Present Authors Research Articles

Piezoinductance in a carbon fiber tow

Piezoinductance (the influence of stress/strain on the inductance) enables inductance-based structural self-sensing and inductance-based investigation of the microstructural effect of loading. The first report of piezoinductance in a carbon material is provided. The piezoinductance of a carbon fiber tow (12K) involves the inductance increasing upon tension. The stress is in the low range of the elastic regime (2.17–10.83 MPa). The piezoinductance is attributed to the tightening of the fiber-fiber contacts in the tow upon tension. The partial nature of the reversibility of the inductance increase means that the tightening is not completely reversible upon unloading. Twisting enhances the inductance (as previously reported by the present authors), because the number of fiber-fiber contact points increases. Twisting strengthens the piezoinductance, with the effect being due to the abovementioned tightening. An increase in either the frequency (1–5 kHz) or voltage (0.9–5.0 V) enhances the inductance, but weakens the piezoinductance. The latter is probably due to the effect of the abovementioned tightening on the inductance being diminished by an increase in the frequency or voltage. Tension reduces the number of fiber-fiber contact points and reduces the effect of twisting on the inductance. Upon combined tension and twisting, the inductance increases, such that the fractional increase in inductance is below that for twisting without tension, but above that for tension without twisting.

Finite element simulation of axial crushing of flat and C-channel composite coupons using a modified mesomodel

The present article investigates the axial crushing behavior of IM7/8552 material system. Two types of specimens are considered-(a) flat coupons with saw-tooth trigger and (b) C-channels with 45° chamfer as a trigger. All the crushing test data (load, displacement and velocity history curves) is shared in the Crashworthiness working group (CWG) consortium of the Composite material handbook (CMH17.org). Flat crush coupons are tested at the University of Utah and the C-channels are tested at the John H. Glenn Research Center of the National aeronautics and space administration (NASA). Three types of layups are considered for flat coupons: QI1-[90/±45/0]2S, HL2-[902/02/±45/02] and HL3-[90/45/02/90/-45/02]. Two types of layups are considered for C-channels: HL2-[902/02/±45/02] and HL3-[90/45/02/90/-45/02]. Both the specimens are subjected to crush loading with an initial velocity and lumped mass attached to the impactor. All the ply and interface material properties come from the current authors' testing campaign except for strength and stiffness in the fiber direction. The crushing test cases are simulated in LS-Dyna software. Ply and interface behavior is described via user-defined material subroutines of LS-Dyna software. Each ply is modeled using 3D solid elements and the interfaces are modeled explicitly using cohesive elements. Intraply behavior is described using a novel modified meso model proposed by the present authors. Intraply model accounts for nonlinear damage, plasticity, ply fracture toughness and finite element mesh size. Interply response is modeled using bi-linear law. Predictions from present FE models showed a good correlation with crush test data. The effect of finite element model parameters e.g., filtering frequency, mesh size, interface properties, ply fracture toughness, element erosion criterion, contact stiffness and friction coefficient on crushing predictions is demonstrated by conducting parametric studies.

The 1952 Ph.D. dissertation of T.F.W. Embleton on nonlinear acoustic propagation and reflection

In his doctoral research Tony Embleton studied nonlinear acoustics experimentally using techniques available in the Physics Department of Imperial College, University of London. Originally developed by V. Timbrell, the interferometric equipment was substantially improved by Tony Embleton, enabling his Ph.D. findings. Studying propagation in a tube and reflection by both closed and open ends, Tony Embleton was able to substantially confirm the theory of the time, developed by Stokes, Earnshaw, Rankine, Rayleigh, and Taylor, many pieces confirmed for the first time. He also found discrepancies regarding the attenuation coefficients regarding the first and second half cycle of each pulse as well as a shortening of a pulse at reflection from a closed end. These findings are still worthy of further investigation. The results of reflection of pulses from the open end of a tube, forming a pressure release boundary condition, still are very interesting to the present author. A pulse propagating toward the open end steepens, but the reflected wave “unsteepens” which is a unique nonlinear acoustic phenomenon. [Work supported by the Penn State College of Engineering and its United Technologies Corporation Professorship.]

