Cylindrical Polar Coordinate System Research Articles

Subsurface flow model for estimating 2D wetting pattern in drip irrigation

AbstractNumerical model using 2D Richard’s equation in cylindrical polar coordinate system was developed to assess the soil wetting patterns in a drip irrigation system. A fully implicit finite‐difference scheme was used to solve equations with suitable initial and boundary conditions. The developed model was validated with an experimental result available in the literature, which shows a high level of concordance with low values of RMSE and R2 greater than 0.93 for all discharge rates. Furthermore, the versatility and applicability of the model were demonstrated for complex subsurface conditions considering homogeneous and heterogeneous soil profiles. Water ponding on the surface was observed in clay loam soil, contrasting with deep percolation in sandy loam soil. This underscores the significance of adjusting both discharge rate and irrigation frequency in accordance with prevailing soil conditions.

Analysis of flow parameters in blood flow through mild stenosis

A buildup of plaque that contracts arteries and decreases blood flow to the heart causes chest pain, difficulties in breathing, or another coronary artery disease, medically called stenosis puts our lives at risk. We have used Navier-Stokes equations in a cylindrical polar coordinate system to study this problem by considering the flow is steady, axially symmetrical, fully developed, and laminar. Flow parameters like velocity profile, pressure drop, shear stress, and volumetric flow rate in the stenosed regions are analyzed after getting analytical solutions. We have focused our study to know the effect of the thickness of the stenosis in different parameters and the effect of the viscosity coefficient on blood flow behavior.

Two-phase binding of drug in a three-layered arterial wall following stent implantation

The present study explores the distribution and the binding of drug eluted from an embedded drug-eluting stent (DES), having struts of the circular cross-section. The arterial tissue is composed of three distinct layers, namely, the intima, the media and the adventitia, with varying diffusivity, and at the interfaces between different layers, a jump condition is imposed. The transport of free drug is modelled by the convection–diffusion-reaction process while the binding of drug is modelled by a nonlinear saturable reversible chemical reaction only. This model incorporates two types of binding mechanisms, namely, the specific binding caused by the interaction of drug and receptors, and the non-specific binding due to the gripping of drug in the extracellular medium (ECM). Furthermore, a constant as well as time-dependent release kinetics have been considered. All the governing equations along with suitable initial, boundary and jump interface conditions in cylindrical polar coordinate system are solved successfully by using the finite difference method. In this study, several clinical key factors like the degree of strut embedment, inter-strut distance (ISD), release mechanism are examined thoroughly. The study reveals that the inclusion of different layers along with jump interface conditions has a remarkable impact on stent-based delivery. Simulated results predict that a higher concentration profile for free drug and rapid saturation of binding sites take place for a half-embedded stent as compared to a quarterly and well-apposed stent. An intriguing feature is to be noted that the inter-strut distance through which the transmural plasma filtration takes place, plays a pivotal role in the distribution and retention of drug in the therapeutic domain considered. Furthermore, the steady-state simulation predicts the influence of strut dimension on the distributions of all drug forms.

Rise of a bubble through a self-rewetting fluid under the combined influence of gravity-driven convection and Marangoni convection

In this study, the influence of gravity-driven convection and Marangoni convection due to the temperature-dependent surface tension gradient on the rise of an axisymmetric bubble moving in another fluid in a self-rewetting system inside a rectangular tube is studied in the presence and absence of a magnetic field. The axisymmetric bubble (fluid 1) moving in another fluid (fluid 2) is considered immiscible. A two-dimensional cylindrical polar coordinate system has been chosen to present the sketch of the problem. Partial differential equations governing the mentioned flow situations are written and converted into non-dimensional forms and their analytical solutions have been obtained. The deformation in the bubble in the form of its radius and length is simulated. The motion of the droplet is also analysed in the microgravity region by graphing the position of the bubble. The graphical results show that there is a decrease in the contribution of the Marangoni effect and gravitational effect when the magnetic field is increased. In the absence of a magnetic field, the contribution of both the Marangoni effect and gravitational effect decrease on increasing the relative viscosity.

