Stationary Disk Research Articles

Thermal performance of a hybrid nanofluid flow through a stretchable stationary disk featuring the Cattaneo-Christov heat flux theory

A hybrid nanofluid is comprised of a (Ethylene glycol) base fluid component and (Copper and Aluminium oxide) nanoparticles, and the nanoparticles are scattered inside the Ethylene glycol. Integrating nanoparticles into a base fluid (Ethylene glycol) can significantly enhance its thermal conductivity, which in turn can boost the base fluid's rate of heat transfer. In addition, the dynamics of viscous fluid together with nanoparticles is quite interesting and has a large of applications in the industrial sector. The current predominately predictive modeling investigates the flow of the hybrid nanofluid via a stretchable stationary disk in the presence of heat source/sink. A progressive modification in he energy equation is done by utilizing the Cattaneo-Christov heat flux expressions. This theory provides predictions for the features of the thermal relaxation time of the liquid on the boundary layer flow. Further, the study focuses on the features of the Lorentz force resulting from the applied of a magnetic field perpendicular to the disk. The similarity approach is used to obtained the dimensionless ordinary differential equations.The bvp4c approach in Matlab is utilized as a numerical method for the solution. All the solutions are obtained through graphical form. According to the results, the thermal profile is reduced by adjusting the thermal relaxation time parameter. Motion of the hybrid nanofluid slows down by enlarging the magnetic force parameter. Additionally, the effect of the heat source significantly increases the thermal profile. It is noted that the temperature field is enhanced in the case of a larger Lorentz force. Moreover, the increase in volume fraction concentration intensifies the thermal distribution, but the velocity is diminished due to the effect of viscosity.

Exploring the possibility of testing the no-hair theorem with Minkowski-deformed regular hairy black holes via photon rings

In this paper, we investigate the optical appearance of a regular static spherically symmetric hairy black hole within the context of Minkowski deformation governed by the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. This black hole describes a hairy black hole with geometric deformations in the radial and temporal metric components, parameterized by α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. The optical appearance of the black hole, illuminated by a static thin accretion disk in three toy emission function models, exhibits distinctive shadows and photon rings. Our findings reveal that for static spherically symmetric hairy Minkowski-deformed regular black holes, the event horizon radius rh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_h$$\\end{document}, photon sphere radius rph\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_{ph}$$\\end{document}, critical impact parameter bph\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b_{ph}$$\\end{document}, and innermost stable circular orbit radius risco\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_{isco}$$\\end{document} all have a positive correlation with α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. These parameters affect the null geodesic trajectories, shadows, and the optical appearance of the photon rings illuminated by a static thin accretion disk around the black hole. Utilizing this, we obtained the optical appearance of this black hole when observed in the forward direction of the optical and geometric thin accretion disk models. In all three models, the radius of the photon ring images also shows a positive correlation with the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Therefore, under the static spherically symmetric hairy Minkowski-deformed regular black hole, there is no degeneracy of photon rings dependent on the parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Theoretically, this allows for the distinction of different spacetime metrics of hairy Minkowski-deformed regular black holes, providing a potential method for testing the no-hair theorem through future observations of black hole photon rings.

Regression modeling of Bödewadt slip flow dynamics involving Reiner-Rivlin nanofluid based on a modified Buongiorno approach

Regression models are useful in analyzing rotational flows as they enable accurate predictions of wall shear and heat transfer coefficient. In addition, Bödewadt flow is of paramount importance in fluid dynamics of rotating systems such as turbomachinery and geophysical flows. Moreover, nanofluid’s enhanced heat transfer properties can improve cooling efficiency in applications involving turbines and electronic systems. This study delves into the Bödewadt boundary layer flow of a Reiner-Rivlin fluid containing nanoparticles over a stationary porous disk under slip conditions. The two-phase Buongiorno model is employed, incorporating temperature-dependent diffusion coefficients for enhanced accuracy. To facilitate numerical simulations, the transport equations are converted into an ordinary differential system comprising four unknowns. In the present work, a highly reliable Keller-Box methodology is adopted which agrees very well with the MATLAB built-in program ‘bvp4c’. The computed 2-D and 3-D streamlines vividly capture the Bödewadt flow scenario with Reiner-Rivlin nanofluid. The principle aim to investigate the impact of non-Newtonian behaviour and slip on the flow pattern, while also examining the behavior of temperature/concentration field for nanoparticle working fluids. As thermophoretic diffusion increases, the thermal boundary layer thickens considerably, leading to a notable decrease in the cooling rate of the disk. In contrast, Brownian diffusion has only a minimal impact on the heat transport. In addition, wall suction effect is observed to significantly boost the disk’s cooling rate, though at the expanse of increasing skin friction coefficients. This study introduces linear and quadratic regression models designed to precisely predict both the surface drag and disk cooling rate, which are crucial factors in engineering processes.