Combining artificial intelligence and computational fluid dynamics for optimal design of laterally perforated finned heat sinks

The efficient design of heat sinks is a severe challenge in thermo-fluid engineering. A creative and innovative way is applying lateral perforations to parallel finned heat sinks. The significance of achieving an optimal design for perforated finned heat sinks (PFHSs) has inspired the present authors to introduce a novel hybrid designing approach that combines computational fluid dynamics (CFD), machine learning (ML), multi-objective optimization (MOO), and multi-criteria decision-making (MCDM). The design variables considered include the size (0.25<φ<0.5) and shape (square, circular, and hexagonal) of the perforations, as well as the airflow Reynolds number (2000<Re<5000). The design objectives have been redefined as dimensionless parameters to assess heat dissipation, pressure drop, and heat sink weight. These modified objectives encompass thermal performance (TP), thermo-hydraulic performance (THP), and thermo-volumetric performance (TVP). The modeling process showed that both stepwise mixed selection (SMS) and GMDH-NN techniques exhibited comparable performance in most modeling scenarios. Nevertheless, the SMS approach demonstrated more reliability in modeling diverse objectives. Furthermore, the optimization results demonstrated that the optimal size of the perforations is strongly dependent on their shapes. In PFHSs with square perforations, approximately 54% of the Pareto points had a φ-value greater than 0.45. Meanwhile, in PFHSs based on circular perforations, more than 50% of the optimal points have φ less than 0.4. The MCDM-based analysis on various real-world scenarios indicated that using PFHSs with square-shaped perforations with Reynolds numbers around 2000 and considering a wide range of perforations' sizes could result in optimal designs.

10 years of the Africa Journal of Management: A bibliometric analysis

ABSTRACT The Africa Journal of Management (AJOM) is devoted to the advancement of management theory, scholarship, practice, service, and education in or about Africa and for the benefit of Africa. The journal was launched at the 2014 Africa Academy of Management (AFAM) Conference in Gaborone, Botswana. Since its launch, AJOM has published 219 documents, including 146 research articles, 26 dialogues/insights/essays, 13 managers’ stories, 13 editorials, and nine papers related to the AFAM. We employed a bibliometric analysis to provide a retrospective overview of the work published in the AJOM throughout these past years. We recognize the most cited papers, conduct a co-citation analysis, present author affiliations, keyword analysis, and a general discussion of the most recurrent topics published in the journal. We end with a synopsis of the future of AJOM and support for its continuous growth.

Does research support the current conservative care recommendations for pregnancy-related diastasis rectus abdominis? A scoping review

Pregnancy-related diastasis rectus abdominis (DRA) is a widespread condition. The current evidence for its conservative management is contradictory, as are the criteria for making a diagnosis, which has stimulated debate among practitioners. The aim of this scoping review was to synthesize the existing peer-reviewed literature to determine whether there is evidence to support or refute recently published recommendations for the conservative care of perinatal DRA. These guidelines suggest that it should be approached from the perspective of considering the linea alba (LA) as an integrative component of the thoracopelvic abdominal system, which assumes functional relationships between the structures of the thorax and pelvis. Specifically, the present authors explored whether relationships exist between: (1) the LA and breathing mechanics; (2) pelvic floor muscle (PFM) function; (3) lumbopelvic pain (LPP) control; and (4) aspects of the structure and function of the abdominal wall. Of the 31 studies included, none were found relating to the LA and breathing mechanics, 11 investigated PFM function, 10 explored LPP, and 18 examined the LA with respect to the structure and/or function of the abdominal wall. The research reviewed does not appear to substantiate several of the recommendations for the conservative care of DRA, but does align with cited gaps in knowledge about this condition. The studies included neither support nor necessarily refute the relationships between breathing, PFM function and LPP. This scoping review also highlights the limitations of the current characterization of DRA and related assessment strategies, particularly the reliance on heterogeneous measurements of inter-recti distance as the primary and sometimes only measurement to inform clinical reasoning with respect to the condition.

Essay: Exploring the Physics of Basic Medical Research.

The 20th century witnessed the emergence of many paradigm-shifting technologies from the physics community, which have revolutionized medical diagnostics and patient care. However, fundamental medical research has been mostly guided by methods from areas such as cell biology, biochemistry, and genetics, with fairly small contributions from physicists. In this Essay, I outline some key phenomena in the human body that are based on physical principles and yet govern our health over a vast range of length and time scales. I advocate that research in life sciences can greatly benefit from the methodology, know-how, and mindset of the physics community and that the pursuit of basic research in medicine is compatible with the mission of physics. Part of a series of Essays that concisely present author visions for the future of their field.

