifaction is briefly presented. Then we turn to thermodynamics and continuous media. The concept of energy and momentum conservation is carefully expounded, and then the equations for fluid flow (continuity equation and Euler equation) are developed from the divergence of the energy tensor. The Bernoulli equation and the equation for hydrostatic equilibrium are obtained. The chapter then goes on to a general discussion of how general relativity operates and how gravitational phenomena are calculated and observed. The relation between gravity and other aspects of physics such as particle physics is discussed, along with the notion of general covariance.