Computational study of magnetohydrodynamic squeeze flow between infinite parallel disks

This paper explores the magnetohydrodynamic (MHD) squeeze flow of an electrically conducting fluid between two infinite parallel disks with a perpendicular magnetic field. The study focuses on the case where the upper disk moves towards a stationary lower disk. By employing similarity variables, we reduce the MHD momentum and continuity equations into a fourth-order linear boundary value problem, solved using a modified operational matrix method. The numerical approach is validated through L2-truncation error analysis, boundary condition comparisons, and by comparing results with other methods like HAM, HPM, and bvp4c that produce analytical and numerical solutions. Graphical analyses reveal the effects of the squeeze number, Hartman number, and the boundary parameter on velocity and flow profile. Results indicate that the Hartman number significantly affects the velocity due to the Lorentz force, while the squeeze number and boundary parameter influence the velocity and flow profile differently in suction and injection cases. The numerical solution demonstrates high accuracy and convergence compared to previous methods in terms of absolute error.

Ambiguous apparent motion in exchanging disks

A large and a small disk with radii R and r, side by side, exchange positions repetitively at 1.33 Hz. The motion is ambiguous. If r/ R < 0.8 then observers perceive a single large disk jumping back and forth across a small static disk. But if r/ R > 0.8, observers perceive two static disks that expand and contract in counter phase, with no motion across the gap between the disks. Observers respond to the movement of the centroid, which is greatest when r/ R < 0.8.

Onset of turbulence in rotor–stator cavity flows

Numerous studies have indicated that turbulence typically initiates along the boundary layer of the stationary disk within a rotor–stator cavity. To describe the transition process to turbulence on the stationary side of a closed rotor–stator cavity, a comprehensive approach combining global linear stability analysis with direct numerical simulation was adopted in the present study. The proposed model aligns with that of Yim et al. (J. Fluid Mech., vol. 848, 2018, pp. 631–647), who investigated the stability characteristics of the rotating-disk boundary layer in a rotor–stator cavity. In order to achieve a stable inflow for the stationary-disk boundary layer, we rotate the shroud together with the rotating disk. Through careful global stability analysis, the predominant spiral mode exhibiting the highest instability in the boundary layer of the stationary disk was discerned, corroborating observations from simulations. Initially, the spiral mode undergoes linear amplification, reaches a state of linear saturation and enters the nonlinear regime. Following nonlinear saturation in the flow field, a circular wave mode arises due to the influence of mean flow distortion. As the Reynolds number attained a sufficiently high level, the interplay between the downstream-propagating circular mode and spiral mode amplified disturbances in the boundary layer of the stationary disk, ultimately leading to the development of localised turbulence at the mid-radius of the rotor–stator cavity. Notably, the present study is the first to elucidate the coexistence of laminar–transitional–turbulent flow states in the stationary-disk boundary layer through direct numerical simulations.

Visualisation of counter-rotating dust disks using ray tracing methods

Physical aspects of stationary axisymmetric vacuum spacetimes given by exact solutions of the Einstein equations are discussed via ray tracing. A detailed study of the spacetime generated by a disk of counter-rotating dust is presented. The spacetime is given in explicit form in terms of hyperelliptic theta functions. The numerical approach to ray tracing is set up for general stationary axisymmetric spacetimes and tested at the well-studied example of the Kerr solution. Similar features as in the case of a rotating black hole, are explored in the case of a dust disk. The effect of the central redshift varying between a Newtonian disk and the ultrarelativistic disk, where the exterior of the disk can be interpreted as the extreme Kerr solution, and the transition from a single component disk to a static disk is explored. Frame dragging, as well as photon spheres, are discussed.

On the sound field of an oscillating elliptical disk in free space with an open and closed back.

