Navier-Stokes Partial Differential Equations Research Articles

Analysis of Heat Transfer for the Copper–Water Nanofluid Flow through a Uniform Porous Medium Generated by a Rotating Rigid Disk

This study theoretically investigates the temperature and velocity spatial distributions in the flow of a copper–water nanofluid induced by a rotating rigid disk in a porous medium. Unlike previous work on similar systems, we assume that the disk surface is well polished (coated); therefore, there are velocity and temperature slips between the nanofluid and the disk surface. The importance of considering slip conditions in modeling nanofluids comes from practical applications where rotating parts of machines may be coated. Additionally, this study examines the influence of heat generation on the temperature distribution within the flow. By transforming the original Navier–Stokes partial differential equations (PDEs) into a system of ordinary differential equations (ODEs), numerical solutions are obtained. The boundary conditions for velocity and temperature slips are formulated using the effective viscosity and thermal conductivity of the copper–water nanofluid. The dependence of the velocity and temperature fields in the nanofluid flow on key parameters is investigated. The major findings of the study are that the nanoparticle volume fraction significantly impacts the temperature distribution, particularly in the presence of a heat source. Furthermore, polishing the disk surface enhances velocity slips, reducing stresses at the disk surface, while a pronounced velocity slip leads to distinct changes in the radial, azimuthal, and axial velocity components. The study highlights the influence of slip conditions on fluid velocity as compared to previously considered non-slip conditions. This suggests that accounting for slip conditions for coated rotating disks would yield more accurate predictions in assessing heat transfer, which would be potentially important for the practical design of various devices using nanofluids.

Hemodynamic factors of spontaneous vertebral artery dissecting aneurysms assessed with numerical and deep learning algorithms: Role of blood pressure and asymmetry

Background and ObjectivesThe pathophysiology of spontaneous vertebral artery dissecting aneurysms (SVADA) is poorly understood. Our goal is to investigate the hemodynamic factors contributing to their formation using computational fluid dynamics (CFD) and deep learning algorithms. MethodsWe have developed software that can use patient imagery as input to recreate the vertebrobasilar arterial system, both with and without SVADA, which we used in a series of three patients. To obtain the kinematic blood flow data before and after the aneurysm forms, we utilized numerical methods to solve the complex Navier-Stokes partial differential equations. This was accomplished through the application of a finite volume solver (OpenFoam/Helyx OS). Additionally, we trained a neural ordinary differential equation (NODE) to learn and replicate the dynamical streamlines obtained from the computational fluid dynamics (CFD) simulations. ResultsIn all three cases, we observed that the equilibrium of blood pressure distributions across the VAs, at a specific vertical level, accurately predicted the future SVADA location. In the two cases where there was a dominant VA, the dissection occurred on the dominant artery where blood pressure was lower compared to the contralateral side. The SVADA sac was characterized by reduced wall shear stress (WSS) and decreased velocity magnitude related to increased turbulence. The presence of a high WSS gradient at the boundary of the SVADA may explain its extension. Streamlines generated by CFD were learned with a neural ordinary differential equation (NODE) capable of capturing the system’s dynamics to output meaningful predictions of the flow vector field upon aneurysm formation. ConclusionIn our series, asymmetry in the vertebrobasilar blood pressure distributions at and proximal to the site of the future SVADA accurately predicted its location in all patients. Deep learning algorithms can be trained to model blood flow patterns within biological systems, offering an alternative to the computationally intensive CFD. This technology has the potential to find practical applications in clinical settings.

