Cylindrical Polar Coordinates Research Articles

Analysis of Effect of Hemodynamic Parameters on Two-layered Blood Flow in a Mild Stenosed Artery

Arterial stenosis is the lessening of the arterial wall due to the growth of aberrant tissues that prevent adequate blood flow in the human circulatory system and induces cardiovascular diseases. Mild stenosis, over time, can lead to serious and permanent damage if it remains uncured. Navier-Stokes equation in cylindrical polar coordinates system has been extended by two-layered blood flow along the axial direction with appropriate boundary conditions.A steady flow through a stenosed artery has been investigated extensively using the analytical approach in the case of two-layered blood flow. Mathematical expressions for two-layer hemodynamic parameters such as velocity profile, volumetric flow rate, and effect of stenosis progression on parameters with the variation of core and peripheral layered coefficient of viscosity are derived. Moreover, pressure drop, and shear stress have been calculated analytically in an artery with and without stenosis. Peripheral viscosity has less contribution to varying velocity distribution than the core and is proportional to stenosis size.The volumetric flow rate decreases with an increasing viscosity coefficient. Pressure drop and shear stress attain maximum value in the region stenosis occur maximum height in core layer.The present work could serve as a model in biomedical engineering for the cure of vascular-related diseases and has the potential in designing the devices of this field.

Elastic response of transversely isotropic and non-homogeneous geomaterials under circular ring concentrated and axisymmetric distributed loads

This paper first develops the closed-form solution of a layered halfspace subject to circular ring concentrated loads. The layered halfspace consists of finite layers and a lower halfspace. All the layers in the layered halfspace are homogeneous and transversely isotropic. Integral transform techniques are utilized to derive the solution in cylindrical polar coordinates. Then, the closed-form solution is used to develop the numerical method of axisymmetric distributed loads over circular areas. The numerical method involves integration along the radius direction and discretization of a loading circular area into one-dimensional elements. Finally, the elastic responses of non-homogeneous geomaterial halfspaces under axisymmetric distributed loads over circular areas are analyzed in detail.

Entropy generation in the ternary nanofluid flow through an atherosclerotic vessel under periodic body acceleration

This article delves into the examination of the entropy generation within the slip flow of a ternary nanofluid flowing through a stenotic blood vessel. To capture the non-Newtonian rheology of blood, the constitutive equation of the Sutterby fluid model is employed. The governing equations are modeled in cylindrical polar coordinates. These equations are made dimensionless by a suitable choice of dimensionless parameters. Additionally, the dimensionless governing equations are simplified by applying the mild stenosis assumption. With the help of prescribed initial and boundary conditions, the numerical solution of the formulated problem is acquired by applying the explicit finite difference scheme. The outcomes of numerous governing parameters on the velocity profile, temperature profile, and entropy generation are explored through graphical forms. The velocity profile rises as the slip parameter and the inclination parameter are increased. The temperature profile rises with the growing values of the source parameter and decreases with enhancing values of the shape size parameter. The increasing strength of the applied magnetic field causes a reduction in the entropy generation. A comparison of the current study with the existing results is also presented. The plot of the Nusselt number shows that the height of stenosis plays an important role in monitoring its temporal variation. The current findings have been refined to provide elasticity-worthy material for researchers in the field of biomedical science who are interested in assessing blood flow under stenosis conditions and developing effective therapies for a wide range of diseases.

Controllability Analysis of Motion of Artificial Satellite Under the Effect of Oblateness of the Earth

In this article we have studied the controllability of artificial satellite under the effect of zonal harmonic J2 in cylindrical polar coordinates systems. Seven different cases of thrusters in various directions have been analyzed and it is found that the system is controllable if we apply thrusters in either r, θ and z or θ and z direction. The equations governing motion of satellite have been linearized and Kalman controllability test is applied to check the controllability of the system. We have also derived controller u for the linearized system. The trajectory of the system has been plotted to show the controllability of the system.

Temperature Fluid Flow of Particles with The Effect of Magnetic Field in A Cylindrical Polar Coordinates of Incompressible Dusty Fluid

The effects of temperature of dusty fluid flow in a weak magnetic field are investigated in this paper. The motion of dusty fluid is affected by a homogeneous magnetic field that runs in the same direction as the fluid. Parameters include longitudinal velocity, Hartmann number; dust particle interactions, stock resistance, Reynolds number, and magnetic Reynolds number fully characterize the topic that is being studied. The obtained partial differential equations are translated into an ode system using appropriate similarity transformations. To solve these equations numerically we use, the Laplace Transform technique. The result shows the temperature of fluid particle reduced significantly in the magnetic field.