Aseptic Silk Ligatures Induce Bone Resorption Around Titanium Implants: A 12-Week Pilot Study in Rabbits.

Marginal bone resorption (MBR) around dental implants may sometimes be a self-limiting condition due to balancing immunologic reactions against utilized materials rather than a progressive bacterial infection. Contrary to previous assumptions from ligature-induced experimental peri-implantitis studies, a recent 8-week experiment by the present authors showed that marginal ligatures trigger an inflammatory immune response, resulting in bone resorption around implants in the absence of plaque. The present study aimed to investigate whether this inflammatory/immunologic reaction attenuates or progresses toward implant failure after a longer healing time (12 weeks). Sterile silk ligatures were placed around the top of titanium (Ti) implants and compressed against the femoral cortical bone plate of six rabbits. A nonligated implant was used as a control. After 12 weeks of submerged healing, ground sections of implants and surrounding tissues were investigated with light microscopy. The marginal soft tissues were also analyzed using selected quantitative polymerase chain reaction (qPCR) markers. Histologically, the ligatures were outlined by immune cells, including multinucleated giant cells (MNGCs), with adjacent fibrous encapsulation and resorbed peripheral bone that contrasted from the osseointegrated nonligated control implants. The difference in expression of qPCR markers was not significant, but > two-fold upregulation of markers CD11b, IL1β, ARG1, NCF1, and CD4 and > twofold downregulation of CD8 indicated a mild, focal inflammatory/immune response against the ligatures compared to controls, with upregulation of M1 and M2 macrophages, neutrophils, and helper T-cells as well as downregulation of killer T cells. Further, the bone formation markers OC and ALPL were > two-fold downregulated (consistent with the lack of osseointegration of the ligatures) compared to control implants. Marginal silk ligatures trigger an inflammatory/immune response and aseptic bone resorption around implants. Compared to the previous 8-week study, the inflammatory reaction against the silk appears to attenuate with time, with only a mild persisting inflammation that may block osseointegration; instead, a fibrous tissue encapsulation-type reaction is maintained. This may explain why traditional ligature experiments have required regular exchange of ligatures for the bone resorption to progress.

STUDY ON SCULPTURAL ART OF TEN JAINA IMAGES FROM THE ODISHA STATE MUSEUM OF BHUBANESWAR

The aim of present article is to focus on the sculptural features as well as religious significance of the ten Jaina images of the Odisha State Museum, Bhubaneswar. Iconography of ten Jaina Tirthankara images of the Odisha State Museum is a major aspect of the Odishan sculptural art in India. Jainism is one of the oldest religions in the state of Odisha. The earliest Jaina sculptures in Odisha date back to the period before Mauryan/Asokan. From the 6th century B.C. Jainism remained one of the major religions of Kalinga (ancient Odisha) until the first century B.C., when the Kharavela was ruling in the state. It is believed that Jainism continued to be a major religion in the state even after the war of Kalinga (261 B.C.). Jainism co-existed with other religions in the state of Odisha. In fact, Jaina Tirthankara images were made all over Odisha during the early medieval era. Some Jain images are noticed at different places of Odisha. After Kharavela, the artists of Odisha continued to create Jain images until the 11th century C.E. Out of the extant Jaina Tirthankara images of Odisha, some of the Jaina sculptures are found to be kept in Odisha State Museum, Bhubaneswar. Among these preserved Jaina Tirthankara images, ten Jaina Tirthankara images are discussed in this piece of work. The names and discovered sites of the ten Jaina images of the State Museum, Bhubaneswar are viz; 1. Risabhanath of Keonjhar, 2. Risabhanatha of Khajuriapada, 3. Mahavir image of Charampa, 4. Parsvanatha of Sisupalgarh, 5. Shantinatha of Charampa, 6. Risabhanath of Charampa, 7. Risabhanatha of Podasingidi, 8. Ajitanatha of Charampa, 9. Risabhanath of Bhubaneswar and 10. Parsvanatha of Baidyakhia. Sculptural features of the above Jaina Tirthankara images of the Odisha State Museum suggest that these images were made during the time period from the 8th century to 11th century C.E. The presence of these Jaina Tirthankara images gives an idea that Jainism was prevalent in eastern and northern Odisha during the medieval period. Artistic point of view, these Tirthankara images are very important and interesting for study. Hence, the present author has made a modest attempt to highlight the sculptural art as well as religious connotation of the ten Jaina images of the Odisha State Museum, Bhubaneswar. Methodologically, the author has written this article by using both primary and secondary sources.

