Parallel Disks Research Articles

Investigations of Fast Vacuum Pump-Down Processes Between Parallel Isothermal Disks

To obtain optimal system throughput for semiconductor device fabrications, fast vacuum pump-down operations are frequently performed to quickly depressurize processing chambers for desirable pressure conditions. However, sudden and drastic temperature variations could occur at the beginning of a fast vacuum pump-down session, resulting in catastrophic consequences such as thermal shock and condensation-induced contamination, profoundly disturbing the otherwise well-conditioned mini-environment. To this end, the vacuum pump-down processes taking place between a pair of parallel isothermal disks resembling typical industrial applications (e.g., high-throughput load lock) are investigated by computational fluid dynamics (CFD) simulation and space-invariant theoretical modeling. It is discovered that the gas temperature first drops rapidly in a manner resembling isentropic gas expansion, and then quickly recovers at a rate governed by the pumping speed and a near-constant Nusselt number. In addition, it is discovered that attenuated temperature variations can be achieved with smaller disk separations and lower pumping speeds. Furthermore, a new empirical formula is proposed to calculate the Nusselt number based on a normalized time parameter, which not only better reflects the heat-transfer characteristics during pump-down than its well-known counterparts based on the Rayleigh number, but also enable efficient evaluations of various pump-down configurations with the space-invariant theoretical modeling.

Numerical study of micropolar nanofluid flow between two parallel permeable disks with thermophysical property and Arrhenius activation energy

A steady-state, incompressible, laminar, and axi-symmetric flow of micropolar nanofluid flowing between two porous disks is investigated. The impacts of variable thermal conductivity and Arrhenius activation energy are incorporated in non-Newtonian fluid model. The electrically conducting fluid experiences the influence of externally applied transverse magnetic field. The parallel porous disks are subjected to a constant uniform injection. A dimensionless form of non-linear flow equations is achieved by the implementation of similarity transformations. The obtained system is then solved by applying Runge-Kutta-Fehlberg (RKF-45) method. The physical parameters on flow, thermal, and concentration fields are described graphically. The three dimensional flow visualization is also presented. The numerical values of shear stresses, couple stresses, heat transportation rate, and mass transportation rate at the upper and lower disks are calculated against the physical quantities in tabular format. It is observed that micropolar fluids are beneficial in the enhancement of couple stresses and in the reduction of shear stresses, which can be helpful for the flow control in polymeric processes. The thermophoretic and Brownian motion parameters have opposite effects on concentration profiles. The concentration field is enhanced by the activation energy parameter.

Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula

Abstract The development of nanofluid technology has become a key research area in physics, mathematics, engineering, and materials science. Nowadays, in many industrial applications, nanofluids are widely used to enhance thermophysical properties such as thermal diffusivity, thermal conductivity, and convective heat transfer. Scientists and engineers have established interests in the direction of flow problems developed via disk-shaped bodies. There are various logics to discuss flow phenomenon due to rotating bodies, but its applications include in thermal power engineering system, gas turbine rotors, air cleaning machines, aerodynamics, etc. Nowadays manufacturing industries have inaugurated to select liquid based on heat transfer properties. Therefore, this article focuses on studying the laminar incompressible nanofluid between two parallel disks. Mathematical formulations of the law of conservation of mass, momentum, and heat transfer are investigated numerically. By using suitable similarities, the flow equations are converted into nonlinear ordinary differential equations. The resulting equations were solved numerically via MATLAB software. The effects of physical parameters of interest, such as Reynolds number, magnetic factor, Brownian parameter, and thermophoresis parameter on normal velocity, streamwise velocity, temperature, and concentration profiles are computed and presented using the graphs. The results revealed that the energy profile significantly rises, and the profile moves closer to the upper disk by enhancing the Brownian motion and thermophoresis parameter. The dynamics behind this is that by increasing the Brownian motion, the boundary layer wideness increases which increases the temperature. Moreover, streamwise velocity increases for large values of Reynolds number. Besides, the thermophoresis profile increases for large values of the thermophoresis factor. It could be observed that shear stress at nonporous/porous disk is adjusted by selecting a suitable value of injection velocity at the porous disk. Also, normal velocity decreases by increasing the parameter M.

