Critical Number Research Articles

Novel experimental data-driven bubble breakage model for universal application in Euler-Lagrange multiphase frameworks

An experimental data driven model for the prediction of bubble breakage in gas–liquid multiphase reactors was developed which fulfills requirements of the Euler-Lagrangian (EL) framework for the analysis of individual bubbles. The critical Weber number (Wecrit) serves as the breakage criterion in combination with the minimum time between two consecutive breakage events. Wecrit indicates when the hydrodynamic forces acting on an individual bubble surmount its surface tension. Once the threshold is passed a binary bubble breakup event occurs, which results in the formation of daughter bubbles assigned to an M−shaped daughter size distribution (DSD). Optimum Wecrit values were obtained by minimizing the difference between simulated and measured Sauter mean diameters d32. The latter were derived from bubble size distributions (BSDs) measurements that cover about 11 – 89 W m−3 power inputs. Using deionized water and a minimum timespan of 30 ms between two consecutive breakages Wecrit = 6.1 was determined. To extend the scope of the model, characteristic media compositions for so-called syngas fermentations with Clostridium ljungdahlii were investigated. To match surface tension and viscosity with those of fresh medium (FM) and late phase medium (LPM), two aqueous solutions containing sodium dodecyl sulfate (SDS) and glycerol were prepared. Notably, the breakage model managed to well predict the impact of altered fluid properties of FM and LPM without requiring further calibration. The sum of Sauter diameter deviation between simulations and experiments was only 0.21 and 0.40 mm for FM and LPM, respectively.

The Significance of Traditional Exegesis in Modern Quranic Sciences (Critical Analysis)

For several centuries, traditional exegesis has played a crucial role in understanding the Quran within Islamic heritage. However, the significance of traditional exegesis has emerged as a profound debate in the era of modern Quranic research. The aim of this research is to conduct a critical analysis of the importance of traditional exegesis in the context of modern Quranic sciences.
 The research findings indicate that traditional exegesis holds historical and intellectual value that cannot be disregarded in comprehending the Quran. Nevertheless, it also faces a number of criticisms that need to be addressed in a contemporary context. This research discusses various efforts aimed at addressing these criticisms, including adaptation, reexamination, and integration with modern scholarly methodologies, highlighting the significant role of Muslim scholars and intellectuals in preserving the importance of traditional exegeses while remaining open to innovation and necessary changes to confront the challenges of the modern era.

The Influence of External Flow Field on the Flow Separation of Overexpanded Single-Expansion Ramp Nozzle

Flow separation and transitions of separation patterns are common phenomena of nozzles working with a wide Mach range. The maximum thrust method is applied to design the single-expansion ramp nozzle (SERN) for specific operating conditions. The nozzle is used to numerically simulate the transition processes of separation patterns under the linear change in the external flow Mach number and the actual trajectory take-off condition of a rocket-based combined cycle (RBCC), to investigate the mechanism through which the external flow field influences the separation pattern transition during acceleration. The computational fluid dynamics (CFD) method is briefly introduced, followed by experimental validation. Then, the design procedure of SERN is described in detail. The simulation results indicate that as the external Mach number increases, the flow field in the nozzle undergoes transitions from RSS (ramp) to FSS, and finally exhibits a no-flow separation pattern. The rate at which the external Mach number varies has little effect on the transition principle of the nozzle flow separation patterns, but it has a significant effect on the critical Mach number of the transition points. The external flow field of the nozzle has an airflow accumulation effect during acceleration, which can delay the transition of the flow separation pattern.

