Non-rectangular Cross-section Research Articles

Schlieren measurements of shock train flow fields in a supersonic cylindrical isolator at Mach 2

AbstractIn a supersonic cylindrical isolator at Mach 2, the structures and frequency characteristics of shock train flow fields were experimentally studied by the schlieren measurement method. According to the design principle of parallel light through schlieren windows in a cylindrical duct, a high-precision conformal optical window pair was designed and integratively processed before. Based on a self-built pipeline structure with conformal windows in a direct-connect wind tunnel under adjustable back-pressure conditions, the shock surfaces in a cylindrical isolator at Mach 2 were first captured by the schlieren method. Then, the schlieren photographs were corrected by a nonlinear image transformation algorithm for the restoration of real shock train structures, and the experimental results were compared with numerical simulation results quantitatively. Finally, the shock train positions were calculated by an image recognition algorithm to analyze the self-excited oscillation frequency characteristics of shock train structures. The methods and experiments in this study enriched optical observation methods of supersonic flows through non-rectangular cross-section isolators in scramjet. Graphical abstract

Database of flanged reinforced concrete squat walls and its utilization based on machine learning

Flanged reinforced concrete (RC) squat walls are widely used in conventional buildings, owing to their capability of meeting certain architectural demands and providing significant lateral strength and stiffness in both principal directions. Despite their popularity and advantages, understanding of these walls is still limited due to the complexities brought by the non-rectangular cross-section and the relatively small shear span ratio. To facilitate future investigations through, a database containing comprehensive information about 182 flanged RC squat walls is presented in this paper. The criteria for selecting data and relevant sources are first introduced. Then items in the database are categorized into five groups, with histograms to assist understanding. The characteristics of these items are also discussed, and suggestions for future research are proposed. Furthermore, potential uses of this database are suggested, and the peak shear strength of these walls is explored based on machine learning as an example of utilization of this database. Compared with the mainstream building codes, this data-based approach produces much better predictions. Researchers are also encouraged to explore the use of this database through other data-driven approaches.

Research on Air Gap Magnetic Field Characteristics of Trapezoidal Halbach Permanent Magnet Linear Synchronous Motor Based on Improved Equivalent Surface Current Method

Accurate analysis of the air gap magnetic field is the focus of research in the field of precision permanent magnet linear synchronous motors. In this paper, the two-dimensional air gap magnetic field of a secondary trapezoidal Halbach permanent magnet array coreless permanent magnet linear synchronous motor (PMLSM) was taken as our research object. On the basis of the equivalent surface current method, we proposed an improved equivalent analytical algorithm with a trapezoidal side length unit. The equivalent analytical model of the magnetic induction vector of the two-dimensional air gap was established, and the air gap magnetic field of the trapezoidal Halbach array coreless PMLSM was calculated. At the same time, we analyzed the influence of the bottom angle α of a trapezoidal permanent magnet equivalent width coefficient αw, pole height coefficient αh, and air gap height coefficient αg on the amplitude (Bpeak) and total harmonic distortion (THDB) of the central magnetic field in the air gap. The results show that α and αw have a significant influence on the Bpeak and THDB of the central magnetic field air gap. With the synergy of α and αw, we identified the “flux convergence” effect, which makes the maximum range of Bpeak α > 90° and αw < 0.5. We also found the “equilateral” effect, which causes the minimum region of THDB to change linearly. The calculation results of the improved equivalent surface current analytical model established in this paper agree with those verified by the finite element method. The calculation is convenient, and the accuracy of the result is high. This research provides a new method for analyzing the air gap magnetic field of a permanent magnet with a nonrectangular cross-section and lays a theoretical foundation for optimizing the PMLSM pole model.

