Flow Field Equations Research Articles

Variable heat source in stagnation-point unsteady flow of magnetized Oldroyd-B fluid with cubic autocatalysis chemical reaction

Understanding the flow behavior of non-Newtonian fluids required more research in terms of various aspects. Because the role of heat transfer in non-Newtonian flows is dominant due to its importance in technology. For these benefits, a theoretical investigation is carried out to analyze the time-dependent flow of non-Newtonian fluid with a focus on thermal and solutal transport. In the present study, we formulate a mathematical model for Oldroyd-B fluid by taking into account the effect of magnetic field. The governing flow field equations are constructed with the help of the rheological expression of Oldroyd-B fluid model. For physical relevance, the effects of non-uniform heat source/sink, Joule heating and thermal radiation with homogeneous-heterogeneous chemical reactions have been incorporated in stagnation point flow towards a stretching cylinder to modify the energy and concentration distributions. The highly non-linear ordinary differential equations are tackled analytically through the optimal homotopy analysis method (OHAM). Study reveals that, the strength of homogeneous and heterogeneous reactions leads to enhanced the concentration of the catalyst at the surface. Furthermore, the non-uniform heat source/sink parameter acts like a major controlling parameter for the heat transportation rate, and heat transport is also influenced by the surface-dependent heat source/sink.

Thermal case study on linearly twisting cylinder: A radial stagnation point flow of nanofluid

The thermal case study is performed to inspect heat transfer aspects for radial stagnation point flow. For this phenomenon, the study assumes a linear twisting cylinder for the heated viscous nanofluid and the arrangement is investigated among the torsionally rotating cylinders and radial stagnation point flow. The dimensionless torsion rate σ , the volume fraction of nanoparticles φ and the Wangs Reynolds number R are three possible parameters that are articulated in the study. The BVP-Midrch routine integrated with the Maple software is used to obtain numerical results of the governing flow field equations. It is found that the axial wall stress f ″ ( 1 ) is a very weak function of σ , a weak function of φ and a strong function of R . Also, the azimuthal wall stress g ′ ( 1 ) is a strong function of R and shows a practically decreasing behavior and is linear with σ for the starting value g ′ ( 1 ) = 0 at σ = 0 . It is seen that the temperature gradient admits trifling variation for the positive iteration in torsional rate. Furthermore, the heat flux at surface admits direct relation towards volume fraction of nanoparticles.

Quantum geometric information flows and relativistic generalizations of G. Perelman thermodynamics for nonholonomic Einstein systems with black holes and stationary solitonic hierarchies

We investigate classical and quantum geometric information flow theories (respectively, GIFs and QGIFs) when the geometric flow evolution and field equations for nonholonomic Einstein systems, NES, are derived from Perelman-Lyapunov type entropic type functionals. In this work, the term NES encodes models of gravitational and matter fields interactions and their geometric flow evolution subjected to nonholonomic (equivalently, non-integrable, anholonomic) constraints. There are used canonical geometric variables which allow a general decoupling and integration of systems of nonlinear partial differential equations describing GIFs and QGIFs and (for self-similar geometric flows) Ricci soliton type configurations. Our approach is different from the methods and constructions elaborated for special classes of solutions characterized by area--hypersurface entropy, related holographic and dual gauge--gravity models, and conformal field theories, involving generalizations of the Bekenstein-Hawking entropy and black hole thermodynamics. We formulate the theory of QGIFs which in certain quasi-classical limits encodes GIFs and models with flow evolution of NES. There are analysed the most important properties (inequalities) for NES and defined and computed QGIF versions of the von Neumann, relative and conditional entropy; mutual information, (modified) entanglement and R\'{e}nyi entropy. We construct explicit examples of generic off-diagonal exact and parametric solutions describing stationary solitonic gravitational hierarchies and deformations of black hole configurations. Finally, we show how Perelman's entropy and geometric thermodynamic values, and extensions to GIF and QGIF models can be computed for various new classes of exact solutions which cannot be described following the Bekenstein-Hawking approach.