Dynamics of a Rail Vehicle in Transition Curve above Critical Velocity with Focus on Hunting Motion Considering the Review of History of the Stability Studies

The most general purpose of the current paper is to trace and discuss the history and state of the art of studies on vehicle motion (dynamics) in a transition curve above the critical velocity, with the aim of potentially increasing the circle of researchers involved in studying this issue and strengthening the will of the authors to continue their studies. This general goal is achieved in two ways: first, through a profiled literature analysis, showing the historical progress and current state of the research; and second, through reference to the history of stability studies as an example of selected studies’ development. In addition, this work has two more specific goals. Together, they consist of collecting the literature in a related field in one place and analyzing it on site to accomplish the review. Both specific goals are attained by dividing the literature into two corresponding parts. In the first part, the current issues of rail vehicle stability are analyzed and divided into four problems. The second part includes works that deal with the subject of the motion and dynamics of a rail vehicle on a transition curve section. Here, the works are divided into five groups and discussed. They are put in order from the closest to the furthest from this paper’s main subject; however, the last group includes the most recent references. In addition, information on the authors’ approach to the problem is provided, including the methods and models used, as well as example results. Based on the analysis of the literature and the state of the art, a summary of the analysis is presented at this paper’s end. It highlights the small number of works on the subject of interest, and based on the review of stability studies, it seeks to encourage present and potential authors to study this field and share their results with society.

Assessment and development of saturated and subcooled heat transfer coefficient correlations for cryogenic flow boiling in tubes

The escalating interest in cryogenic technologies for space-related applications has led to an unprecedented demand for reliable prediction methods for cryogenic two-phase flow and heat transfer. Regrettably, existing heat transfer coefficient (HTC) correlations developed for conventional fluids prove inadequate and provide subpar predictions when applied to cryogenic flow boiling conditions. Therefore, it is imperative to develop a set of new HTC correlations specifically tailored for cryogenic flow boiling. In this study, comprehensive databases are constructed consolidating experimental data from previous studies of the present authors and historical data extracted from open literatures, spanning a wide range of operating conditions, flow orientations, and various cryogenic fluids. Subsequently, based on the constructed databases, an assessment of the predictive accuracy of seminal HTC correlations is conducted, followed by the development of new HTC correlations for cryogenic saturated and subcooled flow boiling. The newly developed correlations demonstrate very good predictive accuracy, with an overall mean absolute error (MAE) of 23.84 % and 21.24 % for LN2 saturated and subcooled HTC, respectively, under terrestrial gravity conditions. When assessed against a microgravity dataset, these correlations exhibit equally good predictive accuracy, yielding MAE values of 16.73 % and 25.99 % for LN2 saturated and subcooled HTC, respectively. Furthermore, the universal applicability of the new HTC correlations is ascertained by assessing the correlations across a multitude of cryogenic fluids, including LN2, LHe, LAr, LCH4, and LH2. Impressively, these correlations display outstanding predictive accuracy, with MAE values of 24.01 % and 21.29 % for saturated and subcooled HTC, respectively, underscoring their superior performance across a wide range of cryogenic fluids, validated against 2,445 saturated cryogenic HTC datapoints and 1,553 subcooled cryogenic HTC datapoints. Overall, the new HTC correlations consistently outperform all prior seminal correlations, yielding good predictive accuracy across a diverse spectrum of operating conditions, irrespective of flow orientation and gravity level, for various cryogenic fluids.

Identification and Characterization of Relict Olive Varieties (Olea europaea L.) in the Northwest of the Iberian Peninsula

Olives (Olea europaea L.) are an important crop in the Mediterranean Basin, but it is not well-known that they have also been grown in other areas, such as Galicia in northwestern Spain. Although commercial production ended long ago in this peripheral growing region, it remains home to olive resources that are well-adapted to the prevailing environmental conditions, providing a valuable but largely undocumented source of genetic variation. Following a survey of Galicia to locate examples of centuries-old olive trees, those detected were subjected to molecular characterization using a set of microsatellite markers, as well as full botanical characterization using the features established by the International Union for the Protection of New Varieties of Plants, along with others proposed by the present authors. These procedures allowed 11 undescribed varieties to be identified, which are new genetic resources that might be of use in olive improvement programs or studies of how the species adapts to different climates. The trees also underwent preliminary health checks, allowing disease-free specimens of each variety to be propagated. The addition of this material to the Community Plant Variety Office’s register of commercial varieties is underway.