Compact analogs of the models of vortex flows generated by aircraft flight

The subject of this work is the development of compact analogs of vortex flows models, which are used in the modeling of vortex structures observed during the flight of an aircraft and the motion of a body in a fluid. In particular, two significant misunderstandings prevailing in this area of science are highlighted. The first misunderstanding is that the stationary motion of fluid parcels in a circle is treated as an inviscid vortex. Therefore, any vortex flow model that does not explicitly contain viscosity is considered to describe inviscid vortex motion. It has been proven that this is not so: the stationary viscous motion of fluid parcels in circular orbits corresponds to the self-balance of one force - the force of viscosity. This conclusion, in an explicit form, was made for the first time. This is very important because it changes our ideas about force balance, where two or more forces of different natures must necessarily be present. Overcoming this misunderstanding opens the way for creating compact analogs of existing models of vortex motions. Along the way, one more - the second general misunderstanding in the field of vortex dynamics was eliminated. Wherever we read it, we can see that the compactness of the vortex flow is associated with the compactness of the vorticity field. This is facilitated by the fact that the equations for vorticity and not for velocity are considered. As a result, except for one or two models of vortices, which correspond to the rotation of the entire space, up to infinity, this violates the fundamental law of physics - the law of conservation and transformation of energy. It is about the fact that, as a second misunderstanding, an error is assumed when calculating the kinetic energy of the vortex flows: the Jacobian in cylindrical (polar) coordinates is not considered. As a result, all the mentioned models of vortex flows, which correspond to the hyperbolic law as their asymptotics in the periphery, have infinite kinetic energy. Certainly, this does not correspond to the formation and evolution of compact vortex structures. Therefore, in this work, based on overcoming the aforementioned misunderstandings, many previously obtained models of compact vortex flows, as well as those obtained for the first time, are presented. In particular, this applies to the turbulent vortex flow during the formation of a vortex sheet, which is a compact analog of the Burgers-Rott vortex - both the classical one corresponding to laminar motion and the one consisting of a laminar flow in the core and a turbulent flow on the periphery of the vortex. Research methods are entirely theoretical. Well-known theorems of theoretical mechanics, mathematical theory of field, and calculus of variations, etc. are used. The obtained solutions are compared with the existing corresponding analogs of non-compact flows. Conclusions. Using the methods of calculus of variation, it was possible to show the possibility of the formation of quasi-solid-like rotational motion in a boundary layer of an incompressible fluid. The very presence of viscosity, or rather its taking into account (boundary layer), indicates a possible transition of the flow from plane-parallel motion to the just-mentioned rotational one due to the Kelvin-Helmholtz instability. In addition, two new models of the Burgers-Rott vortex flow were obtained in this study. The first model uses the general solution obtained by Burgers, but this model corresponds to a combined vortex: although the velocity field is continuous, the vorticity field has a discontinuity - at the point of maximum velocity. It is proved that such an approach is quite possible: the equation of motion is satisfied everywhere, i.e., at every point in space, and the tangential stresses are continuous functions. Since the periphery of the Burgers-Rott vortex is an unstable flow, another model is proposed - with a laminar core and a turbulent periphery. Certainly, the motion of fluid parcels in the peripheral region is described by a velocity distribution other than that of Burgers. Finally, the possible use of known models of compact vortex flows to simulate the von Karman vortex street is considered, with an indication of the advantages of these models.

A balanced model of a hurricane vortex coupled to a boundary layer

AbstractThe boundary layer plays a key role in several aspects of hurricane dynamics. Here we focus on its contribution to the balanced circulation. Previous studies, whilst including centrifugal terms, have not included an explicit balance in the boundary layer. Here, we improve the balanced theory to include an Ekman balance, the so‐called frictional axisymmetric vortex (FAV). This approach is analogous to semigeostrophic theory that includes realistic boundary‐layer diffusion: semigeotriptic (SGT) theory. We formulate the FAV for an axisymmetric system in cylindrical polar coordinates. We then derive a Sawyer–Eliassen equation for the vertical circulation. Example solutions for idealised hurricane‐scale and synoptic‐scale vortices are compared.