Melting heat analysis in squeezed flow of nanofluids (SWCNTs + Water, SWCNTs + Gasoline oil, MWCNTs + Water, MWCNTs + Gasoline oil) with Cattaneo–Cristov heat flux

Abstract This analysis emphasizes impacts of CNTs (Carbon nanotubes) during Darcy–Forchheimer squeezed flow. Melting heat transfer process is addressed. Water and gasoline are treated as baseliquid. By transformation method non-linear system of ODEs are obtained from the flow field equations (PDEs). Further the emerged flow problem is solved via OHAM. Influences of physical involved parameters on velocity, surface friction coefficient, temperature as well as Nusselt number are expressed graphically and interpreted physically.

Electrostatic Potential Analysis in Polyelectrolyte Brush-Grafted Microchannels Filled with Polyelectrolyte Dispersion

In this study, the model framework that includes almost all relevant parameters of interest has been developed to quantify the electrostatic potential and charge density occurring in microchannels grafted with polyelectrolyte brushes and simultaneously filled with polyelectrolyte dispersion. The brush layer is described by the Alexander-de Gennes model incorporated with the monomer distribution function accompanying the quadratic decay. Each ion concentration due to mobile charges in the bulk and fixed charges in the brush layer can be determined by multi-species ion balance. We solved 2-dimensional Poisson–Nernst–Planck equations adopted for simulating electric field with ion transport in the soft channel, by considering anionic polyelectrolyte of polyacrylic acid (PAA). Remarkable results were obtained regarding the brush height, ionization, electrostatic potential, and charge density profiles with conditions of brush, dispersion, and solution pH. The Donnan potential in the brush channel shows several times higher than the surface potential in the bare channel, whereas it becomes lower with increasing PAA concentration. Our framework is fruitful to provide comparative information regarding electrostatic interaction properties, serving as an important bridge between modeling and experiments, and is possible to couple with governing equations for flow field.

An Entropy-stable Ideal EC-GLM-MHD Model for the Simulation of the Three-dimensional Ambient Solar Wind

Abstract The main aim of the current work is to apply the Roe+Lax–Friedrichs (LF) hybrid entropy-stable scheme to the simulation of the three-dimensional ambient solar wind. The governing equations for the solar wind flow and magnetic field utilize the entropy-consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics (MHD) equations, which are symmetric and Galilean invariant with some nonconservative terms proportional to the divergence of magnetic field or the gradient of the Lagrange multiplier ψ. By using solenoidality-preserving and non-negativity-preserving reconstruction, the divergence error is further constrained, and the densities and pressures are reliably guaranteed. Moreover, the entropy is used as an auxiliary equation to completely avoid the appearance of negative pressure, which is independent of any numerical flux and can be retrofit into any MHD equations straightforwardly. All the properties referred to above make the newly developed scheme more handy and robust to cope with the high Mach number or low plasma β situations. After the experiments of the entropy consistency and the robustness of the proposed entropy-stable scheme through two simple tests, we carry out the simulation of the large-scale solar wind structures for Carrington Rotation 2183 (CR 2183) in a six-component grid system with the initial potential field obtained from the Helioseismic and Magnetic Imager magnetogram by retaining spherical harmonics of degree 50. The comparisons of the numerical results with the remote sensing observations and in situ data show that the new model has the capability to produce structured solar wind.

Investigating the effects of a non - uniform magnetic field on heat and flow characteristics of a ferrofluid

A numerical study is performed to investigate the effect of a non - uniform magnetic field from a current carrying wire on the ferrofluid flow. The analysis is carried out for a semi circular annulus with three different locations of wire relative to it, by solving coupled set of flow field equations, energy equations and the Maxwell’s magnetostatics equations. Results from the present study offers better insight about the ferrofluid behaviour and heat transfer mechanism. It also explains the dependency of flow distribution on the location of the electric wire and the magnitude of current flowing through it.