Гидродинамика сетных жестких безузловых конструкций

Rigid netting structures are elements, parts of commercial fishing gear and aquaculture cages. They serve to enclose or filter hydrobionts, they are engineering structures. Rigid netting structures can be used as inserts in commercial fishing gear, as well as selective gratings and elements that prevent hydraulic backwater in trawls. They are also elements of stationary fishing gear, such as winders or other fishing gear or barriers. In aquaculture cages, rigid netting structures, the elements of which have a sufficiently large value of the longitudinal and transverse modulus of elasticity E, are parts that ensure the strength of the netting structures. Mesh rigid structures are made of plastic or aluminum rods in the form of cylinders, which can be either smooth or twisted. The article considers the application of a numerical method for determining the hydrodynamic characteristics of mesh rigid structures using the software developed by the authors. A schematization of a rigid netting structure, a mathematical model based on Navier-Stokes partial differential equations, a computational domain, initial and boundary conditions are proposed. The calculation was carried out on a regular computational grid using an implicit finite-difference scheme using the methods of coordinate-wise splitting, linearization of nonlinear equations with subsequent correction of nonlinear coefficients, and solution of the obtained tridiagonal systems by the sweep method. The results of numerical experiments are presented in the form of visualization of pressure on the surface of various mesh structures at various angles of attack.

Peristaltic flow of a viscous fluid in a curved duct with a rectangular cross section

Most flow systems in the human body are duct shaped, such as the pancreatic, bile, and gallbladder ducts. Such flow systems are also common in industrial applications like HVAC systems. This study presents a novel mathematical model to analyze the peristaltic motion of a viscous fluid in a three-dimensional curved duct with a rectangular cross section; specifically, such geometries are used more in industrial and medical applications. In the current investigation, the constraints of lubrication theory are considered, and a perturbation technique is used to solve the Navier–Stokes partial differential equations. The major focus of this work is on the aspect ratio of the duct and curvature of the flow axis. Curvilinear coordinates of cylindrical systems are considered for the derivations because of the curved geometry; homogeneous no-slip boundary conditions are proposed at the flexible surfaces, and the expression for pressure increase is found numerically using the NIntegrate tool of computing software Mathematica. A comprehensive graphical discussion is presented to determine the effects of all salient physical factors related to the problem. The results show that the large curvature and aspect ratio reduce the fluid speed gradually but that the flow rate promotes fluid velocity. The pumping rate is a decreasing function of the curvature and aspect ratio; however, reverse pumping can occur for large curvature values. Streamline evaluations suggest that large wave amplitudes increase the number of circulating boluses.

Reduction of Asymptotic Approximate Expansion of Navier–Stokes Equation and Solution of Inviscid Burgers Equation by Similarity Transformation

Symmetry methods for differential equations are a powerful tool for the solutions of differential equations. It linearizes nonlinear differential equations, reduces the order of differential equations, reduces the number of independent variables in partial differential equations, and solves almost all those differential equations for which the other analytic methods fail to solve them. Similarity transformation is a particular case of symmetries, but it is easy and often used to deal with differential equations. The similarity transformation can do all the aforementioned works. In this research, we use the similarity transformation to solve different nonlinear differential equations. Particularly, we will apply this transformation to the nonlinear Navier–Stokes partial differential equations to reduce them to ordinary differential equations. Ordinary differential equations are easy to deal with than partial differential equations. Some nonlinear physical examples of ODEs and PDEs are given to show that the similarity transformation solves those problems where the other analytic methods fail to work.

Insight into the dynamics of the Non-Newtonian Casson fluid on a horizontal object with variable thickness

With quick variations and growths in engineering technology, the activation and binary chemical reaction have sparked vast interest for engineers and scientists due to its immense applications in chemical engineering, food processing, processes of transportation, reservoir of geothermal, etc. In the energy activation, the least amount of energy is required for stimulation of reactants, wherever a chemical change undergoes. Internal energy change of the viscoelastic fluid is the fundamental part of thermophysical properties determining their enactment is a subject of extensive debates over the years. With this significance, we present the steady-state momentum heat and mass transfer flow of a viscoelastic fluid flow in the existence of pre-exponential factor. The velocity of the fluid over the horizontally stretched pin is changed linearly with the axial distance while Casson fluid is supposed as a fluid model. A similarity transformation eases the Navier–Stokes partial differential equations that are transformed into ordinary differential equations and solved numerically through bvp4c solver for the velocity, concentration and energy fields. Moreover, viscosity and conductivity are assumed to be dependent on temperature. Results are discussed near the boundary layer of the pin, while diffusivity is dependent on concentration. A reaction in the form of pre-exponential factor is taken on the surface of pin. Parameters like the ratio parameter, viscosity parameter and viscoelastic parameter are used to control the flow field. We also find that velocity field declines for growing values of viscoelastic parameterγ. Stripe of temperature field shows increasing behavior with positive values of heat generation parameter b but shows adverse behavior with negative values of b. Small values of Dramkohler number Da gives larger values of Prandtl number but in case of Eckert number we saw an opposite behavior. The fitted rate n and temperature difference parameter have conflicting influence on concentration profile. Activation energy E and ε1 causes increment in the behavior of temperature profile. Moreover, numerical data of current paper is compared with previous data.