DEFORMATION MODEL AND ALGORITHM FOR CALCULATION OF REINFORCED CONCRETE STRUCTURES OF ROUND CROSS-SECTION UNDER TORSION WITH BENDING

Despite a fairly long period of research and a significant number of publications around the world on the problem of the complex resistance of reinforced concrete, the existing calculation models still remain far from perfect. This is especially true for structures with a non-rectangular cross section. The article presents a version of the model and an algorithm for the analytical calculation of reinforced concrete structures of a circular cross section in torsion with bending, which most fully reflects the specifics of the power resistance of such structures. The model takes into account all the components of external forces in a rod element of a circular cross section, the spatial nature of cracks, with the combined action of moments, various cases of the location of the compressed concrete zone, depending on the ratio of the acting forces in the calculated structure. For a spatial crack, calculated sections are taken in the form of diagonal large and small ellipses and a spatial surface bounded by concave and convex spatial parabolas. In compressed and stretched concrete, a broken section of three sections is considered, two in the form of longitudinal trapezoid and the third, middle section in the form of a small ellipse rotated at an angle to the longitudinal axis of the structure. The obtained analytical dependencies allow one to determine interconnected design parameters, such as stresses in the concrete of the compressed zone, the height of the compressed concrete, stresses in the longitudinal and transverse reinforcement, deformations in concrete and reinforcement, the length of the projection of a spatial inclined crack, and others. The deformation model and algorithm can be used in the design of reinforced concrete structures of circular and annular cross-section, working in bending with torsion.

A New Method for the Calculation of Characteristics of Disc Springs with Trapezoidal Cross-Sections and Rounded Edges.

In the European standards specifying disc spring manufacturing, geometry, shape and characteristic, an edge rounding is prescribed. Common methods for the calculation of disc spring characteristics, even in these standards, are based on a rectangular cross-section. This discrepancy can lead to a considerable divergence of the computed characteristic from the characteristic determined by testing. In literature, this divergence has not yet been examined with regard to rounded edges. In this paper, a new method addressing this problem is introduced. For this purpose, the geometry of idealized disc springs is parameterized. Based on four edge radii and two angles of the inner and outer faces, equations to compute the initial cone angle and the lever arm are introduced. These equations are used to formulate an algorithm to adapt other computation methods to non-rectangular cross-sections and rounded edges. The method is applied to the formulas by Almen–Laszlo, Curti–Orlando, Zheng and those by Kobelev. FE simulations of disc springs with rounded edges and a non-rectangular cross-section were used to verify the new formulas. The results show that the introduced method can be applied to known characteristic computation methods and result in a model expansion taking cross-section variations into account. The adjusted characteristics show more accurate alignment to the FE simulation for the cross-section variations investigated. These findings not only close the geometric gap between the manufacturing guidelines and the computation on an analytical basis, they also define a new parameter space for designs of disc springs and a corresponding force computation method to optimize spring characteristics.

Particle Focusing in a Straight Microchannel with Non-Rectangular Cross-Section.

Recently, studies on particle behavior under Newtonian and non-Newtonian fluids in microchannel have attracted considerable attention because particles and cells of interest can be manipulated and separated from biological samples without any external force. In this paper, two kinds of microchannels with non-rectangular cross-section were fabricated using basic MEMS processes (photolithography, reactive ion etching and anisotropy wet etching), plasma bonding and self-alignment between two PDMS structures. They were used to achieve the experiments for inertial and elasto-inertial particle focusing under Newtonian and non-Newtonian fluids. The particle behavior was compared and investigated for different flow rates and particle size in the microchannel with rhombic and equilateral hexagonal cross section. We also investigated the influence of Newtonian fluid and viscoelastic fluid on particle migration in both microchannels through the numerical simulation. The experimental results showed the multi-line particle focusing in Newtonian fluid over a wide range of flow rates, but the single-line particle focusing was formed in the centerline under non-Newtonian fluid. The tighter particle focusing appeared under non-Newtonian fluid in the microchannel with equilateral hexagonal cross-section than in the microchannel with rhombic cross section because of the effect of an obtuse angle. It revealed that particles suspended in the channel are likely to drift toward a channel center due to a negative net elasto-inertial force throughout the cross-sectional area. Simulation results support the present experimental observation that the viscoelastic fluid in the microchannel with rhombic and equilateral hexagonal cross-section significantly influences on the particle migration toward the channel center owing to coupled effect of inertia and elasticity.