An oscillating elliptical disk in free space has a uniform surface velocity distribution and is therefore a rigid sound radiator. The radiated sound also represents that scattered from a stationary disk in the presence of a normal plane wave. Using the Fourier (Hankel) Green's function in cylindrical coordinates, together with the monopole Kirchhoff-Helmholtz boundary integral, rigorous analytical formulas are derived for calculating the surface pressure distribution, radiation impedance, on-axis pressure, and, directivity pattern of an elliptical open-back disk in free space, and these are plotted over a range of normalized frequencies. The directivity is compared with that of a rigid circular disk in free space, and asymptotic low-frequency approximations are derived for the radiation impedance and on-axis pressure. Using the Gutin concept, the radiated sound is combined with that from a rigid elliptical disk in an infinite baffle to obtain the radiation impedance, on-axis pressure, and, directivity pattern of a closed-back elliptical disk in free space. The latter has a uniform velocity distribution on the front surface and zero velocity over the entire back surface.

Rotationally induced ingress in rotor–stator systems

The presence of a rotating disk adjacent to a stationary disk forms a rotor–stator cavity known as a wheel-space. It is necessary for gas turbine wheel-spaces to be purged with sealing flow bled from the compressor to counteract the harmful effects of ingress. This paper presents a combined experimental, theoretical, and computational study of rotationally induced ingress in rotor–stator systems. Measurements were made in a wheel-space with an axial clearance rim seal under axisymmetric conditions in the absence of a mainstream annulus through-flow. Ingress was quantified using a gas concentration technique and the flow structure in the cavity was explored with static and total pressure measurements to determine the swirl ratio. A low-order theoretical model was developed based on the boundary layer momentum-integral equations. The theory gave excellent results when predicting the effects of ingress and purge flows on the radial pressure and swirl gradients. Unsteady Reynolds-Averaged Navier–Stokes computations were conducted to provide greater fluid dynamic insight into the wheel-space flow structure and ingress through the rim seal. The computational results demonstrated some of the closest agreement with experimental measurements of ingress available in the literature, showing that rotationally induced ingress is dominated by unsteady large-scale structures in the rim seal gap instead of the previously ascribed disk-pumping effect. The study serves as an important validation case for investigations of ingress in rotor–stator systems in more complex environments.

Discrete Element Modeling of the Breakage of Single Polyhedral Particles in the Rotary Offset Crusher

Innovation in comminution is expected to continue unabated to address the inefficiencies that are inherent in comminution circuits. The rotary offset crusher (ROC) is a new comminution device with a promising performance potential in terms of throughput due to the enhanced speed of transportation induced by the centrifugal force of the discs. However, the processes driving the comminution of particles trapped in the conical space between the two discs of the crusher are not fully understood. To gain a better insight into the comminution process in this device, discrete element modeling (DEM) simulations were conducted to study the breakage of a single particle for the crusher operated under two different dynamic conditions, i.e., (1) a stationary top disc and (2) both discs rotating at the same speed. For both scenarios, the speed of the discs was varied between 550 and 2350 rpm. Experimental testwork was also conducted with the laboratory prototype to generate the data that were used to calibrate the breakage parameters of the Ab × t10 breakage model. Simulations were performed using polyhedral UG2 ore particles that were generated with the in-built particle generator in the DEM simulator. The simulated ROC, which is operated with both discs rotating, outperformed the ROC with a stationary top disc in terms of the specific input energy and throughput. The crusher with a stationary top disc is characterized by high shear forces (suggesting a higher wear rate), specific input energies greater than 1 kWh/t, and low throughputs (<50 kg/h). The ROC operated with a stationary disc is not recommended for hard rock applications due to expected excessive wear of crushing surfaces and higher energy consumption. The freewheeling discs are recommended, but there is scope to optimize the crusher performance in terms of the power draw, size reduction, and throughput by manipulating the difference between the speeds of the discs. There is also scope to optimize the crusher performance when it is simulated with many particles. Once the full performance potential of the ROC is established, it will then be important to benchmark it against the existing crushers in the minerals industry as well as other industries where crushers are used.