Influence of the chamber inclination angle and heat-generating element location on thermal convection of power-law medium in a chamber

PurposeThis paper aims to study the mathematical modeling of passive cooling systems for electronic devices. Improving heat transfer is facilitated by the correct choice of the working fluid and the geometric configuration of the engineering cavity; therefore, this work is devoted to the analysis of the influence of the position of the heat-generating element and the tilted angle of the electronic cabinet on the thermal convection of a non-Newtonian fluid.Design/methodology/approachThe area of interest is a square cavity with two cold vertical walls, while the horizontal boundaries are adiabatic. An element of constant volumetric heat generation is placed on the lower wall of the chamber. The problem is described by Navier–Stokes partial differential equations using dimensionless stream function and vorticity. The numerical solution is based on the developed computational code using the finite difference technique and a uniform rectangular grid.FindingsThe key conclusions of this work are the results of a detailed analysis of streamlines and isotherms, the average Nusselt number and profiles of the average heater temperature. It was found that more intensive cooling of the heat-generating element occurs when the cavity is filled with a pseudoplastic fluid (n < 1) and not inclined (α = 0). The Rayleigh number of Ra = 105 and the thermal conductivity ratio of k = 100 are characterized by the most positive effect.Originality/valueThe originality of the research lies in both the study of thermal convection in a square chamber filled with power-law fluid under the influence of a volumetric heat production element and the analysis of the influence of geometric and thermophysical parameters characterizing the considered process.

Thermophyical properties and internal energy change in Casson fluid flow along with activation energy

Abstract This paper examines the steady-state three dimensional momentum and internal energy change in rotating viscoelastic fluid flow along with convective boundary conditions. In most of the literature, the thermophysical properties of the fluid are assumed to be constant. But current analysis fills this gap by taking viscosity, conductivity and diffusivity to be temperature dependent. In order to create a chemical reaction in a system activation energy is added. Velocity of the fluid over an exponential three dimensional surface is varied exponentially while a Casson fluid model is assumed for temperature dependent viscosity. A similarity transformation diminishes the Navier-Stokes partial differential equations into ordinary differential equations and then solved numerically using Bvp4c for the velocity, temperature and concentration distributions. Moreover, drag forces, heat and mass transfer rates near the surface are calculated numerically in tabular form. Outcomes are discussed for parameters appearing in dimensionless system like rotating parameter, the viscoelastic parameter, Eckert number, Prandtl number, temperature distinction parameter, Schmidt number, thermal and concentration Biot numbers and variable viscosity, conductivity and diffusivity parameters. We found that the minimum force required to initiate the fluid motion increases with an increment in local rotation parameter Ω. An accretion in Ω exhibits curiously oscillatory pattern in velocity profile. Casson fluid β and viscosity parameter θ r has adverse influence on temperature profile. The fitted rate n and temperature difference parameter δ have conflicting influence on concentration profile. Activation energy E and Eckert number Ec causes increment in the behavior of temperature profile. In addition, numerical data of previous papers are matched with current data.