The Influence of Effective Prandtl Number Model on the Micropolar Squeezing Flow of Nanofluids between Parallel Disks

A mathematical model of micropolar squeezing flow of nanofluids between parallel planes is taken into consideration under the influence of the effective Prandtl number using ethyl glycol (C2H6O2) and water (H2O) as base fluids along with nanoparticles of gamma alumina (γAl2O3). The governing nonlinear PDEs are changed into a system of ODEs via suitable transformations. The RKF (Range–Kutta–Fehlberg) technique is used to solve the system of nonlinear equations deriving from the governing equation. The velocity, temperature, and concentration profiles are depicted graphically for emerging parameters such as Hartmann number M, micronation parameter K, squeeze number R, Brownian motion parameter Nb, and thermophoresis parameter Nt. However, physical parameters such as skin friction coefficient, Nusselt number, and Sherwood number are portrayed in tabulated form. The inclusion of the effective Prandtl number model indicated that the effect of the micropolar parameter K on angular velocity h(ξ) in both suction and injection cases is opposite for both nanofluids. It is observed that the increase in angular velocity is rapid for γAl2O3−C2H6O2 throughout the study.

Oscillatory behavior with fiber orientation in the flow–vorticity plane for fiber suspension under large amplitude oscillatory shear flow

In fiber suspensions, fibers with three-dimensional orientation states are dramatically flow-oriented in the flow–flow gradient plane when the flow starts. In contrast, large strains are required for flow orientation in the flow–vorticity plane. Under oscillatory shear flow, when the strain amplitude is small, the flow orientation in the flow–vorticity plane is weakly induced, unlike that in the flow–flow gradient plane. The orientation in the flow–vorticity plane increases with the strain amplitude. At large strain amplitudes, fibers are oriented in the flow–flow gradient plane; thus, the rotational motion of fibers in the flow–flow gradient plane is dominant, i.e., fibers are almost flow-oriented in the flow–vorticity plane. However, contributions of the oscillatory behavior of fiber orientation in the flow–flow gradient and flow–vorticity planes to complex viscosity are unclear. In this study, the objective is to determine the contributions of fiber orientation in each plane to complex viscosity with the two initial orientation states (random and flow-oriented states) for the strain sweep test. The two initial orientation states were controlled by manipulating the upper disk of the parallel disk flow path. The initial random state was induced by sliding the upper disk up and down. The initial flow-oriented state was induced by rotating the upper plate. Furthermore, phase transition behaviors from the random to flow-oriented state in the flow–vorticity plane with increasing strain amplitude were qualitatively estimated as the orientation angle via visualization. Consequently, when the initial orientation was random, the fibers gradually vibrated in a medium strain amplitude region, and complex viscosity was higher than that of the initial flow-oriented state. In the conclusion, we divided the complex viscosity behavior of the strain sweep test into five strain amplitude regions and clarified the dominant orientation state in each region. It is conceivable that this result will help in understanding the relationship between fiber orientation and dynamic viscoelastic properties of fiber suspension like fiber-reinforced thermoplastics.

Entropy generation minimization of Ag-Fe_3O_4/water-ethylene glycol squeezed hybrid nanofluid flow between parallel disks

PurposeThis paper aims to examine the squeezing flow of hybrid nanofluid within the two parallel disks. The 50:50% water–ethylene glycol mixture is used as a base fluid to prepare Ag–Fe_3O_4 hybrid nanofluid. Entropy generation analysis is examined by using the second law of thermodynamics, and Darcy’s modal involves estimating the behavior of a porous medium. The influences of Viscous dissipation, Joule heating and thermal radiation in modeling are further exerted into concern.Design/methodology/approachFor converting partial differential systems to ordinary systems, a transformation technique is used. For the validation part, the numerical solution is computed by embracing a fourth-order exactness program (bvp4c) and compared with the analytical solution added by the homotopy analysis method (HAM). Graphical decisions expose the values of miscellaneous-arising parameters on the velocity, temperature and local-Nusselt numbers.FindingsHybrid nanofluid gives significant enhancement in the rate of heat transfer compared with nanofluid. The outcomes indicate that the average Nusselt number and entropy generation are increasing functions of the magnetic field, porosity and Brinkman number. When the thermal radiation rises, the average Nusselt number diminishes and the entropy generation advances. Furthermore, combining silver and magnetite nanoparticles into the water–ethylene glycol base fluid significantly enhances entropy generation performance.Originality/valueEntropy generation analysis of the magneto-hydrodynamics (MHD) fluid squeezed between two parallel disks by considering Joule heating, viscous dissipation and thermal radiation for different nanoparticles is addressed. Furthermore, an appropriate agreement is obtained in comparing the numerical results with previously published and analytical results.