Melting heat transfer features in a B[formula omitted]dewadt flow of hybrid nanofluid ([formula omitted]/water) by a stretching stationary disk

The melting impacts on the three-dimensional hybrid nanofluid flow by a stationary stretching disk is examined. The flow configuration is due to the stretching of the disk. The magnetic field is applied normal to the disk in order to generate the resistance in the liquid. The performance of heat transfer in the liquid, the thermal radiation and heat generation/absorption impacts are taken into consideration. The melting heat transfer impact on the surface is analyzed here. The impact of stretching and rotating mechanisms along with applied physical impacts on the hybrid nanofluid flow and heat features are of the present concern for this physical problem. A conventional transformation is presented to lead to a set of coupled similarity equations which are highly non-linear in nature. A numerical method subject to bvp4c technique is adopted for the solution process. Results reveal that increasing trend of stretching of the disk lowers the boundary layer thickness and consequently highers the rate of heat transfer. It is found the reduction in thickness of thermal boundary layer by the strength of stretching. Additionally, the temperature distribution rises by the influence of melting temperature. Further, the skin fraction coefficient seen to be enhanced with the enlargement of nanoparticles volume fraction.

A Generalized Empirical Model for Velocity Deficit and Turbulent Intensity in Tidal Turbine Wake Accounting for the Effect of Rotor-Diameter-to-Depth Ratio

Commercial scale tidal stream turbines (TST) are expected to be deployed in shallow water where the depth varies from 1.5 to 3 turbine diameters. In this study, numerical simulation is conducted at realistic hydrodynamic conditions of potential tidal sites using the stationary actuator disc method at ambient turbulence varying from 5% to 20%, a range of rotor realistic rotor thrust coefficient from 0.64 to 0.98 and a rotor-diameter-to-depth ratio of 20% to 60%. The result shows that the TST wake is affected by the rotor-diameter-to-depth ratio, ambient turbulence, and thrust coefficient. The new empirical model is in accordance with the numerical simulation of a full-scale turbine and is validated with the TST experiment at different rotor-diameter-to-depth ratios with reasonable results in the far wake. This low computational model can benefit the investigation of tidal turbine parks at different configurations where the far wake is pertinent.

Characterization of the friction-induced attractor trajectories during the running-in process in ball-on-disk tribosystem

PurposeThe purpose of this study is to solve the issues of time-consuming and complicated computation of traditional measures, as well as the underutilization of two-dimensional (2D) phase-trajectory projection matrix, so a new set of features were proposed based on the projection of attractors trajectory to characterize the friction-induced attractors and to reveal the tribological behavior during the running-in process.Design/methodology/approachThe frictional running-in experiments were conducted by sliding a ball against a static disk, and the friction coefficient was collected to reconstruct the friction-induced attractors. The projection of the attractors in 2D subspace was then mapped and the distribution of phase points was adapted to conduct the feature extraction.FindingsThe evolution of the proposed moment measures could be described as “initial rapid decrease/increase- midterm gradual decrease/increase- finally stable,” which could effectively reveal the convergence degree of the friction-induced attractors. Moreover, the measures could also describe the relative position of the attractors in phase–space domain, which reveal the amplitude evolution of signals to some extent.Originality/valueThe proposed measures could reveal the evolution of tribological behaviors during the running-in process and meet the more precise real-time running-in status identification.

Images of hairy Reissner–Nordström black hole illuminated by static accretions

In this paper, we investigate the shadow and optical appearance of the hairy Reissner–Nordström (RN) black hole illuminated by two toy models of static accretion. The hairy RN black hole was constructed in the gravitation decoupling approach to describe the deformation of a Schwarzschild black hole due to the inclusion of additional arbitrary source (scalar field, tensor field, fluidlike dark matter, etc.). So it is characterized by the parameters: mass (M), deformation factor (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}), electric charge (Q) and the additional hairy charge (lo\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_o$$\\end{document})., differentiating from the case in RN black hole. Though the specific background theory that results in this hairy RN black hole is still tricky, here we shall focus on the novel observable features introduced by the hair of this black hole. First, we find that for the hairy RN black hole, the event horizon, radius of photon sphere and critical impact parameter all increase as the increasings of Q and lo\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_o$$\\end{document}, but decrease as α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} grows. Furthermore, the three characterized parameters are found to have significant effects on the photon trajectories, and shadows as well as images of the hairy RN black hole surrounded by the static accretion disk and spherical accretion, respectively. In particular, both Q and lo\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_o$$\\end{document} have mutually reinforcing effects on the optical appearance and shadows of the hairy RN black hole, which implies that we may not distinguish the electric charge and hairy charge from the shadow and image of black hole in this scenario. Additionally, because of the competing effects of the charge parameters (Q,lo\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q, l_o$$\\end{document}) and the deviation parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} on the observed intensities of brightness, the optical appearance between the hairy RN black hole and RN black hole could have degeneracies, indicating the indistinguishability. Our current results contribute more to the phenomenal aspects which could be helpful to build the background theory of this hairy RN black hole.