Internal energy change and activation energy effects on Casson fluid

This paper examines the steady-state momentum heat and mass transfer flow of a Casson fluid flow in the existence of a pre-exponential factor. The velocity of the fluid over a vertical stretched pin changes linearly with the axial distance when a Casson model is supposed for the viscosity. A similarity transformation eases the Navier–Stokes partial differential equations that are converted into ordinary differential equations and solved numerically for concentration, velocity, and temperature fields. Moreover, viscosity and conductivity are assumed to be dependent on the temperature profile. Results are discussed for two boundary conditions of the pin, while diffusivity is dependent on concentration. A reaction in the form of a pre-exponential factor is taken on the surface of the pin. Parameters such as the mixed convection parameter, viscosity parameter, and viscoelastic parameter are considered for the control of the flow field. In addition, the internal energy change and the Prandtl number are found to examine the temperature field inside the stretched pin, while the Schmidt number, temperature relative parameter, concentration buoyancy parameter, activation energy parameter, and chemical reaction parameter control the concentration field.

Numerical investigation of the heat transfer through woven textiles by the jet system theory

The paper presents a numerical study on the heat transfer in through-thickness direction of single woven layers, based on the jet system theory. A mathematical model, involving the Reynolds-Averaged Navier-Stokes partial differential equations is used, and two turbulence models (k − ɛ and RSM) are applied to solve the closure problem. Numerical results for the temperature distribution, heat rate, heat flux and thermal resistance of the samples are obtained, analyzed, and validated by experimental data. The presented approach for modeling the heat transfer through woven macrostructures is concluded to be a working numerical tool that has the potential to replace costly design iterations and experiments to produce woven textiles with desired performance.

Analytical Reduced Models for the Non-stationary Diabatic Atmospheric Boundary Layer

Geophysical boundary-layer flows feature complex dynamics that often evolve with time; however, most current knowledge centres on the steady-state problem. In these atmospheric and oceanic boundary layers, the pressure gradient, buoyancy, Coriolis, and frictional forces interact to determine the statistical moments of the flow. The resulting equations for the non-stationary mean variables, even when succinctly closed, remain challenging to handle mathematically. Here, we derive a simpler physical model that reduces these governing unsteady Reynolds-averaged Navier–Stokes partial differential equations into a single first-order ordinary differential equation with non-constant coefficients. The reduced model is straightforward to solve under arbitrary forcing, even when the statistical moments are non-stationary and the viscosity varies in time and space. The model is successfully validated against large-eddy simulation for, (1) time-variable pressure gradients, and (2) linearly time-variable buoyancy. The new model is shown to have a superior performance compared to the classic Blackadar solutions (and later improvements on these solutions), and it covers a much wider range of conditions.

Design of bio-inspired computing technique for nanofluidics based on nonlinear Jeffery–Hamel flow equations

In this study, stochastic numerical treatment is presented for boundary value problems (BVPs) arising in nanofluidics for nonlinear Jeffery–Hamel flow (NJ-HF) equations using feed-forward artificial neural networks (ANNs) optimized with bio-inspired computing based on genetic algorithms (GAs) integrated with the active-set method (ASM). NJ-HF equations associated with both convergent and divergent channels, involving nanoparticles, are derived from the transformation of Navier–Stokes partial differential equations to nonlinear BVPs of third-order ordinary differential equations. The mathematical model of the transformed BVPs is developed with the help of ANNs in an unsupervised manner and the design parameters of these networks are trained with GAs, ASM, and GA–ASM. The design scheme is evaluated for NJ-HF by taking water as a base fluid containing three different types of nanomaterials: copper (Cu), alumina (Al2O3), and titania (TiO2) under various scenarios based on the angle of the channels and Reynolds numbers. Accuracy and convergence of the designed scheme are validated through comparison with standard numerical results using the Adams method.