荞麦剥壳机剥壳性能影响因素试验研究

Buckwheat huller is the key equipment of the buckwheat rice processing and its performance is directly related to the quality and yield of buckwheat rice. The key working parts of the sand disc sheller are a pair of parallel sand discs. The structure and the working parameters of the sand discs are the key factors affecting the shelling performance. In this paper, the effect of rotating speed of sand disc, shelling clearance, grain size of sand disc and width of the working face on the shelling performance are experimentally investigated. The results show that the rice yield and relative broken rice rate increase with the increase of the rotating speed of the lower sand disc, and the relative broken rice rate increases sharply when the speed reaches a certain value. With the linear increase of the shelling clearance, the rice yield and the relative broken rice rate are first decreased rapidly, and then decrease slowly. The smaller the particle size of the sand disc is, the higher the rice yield and the relative broken rice rate are, and the influence of the lower sand disc is obviously greater than that of the upper one. Properly widening the width of working face will increase the rice yield but the width tends to have little influence on the rice yield when it increases to a certain extent, and its change has a small impact on the relative broken rice rate. According to the results of single factor test and orthogonal test, the sheller can achieve a better shelling performance and work efficiency when the grain size of the upper sand disc is F24, the grain size of the lower sand disc is F36, the shelling clearance is 5 mm, the rotating speed of the sand disc is 950 r/min, and the width of the working face is 2 cm.

Parallel-Disk Viscometry of a Viscoplastic Hydrogel: Yield Stress and Other Parameters of Shear Viscosity and Wall Slip.

The rheology, i.e., the flow and deformation properties, of hydrogels is generally a very important consideration for their functionality. However, the accurate characterization of their rheological material functions is handicapped by their ubiquitous viscoplasticity and associated wall slip behavior. Here a parallel-disk viscometer was used to characterize the shear viscosity and wall slip behavior of a crosslinked poly(acrylic acid) (PAA) carbomer hydrogel (specifically Carbopol® at 0.12% by weight in water). It was demonstrated that parallel-disk viscometry, i.e., the steady torsional flow in between two parallel disks, can be used to unambiguously determine the yield stress and other parameters of viscoplastic constitutive equations and wall slip behavior. It was specifically shown that torque versus rotational speed information, obtained from parallel-disk viscometry, was sufficient to determine the yield stress of a viscoplastic hydrogel. Additional gap-dependent data from parallel-disk viscometry could then be used to characterize the other parameters of the shear viscosity and wall slip behavior of the hydrogel. To investigate the accuracy of the parameters of shear viscosity and apparent wall slip that were determined, the data were used to calculate the torque values and the velocity distributions (using the lubrication assumption and parallel plate analogy) under different flow conditions. The calculated torques and velocity distributions of the hydrogel agreed very well with experimental data collected by Medina-Bañuelos et al., 2021, suggesting that the methodologies demonstrated here provide the means necessary to understand in detail the steady flow and deformation behavior of hydrogels. Such a detailed understanding of the viscoplastic nature and wall slip behavior of hydrogels can then be used to design and develop novel hydrogels with a wider range of applications in the medical and other industrial areas, and for finding optimum conditions for their processing and manufacturing.

A modified expression for breakdown voltage of Townsend discharge in a non-uniform electric field and uniform axial magnetic field

Townsend breakdown mechanism for direct current discharge in argon gas by considering the effects of the non-uniformity of electric field, the loss of electrons due to radial diffusion, and the applied axial magnetic field has been discussed and an analytical expression has been derived for the breakdown voltage. It was Shown that the non-uniformity of electric field, between two parallel disc electrodes with radius R and separation of d, significantly affects the electron multiplication for d≥2.7R and that effect was disappeared for the gas pressure higher than 0.35 Torr. The loss coefficient δ was used to quantify the effects of the electron radial diffusion and axial magnetic field and the PIC-MCC numerical simulations were used to determine it. The results show that increasing the gap size above the electrode radius leads to dramatic increase of the δ coefficient while applying an axial magnetic field can effectively decrease it.