Confinement Effects on Coaxial Jet Diffusion Flame

ABSTRACT A combined experimental and numerical study has been conducted to investigate the characteristics of confined coaxial jet diffusion flames in air coflow, using methane as fuel. The main concerns were focused on the unequal fuel velocity (0.1–72 m/s)/air velocities (0.045–0.53 m/s) and confinement ratio (δ*, defined as the ratio of the confinement tube diameter and the jet diameter), and most of the flow remains laminar while the ranges of δ* and gas velocities vary widely. The flame height linearly increases with the central fuel jet velocity, until the flame tip-opening or liftoff occurs, which is similar to that of the free jet flame. The critical fuel jet velocity for flame liftoff decreases with the ratio of air velocities to δ * . The critical fuel jet velocity of flame tip-opening increases with δ*, and the critical volumetric flow rate of air increases to a maximum, then decreases, indicating a combined effect of the oxygen supplement and the mixing with the fuel. Under smaller δ* with high jet flow velocity, a circular plane flame is observed before flame extinction. The formation of a recirculation zone in a confinement with combustion reaction has a significant influence on flame behaviors. A critical nondimensional Craya-Curtet number is used to predict the onset of flow recirculation with combustion reaction. At a certain coflow air velocity, the critical number initially increases and then remains constant with δ*, while at a certain δ*, the critical number decreases with air velocity. When the Craya-Curtet number is higher than the critical point, recirculation cannot be established, resulting in the formation of a tip-closed flame. With the decrease of the Craya-Curter number, the recirculation evolves to the upstream of the flame, then, the flame may be lifted off, along with the recirculation zone moving downstream. With the effect of recirculation, the stoichiometric mixture fraction contour shrinks toward the central axis where there is a higher local velocity. Therefore, the flame will be lifted off under a lower jet velocity in a smaller δ * . With the increase of the jet velocity, the Craya-Curtet number further decreases, and the recirculation becomes much stronger than the upstream inflow, resulting in the flame being compressed into a circular plane flame at a small δ* with a relatively high jet flow velocity or blowout directly at a relatively large δ * .

Subverting the Civilising Mission in Conrad’s Heart of Darkness

Conrad's Heart of Darkness (1899) is often presented as one of the harshest indictments of the colonial entreprise in Africa. Yet, a large number of critics consider that it perpetuates the colonial discourse in several ways. The aim of this paper is to critically look at the colonial discourse in Conrad’s novella from a postcolonial and Marxist standpoint. To be specific, we demonstrate the limitations and contradictions of the theory of the civilising mission, which portrays Africa as the “heart of darkness” and Africans as uncivilised individuals who need to be enlightened by Europeans. The analysis concludes that, above their moral corruption, the colonists, through the atrocity of their actions and their greed, are a perfect reflection of a Western civilisation corrupted by the capitalist spirit that underpins it. In this sense, they are proof that the theory of the civilising mission is a fallacy. The real savages are those who think that they bear the light of civilisation. Black people are savages only in the Western imagination. In reality, they are the victims of the savagery, the darkness inherent in Western materialistic culture.

Minimal Crew Size and Sensitive Reproductive Parameters on Multigenerational Interstellar Travel

Multigenerational interstellar travel to exoplanets, as well as manned missions to the Moon and Mars, has been investigated in the world. Defining the minimum crew size of interstellar ships is one of the most important research areas for interstellar travel, because designing multigenerational interstellar ships requires defining a critical crew size, which factors into many variables, including food production, air/water control, and propulsion. Anthropologists and astrophysicists have recently tackled with the minimum crew size of interstellar ships in the field of “Space anthropology”. Previously published computations in the field of space anthropology provided a critical crew number of 1,900- 2,000 (Sano, 2021) with constant reproductive parameters, but did not fluctuate reproductive parameters such as infertility, initial genetic diversity and number of children per woman. These parameters would fluctuate during the real multigenerational journey. Then, a more accurate estimate of the critical crew size would be obtained from the fluctuating parameters. This paper provides a critical crew size of 1,400-6,800 for interstellar travel using the fluctuating parameters, and clarifies which anthropological parameters are sensitive and how critical to be controlled aboard interstellar ships. It indicates that measuring infertility and controlling the number of children per woman for many generations onboard interstellar ships are essential to prevent extinction, and firstly quantifies the importance of high diversity of initial crew and procreation window for interstellar travel.