A new perspective of higher order compact nonuniform Padé approximation based finite difference scheme for solving incompressible flows directly on polar grids

In this paper, we propose a compact scheme directly on polar grid by combine virtues of computing implicit form of first order derivatives on nonuniform grids with variable coefficients and compact discretization of higher-order derivatives in terms of first order derivatives of the unknown variables. Our present scheme is third order accurate in space. We have solved the two dimensional (2D) fluid flow problems in cylindrical polar coordinates on non-uniform grids. The main efforts are focused on curvilinear grids. Our newly developed scheme is used to solve four different problems, namely a problem having analytical solution, Stokes flow in a half filled annulus, the problem of driven polar cavity and finally the flow past an impulsively started circular cylinder problem. The Navier-Stokes (N-S) and the Stokes equations are efficiently solved with Dirichlet as well as Neumann boundary conditions. For all the test cases, detailed comparison data are reproduced by the proposed scheme and compared with analytical as well as established numerical results available in the literature.

A single-domain spectral solver for spatially nonsmooth differential equations of quasistatic solid mechanics in polar coordinates

In the present work, a spectral solver is developed for integration of certain differential equations of solid mechanics, namely static stress equilibrium in composite materials, described in cylindrical or spherical polar coordinates. The spectral approach is encompassed in approximating the displacement field using expansion into a series of Chebyshev polynomials in the radial coordinate and complex exponents in the angular direction. Consequently, differential operators in real space become algebraic operators in spectral space. The spatial heterogeneity and metric non-flatness pertinent to polar geometry are addressed by an iterative strategy, employing both second-order and first-order iterative solvers. The essence of the new contribution is in addressing the difficulty posed by the inherent nonsmoothness present in composite materials and the polar singularity. The interplay of the two produces instability, which is resolved in the proposed approach, specifically by using a new efficient linesearch algorithm, appropriate for the studied class of problems. The method is illustrated by analysis of 1D and 2D linear-elastic and linear-elastic–perfectly plastic response of composites to prescribed radial surface displacement. The developed method allows performing stress homogenization on polar representative volume elements, which has its conceptual advantages, while allowing similar runtime (for sufficient computing resources and an iterative strategy) to the one exhibited by spectral analysis in Cartesian coordinates.

A study on dual solutions for water-based hybrid nanofluids flowing through convergent channel with dissipative heat transport

Over the years, the fluid flows in conjunction with thermal transport between non-parallel surfaces having converging nature is of great significance due to their broad spectrum of applications, which include fluid flows through nozzles in petroleum engineering, blood flow in arteries, lubrication systems, automobile radiators, thermal pumps, and water purification processes. Additionally, hybrid nanofluid is a prolific topic because of its thermal properties and potentials which provide a better performance even compared with common nanofluid in optimizing heat transfer. Therefore, this article presents a numerical simulation to investigate the heat transport characteristics of hybrid nanofluids in Jeffery-Hamel flow through a convergent channel. The considered hybrid nanofluids are composed of Copper (Cu) and Graphene-oxide (Go) as suspended nanoparticles and water as base fluid. This analysis further includes the impacts of viscous dissipation and magnetic field. A mathematical model for fluid flow and heat transfer are constructed with the help of cylindrical polar coordinates. The governing equations are converted into a system of ordinary differential equations (ODEs) by Lie symmetry group transformation. A MATLAB code is exercise to get the numerical solutions for flow and thermal distributions. An interesting phenomenon is that dual solutions are obtained in the computation. Thus, a comprehensive discussion is included on the dual solutions for various involved variables. The current findings may be employed in petroleum science, r biomedical scientists, polymer industry, etc.

The ciliated flow of water-based graphene oxide and copper nanoparticles (GO-Cu/H2O) in a complex permeable tube with entropy generation phenomenon

In recent years, thermal enhancement using the suspension of hybrid nanofluid has become useful to improve the heat transportation phenomenon in various engineering and industrial processes. The current investigation presents the thermal impacts of hybrid nanofluid with applications of ciliated motion through a cylindrical tube. The concentration of graphene oxide and copper nanoparticles is used in current formulations. Additionally, the physical influences of porosity parameters on the rheology and the heat analysis are examined. The rheological equations are modeled using cylindrical polar coordinates in the laboratory frame. The problem is simplified using two distinct assumptions: the lubrication theory and the long-wavelength estimate. The solutions of rheological equations are attained analytically using the integration technique. The obtained expressions for velocity profile, pressure gradient, temperature profile, and entropy production are described through graphs for numerous embedded parameters. The boundary layer (BL) occurrences are predicted in the axial velocity under higher porosity impacts. The transportation of the hybrid fluid is controlled with the help of the ciliated length. The present research is useful in the chemical processes of thermo-fluidic micro-pumps and devices of nanomaterials.