Temperature Gradient Effects on Stator Boundary-Layer Stability in a Rotor–Stator Cavity

A comprehensive approach, combining theoretical analysis and direct numerical simulation, is employed in this study to investigate the influence of temperature gradient on the stability phenomenon of the stator boundary layer in a rotor–stator cavity. In contrast to previous studies, a temperature term is introduced to account for centrifugal buoyancy within the cavity. The focus is on analyzing the transitional behavior and the effects of centrifugal buoyancy on the boundary layers of the stationary disk under operating conditions characterized by a Reynolds number of Re=2.5×104. The investigation reveals that this temperature gradient significantly affects the base flow and alters the instability governing the boundary-layer transition on the stationary disk. Specifically, the centrifugal buoyancy induced by the higher temperature on the stationary side weakens the spiral mode perturbations without inducing changes in the azimuthal wavenumber of the spiral mode. However, when the centrifugal buoyancy effect exceeds a certain threshold, it directly suppresses the generation of the spiral mode and induces the formation of low-radius circular waves, thereby promoting a more stable boundary layer. This research emphasizes the importance of considering temperature variations in the rotor–stator cavity for improved control of stability within the boundary-layer flow.

Thermal examination for double diffusive MHD Jeffrey fluid flow through the space of disc and cone apparatus subject to impact of multiple rotations

Non-Fourier’s and non-Fick’s models enhance predictions of transient heat transfer and mass transport, especially in situations where local equilibrium assumptions break down. In such fluids, the deformation rate and shear stress display a natural parabolic structure. In this study, we explore the flow of a Jeffrey fluid with dual diffusions, incorporating both non-Fourier's and non-Fick's assumptions, through the gap of disk-cone devices. Four scenarios have been investigated for fluid flow and heat transformation in this work comprising of (i) both disc and cone are rotating in opposite directions (ii) both disc and cone are rotating in same direction, (iii) cone is rotating while disc remains stationary, (iv) disc is rotating while cone remains stationary. After deriving the basic equations, a set of appropriate variables is used to convert them into dimension from form. RK-4 (Runge-Kutta 4th order) technique has been employed for the solution of nonlinear equations. As the magnetic and Maxwell factors experience a notable increase, whether the disk and cone move in tandem, in opposite directions, or involve a static cone with a gyrating disk, or a static disk with a gyrating cone, the transverse velocity panels consistently exhibit retardation in all scenarios. Additionally, a noteworthy observation is made when evaluating changes in both the Nusselt number and the Sherwood number, with significantly more pronounced variations detected at the surface of the disk compared to that of the cone. There has been a 35% increase observed in the thermal flow rate with variations in the thermophoresis factor.

Impact of solid–liquid interfacial layer in the nanofluid flow between stretching stationary disk and a rotating cone

The present study explores the flow of nanofluid between a cone and a disk with the thermophoresis and Brownian motion impact. The flow is due to an expanding disk and a rotating cone. In the current study, radiative heat flux is considered. Using proper similarity transformations, the governing differential equations of momentum, temperature and concentration are transformed into a set of nonlinear ordinary differential equations (ODEs). The Runge-Kutta Fehlberg fourth fifth-order (RKF-45) technique and the shooting strategy are employed to solve the reduced ODEs. The impact of various parameters is illustrated through graphs and used to comprehend flow profile behaviour easily. The obtained results demonstrate that the velocity profiles enhance as improve in values of the Reynolds number. The rise in radiation, thermophoresis and Brownian motion parameters increases the thermal profile due to the potency of Brownian movement which results in a compelling nanoparticle interaction and boosts the fluid's thermal efficiency. Increasing values of the thermophoresis parameter reduces the heat transmission rate at the disk. As the Brownian motion parameter elevates, the concentration profile reduces, but the concentration profile increases as the thermophoresis parameter is raised.