The influence of soil fabric on the monotonic and cyclic shear behaviour of consolidated and compacted specimens

While the fabric of soil can significantly influence its behaviour, the effect of varying fabric parameters on the subgrade shear response is still not well understood. This study creates soil specimens with different fabrics which are then captured through X-ray microscopic-computed tomography scanning and quantified by image processing techniques. A comprehensive laboratory investigation is conducted to understand how the soil fabric affects its monotonic and cyclic shear behaviour. The results indicate that the consolidation method creates a more homogeneous fabric with mainly small-to-medium interconnected pores, whereas the compaction technique creates significantly large and mostly inter-aggregate pores with lower connectivity. In this regard, the consolidated specimens exhibit an elastic-perfectly plastic behaviour, while the compacted specimens show strain-hardening transformation during isotropic monotonic shearing. Under anisotropic conditions, the compacted specimens exhibit a greater strain softening response and excess pore pressure than the consolidated specimens because they have a weaker fabric. Furthermore, the compacted specimens show a smaller threshold strain at a lower critical number of cycles due to the collapse of large pores. These current findings prove the decisive role that soil fabric plays in determining the shear response and failure of subgrade soils.

Variable gravity effects on penetrative porous convection

The effect of variable gravity on the onset of penetrative convection in a horizontal fluid-saturated porous layer is studied, performing the linear instability analysis and the nonlinear stability analysis (via the weighted energy method) of the thermal conduction solution. The linear and nonlinear critical Rayleigh numbers for the onset of penetrative porous convection are determined and their behaviour with respect to the fundamental physical parameters of the model is studied. In particular, we found that there exists a region of subcritical instabilities and that the increase of the upper bounding plane temperature and the decrease of gravity have a stabilizing effect on the onset of convection.

An experimental investigation on the nanofluids in a cavity under natural convection with and without the rotary magnetic field

This experimental study involves the use of two distinct categories of nanofluids, namely ferromagnetic and non-magnetic, within a square cavity that facilitates natural convection. There are five distinct concentrations associated with each nanofluid. Natural convection arises as a consequence of the thermal gradient between the opposing surfaces of the copper cavity, which has a thickness of 18 gauge. The purpose of utilizing the constructed electro-magnet assembly is to investigate the impact of the rotational magnetic field on the process of heat transfer. The manipulation of magnetic strength can be achieved by regulating the magnetic power and direct current (DC) power. The manipulation of the electromagnet’s spin can also be regulated. In the context of a rotational magnetic field, it is seen that the magnetic flux undergoes a transition from a positive value to an almost identical negative value throughout a full rotation. The optimal heat transfer performance is observed at a nanoparticle concentration of 0.1% by volume (φ) for both nanofluids. In the absence of a rotating magnetic field, the ferromagnetic nanofluid exhibited superior performance. When the Rayleigh number (Ra) is one order smaller than the critical Rayleigh number value, the heat transfer performance is often superior with nanofluid compared to demineralized water.

Jeans Gravitational Instability of a Rotating Collisionless Magnetized Plasma with Anisotropic Pressure

The problem of self-gravitational instability of an astrophysical rotating plasma in a strong magnetic field with an anisotropic pressure tensor is studied on the basis of the Chew–Goldberger–Low (CGL) quasi-hydrodynamic equations modified by generalized polytropic laws. Using the general form of a dispersion relation obtained by the normal-mode perturbation method, a discussion is provided of the propagation of small-amplitude perturbation waves in an infinite homogeneous plasma medium for transverse, longitudinal, and oblique directions with respect to the magnetic field vector. It is shown that different polytropic indices and anisotropic pressures not only change the classical Jeans instability condition but also cause the appearance of new unstable regions. Modified Jeans instability criteria are obtained for isotropic MHD equations and anisotropic CGL equations owing to the influence of the polytropic indices on gravitational and firehose instabilities for astrophysical plasma. It is shown that in the case of a longitudinal mode of perturbation wave propagation, the Jeans instability criterion does not depend on uniform rotation. In the case of the transverse propagation regime, the presence of rotation reduces the critical wave number and exerts a stabilizing effect on the growth rate of the unstable regime.