Effects of non-linear radiation and chemical reaction on Oldroyd‐B nanofluid near oblique stagnation point flow

The oblique stagnation point flow of a viscoelastic nanofluid over a stretching cylinder is entertained in presence of non-linear thermal radiation and chemical reaction. Buongiorno's model is used to incorporate the significance of Brownian motion and thermophoresis diffusion. The governing partial differential equations in cylindrical polar coordinates are first transformed to system of ordinary differential equations using suitable transformations and dimensionless variables and then simulated numerically. Impact of interesting parameters on flow and heat transfer features is analyzed graphically and numerically. It is noticed that axial velocity increases with the retardation parameter while tangential velocity reduces. The magnitude of both axial and tangential velocities is large over stretching cylinder as compared to stretching sheet. Radiation parameter enhanced temperature profile and chemical reaction parameter rise the concentration profile.

Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

We propose a numerical solution to the heat equation in polar cylindrical coordinates by using the meshless method of lines approach. The space variables are discretized by multiquadric radial basis function, and time integration is performed by using the Runge-Kutta method of order 4. In radial basis functions (RBFs), much of the research are devoted to the partial differential equations in rectangular coordinates. This work is an attempt to explore the versatility of RBFs in nonrectangular coordinates as well. The results show that application of RBFs is equally good in polar cylindrical coordinates. Comparison with other cited works confirms that the present approach is accurate as well as easy to implement to problems in higher dimensions.

A New Exponential Compact Scheme for the Two-Dimensional Unsteady Nonlinear Burgers’ and Navier-Stokes Equations in Polar Cylindrical Coordinates

A New Exponential Compact Scheme for the Two-Dimensional Unsteady Nonlinear Burgers’ and Navier-Stokes Equations in Polar Cylindrical Coordinates

Finite Element Analysis of Mixing Flow in a Circular Vessel with Concentric Three Blade Agitator

This study is to investigate the optimisation of the mixing process in two folds, such as, homogenisation of material and to predict the power consumption. This research work is extension of other studies conducted by different researchers. In all other research investigations, no one had employed agitator. The geometry features a cylindrical vessel fixed with a mechanically revolving stirrer along with fixed agitator. The flow is modelled for incompressible constant viscosity Newtonian fluid with isothermal condition. Simulated numerical predictions are achieved through so called a finite element algorithm. The governing equations considered here are two–dimensional equation continuity and time–dependent Navier–Stokes equation in cylindrical polar coordinates. Employed numerical scheme is constructed in multiple–stages. Where, time derivative is discretised in two step quadratic approximation of Taylor series expansion. Whereas, Galerkin approximation is employed for spatial discretisation. Whilst, at second step pressure correction is adopted through projection method. Whilst, implicitness is applied on only diffusion term to make algorithm in semi– implicit form of TGPC. The influences of inertia will be analysed through fluid inertia using dimensionless Reynolds number. The effects of rotational speed of stirrer with agitator will be explored. The computed results will be illustrated for pressure by isobars and flow structure through streamline contours plots. The keyaim of the numerical study is to estimate the improved possible design of the blenders, that enhance the mixing process

Fisher Information of Free-Electron Landau States.

An electron in a constant magnetic field has energy levels, known as the Landau levels. One can obtain the corresponding radial wavefunction of free-electron Landau states in cylindrical polar coordinates. However, this system has not been explored so far in terms of an information-theoretical viewpoint. Here, we focus on Fisher information associated with these Landau states specified by the two quantum numbers. Fisher information provides a useful measure of the electronic structure in quantum systems, such as hydrogen-like atoms and under some potentials. By numerically evaluating the generalized Laguerre polynomials in the radial densities, we report that Fisher information increases linearly with the principal quantum number that specifies energy levels, but decreases monotonically with the azimuthal quantum number m. We also present relative Fisher information of the Landau states against the reference density with , which is proportional to the principal quantum number. We compare it with the case when the lowest Landau level state is set as the reference.