Electroviscous Effect of Water-Base Nanofluid Flow between Two Parallel Disks with Suction/Injection Effect

This article, investigates the behaviour of an ionized nanoliquid motion regarding heat transmission between two parallel discs. In the proposed model, the squeezing flow of Cu-water nanofluid with electrical potential force is analysed for studying the flow properties and an uniform magnetic field is applied to that fluid, by taking the surface of the bottom disc porous. We have also studied the effects of different nanomaterials on the transmission of heat through nanofluids. Furthermore, the influence of various physical parameters in the proposed model of nanofluids flow like volume fraction of nanomaterials, squeezing number, Hartmann number, Eckert number, and Prandtl number are analysed and discussed quantitatively through various tables and graphs. The system of nonlinear partial differential equations (PDE’s) has been used to formulate the proposed flow model and later converted to a set of nonlinear ODE’s by mean similarity transformation. Further, the reduced form of ODEs has been solved by Parametric Continuation Method (PCM), which is a stable numerical scheme. The outcomes obtained from the proposed model could also be used to analyse nanofluid flow in several fields, such as polymer processing, power transfer and hydraulic lifts.

Squeezing flow analysis of time dependent AA7072 and AA7075 water-based nanofluids through parallel disks with velocity and thermal slip conditions

Present investigation based on the axisymmetric squeezing flow of various alloy nanofluids between parallel disks embedding with porous medium. To enhance the heat transfer properties we have considered two special type of alloy nanoparticles such as AA7072 which contains 98% of Aluminum (Al), 1% of Zink (Zn), rest Silica (Si), Iron (Fe), Copper (Cu), and AA7075 contains 90% of Al, 5%–6% of Zn, 3% of manganese (Mg), 1%–2% of Cu with additive Si, Fe, and Mn. The effects of velocity slip and temperature jump boundary conditions are also considered. Suitable similarity transformation along with Maxwell model physical properties for nanofluid are use to formulate the dimensionless governing ordinary differential equations. Physical significance of characterizing parameters are presented graphically and discussed. The numerical computations for engineering coefficients are shown in tabular form. In a comparative study, the current result shows a good correlation with the earlier established result that confirms the convergence of the solution methodology. Further, the major outcomes are laid down as: The velocity is more prominent for AA7075-water nanofluid in the first region for applied magnetic field whereas reverse impact is observed in the second region. The fluid temperature retards when there is a rise in thermal slip parameter. The shear rate of AA7072-water based nanofluid is higher than other nanofluid AA7075.

Time-dependent squeezing bio-thermal MHD convection flow of a micropolar nanofluid between two parallel disks with multiple slip effects

We performed numerical analysis on fluid model for incompressible, time dependent electrically conducting squeezing flow of micropolar fluid. The non-Newtonian fluid is confined across two disks in which lower disk is fixed while upper disk moves upward and downward. A uniform suction is made at the lower fixed disk. The Buorngiorno nanofluid theory is implemented to discuss the thermal performance by thermopherosis and Brownian motion phenomenon. Microorganism theory is applied to sustain the suspended nanoparticles by bioconvection induced through combined influence of magnetic field and buoyancy forces. The nonlinear system is reduced to a set of ordinary differential equation by similarity variables. The dimensionless quantities are discussed in pictorial way on velocity, temperature, concentration and microorganism fields. It is perceived that micropolar and magnetic parameters have opposite trend on micro-rotation field. Brownian motion and Thermopheresis parameters have similar kind of influence on temperature profiles and the thermal fields is reduced by suction parameter. Radial velocity curves declined by the squeezing Reynolds number. Numerically values of the physical parameters are also calculated and described through tabular format. It is calculated that slip parameters are reduced the Nusselt and Sherwood numbers values at the both disks respectively. At the center of the plane between two disks, the radial velocity curves are declined, whereas such profiles are enhanced at the lower and upper disks against the magnetic parameter. The increased suction parameter reduces the radial field of f′η. Moreover the radial profiles are of parabolic nature.Microrotation and magnetic parameters tend to rotate the microrotation field in opposite way. In the left region of the central plane, the profiles are enhanced, while in the right half plane the micorotation profiles are reduced. Furthermore, the microrotation curves are intersecting at η = 0.5, where the profiles turn into opposite way.