Three-dimensionality effects on the flow past a horizontal heated cylinder under mixed convection heat transfer

The three-dimensionality has a significant effect on the flow-field and heat transfer, especially when the geometry is affected by thermal buoyancy. In the present research, the interaction of three-dimensionality, thermal buoyancy, and vortex shedding on the flow around an isothermal horizontal cylinder and its mixed convection heat transfer were numerically investigated. Both the secondary flow due to the thermal buoyancy and the free stream had an upward direction. The problem was simulated in a constant Prandtl number of 0.7, a range of Reynolds numbers (based on the cylinder's diameter) between 100 and 300, and Richardson numbers ranging between 0 and 0.5. The aspect ratio of the cylinder was 6. The numerical simulation of the problem was based on the discretization of the Navier–Stokes, energy, and k–ω SST turbulence equations. A finite-volume and pressure-based method with second-order spatial and temporal accuracy was utilized. The momentum, energy, and turbulence equations were explicitly solved, while the pressure equation was solved, implicitly. Comparison of the two-dimensional and three-dimensional simulations showed that the thermal buoyancy had a significant impact on the flow three-dimensionality, so that the flow could become three-dimensional even at low Reynolds numbers such as Re = 130. In addition, the results demonstrated that for each Reynolds number, there was a critical Richardson number where the mean drag coefficient and Nusselt number reached their minimum value. Moreover, for Richardson numbers higher than the critical value, the flow entered a new low amplitude vortex shedding mode.

Schmidt number's impact during the growth of mc-silicon ingot through directional solidification furnace for photovoltaic applications: A computational investigation

The use of multi-crystalline silicon for photovoltaic solar cell applications is highly advantageous due to its low manufacturing costs. In this study, numerical simulations were conducted to investigate mass transport phenomena during the directional solidification of molten silicon for different Schmidt values at the critical Prandtl number of 0.01. The simulations were performed using the diffusion model and convection-conduction equations as the framework for the incompressible Navier-Stokes equation. The finite-volume method was used to carry out computations in a two-dimensional (2D) axisymmetric model. The diffusion flux of impurities (oxygen, carbon, and nitrogen) and dopants (phosphorus and boron) during the mc-silicon growth process was examined for Schmidt numbers Sc=0.5, Sc=5, and Sc=10. It was found that the Schmidt number Sc=0.5 is suitable for producing higher-grade mc-silicon ingots by regulating the production of impurities and dopants. The simulation results obtained in this study may provide a better understanding of the mc-silicon growth process and can be used to improve the production of high-quality mc-silicon ingots for photovoltaic applications.

Weakly nonlinear bifurcation behaviour at the onset of instability in horizontal convection using an advective buoyancy approximation

Horizontal free convection is studied under an extended buoyancy approximation constructed to capture centrifugal effects in strongly rotating regions within buoyancy driven flows. Under this approximation, the Gay-Lussac parameter (Ga) arises in the momentum equation to characterise non-Boussinesq behaviour. The scaling of average Nusselt numbers (Nuavg) are determined, as is its dependence on Ga. Increasing Ga is found to decrease Nuavg when convection dominates the flow, while no influence is detected in the low Rayleigh number regime dominated by conduction. The influence of Ga on the onset of the time-dependent regime is investigated and the critical Rayleigh number corresponding to different Ga are determined. Results demonstrate that increasing Ga stabilises the flow, delaying transition to the time-dependent regime. An Orr–Sommerfeld type stability analysis is performed to determine the local stability at different horizontal stations. A zone of convective instability is first detected, growing from the hot end of the enclosure, at Rayleigh numbers three orders of magnitude lower than the predicted global stability threshold at the maximum investigated Ga value. Ga is found to play a key role in determining the preferred orientation of instability roll structures, with increasing Ga being accompanied by a bias toward transverse-roll instabilities over longitudinal rolls. The Stuart—Landau model is used to reveal that the non-linear characteristics of this unsteady transition are consistent with a supercritical Hopf bifurcation.