Phase field feature of inclined hydraulic fracture propagation in naturally-layered rocks under stress boundaries

The phase field feature of inclined hydraulic fracture propagation in naturally-layered rocks under stress boundaries is investigated by using a numerical method. Based on a phase field fracture model, the coupled governing equations of displacement field, phase field, and flow field concerning hydraulic fracturing in naturally-layered rocks were established. The equations were solved using the finite element method and the influences of initial stress field and stiffness contrast on the fracture patterns were deeply studied. The numerical simulations indicate that: 1) The ratio of vertical in-situ stress to horizontal in-situ stress (S v/S h) has a significant effect on propagation of the hydraulic fracture. With the increase in S v/S h, the hydraulic fracture deflects and propagates along the direction of S v, which is also the maximum in-situ stress; 2) The stiffness contrast of the two layers (E 1/E 2) has great influence on fracture penetration into the adjacent layer. For a low E 1/E 2, a singly-deflected scenario is observed because the fracture propagation is depressed by the stiff rock. With the increase in E 1/E 2, the hydraulic fracture more tends to penetrate into the adjacent layer.

Numerical study of entropy generation in Darcy-Forchheimer (D-F) Bödewadt flow of CNTs

Three-dimensional Bödewadt flow (fluid rotates at a large enough distance from the stationary plate) of carbon nanomaterial is examined. Single walled and multi walled CNTs are dissolved in water and gasoline oil baseliquids. Darcy-Forchheimer porous medium is considered. Stationary disk is further stretched linearly in radial direction. Heat transfer effect is examined in presence of radiation and convection. Effect of viscous dissipation is accounted. Entropy generation rate is studied. By using adequate transformation (von Kármán relations), the flow field equations (PDEs) are transmitted into ODEs. Solutions to these ODEs are constructed via implementation of shooting method (bvp4c). In addition to Entropy generation rate, Bejan number, heat transfer rate (Nusselt number), skin friction and temperature of fluid are examined through involved physical parameters. Axial component of velocity intensifies with increment in nanoparticles volume fraction and ratio of stretching rate to angular velocity parameter while it decays with higher porosity parameter. Higher nanoparticles volume fraction and porosity parameter lead to decay in radial as well as tangential component of velocity. However it enhances with higher ratio of stretching rate to angular velocity parameter. Temperature of fluid directly varies with higher ratio of stretching rate to angular velocity parameter, radiation parameter, Eckert number, Biot number and nanoparticles volume fraction. Rate of Entropy generation is reduced with higher estimations of porosity parameter, nanoparticles volume fraction and radiation parameter. Skin friction coefficient decays with higher porosity parameter and ratio of stretching rate to angular velocity parameter. Intensification in porosity parameter, nanoparticles volume fraction and Biot number leads to higher Nusselt number. Prominent impact is shown by multiple-walled CNTs with gasoline oil basefluid than single-walled CNTs with water basefluid.

The numerical modeling and study of gas entrapment phenomenon in non-isothermal polymer filling process

• The problem of gas entrapment in the non-isothermal polymer filling process is simulated. • The finite element method coupled with implicit discontinuous Galerkin method is developed for the simulation. • The influence of injection velocity and gate size on the gas entrapment phenomenon have been analyzed. In this paper, the intricate gas entrapment phenomenon in the non-isothermal polymer filling process is investigated via a combined finite element and discontinuous Galerkin numerical algorithm. The evolving melt interface is captured via level set method and the eXtended Pom-Pom (XPP) constitutive model is employed to describe the viscoelastic fluid. The governing equations of flow field are solved via a coupled continuous and discontinuous Galerkin method. The implicit discontinuous Galerkin method is utilized for the solution of the XPP constitutive equation, level set and its re-initialization equations. The good agreement of experimental and simulation results illustrates the validity of this combined computational algorithm. The influence of injection velocity and the gate size on the gas entrapment phenomenon are both studied in the irregular cavity with complex insert. We have found that the entrapped gas is easy to appear for the case of higher injection velocity and smaller gate size. The distribution of temperature field in different cases are also shown.