Specific and nonspecific binding of drug eluted from a half-embedded stent in presence of atherosclerotic plaque

This study is dealt with the two-phase binding (specific and nonspecific) of drug eluted from a half- embedded drug-eluting stent in presence of atherosclerotic plaque. The specific binding due to the interaction of drug molecules with specific receptors and nonspecific binding caused by the trapping of drug in the extra-cellular matrix have been paid due attention. An idealised wall consisting of a plaque and a healthy tissue region has been considered. Moreover, a Dirichlet release condition is imposed on the strut surface. In this investigation, a two-dimensional model governing drug transport and its two-phase binding in cylindrical polar coordinate system has been solved numerically by a finite-difference method. Our simulation predicts that plaque behaves like a physical barrier in two types of the binding process and there is an inverse relationship between bound drug concentration and plaque thickness. Simulations show that a single peak profile of drug is noted when the struts are situated one-strut radius apart and as the inter-strut distance increases, the peak concentration falls and distinct peak profiles over each strut are visualised. The model also reveals that in the region downstream of a strut, the concentration of both bound drug forms in the plaque and healthy regions increases, and eventually, the saturation length of binding sites increases. Predicted results show for smaller Damköhler number, the rapid saturation of binding sites takes place and the stent having thinner strut may perform well in terms of effectiveness as well as efficacy in the stent-based delivery.

Thermal Memory Response in Magneto-thermoelastic Medium Having Long Cylindrical Cavity

The present paper deals with the memory response on thermal disturbances emanating from a cylindrical cavity in an unbounded thermoelastic solid. Here we have theoretically demonstrated the memory response of thermal disturbances in the generalized magneto-thermo-elastic materials. Firstly, the characteristics of thermoelastic disturbances originated from the cavity in an unbounded elastic solid under the light of generalized magnetothermoelasticity theory with memory dependent derivatives (MDD). For numerical computation, cylindrical-polar coordinate system with radial symmetry subjected to two different types of heat sources into the cavity are considered. An integral transform method and, while in inverse transformation, an efficient and pragmatic NILT (Numerical Inverse Laplace Transform) is adopted. Finally, parameter studies are performed to evaluate the effect of the kernel function and time delay. For thermal wave the results show appreciable differences with those in the usual magneto-thermoelasticity theory.

Memory response on thermoelastic deformation in a solid half-space with a cylindrical hole

The present article deals with the memory response on thermal wave propagation emanating from a cylindrical hole in an unbounded thermoelastic solid. Here, we have theoretically demonstrated two aspects of thermal waves in elastic materials with memory. First, the characteristics of thermoelastic disturbances originated from the hole in an unbounded elastic solid under the light of generalized thermoelasticity theory with memory-dependent derivatives (MDDs). For numerical implementation, cylindrical polar coordinate system with radial symmetry is considered. An integral transform method and, while in inverse transformation, an efficient and pragmatic NILT (numerical inverse Laplace transform) is adopted. Second, a comparison study is being made on the memory response of thermal waves with and without energy dissipation into the medium. Finally, parameter studies are performed to evaluate the effect of the kernel function and time delay. For thermal wave, the results show appreciable differences with those in the usual thermoelasticity theory.

Adomian decomposition method simulation of von Kármán swrling bioconvection nanofluid flow

The study reveals analytically on the 3-dimensional viscous time-dependent gyrotactic bioconvection in swirling nanofluid flow past from a rotating disk. It is known that the deformation of the disk is along the radial direction. In addition to that Stefan blowing is considered. The Buongiorno nanofluid model is taken care of assuming the fluid to be dilute and we find Brownian motion and thermophoresis have dominant role on nanoscale unit. The primitive mass conservation equation, radial, tangential and axial momentum, heat, nano-particle concentration and micro-organism density function are developed in a cylindrical polar coordinate system with appropriate wall (disk surface) and free stream boundary conditions. This highly nonlinear, strongly coupled system of unsteady partial differential equations is normalized with the classical Von Karman and other transformations to render the boundary value problem into an ordinary differential system. The emerging 11th order system features an extensive range of dimensionless flow parameters i.e. disk stretching rate, Brownian motion, thermophoresis, bioconvection Lewis number, unsteadiness parameter, ordinary Lewis number, Prandtl number, mass convective Biot number, Peclet number and Stefan blowing parameter. Solutions of the system are obtained with developed semi-analytical technique i.e. Adomian decomposition method. Validation of the said problem is also conducted with earlier literature computed by Runge-Kutta shooting technique.