Polarization characteristics of shallow-etched SOI ridge waveguide with non-rectangular cross section

Polarization characteristics of shallow-etched SOI ridge waveguide with non-rectangular cross section

Gravity currents with internal stratification in channels of non-rectangular cross-section

We investigate by theoretical means the motion of a gravity current (GC) released from a lock into an ambient fluid in a horizontal channel of non-rectangular cross-section area (CSA) like triangle, trapezoid or semicircle. The lock contains an internal stratification, in particular a layer of density ρL overlain by a layer of density ρU, where ρo<ρU<ρL and ρo is the density of the ambient fluid. Assuming a Boussinesq and large Reynolds flow, we develop a new self-contained model for the prediction of the motion of the GC, in particular the position and speed of the nose xN(t) and uN(t), where t is the time from release. Upon an appropriate scaling the system of equations and boundary conditions is reduced to a form amenable to analysis and solution by the methods employed for the non-stratified flow. The solution reveals an initial slumping with constant uN, and eventually the transition at xV to the viscous regime. A self similar phase of the form xN∼tβ can develop only for CSA of power-law shape, in which case the constant β depends on the shape but not on the stratification. The stability is also considered, and we show that the flow is stable with few exception. Unfortunately, experimental data for comparisons are not available, and we hope that the sharp present results will motivate the performance of needed experimental tests.

Fin Cross-Section Shape Influence on Short Channel Effects of MuGFETs

Multiple-gate FETs is normally constructed on pre-etched silicon fins. These devices often present casual width variations along the silicon height; mostly caused by technological limitations of the fin definition process, due to non-ideal anisotropic etch. The resulting devices have, consequently, non-rectangular cross-sections, which can affect their electrical behavior. This work addresses the dependence of fin width non-uniformity on the occurrence of short-channel effects through comparative analysis, based on threedimensional numeric simulation of non-rectangular cross-section devices. The influence of the fin crosssection shape on electrical parameters showed to be dependent on channel length, becoming more sensible to the fin shape as the channel length is reduced, with better DC performance present on devices with bottom fin width smaller than top fin width due to the higher transconductance and lower output conductance, resulting on higher intrinsic voltage gain. For opposite fin shapes the total gate capacitance present higher values, beneficiating AC analog parameters, such as unit gain frequency.

Influence of Fin Shape and Temperature on Conventional and Strained MuGFETs’ Analog Parameters

This work evaluates two important technological variations of Triple-Gate FETs: the use of strained silicon and the occurrence of non-rectangular body cross-section. The anaysis is focused on the electrical parameters for analog applications, and covers a temperature range from 150 K to 400 K. The comparison of the intrinsic voltage gain between the different trapezoidal fin shapes showed that the fin shape can have a major role in some analog parameters than the use of the strained silicon technology, helping to improve those parameters under certain circumstances. The highest intrinsic voltage gains were obtained for strained devices with top fin width larger than bottom at low temperature. Besides the intrinsic voltage gain, were also studied: the threshold voltage, subthreshold swing, drain induced barrier lowering, channel resistance, total harmonic distortion, transconductance, transconductance to drain current ratio, output conductance, Early voltage, drain voltage saturation and unity gain frequency.

Refined Zigzag Theory: an appropriate tool for the analysis of CLT-plates and other shear-elastic timber structures