Thermal analysis of buoyancy‐motivated Casson fluid flow with time‐independent chemical reaction under Lorentz forces

AbstractThe present research study examines the magneto‐hydrodynamic natural convection visco‐elastic boundary layer of Casson fluid past a nonlinear stretching sheet with Joule and viscous dissipation effects under the influence of chemical reaction. To differentiate the visco‐elastic nature of Casson fluid with Newtonian fluids, an established Casson model is considered. The present physical problem is modeled by utilizing the considered geometry. The resulting system of coupled nonlinear partial differential equations is reduced to a system of nonlinear ordinary differential equations by applying suitable similarity transformations. Numerical solutions of these reduced nondimensional governing flow field equations are obtained by applying the Runge‐Kutta integration scheme with the shooting method (RK‐4). The physical behavior of different control parameters is described through graphs and tables. The present study describes that the velocity and temperature profiles decreased for increasing values of Casson fluid parameter. Velocity field diminished for the increasing nonlinear parameter whereas velocity profile magnified for increasing free convection parameter. Thermal field enhanced with increasing magnetic parameter in the flow regime. The concentration profile decreased for the rising values of the chemical reaction parameter. The magnitude of the skin‐friction coefficient enhanced with increasing magnetic parameter. Increasing Eckert number increases the heat transfer rate and increasing chemical reaction parameter magnifies the mass transfer rate. Finally, the similarity results presented in this article are excellently matched with previously available solutions in the literature.

3D numerical study of heat and mass transfer of moving porous moist objects

Heat and moisture transfer of two moving moist objects in a channel with convection of hot dry air is investigated numerically. A moving mesh approach is used for providing movement by means of the Arbitrary Lagrangian-Eulerian (ALE) framework. Governing equations for flow field and porous objects are solved by finite element discretization method for the 3D transient problem. Effects of drying air velocity and temperature are simulated by means of temperature, moisture content and heat transfer distributions. Results showed the importance of varying parameters on drying behavior. It was observed that the air velocity also has an increasing effect on evaporation and most reduction in moisture content is obtained at 0.8 m/s air velocity with 63.12% loss. Besides, there are insignificant differences between heat transfer values for fixed and moving objects, differences in average heat transfer values calculated between 6 and 13% for static bed and moving bed.

Wetting dynamics of a sessile ferrofluid droplet on solid substrates with different wettabilities

There are several numerical approaches to define a permanent magnet in terms of mathematical equations, and each approach has progressed since its inception, but still endures some limitations on specific numerical phenomena. This study seeks to propose a novel numerical representation of a permanent magnet without incorporating its effect through boundary conditions, which overcomes the limitations of previous studies and enables us to introduce a magnetic field of desired strength at any location. A self-correcting method is modified to incorporate the magnetic field effects, while a simplified lattice Boltzmann method is utilized to solve the governing equations for flow field and interface. The validity of the proposed method is ensured by simulating some benchmark phenomena with and without the external magnetic field. This study also investigates the wetting dynamics of a sessile ferrofluid droplet deposited on solid substrates with different wettabilities. The influence of uniform and non-uniform magnetic fields on droplet spreading is discussed in detail. It is observed that for a non-uniform magnetic field in vertical direction, the ferrofluid droplet on a hydrophilic surface does not observe the spherical cap approximation unless the magnetic field strength is below saturation magnetization. Moreover, if the magnet is located above, the droplet undergoes large deformations and achieves pointy shapes with sharp tips on less wettable surfaces.