The Effect of Slip Velocity on Unsteady Peristalsis MHD Blood Flow through a Constricted Artery Experiencing Body Acceleration

In this analysis, we present a theoretical study to examine the combined effect of both slip velocity and periodic body acceleration on an unsteady generalized non-Newtonian blood flow through a stenosed artery with permeable wall. A constant transverse magnetic field is applied on the peristaltic flow of blood, treating it as an elastico-viscous, electrically conducting and incompressible fluid. Appropriate transformation methods are adopted to solve the unsteady non-Newtonian axially symmetric momentum equation in the cylindrical polar coordinate system with suitably prescribed conditions. To validate the applicability of the proposed analysis, analytical expressions for the axial velocity, fluid acceleration, wall shear stress and volumetric flow rate are computed and for having an adequate insight to blood flow behavior through a stenosed artery, graphs have been plotted with varying values of flow variables, to analyse the influence of the axial velocity, wall shear stress and volumetric flow rate of streaming blood.

Axisymmetric lattice Boltzmann method for non‐Newtonian flows

SummaryA new axisymmetric lattice Boltzmann model is proposed for non‐Newtonian fluid flows. The model incorporates the additional axisymmetric contributions into the evolution equation by introducing a force term. The precise form of the Navier‐Stokes equations in the cylindrical polar coordinate system is derived through Chapman‐Enskog analysis, and the consistency in dimension with the lattice Boltzmann equation is retained in the introduction of force terms. The model is used to simulate the flows in a straight pipe and an annulus channel for various power‐law indexes and Reynolds number. The velocity profiles in the straight pipe and annulus channel are presented and shown to be in excellent agreement with analytical results. The 4:1 axisymmetric contraction flow was studied as well. The results show that the reattachment length increases as the power‐law indexes for a shear‐thickening fluid. The opening angle increases substantially with the increase of the power‐law index for both a shear‐thinning fluid and a shear‐thickening fluid. The velocity profiles and vortex sizes of contraction flows agree very well with those of the finite element analysis. Results demonstrate that the proposed method is very effective to analyze the axisymmetric flows of a non‐Newtonian fluid.

Effect of flow pulsatility and time-dependent release kinetics on stent-based delivery through atherosclerotic plaque

The present work is concerned with an investigation of the effect of flow pulsatility and time-dependent release kinetics on the transport and retention of drug in a stent-based delivery through atherosclerotic plaque. Keeping the relevance of the physiological situation in view, an axi-symmetric two-dimensional model of drug delivery from a strut has been developed in cylindrical polar coordinate system. Here, the luminal drug transport is considered as an unsteady convection–diffusion process, while the drug transport within the arterial tissue is considered as a two-species model (both free and bound), capable of predicting the delivery of drug and its retention in the arterial tissue, in which the specific binding site action is modelled using a reversible chemical reaction. The transport of free drug is modelled as an unsteady convection–diffusion–reaction process, while the bound drug, assuming completely immobilised in the tissue, as an unsteady reaction process. An explicit finite-difference scheme in staggered grids has been developed to tackle numerically the governing equations of motion in order to compute the physiologically significant quantities with desired degree of accuracy. Model results predicted that, unlike within the tissue, the averaged free drug concentration increases with the increase of Re in the lumen and also the averaged concentrations of both drug forms decrease with increasing plaque thickness. Moreover, higher Peclet number in the tissue and coating results in lowering down the averaged concentrations of free as well as bound drug in the tissue and lumen. The present results predict that the presence of plaque dramatically affects the local pharmacokinetics and provides higher resistance to drug transfer in the arterial wall. The averaged concentration of free drug increases in the lumen due to flow pulsatility while a reverse trend is observed in the tissue. Additionally, modelling results also indicate that time-dependent release of drug in stent-based delivery diminishes the averaged concentration of both drug forms as compared to constant release kinetics. All the above observations are consistent with several existing results in the literature.