Fourteen intracranial aneurysms in a single patient—the largest number ever reported in the literature

A 52-year-old female came with complaints of intense headaches for the past 10 days. The CT brain revealed SAH in the bilateral Sylvain fissure (left more than right) with Hunt and Hess Grade 1. An urgent CT cerebral angiogram was done, which revealed multiple bilateral MCA, ACA, and basilar top aneurysms. DSA was done the next day, which revealed multiple aneurysms, a total of fourteen in number. Such multiple aneurysms in a single patient were never reported in the literature until now, according to the present author's knowledge.

Therapeutic potential of vasopressin in the treatment of neurological disorders

Vasopressin (VP) is a nonapeptide made of nine amino acids synthesized by the hypothalamus and released by the pituitary gland. VP acts as a neurohormone, neuropeptide and neuromodulator and plays an important role in the regulation of water balance, osmolarity, blood pressure, body temperature, stress response, emotional challenges, etc. Traditionally VP is known to regulate the osmolarity and tonicity. VP and its receptors are widely expressed in the various region of the brain including cortex, hippocampus, basal forebrain, amygdala, etc. VP has been shown to modulate the behavior, stress response, circadian rhythm, cerebral blood flow, learning and memory, etc. The potential role of VP in the regulation of these neurological functions have suggested the therapeutic importance of VP and its analogues in the management of neurological disorders. Further, different VP analogues have been developed across the world with different pharmacotherapeutic potential. In the present work authors highlighted the therapeutic potential of VP and its analogues in the treatment and management of various neurological disorders.

A string-theoretical analog of non-maximal chaos in some Sachdev-Ye-Kitaev-like models

Very recently two of the present authors have studied the chaos exponent of some Sachdev-Ye-Kitaev (SYK)-like models for arbitrary interaction strength [1]. These models carry supersymmetric (SUSY) or SUSY-like structures. Namely, bosons and Majorana fermions are both present and each of them interacts with (q − 1) particles, but the model is not necessarily supersymmetric. It was found that the chaos exponents in different models, no matter whether they carry SUSY(-like) structures or not, all follow a universal single-parameter scaling law for large q, and by tuning that parameter continuously a flow from maximally chaotic to completely regular motion results. Here we report a string-theoretical analog of this chaotic phenomenon. Specifically, we consider closed string scattering near the two-sided AdS black hole, whose amplitude grows exponentially in the Schwarzschild time, with a rate determined by the Regge spin of the Pomeron exchanged during string scattering. We calculate the Pomeron Regge spin for strings of different types, including the bosonic string, the type II superstring and the heterotic superstring. We find that the Pomeron Regge spin also displays a single-parameter scaling behavior independent of string types, with the parameter depending on the string length and the length scale characterizing the spacetime curvature; moreover, the scaling function has the same limiting behaviors as that for the chaos exponent of SYK-like models. Remarkably, the flow from maximally chaotic to completely regular motion in SYK-like models corresponds to the flow of the Pomeron Regge spin from 2 to 1.

Participatory action research: advocacy and activism for promoting social justice in and through CALL

Participatory action research (PAR)—the inquiry procedure adopted by researchers who collaborate with communities experiencing injustices to identify workable and evidence-based solutions to existing problems—is not well-known in the area of CALL. Taking the social justice challenges existing in CALL, PAR methodology may be of relevance and importance to CALL researchers-educators committed to initiating change and cultivating social justice in and through CALL. Accordingly, this methodological paper sets out to bring PAR methodology closer to the CALL research community as a type of socially engaged research inquiry. It (re)introduces the notion of social justice in CALL and indicates a range of equity-related challenges and opportunities existing in the field, provides an overview of PAR methodological issues, its foundational principles, and the essential phases, and articulates the relevance of PAR for the CALL research community. The paper ends with an illustration of PAR using an advocacy project conducted by the present author for social justice in L2 education through CALL at her university.

Global Universality of the Two-Layer Neural Network with the k-Rectified Linear Unit

This paper concerns the universality of the two-layer neural network with the k-rectified linear unit activation function with k=1,2,… with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for k=1 by the present authors. This paper covers k-sigmoidal functions as an application of the fundamental result on k-rectified linear unit functions.