Conjugate Analysis of Silica-Phenolic Charring Ablation Coupled with Interior Ballistics

Because of its excellent insulation capability, the usage of a silica-phenolic charring ablator as a nozzle liner is a common practice in the solid rocket motor industry. During the design of a solid rocket motor employing a silica-phenolic nozzle liner, it is desired to conduct an accurate analysis yielding in-depth thermal response and recession characteristics. As the interior ballistics and nozzle recession rate mutually interact, the best practice is to perform a coupled solution to both. Commonly used one-dimensional analysis tools with empirical approaches for estimation of convective heat transfer rate and blowing effect generally lack sought accuracy and do not model the transient shape-change phenomenon, which affects the nozzle performance. This Paper considers governing equations for charring, including pyrolysis gas injection and surface energy balance for melting ablation, along with a boundary condition governed by interior ballistics, and demonstrates a framework in which these equations are solved with governing equations for the nozzle flowfield in a coupled manner. Development and validation of a one-dimensional material response solver based on the same governing equations is also demonstrated. Also, results from a static firing test conducted with a small-scale ballistic evaluation motor employing a silica-phenolic nozzle insert are provided. Results from both investigations are compared and discussed.

Magnetized couple stress fluid flow past a vertical cylinder under thermal radiation and viscous dissipation effects

Abstract Contemporary investigation studies the silent features of the dissipative free convection couple stress fluid flow over a cylinder under the action of magnetic field, thermal radiation and porous medium with chemical reaction effect. Present two-dimensional viscous incompressible physical model is designed based on the considered flow geometry. Present physical problem gives the highly complicated nonlinear coupled partial differential equations (PDE's) which are not amenable to any of the known techniques. Thus, unconditionally stable, most accurate and speed converging with flexible finite difference implicit technique is utilized to simplify the dimensionless flow field equations. It is apparent from the current results that; the velocity profiles are diminished with enhancing values of magnetic field. Temperature profile increases with enhancing values of thermal radiation parameter. Velocity contours deviates away from the wall with enhancing magnetic parameter. Also, the effects of magnetic field, porous medium, thermal radiation, chemical reaction, buoyancy ratio parameter and Eckert number on couple stress flow velocity, temperature, and concentration profiles are studied. However, the present study has good number of applications in the various fields of engineering such as; polymer processing, solidification of liquid crystals, colloidal solutions, synovial joints, geophysics, chemical engineering, astrophysics and nuclear reactors etc. Finally, the current solutions are validated with the available results in the literature review and found to be in good agreement.

Mixed convective nanofluid flow over a non linearly stretched Riga plate

The present article concentrates on the heat and mass transfer characteristics of a mixed convective flow of an electrically conducting nanofluid past a slender Riga plate in the presence of viscous dissipation and chemical reaction. The heat and mass transfer characteristics are analyzed by a zero nanoparticle mass flux and convective boundary conditions. The thermophoretic and Brownian aspects for nanoliquid are proposed using Buongiorno's relations. The similarity quantities are utilized to obtain dimensionless forms of flow field equations. The resulting coupled nonlinear equations associated with the constituted boundary assumptions analytically proceed via optimal homotopy analysis method (OHAM). Various parameters causing the change in distributions of velocity, temperature, and concentration are graphically explained with relevant physical references. One of the important observations of the present study is, for higher values of modified Hartmann number the velocity profile increases and temperature profile supreses and it is interesting to note that the contemporary numerical simulations offer confirmable accuracy in the existing literature, which authenticates the novelty of current investigation.

Thermocapillary motion of a solid cylinder near a liquid–gas interface

The motion of a solid, infinitely long cylinder perpendicular to a convective liquid–gas interface due to thermocapillarity is investigated via an analytical model. If the cylinder temperature differs from the bulk temperature, a temperature gradient exists along the liquid–gas interface. This results in surface tension gradients at the liquid–gas interface, causing fluid flow around the particle, which induces propulsion. For small particles and, thus, small Péclet and Reynolds numbers, the steady-state equations for temperature and flow fields are solved exactly using two-dimensional bipolar cylindrical coordinates. The velocity of the cylinder as a function of separation distance from the liquid–gas interface is determined for the case of a constant temperature or a constant heat flux on the surface of the cylinder. A larger temperature gradient at the liquid–gas interface in the latter system leads to a larger cylinder velocity and a higher propulsion efficiency. The thermocapillary effect results in larger force on a cylinder than forces arising from other self-propulsion mechanisms.