Impact of charged soft layers on electroosmosis of Maxwell fluids in soft nanochannels

In the present paper, we theoretically study the transient electroosmotic flow of general Maxwell fluids through the polyelectrolyte grafted nanochannel with a layered distribution of charges. By applying the method of Laplace transform, we semi-analytically obtain the transient electroosmotic flow from the Cauchy momentum equation and the Maxwell constitutive equation. For the nanochannels grafted with polyelectrolyte layers having different layered distributions of charges, we consider the influence of the dimensionless relaxation time, the dimensionless polyelectrolyte layer thickness, and the dimensionless drag coefficient on the transient electroosmotic flow. We present the results for some particular cases. First, we unravel that for the case of polyzwitterionic brush that the sum of positive and negative structural charges is zero, the total electroosmotic flow is non-zero. In particular, depending on charge distribution within the end part of polyelectrolyte layers, the direction of the electroosmotic flow can be reversed critically. Second, in order to quantitatively evaluate a reversal of electroosmotic flow for two polyelectrolyte layers of opposite signs, we introduce a critical number ks as the ratio between the layered charge densities of two polyelectrolyte layers. Increasing ks allows the electroosmotic flow to be reversed easily. We verify that adjusting charge distributions of the layer can intentionally control the direction of the flows as well as the strength of electroosmotic flow.

Nucleation Work on Curved Substrates

Nucleation is the initial phase transition process when nuclei of a new phase form within an undercooled or supersaturated parent phase under appropriate conditions. Nucleation most often occurs through a heterogeneous process on active centers on which the probability of nucleus formation is high. In general, the origin of active centers may be difficult to distinguish. In this work, we consider the formation of crystalline nuclei in a melt on various curved substrates. Knowledge of excess free energy plays a key role in understanding the process of formation of clusters and it is not easy to express this quantity in a considered system. Excess free energy is often approximated within the framework of capillarity approximation based on interfacial energy, which depends on interatomic interactions near the interface, as well as the misfit between melts, surface roughness, temperature, composition, etc., near the phase interface. The formation of nuclei requires overcoming a certain energy (nucleation) barrier that is a consequence of balancing the volume and the interfacial free energy. Knowing the nucleation barrier (W) is crucial for understanding this process, as nuclei predetermine the physical properties of a newly formed phase. W is typically expressed as a function of the nucleus radius; however, in nucleation kinetics, one needs to determine (W) as a function of the number of molecules forming the nucleus. We analyze nucleation work on various substrates (flat, convex, and concave) for crystallization from an aluminum melt to show that the formation of nuclei is the most probable on concave substrates. An analytical expression for W can be easily applied to other systems under consideration. We show that under the same conditions, the critical radius of nuclei is identical for various substrate, in contrast with the critical number of molecules forming a nucleus.

Lorenz model of instability in porous media for van der Waals gas

The article is devoted to the study of the nonlinear instability of the van der Waals gas in a gap with a porous medium of finite thickness heated from below. A fixed temperature difference is set between the upper cold surface and the lower hot one. The Lorenz approach was used to solve the problem. During a numerical solution, criteria for monotonic instability Racr1 and oscillating instability Racr2 were obtained. The results of calculations of the criterion Racr3 for the appearance of a strange attractor in the phase space are presented, which can be interpreted as a criterion for the generation of undamped turbulent fluctuations. The nature of the dependences of the critical Rayleigh numbers on the physical properties of the gas, the porosity of the medium, and the parameters of the van der Waals equation of state Waa and Wab is analyzed.The phenomenon of instability (formation of vortices in liquids or gases) in the space between two horizontal boundaries, when the lower one is heated, is widespread in technical applications. Lorenz developed a non-linear approach to the study of this phenomenon. This approach makes it possible to determine the criteria for the emergence of vortices and their transformation. This article uses the Lorenz approach to obtain instability criteria for a real gas whose properties are described by the van der Waals equation of state. This gas is in a porous medium. This combination of gas and medium has a wide technological application in the food, chemical and other industries. That is why the present study focused on this problem.