Effects of Interstrut Distance on the Transport of Drug and its Retention in the Arterial Tissue

The present investigation is dealt with the effect of interstrut distance on the transport and retention of drug eluted from a well-apposed drug-eluting stent (DES). We consider a two-species model (free and bound), capable of predicting the delivery of drug and its retention in the arterial tissue, in which the specific binding site action is modelled using a reversible chemical reaction. The model also takes into account a time-dependent release kinetics of DES. An axi-symmetric model of drug delivery from a pair of stent struts has been developed in cylindrical polar coordinate system where the transport of free drug is modelled as an unsteady convection–diffusion-reaction process, while the bound drug as an unsteady reaction-diffusion process. An explicit finite-difference scheme has been used to tackle numerically the governing equations of motion together with the physiologically realistic boundary conditions. The effects of quantities of significance such as Peclet number (Pe), equilibrium coefficient (\(K_{eq}\)) and interstrut distance on the transport of free as well as bound drugs are quantitatively investigated graphically. The present results predict a single peak profile of drug concentration when the pair of struts are placed one strut-width apart and also as the interstrut distance increases, discrete peaks form over each strut which is consistent with several existing results in the literature.

Analytical Solution for Three-Dimensional, Unsteady Heat Conduction in a Multilayer Sphere

An analytical solution has been obtained for the transient problem of three-dimensional multilayer heat conduction in a sphere with layers in the radial direction. The solution procedure can be applied to a hollow sphere or a solid sphere composed of several layers of various materials. In general, the separation of variables applied to 3D spherical coordinates has unique characteristics due to the presence of associated Legendre functions as the eigenfunctions. Moreover, an eigenvalue problem in the azimuthal direction also requires solution; again, its properties are unique owing to periodicity in the azimuthal direction. Therefore, extending existing solutions in 2D spherical coordinates to 3D spherical coordinates is not straightforward. In a spherical coordinate system, one can solve a 3D transient multilayer heat conduction problem without the presence of imaginary eigenvalues. A 2D cylindrical polar coordinate system is the only other case in which such multidimensional problems can be solved without the use of imaginary eigenvalues. The absence of imaginary eigenvalues renders the solution methodology significantly more useful for practical applications. The methodology described can be used for all the three types of boundary conditions in the outer and inner surfaces of the sphere. The solution procedure is demonstrated on an illustrative problem for which results are obtained.

Electromagnetic Wave Propagation Through Two Coaxial Transformation-based Cylindrical Media

Maxwell’s wave equation is solved analytically in a general medium designed by transformation optics theory with a general form of transformation function on radial coordinate in a cylindrical polar coordinate system. Some well-known examples such as interior invisibility cloak, concentrator and external cloak are solved according to this approach. We particularly concentrate on the external invisibility cloak which was not considered before. In order to investigate the functionality and correctness of the solutions, some simulations with different form of transformation functions are created according to this approach and the simulations are compared with the ones obtained by another method (finite element). Comparison of the results extracted from the two methods show a very good coincidence in all cases.

Nested numerical scheme in a polar coordinate shallow water model for the coast of Bangladesh

This study is concerned with the development of storm surge model to predict water level accurately due to surge associated with a cyclone along the coast of Bangladesh. Considering the funnel shape of the Bay of Bengal, coastal bending and the existence of off-shore islands, a vertically integrated model in cylindrical polar coordinate system is developed using nested numerical scheme. A fine mesh scheme capable of incorporating coastal bending and offshore islands properly in the numerical scheme is nested into a coarse mesh scheme covering up to 15° N latitude of the Bay of Bengal. The developed model is used to estimate water levels at different coastal and island stations associated with a few storms that hit the coast of Bangladesh. The computed surge levels compared well with observed ones and predicted surge levels investigated with different approaches.