Cross laminated timber (CLT), as a structural plate-like timber product, has been established as a load bearing product for walls, floor and roof elements. In a bending situation due to the transverse shear flexibility of the crossing layers, the warping of the cross section follows a zigzag pattern which should be considered in the calculation model. The Refined Zigzag Theory (RZT) can fulfill this requirement in a very simple and efficient way. The RZT, founded in 2007 by A. Tessler (NASA Langley Research Center), M. Di Sciuva and M. Gherlone (Politecnico Torino) is a very robust and accurate analysis tool, which can handle the typical zigag warping of the cross section by introducing only one additional kinematic degree of freedom in case of plane beams and two more in case of biaxial bending of plates. Thus, the RZT-kinematics is able to reflect the specific and local stress behaviour near concentrated loads in combination with a warping constraint, while most other theories do not. A comparison is made with different methods of calculation, as the modified Gamma-method, the Shear Analogy method (SA) and the First Order Shear Deformation Theory (FSDT). For a test example of a two-span continuous beam, an error estimation concerning the maximum bending stress is presented depending on the slenderness L/h and the width of contact area at the intermediate support. A stability investigation shows that FSDT provides sufficiently accurate results if the ratio of bending and shear stiffness is in a range as stated in the test example. It is shown that by a simple modification in the determination of the zigzag function, the scope can be extended to beams with arbitrary non-rectangular cross section. This generalization step considerably improves the possibilities for the application of RZT. Furthermore, beam structures with interlayer slip can easily be treated. So the RZT is very well suited to analyze all kinds, of shear-elastic structural element like CLT-plate, timber-concrete composite structure or doweled beam in an accurate and unified way.

Modeling of double-layer triangular microchannel heat sink based on thermal resistance network and multivariate structural optimization using firefly algorithm

The micro-channel heat dissipation system has minor specifications and good thermal conductivity per unit, which is the best choice for heat dissipation of micro-chips. By optimizing the cross section of microchannel, the heat exchange efficiency and temperature uniformity can be effectively improved. In this article, a double-layer triangular microchannel heat sink is proposed, which uniquely combines triangular cross section and double-layer structure to obtain a better heat dissipation performance. A new thermal resistance network model is established. At the same time, the model of pressure drop in microchannel heat sink is obtained by use of fluid theory. Taking thermal resistance and pressure drop as optimization objectives, the thermal resistance of double-layer triangular microchannel heat sink is 0.284 K/W and the pressure drop is 1386.89 Pa by using the firefly algorithm based on the Pareto optimal solution set, obtaining the optimal structural parameters. The thermal-flow-solid coupling simulation analysis shows that the thermal resistance and theoretical analysis error is 5.19%, and the pressure drop and theoretical analysis error is 9.49%, which can verify the accuracy of the thermal resistance network model. This article has a guiding significance for the thermal resistance analysis and heat dissipation improvement of non-rectangular cross section microchannel heat sinks.

Inertial microfluidics: A method for fast prediction of focusing pattern of particles in the cross section of the channel

Inertial microfluidics is utilized as a powerful passive method for particle and cell manipulation, which uses the hydrodynamic forces of the fluid in the channel to focus particles in specific equilibrium positions in the cross section of the channel. To achieve high performance manipulation, knowledge of focusing pattern of particles in the cross section of channel is essential. In this paper, we propose a method to address this important issue. To this end, firstly inertial microfluidics is analyzed in rectangular cross section channels. The results indicate that fluid flow velocity and channel's cross-sectional profiles have great impacts on the forces exerted on particles. Next, these results are utilized to propose a method to predict equilibrium positions in non-rectangular cross section channels through some simple calculations. This method is based on approximating the velocity profile of a non-rectangular cross section channel by utilizing portions of velocity profiles of different rectangular cross section channels. To analyze the method's performance, results obtained from the proposed method are compared with Direct Numerical Simulation (DNS) and experimental studies of seven non-rectangular channels. It is observed that the proposed approach accurately predicts particles trajectories and their equilibrium positions in the cross section of channels.

Performance and implementation of centrifugal serpentine micromixers with non-rectangular cross-section

The fluid flow and mixing characteristics of planar serpentine microchannels consisting of repeating S-shaped units was characterized both computationally and experimentally for Reynolds numbers Re < 100. In particular, it was found that using non-rectangular cross-sections with changing orientation for the geometrical units greatly enhances their mixing performance relative to standard rectangular profile serpentine channels. The optimization of the cross-section enables efficient mixing of two-fluid components at Reynolds numbers as low as 20, making these designs practical for chemical reaction engineering and biological assays platforms. A computational analysis of the cross-sectional flow structures present in this type of channels indicates that their enhanced performance is correlated with the formation of secondary transversal vortexes. In the optimal designs, their presence increases the mass transfer in the microchannel by both stretching the interface between the fluid components to be mixed, as well as by promoting chaotic advection.