Optimal Transient Energy Growth of Two-Dimensional Perturbation in a Magnetohydrodynamic Plane Poiseuille Flow of Casson Fluid

Abstract The study investigates the influence of the Casson fluid parameter and the spanwise uniform magnetic field on the onset of instability against infinitesimal disturbances in an electrically conducting fluid flow between two parallel nonconducting rigid plates. The fourth-order linearized disturbance equation governing stability is solved using the spectral collocation method with Chebyshev-based polynomials. The aim is to analyze in detail the effect of the parameters involved in the problem using both modal and nonmodal linear stability analysis. The modal analysis provides accurate values of the critical Reynolds number, critical wave number, and critical wave speed, denoted as critical triplets (Rc, αc, cc). Additionally, it examines the eigen-spectrum, growth rate curves, and neutral stability curves. On the other hand, the nonmodal analysis investigates the transient energy growth G(t) of two-dimensional (2D) optimal perturbations, the pseudospectrum of the non-normal Orr–Sommerfeld (O–S) operator (ℒ), and the regions of stability, instability, and potential instability of the fluid flow system. The extensive examination of both long-term behavior through modal analysis and short-term behavior through nonmodal analysis reveals that the Hartmann number (Ha) acts as a stabilizing agent, delaying the onset of instability. Conversely, the Casson parameter (η) acts as a destabilizing agent, advancing the onset of instability. The results obtained here are verified to be in good agreement with the existing literature in the absence of the Casson fluid flow parameter.

Dynamics of a droplet in shear flow by smoothed particle hydrodynamics

The behavior of a droplet under shear flow in a confined channel is studied numerically using a multi-phase smoothed particle hydrodynamics (SPH) method. With an extensive range of Reynolds number, capillary number, wall confinement, and density/viscosity ratio between the droplet and the matrix fluid, we are able to investigate systematically the droplet dynamics such as deformation and breakup. We conduct the majority of the simulations in two dimensions due to economical computations, while perform a few representative simulations in three dimensions to corroborate the former. Comparison between current results and those in literature indicates that the SPH method adopted has an excellent accuracy and is capable of simulating scenarios with large density or/and viscosity ratios. We generate slices of phase diagram in five dimensions, scopes of which are unprecedented. Based on the phase diagram, critical capillary numbers can be identified on the boundary of different states. As a realistic application, we perform simulations with actual parameters of water droplet in air flow to predict the critical conditions of breakup, which is crucial in the context of atomization.

Drop Impact on Submillimeter-Structured Surfaces with Different Wetting Behaviors.

Droplet impact behaviors are crucial in controlling infectious diseases, inkjet printing, and anti-icing applications. The wettability and microstructure of the material surface are critical factors in this regard. Compared to microstructures, submillimeter structures are more damage-resistant, thereby ensuring droplet impact behaviors' stability. Herein, submillimeter-structured PDMS surfaces with varying wetting properties were prepared to investigate droplet impact behaviors. Experimental results indicate that submillimeter-structured surfaces are more prone to droplet splashing than flat surfaces, which can be suppressed by increasing surface hydrophilicity. An increase in the submillimeter pillar height and a decrease in spacing result in an increased critical Weber number. Additionally, the capillary forces of the superhydrophilic surface lead to droplet impact, accompanied by deposition. This study supports the long-term stable use of the droplet impact effect to achieve fluid separation.

Nonlinear magnetoconvection of a Maxwell fluid in a porous layer with chemical reaction

ABSTRACT In this paper, the linear and nonlinear instability of magnetoconvection in a Darcy-Benard setup saturated by a Maxwell fluid with chemical reactions is studied. The governing non-dimension equations are solved using the normal modes, and we obtain the expressions for steady and oscillatory thermal Rayleigh numbers. The effects of different physical parameters such as the Damkohler number (), Hartmann number (), Solute Rayleigh number (), relaxation parameter (), Magnetic Prandtl number (), Lewis number () on stationary and oscillatory critical thermal Rayleigh numbers are presented and described. Enhancing the values of the Solute Rayleigh number and Lewis number makes the system unstable. Also, the Hartmann number and Damkohler number have a contrasting effect on stationary and oscillatory convection. Enhancing the value of the relaxation parameter makes the system more stable. In order to study heat transport by convection, the well-known equation, the Landau-Ginzburg equation, is derived.