Impact of finite ion size, Born energy difference and dielectric decrement on the electroosmosis of multivalent ionic mixtures in a nanotube

The electroosmotic flow (EOF) in a nanotube is analyzed by considering the finite ion size and dielectric decrement effects. The ion transport is governed by the modified Nernst-Planck equation, which takes into account the ion steric repulsion, Born energy difference, and dielectrophoretic force. The dielectric permittivity is considered to depend on the ionic concentration and ion size. The governing equations for fluid flow, ion transport, and electric field are solved numerically in their full form. The steric interaction is accounted, based on the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) model, which considers the nonuniform ion size for monovalent as well as multivalent salts. We made a quantitative comparison with the standard PNP model for the lower range of charge density and highlighted the impact of ion size and dielectric decrement effects. For a highly charged surface, the impact due to finite ion size is non-negligible when the Debye length is in the order of the nanotube radius. However, the dielectric decrement is found to have a strong impact on a higher range of surface charge density or bulk ionic concentration. The EOF and the ion selectivity of the mixed electrolyte solution of different counterion size are considered in the present study. Present study provides an in-depth analysis of EOF through a cylindrical nanotube beyond the dilute solution and low charge density limits, which has not been addressed before.

Regional deposition of the allergens and micro-aerosols in the healthy human nasal airways

The nasal cavity is the inlet to the human respiratory system and is responsible for the olfactory sensation, filtering pollutant particulate matter, and humidifying the air. Many research studies have been performed to numerically predict allergens, contaminants, and/or drug particle deposition in the human nasal cavity; however, the majority of these investigations studied only one or a small number of nasal passages. In the present study, a series of Computed Tomography (CT) scan images of the nasal cavities from ten healthy subjects were collected and used to reconstruct accurate 3D models. All models were divided into twelve anatomical regions in order to study the transport and deposition features of different regions of the nasal cavity with specific functions. The flow field and micro-particle transport equations were solved, and the total and regional particle deposition fractions were evaluated for the rest and low activity breathing conditions. The results show that there are large variations among different subjects. The standard deviation of the total deposition fraction in the nasal cavities was the highest for 5×104<impaction parameter (IP)<1.125×105 with a maximum of 20%. The achieved results highlighted the nasal cavity sections that are more involved in the particle deposition. Particles with IP = 30,000 deposit more in the middle turbinate and nasopharynx areas, while for particles with IP = 300,000, deposition is mainly in the anterior parts (kiesselbach and vestibule regions). For small IP values, the amounts of deposition fractions in different regions of the nasal cavity are more uniform.

Numerical study of Newtonian heating in flow of hybrid nanofluid (SWCNTs + CuO + Ethylene glycol) past a curved surface with viscous dissipation

Presented work concerns with investigation of hybrid nanomaterial (SWCNTs + CuO + Ethylene glycol) flow over curved nonlinear stretched sheet. Heat transfer features are emphasized through Newtonian heating and viscous dissipation. Coupled nonlinear ODEs are constructed from the flow field equations (Continuity Eq., Momentum Eq. and Energy Eq.) by mean of adequate transformations. Such resulting nonlinear ODEs are then converted into system of 1st-order ODEs and solved via shooting technique using RK-4 algorithms (bvp4c). Impacts of emerging flow parameters on temperature, skin friction coefficient, velocity and Nusselt number are illustrated graphically. Also comparison among hybrid nanofluid (SWCNTs + CuO + Ethylene glycol), nanofluid (SWCNTs + Ethylene glycol) and basefluid (Ethylene glycol) is performed.