NETWORK NUMERICAL ANALYSIS OF OPTICALLY THICK HYDROMAGNETIC SLIP FLOW FROM A POROUS SPINNING DISK WITH RADIATION FLUX, VARIABLE THERMOPHYSICAL PROPERTIES, AND SURFACE INJECTION EFFECTS

The steady, incompressible, laminar Newtonian magnetohydrodynamic slip flow with heat transfer from an impulsively started, spinning porous disk is investigated when strong injection (blowing) and significant thermal radiation heat transfer are present. The properties of the fluid, i.e., density, viscosity, and thermal conductivity, are assumed to vary with temperature. Using appropriate transformations, the axisymmetric flow conservation equations for mass, momentum, and energy in a cylindrical polar coordinate system (r, ϕ, z) are normalized to yield a series of highly nonlinear, coupled ordinary differential equations that are solved under appropriate boundary conditions with the network simulation method (NSM). Comparisons are made with an earlier study for the case of Prandtl number = 0.64 with suction present and found to be in excellent agreement. The effects of the radiation-conduction parameter (Nr), hydromagnetic parameter (Nm), slip factor (γ, which is related to Knudsen number), uniform injection parameter (W > 0), and temperature difference parameter (ϵ) on the axial, radial, and tangential velocity components, Nusselt number, and temperature function are investigated in detail. Applications of the study include turbine blade systems, magnetic field control of chemical engineering processes, and electronic computer disk drive cooling.

Using Stress Wave Based Technology for Wood Material Nondestructive Testing

Stress wave based technology has its special advantages in the field of wood nondestructive testing and quality evaluation. To analysis the propagating properties of stress wave in the wood material, a transmitting model in anisotropic material was introduced in rectangular and cylindrical polar coordinate system. The relationship between stress wave propagation and elastic constants was shown. The propagating of stress wave in wood material and the possible solutions with different parameters were studied. The transmitting character of stress wave in symmetry planes was analyzed, and the elastic constants of woods were used to compute the stress wave propagation in wood material.

A numerical simulation of unsteady blood flow through multi-irregular arterial stenoses

An unsteady mathematical model to study the characteristics of blood flowing through an arterial segment in the presence of a couple of stenoses with surface irregularities is developed. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mildly stenosed artery [1], so that the flow effectively becomes two-dimensional. The governing equations of motion accompanied by appropriate choice of boundary and initial conditions are solved numerically by MAC (Marker and Cell) method in cylindrical polar coordinate system in staggered grids and checked numerical stability with desired degree of accuracy. The pressure-Poisson equation has been solved by successive-over-relaxation (SOR) method and the pressure–velocity correction formulae have been derived. The flexibility of the arterial wall has also been accounted for in the present investigation. Further, in-depth study in the flow pattern reveals that the separation Reynolds number for the multi-irregular stenoses is lower than those for cosine-shaped stenoses and a long single irregular stenosis. The present results predict the excess pressure drop across the cosine stenoses than the irregular ones and show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.

Interception of two spherical drops in linear Stokes flow

A boundary-integral method is developed for computing the interception of two spherical drops with arbitrary radii and viscosities in infinite linear Stokes flow. At any instant, the flow is computed in a frame of reference with origin at the center of one drop, using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second drop. Taking advantage of the axial symmetry of the interfaces in the drop coordinates, the problem is formulated as a system of integral equations for the zeroth, first, and second Fourier coefficients of the normal component of the jump in the interfacial traction and for the meridional and azimuthal components of the interfacial velocity with respect to the meridional angle. The integral equations are solved with high accuracy using a boundary-element method featuring adaptive boundary-element distribution and automatic time-step adjustment according to the interfacial gap. Simulations of two drops intercepting in uniaxial straining flow provide accurate data on the drop collision velocity and particle stress tensor for gaps as small as 10−4 times the drop radius. Simulations of two drops intercepting in simple shear flow confirm that slightly offset drops collide during the interception. Accurate data are presented for Batchelor’s relative mobility functions in linear Stokes flow used to describe the relative droplet motion.

INTERPRETING THE M2-BRANE ACTION

The world-volume theory of multiple M2-branes proposed recently has a free scalar field. For large vev of this scalar field the world-volume action reduces to that of multiple D2-branes with Yang–Mills coupling proportional to the vev. We suggest that the correct interpretation of this scalar field is as the radial position of the M2-brane center of mass in a cylindrical polar coordinate system. Regarding the azimuthal angle as compact we can regard this as a set of coincident D2-branes in type IIA string theory with varying dilaton and metric. We find that the effective world-volume theory on the D2-branes has Yang–Mills coupling proportional to the radial coordinate; furthermore the radial coordinate satisfies free field equations of motion. This agrees with the corresponding results derived from the M2-brane world-volume theory.