Heat transfer and fluid flow of a non-Newtonian nanofluid over an unsteady contracting cylinder employing Buongiorno’s model

Purpose – The purpose of this paper is to discuss a combined similarity-numerical approach that is used to study the unsteady two-dimensional flow of a non-Newtonian nanofluid over a contracting cylinder using Buongiorno’s model and the Casson fluid model that is used to characterize the non-Newtonian fluid behavior. Design/methodology/approach – Similarity transformations are employed to transform the unsteady Navier-Stokes partial differential equations into a system of ordinary differential equations. The transformed equations are then solved numerically by means of the very robust symbolic computer algebra software MATLAB employing the routine bvpc45. Findings – The effect of increasing values of the Casson parameter is to suppress the velocity field (in absolute sense), the temperature and concentration decrease as Casson parameter increase. The heat and mass transfer rates decrease with the increase of unsteadiness parameters and Brownian motion parameter. In addition, they increase as the Casson parameter and the thermophoresis parameter increase. Originality/value – The problem is relatively original and represents a very important contribution to the field of non-Newtonian nanofluids.

Reduced-order model for the analysis of mass transfer enhancement in membrane channel using electro-osmosis

Flow control has the potential to mitigate concentration polarization and fouling in membrane systems by enhancing mixing near the membrane surface. Although Computational Fluid Dynamics (CFD) modeling has been used to study the effect of externally induced unsteady flow on mass transfer enhancement, the analysis based on CFD results is computationally expensive and cannot be performed systematically. Existing systematic approaches to quantify mixing enhancement only consider hydrodynamics but not the direct effect on mass transfer improvement, due to the difficulties caused by the non-spatially invariant nature of the mass transfer phenomenon. This paper presents a reduced-order model that combines the discretized mass transfer and linearized Navier–Stokes partial differential equations. The proposed model can be used to simulate and systematically analyze mass transfer enhancement caused by the flow induced by a pair of electrodes. When the Reynolds number and temporal frequency of the external field are low (Re<2000), the effect of a forced wall slip velocity on the overall flow profile in a 2D channel can be approximated by its instantaneous component. This allows mass transfer enhancement to be analyzed explicitly using a discretized mass transfer equation. The results predicted by the reduced-order model are in good agreement with CFD simulations. The benefit of the proposed reduced-order model is demonstrated by the frequency response analysis to identify the temporal frequency that has the maximum effect on mass transfer enhancement.

Nonlinear Interaction of &amp;lt;i&amp;gt;N&amp;lt;/i&amp;gt; Conservative Waves in Two Dimensions

Kinematic Fourier (KF) structures, exponential kinematic Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with constant and space-dependent structural coefficients are developed in the current paper to treat kinematic and dynamic problems for nonlinear interaction of N conservative waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The computational method of solving partial differential equations (PDEs) by decomposition in invariant structures, which continues the analytical methods of undetermined coefficients and separation of variables, is extended by using an experimental and theoretical computation in Maple?. For internal waves vanishing at infinity, the Dirichlet problem is formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains. Exact solutions for upper and lower cumulative flows are discovered by the experimental computing, proved by the theoretical computing, and verified by the system of Navier-Stokes PDEs. The KEF and KEF-DEF structures of the cumulative flows are visualized by instantaneous surface plots with isocurves. Modeling of a deterministic wave chaos by aperiodic flows in the KEF, DEF, and KEF-DEF structures with 5N parameters is considered.

Numerical modeling of electrowetting transport processes for digital microfluidics

Electrical actuation and control of liquid droplets in Hele-Shaw cells have significant importance for microfluidics and lab-on-chip devices. Numerical modeling of complex physical phenomena like contact line dynamics, dynamic contact angles or contact angle hysteresis involved in these processes do challenge in a significant manner classical numerical approaches based on macroscopic Navier–Stokes partial differential equations. In this paper, we analyze the efficiency of a numerical lattice Boltzmann model to simulate basic transport operations of sub-millimeter liquid droplets in electrowetting actuated Hele-Shaw cells. We use a two-phase three-dimensional D3Q19 lattice Boltzmann scheme driven by a Shan–Chen-type mesoscopic potential in order to simulate the gas–liquid equilibrium state of a liquid droplet confined between two solid plates. The contact angles at the liquid–solid–gas interface are simulated by taking into consideration the interaction between fluid particles and solid nodes. The electrodes are designed as regions of tunable wettability on the bottom plate and the contact angles adjusted by changing the interaction strength of the liquid with these regions. The transport velocities obtained with this approach are compared to predictions from analytical models and very good agreement is obtained.