Adaptation of rectangular and trapezoidal time functions to simulate the rotational motion of the brake disc

Numerical modeling of the rotational motion of the brake disc during frictional heating requires the use of specialized software. In this paper, an alternative approach was proposed consisting in the simulation of the displacement of the heating region contained within the boundaries of the friction surface of the brake pad relative to the stationary disc. The moving heat source effect was obtained using rectangular time functions generated based on the operating parameters and dimensions of the brake components. The first part of the article discusses the mathematical basis for the construction of time curves in relation to the solid brake disc combined with a pad in the shape of the angular section of a ring. Braking simulations were carried out using the finite element method (FEM) based software for 3D, 2D axisymmetric and 1D models of the disc. The calculated temperature filed changes were confronted with the results obtained using the 3D model developed based on standard methods. The temperature on the contact surface of the disc in the position of the mean radius at the end of braking, determined using the 2D axisymmetric and the 3D model with adaptation of rectangular time functions as well as the 3D reference model agreed well and were equal to 157.83 °C, 157.87 °C, and 157.90 °C, respectively. In the second part of the study, the 3D model of periodic heating was extended to the disc brake of a railway vehicle. The results of the calculations for the ventilated disc and segmented brake pad were compared with the experimental data obtained during testing on a full-scale dynamometer test stand. The maximum temperature values obtained on the basis of the 3D-pulse, 3D numerical models and measured on the test stand were equal to 73.3 °C, 74.4 °C and 75.6 °C, respectively. The insignificant temperature difference, i.e. below 3 %, obtained using the 3D-pulse heating model and measured using a thermocouple, confirms the correctness of the developed algorithm for simulating the rotational motion of the disc as well as the calculation methodology incorporating the heat partition coefficient (HPC).

Self-propulsion of an elliptical phoretic disk emitting solute uniformly

Self-propulsion of chemically active droplets and phoretic disks has been studied widely; however, most research overlooks the influence of disk shape on swimming dynamics. Inspired by experimentally observed prolate composite droplets and elliptical camphor disks, we employ simulations to investigate the phoretic dynamics of an elliptical disk that emits solutes uniformly in the creeping flow regime. By varying the disk's eccentricity $e$ and the Péclet number $Pe$ , we distinguish five disk behaviours: stationary, steady, orbiting, periodic and chaotic. We perform a linear stability analysis (LSA) to predict the onset of instability and the most unstable eigenmode when a stationary disk transitions spontaneously to steady self-propulsion. In addition to the LSA, we use an alternative approach to determine the perturbation growth rate, illustrating the competing roles of advection and diffusion. The steady motion features a transition from a puller-type to a neutral-type swimmer as $Pe$ increases, which occurs as a bimodal concentration profile at the disk surface shifts to a polarized solute distribution, driven by convective solute transport. An elliptical disk achieves an orbiting motion through a chiral symmetry-breaking instability, wherein it repeatedly follows a circular path while simultaneously rotating. The periodic swinging motion, emerging from a steady motion via a supercritical Hopf bifurcation, is characterized by a wave-like trajectory. We uncover a transition from normal diffusion to superdiffusion as eccentricity $e$ increases, corresponding to a random-walking circular disk and a ballistically swimming elliptical counterpart, respectively.

Laminar Torsional Couette Flow Over a Wavy Disk

Abstract A parametric study of laminar torsional Couette flow between a rotating flat disk and a stationary wavy disk is conducted using computer simulations. The waviness of the stationary disk, characterized by the parameter wave slope (WS), is increased across four cases to evaluate its influence. The resulting flow field is significantly different from torsional Couette flow between flat disks. Shear stress and pressure on the flat and wavy disks are sinusoidal with the same spatial frequency as the undulation. Their averages over one wavelength remain unaffected by the increase in WS. However, this is not the case with their amplitudes. A critical layer is observed approximately at a distance of 70% radius of the disk across which the effect of WS on shear amplitude is inverted. Furthermore, the sinusoidal shear on the flat disk is observed to be in destructive phase with respect to the shear on wavy disk. This is corroborated by the concave/convex distortion of the height-wise tangential velocity profile across the undulation. Alternately, the net radial velocity becomes biased either toward or away from the axis depending on angular location with respect to the undulation. The effect of increasing WS on the tangential velocity is in the form of small increments to the curvature of its height-wise profile, while the axial velocity, though it is orders of magnitude smaller than the tangential velocity, increases substantially with increase in WS.