Coriolis force effects on radial convection in a cylindrical annulus

Thermal convection under the influence of rotation appears in a wide range of engineering and geophysical applications, and has been investigated extensively both experimentally and numerically utilizing different canonical configurations. The problem of radial convection in a cylindrical annulus, studied in the present work, has received less attention than other configurations, such as rotating Rayleigh-Bénard convection. It has long been known that rotation has a stabilizing effect and, in the limit of strong rotation, a two-dimensional flow emerges, resulting in reduced heat transfer rates. Here, direct numerical simulations are performed for buoyancy-driven flows in a cylindrical annulus bounded by two parallel disks with and without rotation. By maintaining a radial gravity term analogous to a centrifugal acceleration even in the stationary cases, it is possible to isolate the effect of the Coriolis force on the flow. As expected, when rotation is imposed the flow is characterized by large-scale structures and the heat transfer rates are significantly reduced. Turbulence statistics are presented for the mean temperature profiles, temperature and velocity fluctuations, turbulent kinetic energy budgets, and Reynolds stress anisotropy tensor. It is shown that rotation significantly affects the distribution of turbulent kinetic energy throughout the radial direction, in particular away from the cylindrical surfaces. The anisotropy maps reveal that the turbulence is quasi-two-dimensional in the rotating cases, even for the highest value of Ra considered, whereas without rotation the core region approaches the isotropic limit with increasing Ra.

High gain wide band grid array antenna for short range radar and vehicle-to-satellite communications

This paper presents the design of a Grid Array Antenna that covers both Short Range Radar applications at 24 GHz, the spectrum requirement of Ku-band transmit mode for Fixed Satellite Service and Direct Broadcast Service for Vehicle to Satellite communications. The proposed Grid Array Antenna design has vertical short radiating wire segments and long radiating wire segments loaded with parallel reactive disc structures to enhance the bandwidth. Taylor synthesis method of amplitude-tapering reduces the Side Lobe Level to −16.3 dB. The simulation and measurement results are in alignment with each other covering the bandwidth of 12.58 GHz (VSWR ≤ 2), Half Power Beam Width of 7.8° in E-plane, and 101.6° in H-plane at 24 GHz and broadside radiation for Vehicle-to-Satellite communication. The antenna parameters are analyzed for housing effect and mounted on a virtual model of a vehicle. The aforementioned results show that the proposed antenna is well suited for the spectral coverage of Vehicle-to-Satellite communication, Blind Spot Detection, and pre-impact sensing in Short Range Radar automotive communication.

Analytical Solution of Electromagnetic Force on Nanofluid Flow with Brownian Motion Effects Between Parallel Disks

The innovation of the present paper is the analytical study of Brownian motion effects on nanofluid flow and electromagnetic force between parallel disks with a heat source. Nanoparticle effects on nondimensional temperature field and velocity of fluid flow were analyzed using Akbari-Ganji’s Method and radial basis function approximation based on Hardy multiquadric function. Akbari-Ganji’s Method (AGM) is a strong analytical method that solves any linear and nonlinear differential equation with any degree of variables. Radial basis functions is an approximation method for analyzing functions and equations at high degrees, especially when it is necessary to apply the interpolation problem for scattered data on irregular geometry. The results signified that the maximum difference between AGM and RBF methods, for nondimensional horizontal velocity on CuO nanofluid at and is 0.2251 and the minimum difference for the nondimensional vertical velocity of Al2O3 nanofluid at is equal to 0.0018. Also, the effects of the Hartmann number (Ha) on nondimensional horizontal and vertical velocities field for Al2O3 nanoparticles at have a slight difference from the other Hartmann values using the AGM method. The maximum of nondimensional horizontal velocities at and is equal to 1.9354.

INVESTIGATION OF THE HEAT PERFORMANCE FOR SQUEEZED HYBRID NANOFLUID FLOW BETWEEN PARALLEL DISKS EMBEDDED IN POROUS MEDIUM WITH THERMAL RADIATION

This study's primary contribution determines the heat transfer performance and flow characteristics of a hybrid nanofluid between two parallel disks. The squeezing aspects of the hybrid nanofluid are investigated considering when the plates shift apart and come together. The impacts of thermal radiation and suction/injection are further investigated. Also, the Darcy-Forchheimer model is used for porous medium. The mathematical formulation for the transport of momentum and heat is portrayed by a partial differential equation set, which is then diagnosed by embracing a fourth-order exactness program (Bvp4c). Graphical findings expose the importance of miscellaneous arising parameters on the velocity, temperature, and local Nusselt numbers. The dominant value of porosity constraint and magnetic parameter reduces the temperature distribution of fluid. Furthermore, increased squeeze number causes the temperature profile to exhibit contrary behavior. Hybrid nanofluid gives notable enhancement in heat transfer rate compared to nanofluid.