Effects of strain gradient and electromagnetic field gradient on potential and field distributions of multiferroic fibrous composites

The objective of this work is to study the effect of strain gradient, electric field gradient, and magnetic field gradient on the potential and field distributions of multiferroic fibrous composites subjected to generalized anti-plane shear deformation. A detailed energy variational formulation with strain, strain gradient, electric field, electric field gradient, magnetic field, and magnetic field gradient as independent variables is provided, and the equilibrium equations with the complete boundary conditions are simultaneously determined. Three internal characteristic lengths of the underlying microstructure are introduced in the constitutive equations. The general solutions in cylindrical polar coordinates consisting of the modified Bessel functions of first and second kind are derived. One inclusion problem’s solution is also obtained. The derived solutions are applied to a $$\hbox {CoFe}_{2}\hbox {O}_{4}$$ -PZT-4 composite to demonstrate the effect of strain gradient, electric field gradient, and magnetic field gradient. Numerical results provide insights into the role of characteristic lengths on the potential and field distributions of a multiferroic fibrous composite.

Von-Karman rotating flow in variable magnetic field with variable physical properties

The steady incompressible slip flow with convective heat transport under the impact of a variable magnetic field has been taken into an account over a revolving disk. The temperature dependent viscosity, density, and thermal conductivity has been scrutinized. The obtained system of nonlinear differential equations governing the induced magnetic field, steady flow, and heat transmission has put down in polar cylindrical coordinates. The subsequent arrangement of nonlinear PDEs are subside into dimensionless system of ordinary equations, while making use of similarity abstraction. The modeled equations are tackled through Homotopy Analysis Method (HAM). The skin fraction coefficient, heat transmission rate, and Nusselt number (skin effects coefficient) are deliberated. From the results, It can be perceived that the slip factor effectively controls the heat and the flow characteristics. The influence of dimensionless numbers such as Batcheler number [Formula: see text] and magnetic strength [Formula: see text] and [Formula: see text] are explored and shown graphically. Further the out-turn of Prandtl number, relative temperature difference, suction parameter, and slip factor on the temperature fields and velocity profile are discussed.

Stream Function Solution of the Brinkman Equation in Parabolic Cylindrical Coordinates

The present work concerns the general stream function solution of the Brinkman equation in parabolic cylindrical coordinates, arising in the study of fluid flow through porous medium. Analytical stream function solutions of this equation are available in the coordinates (Cartesian, cylindrical polar, spherical polar and prolate spheroidal coordinates). Stream function solution of the Stokes equation in parabolic cylindrical coordinates is also investigated analytically. The parabolic cylinder functions are a class of functions which are the solution of Weber differential equation. A transformation of parabolic cylinder function into the Whittaker function is used. Method of inverse operator is applied to obtain particular integral in solving the Stokes equation. Explicit expressions of velocity components and vorticity are also reported.

An accurate SPH Volume Adaptive Scheme for modeling strongly-compressible multiphase flows. Part 2: Extension of the scheme to cylindrical coordinates and simulations of 3D axisymmetric problems with experimental validations

The present work is dedicated to extending the strongly-compressible multiphase SPH Volume Adaptive Scheme (see [22]) from Cartesian to cylindrical polar coordinates for addressing axisymmetric problems. By omitting the gradient in the circumferential direction, an Axisymmetric-SPH model is developed. Three-dimensional axisymmetric problems including rising bubbles, expanding or collapsing bubbles are conveniently and efficiently simulated using the proposed Axisymmetric-SPH model. Contrary to the purely three-dimensional SPH model established in Cartesian coordinates, with the axisymmetric model, sufficient particle resolutions can be easily adopted to reach converged simulations of complex problems. Axisymmetric-SPH results are validated either using experimental data or other numerical results. In the Axisymmetric-SPH model, a convenient and effective way of avoiding singularity at the axis is presented. In addition, the Volume Adaptive Scheme (VAS), originally developed for compressible flow simulations with large volume variations in Cartesian coordinates, is shown to be a crucial tool to adjust particle volumes in the Axisymmetric-SPH model for all flow cases, including both weakly-compressible and strongly-compressible flows.