Dynamic optimisation of an industrial web process

An industrial web process has been studied and it is shown that theunderlying physics of such processes governs by the Navier-Stokes partialdifferential equations with moving boundary conditions, which in turn have tobe determined by the solution of the thermodynamics equations. Thedevelopment of a two-dimensional continuous-discrete model structurebased on this study is presented. Other models are constructed based onthis model for better identification and optimisation purposes. Theparameters of the proposed models are then estimated using real dataobtained from the identification experiments with the process plant. Varioussimulation tests for validation are accompanied with the design, developmentand real-time industrial implementation of an optimal controller for dynamicoptimisation of this web process. It is shown that in comparison with thetraditional controller, the new controller resulted in a better performance, animprovement in film quality and saving in raw materials. This demonstrates theefficiency and validation of the developed models.

Detailed Flow Field Structure in Rapid Valve Closure.

In this paper the detailed structure of velocity and pressure fields during and subsequent to instantaneous valve closure are presented. The full Navier-Stokes partial differential equations, formulated for a two-dimensional axi-symmetric flow geometry, have been solved to obtain the transient solutions. The Finite Volume Integration Technique together with a procedure involving splitting of operators is employed instead of the Method of Characteristics which is commonly used in the numerical solution of transient flows in pipelines. A solution in two dimensions, viz., in both the pipe axial and radial axis, is obtained for velocity and pressure. The well known oscillations of both the pressure and velocity components have been confirmed. However, it has been revealed, in this work, that the velocity field has a lot of structure in the radial axis which includes a region of re-circulation flow immediately upstream of the closed valve.

Numerical Analysis on the Similarity between Steel Ladles and Hot-water Models Regarding Natural Convection Phenomena.

The similarity between steel ladles and hot-water models regarding natural convection phenomena has been systematically analysed through examination of the numerical solutions of turbulent Navier-Stokes partial differential equations governing the phenomena in question. The numerical solutions have been obtained by using a computational fluid dynamics (CFD) simulation method. Key similarity criteria for non-isothermal physical modelling of steel ladles with hot-water models have been derived as Fr m =Fr p and (βΔT) m =(βΔT) p where the subscripts m and p stand for the water model and the prototype steel ladle, respectively. Accordingly, appropriate conditions fulfilling the above criteria, such as model size, water temperature, time scale factor and the scale factor of boundary heat loss fluxes, have been proposed and discussed. As a result, water models with geometry scales between 1/5 and 1/3 and using hot-water of temperature higher than 45°C are appropriate for simulating natural convection phenomena in steel ladles.

NUMERICAL EVALUATION OF HEAT AND FLUID FLOW WITHIN ROTATING DISKS

Convective heat transport affects the thermal field and drag coefficient in a head-disk assembly significantly. The flow structure and the temperature distribution in ventilated disk passage are being investigated thoroughly in the present work. Two computational methods are used namely: finite volume method (FVM) and finite element method (FEM). Both are based on the Navier–Stokes partial differential equations but with different algorithms and meshes. With different settings of operating conditions on disk, the following important parameters are considered: inlet Reynolds number, rotational speed, shroud clearance and wall temperature. Both co-rotating and counter-rotating disks are also being studied. The former is based on FORTRAN code and the latter on FASTFLO PDE calculator. All the codes are run on a SGI UNIX workstation. It is found that both the Corolis force and centrifugal-buoyancy have important effects on the flow structure and heat transfer due to the rotatinal speed and inlet velocity. Nusselt number decreases along the radius, especially near the disk outer edge. It decreases more rapidly along the radius and the absolute value is much greater for nonventilated disks. The predicted friction coefficients (Cf) are also very different in both cases. For the ventilated case, Cf varies rapidly. While in the nonventilated case, Cf is rather constant when rotational speed (Ω) is low ( 800 rpm), Cf varies appreciably along the radius. Finally, the shroud clearance has the pronunced effects only in the high-Re flow. The results are compared, discussed and their agreement is good.