FLOW OF HYDROMAGNETIC MICROPOLAR-CASSON NANOFLUID OVER POROUS DISKS INFLUENCED BY CATTANEO-CHRISTOV THEORY AND SLIP EFFECTS

This study elaborates the incompressible and steady flow phenomenon of non-Newtonian (micropolar-Casson) fluids confined in two parallel porous disks. The fluid motion is generated due to the uniform injection along the axial direction at the stationary disks. The thermal execution of the system is augmented by the Buongiorno and Cattaneo-Christov theories. The velocity and thermal slip effects are also encountered through boundary conditions. To perform the numerical solutions, the flow governing system is first converted into a system of ordinary differential equations and then these reduced equations are solved by implementing the Runge-Kutta-Fehlberg's fourth-fifth (RKF-45) order method accompanied with a shooting scheme. The major outcomes of the study reveal that for increasing levels of magnetic factor and vortex viscosity, radial velocity decreases in the center plane and increases closer to the upper and lower disks. For varied magnetic parameter and vortex viscosity factors, the microrotation profile upswings at the bottom and upper disks, while the central plane shows an opposing tendency. The concentration profiles exhibit rising behavior for enhanced values of Schmidt number, whereas a reverse trend can be seen for the thermophoretic parameter.

Thermophysical impact on the squeezing motion of non-Newtonian fluid with quadratic convection, velocity slip, and convective surface conditions between parallel disks

The present paper considers suction/injection, constant and variable thermophysical influence in squeezing flow of magnetized blood rheological (Casson) fluid model between two parallel disks subjected to nonlinear convection process, slip and convective surface conditions. The unsteady, squeezing, magnetized, and chemically reacting dissipative fluid flowing between parallel disks is modeled and non-dimensionalized via an applicable similarity transformation. An approximate solution of the flow distributions is sought by employing the Chebyshev Collocation Approach (CCA). In the limiting scenario, an excellent agreement in the present results with the existing literature is obtained. The analysis reveals the dominance of constant thermophysical effect over the variable thermo-properties on velocities and temperature field. Higher thermal/solutal Biot number highlighted the dominance of positive squeezing number while its minimal values showcase the dominance of negative squeezing number on flow and energy field. Therein, the squeezing effect remains invariant for some range values of thermal/solutal Biot number, while a rise in velocity slip and suction parameter enhanced the fluid velocities and concentration profiles respectively. Moreover, the current analysis is applicable in moving engines such as in parallel disk gate valve, locomotion of piston in ring where oil is viewed as Casson fluid, thermoforming, injection molding, biomedical use among others.

A multi-dimensional Child–Langmuir law for any diode geometry

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

Effects of variable thermal conductivity and electrical conductivity on flow of Williamson nanofluid between two parallel disks

The variable nature of the thermal conductivity of nanofluid with respect to temperature plays an important role in many engineering and industrial applications including solar collectors and thermoelectricity. Thus, the foremost motivation of this article is to investigate the effects of thermal conductivity and electric conductivity due to variable temperature on the flow of Williamson nanofluid. The flow is considered between two stretchable rotating disks. The mathematical modeling and analysis have been made in the presence of magnetohydrodynamic and thermal radiation. The governing differential equations of the problem are transformed into non-dimensional differential equations by using similarity transformations. The transformed differential equations are thus solved by a finite difference method. The behaviors of velocity, temperature and concentration profiles due to various parameters are discussed. For magnetic parameter, the radial and tangential velocities have showed decreasing behavior, while converse behavior is observed for axial velocity. The temperature profile shows increasing behavior due to an increase in the Weissenberg number, heat generation parameter and Eckert number, while it declines by increasing electric conductivity parameter. The nanoparticle concentration profile declines due to an increase in the Lewis number and Reynolds number.