Dr John Grant Malcolmson and a reconciliation of the Middle Devonian Lethen Bar and Lethen House fish-bearing nodule localities, with notes on the Middle Devonian nodule beds of the Moray Firth area

The area known as Lethen Bar and Clune, SE of Nairn, Scotland, is a classic Middle Devonian locality, which has yielded nodules or concretions, some of which contain fossil fishes with the highest quality of preservation. The locality was largely centred on farm limestone quarries situated around the perimeter of an isolated outlier of the main fishbed. It was first described in the 19th century, although the upsurge in collecting fossil fishes only occurred some 12 years after the quarries were first mentioned in the scientific literature. Our knowledge of the provenance of the locality is based on very limited accounts, which have never been scientifically tested; these accounts also contain apparent anomalies that have never been adequately addressed. Based on these anomalies, the author of a paper published in 1983 proposed that the locality had been untraceable since the late 19th century and that the outcrop was quarried out. In 2005, the present authors recorded the first scientifically detailed stratigraphical section of the fishbed, followed in 2021–23 by detailed field surveys and by a reappraisal of the literature. This work has resolved the 19th century anomalies and allowed us to confirm the locations of old quarries, to give affirmation to two previously unrecognized sites and to show that the outcrop is still present. Thematic collection: This article is part of the Geology of Scotland collection available at: https://www.lyellcollection.org/topic/collections/geology-of-scotland

Numerical solution of entropy generation in nanofluid flow through a surface with thermal radiation applications

The main purpose of this research is to investigate a mathematical model of nanofluid passing through a stretching sheet having titanium dioxide (TiO2), copper oxide (CuO), and carbon nanotubes (CNTs) as nanoparticles, and sodium alginate is used as a base fluid. The present model can be very useful for numerous fields like Bioengineering, science, and technology. The main reason for using titanium dioxide, copper oxide, and carbon nanotubes nanoparticles as nanomaterials they have many applications in several areas. Moreover, impacts of non-linear thermal radiation and entropy generations are taken into account. By utilizing the similarity transformation technique given system of governing PDEs are transformed into a set of ODE. Then dimensionless equations are numerically tackled via the Keller box method a built-in function in the computer tool MATLAB. To support the present study, physical and numerical interpretations of involved parameters against velocity, temperature, and entropy generation profiles are also presented. From the analysis it is concluded that the combination of a magnetic field and suction reduces the rate of fluid mobility due to the synchronization of the magnetic and electric fields generated by the formation of the Lorentz force. In addition, boosting the thermal radiation parameter raises the present authors' heat transfer rate. The velocity profile drops when the Casson parameter is raised, but the heat and entropy production profiles grow. When the porosity parameter is increased, the velocity profile falls, and the heat and entropy production profiles rise. Additionally the temperature distribution is boosted up with increasing the values of nanoparticles volume fraction. The good agreement between current results and published data is observed. It is necessary to investigate the thermophysical characteristics of the magnetite nanofluid (titanium dioxide, copper oxide, and carbon nanotubes-sodium alginate nanofluid) under various situations. The MHD flow of nanofluid is significant because of its wide range of industrial uses, including flows over the tips of rockets, planes, submarines, and oil tankers. The novelty of this work is to scrutinize the MHD flow and heat transfer analysis of a nanofluid towards a stretched surface. The features of magnetite nanofluid (titanium dioxide, copper oxide, and carbon nanotubes-sodium alginate nanofluid) under entropy generation, thermal radiation and velocity slip are investigated. All of this contributes to the work's uniqueness and novel.

REMARKS ON THE PHYSICAL BASIS FOR THE CONSTRUCTION OF DIFFUSION FLUX TERMS IN FINITE-VOLUME EQUATIONS

The formulation for the diffusion terms in unstructured meshes is compared to that previously derived by the present authors and others for structured body-fitted meshes in terms of direction cosines between the normal and tangential directions. The so-called 'over-relaxed' approach is entirely equivalent/identical, subject to the caveat that the gradient of a scalar field is computed as the product of the scalar and the local surface area vector per unit volume summed (integrated) over the cell volumes, rather than by bilinear field interpolation. The physical/mathematical basis for the 'minimum correction' and 'orthogonal correction' approaches is not consistent with the present authors' derivation, which is in agreement with previous observations. The derivation for a structured mesh here differs from previous work in that physical, rather than mathematical components/projections of vectors are considered, thus filling a significant historical gap in the literature.