Benjamin’s gravity current into an ambient fluid with an open surface in a channel of general cross-section

We present the solution of the idealized steady-state gravity current of height $h$ and density $\unicode[STIX]{x1D70C}_{1}$ that propagates into an ambient motionless fluid of height $H$ and density $\unicode[STIX]{x1D70C}_{2}$ in a channel of general non-rectangular cross-section, with an upper surface open to the atmosphere, at high Reynolds number. The current propagates with speed $U$ and causes a depth decrease $\unicode[STIX]{x1D712}$ of the top surface. This is a significant extension of Benjamin’s (J. Fluid Mech., vol. 31, 1968, pp. 209–248) seminal solution for the gravity current in a rectangular (or laterally unbounded) channel with a fixed top ($\unicode[STIX]{x1D712}=0$). The determination of $\unicode[STIX]{x1D712}$ is a part of the problem. Supposing that the direction of propagation is $x$ and gravity acceleration $g$ acts in the $-z$ direction, the sidewalls are specified by $y=-f_{I}(z)$ and $y=f_{II}(z),~z\in [0.H]$, and the width is $f(z)=f_{I}(z)+\,f_{II}(z)$. The dimensionless parameters of the problem are $a=h/H\in (0,1)$ and $r=\unicode[STIX]{x1D70C}_{2}/\unicode[STIX]{x1D70C}_{1}\in (0,1)$. We show that a control-volume analysis of the type used by Benjamin produces a system of algebraic equations for $\tilde{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D712}/H$ and $Fr=U/(g^{\prime }h)^{1/2}$ as functions of $a$ and $r$, where $g^{\prime }=(r^{-1}-1)g$ is the reduced gravity. The geometry enters the equation via the width function $f(z)$. We present solutions for typical $f(z)$: rectangle, semi-circle, $\vee$ triangle and trapezoid $\text{}\underline{/~\backslash }$ . The results are physically acceptable and insightful. The non-negative dissipation condition defines the domain of validity $a\leqslant a_{max}(r)$ (also depending on $f(z)$); the equality sign corresponds to energy-conserving cases. The critical speed limitation (with respect to the characteristics) is also considered briefly and suggests a slightly smaller $a\leqslant a_{crit}(r)$. The open-top results in the Boussinesq limit $r\rightarrow 1$ coincide with the fixed-top solution. Upon the reduction of $r$, for a fixed thickness $a$, the value of $Fr$ decreases and $\unicode[STIX]{x1D712}$ increases, until the point of energy-conserving (non-dissipative) flow; for smaller $r$, a negative non-physical dissipation appears. The trends are more pronounced for a converging cross-section geometry (like $\text{}\underline{/~\backslash }$ ) than for the opposite shape (like $\vee$ triangle). The previously investigated Benjamin-type steady-state $Fr$ and dissipation results are particular cases of the new formulation: $f(z)=$ const. reproduces the two-dimensional results, and $\unicode[STIX]{x1D712}=0$ recovers the fixed-top solution.

Asymptotic similarity solutions for particle-driven gravity currents in non-rectangular cross-section channels of power-law form

Abstract We consider a high-Reynolds particle-driven gravity current (GC) propagating in channel above a horizontal boundary. The bottom and the top of the channel are at z = 0 , H and its cross-section is given by − f ( z ) ≤ y ≤ f ( z ) for 0 ≤ z ≤ H , where f ( z ) is a power-law function f ( z ) = z α . We focus attention on the similarity stage of propagation of the current using a shallow-water (SW) model. While it is possible to derive the analytical similarity solutions for the homogeneous GC (for which the density of the current remains constant during the propagation), such similarity-solutions do not exist for the particle-driven GCs. We extend the similarity solutions of homogeneous GC to particle-driven currents by developing an asymptotic expansions and derive approximations of such solutions for channels of typical rectangular, triangular and parabolic forms. Comparison with the numerical solution of the SW equations shows that the leading-order asymptotic results are a good approximation in the domain of expected validity. However, out of bounds of this domain the numerical solutions systematically depart from the first-order asymptotic solutions, suggesting the need to include higher-order terms in the asymptotic series.

Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section.

In this numerical study, a new type of serpentine micromixer involving mixing units with a non-rectangular cross-section is investigated. Similar to other serpentine/spiral shaped micromixers, the design exploits the formation of transversal vortices (Dean flows) in pressure-driven systems, associated with the centrifugal forces experienced by the fluid as it is confined to move along curved geometries. In contrast with other previous designs, though, the use of non-rectangular cross-sections that change orientation between mixing units is exploited to control the center of rotation of the transversal flows formed. The associated extensional flows that thus develop between the mixing segments complement the existent rotational flows, leading to a more complex fluid motion. The fluid flow characteristics and associated mixing are determined numerically from computational solutions to Navier–Stokes equations and the concentration-diffusion equation. It is found that the performance of the investigated mixers exceeds that of simple serpentine channels with a more consistent behavior at low and high Reynolds numbers. An analysis of the mixing quality using an entropic mixing index indicates that maximum mixing can be achieved at Reynolds numbers as small as 20 in less than four serpentine mixing units.

Size-Dependent Inertial Focusing Position Shift and Particle Separations in Triangular Microchannels.

A recent study of inertial microfluidics within nonrectangular cross-section channels showed that the inertial focusing positions changes with cross-sectional shapes; therefore, the cross-sectional shape can be a useful control parameter for microfluidic particle manipulations. Here, we conducted detail investigation on unique focusing position shift phenomena, which occurs strongly in channels with the cross-sectional shape of the isosceles right triangle. The top focusing positions shift along the channel walls to the direction away from the apex with increasing Reynolds number and decreasing particle size. A larger particle with its center further away from the side walls experiences shear gradient lift toward the apex, which leads to an opposite result with changes of Reynolds and particle size. The focusing position shift and the subsequent stabilization of corner focusing lead to changes in the number of focusing positions, which enables a novel method for microparticle separations with high efficiency (>95%) and resolution (<2 μm). The separation method based on equilibrium focusing; therefore, the operation is simple and no complex separation optimization is needed. Moreover, the separation threshold can be easily modulated with flow rate adjustment. Rare cell separation from blood cell was successfully demonstrated with spiked MCF-7 cells in blood by achieving the yield of ∼95% and the throughput of ∼106 cells/min.

Polydisperse particle-driven gravity currents in non-rectangular cross section channels

We consider a high-Reynolds-number gravity current generated by polydisperse suspension of n types of particles distributed in a fluid of density ρi. Each class of particles in suspension has a different settling velocity. The current propagates along a channel of non-rectangular cross section into an ambient fluid of constant density ρa. The bottom and top of the channel are at z = 0, H, and the cross section is given by the quite general form −f1(z) ≤ y ≤ f2(z) for 0 ≤ z ≤ H. The flow is modeled by the one-layer shallow-water equations obtained for the time-dependent motion. We solve the problem by a finite-difference numerical code to present typical height h, velocity u, and mass fractions of particle (concentrations) (ϕ( j), j = 1, …, n) profiles. The runout length of suspensions in channels of power-law cross sections is analytically predicted using a simplified depth-averaged “box” model. We demonstrate that any degree of polydispersivity adds to the runout length of the currents, relative to that of equivalent monodisperse currents with an average settling velocity. The theoretical predictions are supported by the available experimental data. The present approach is a significant generalization of the particle-driven gravity current problem: on the one hand, now the monodisperse current in non-rectangular channels is a particular case of n = 1. On the other hand, the classical formulation of polydisperse currents for a rectangular channel is now just a particular case, f(z) = const., in the wide domain of cross sections covered by this new model.

Thin-layer models for gravity currents in channels of general cross-section area, a review

We present a brief review of the recent investigations on gravity currents in horizontal channels with non-rectangular cross-section area (such as triangle, $$\bigvee $$ -valley, circle/semi-circle, trapezoid) which occur in nature (e.g., rivers) and constructed environment (tunnels, reservoirs, canals). To be specific, we discuss the propagation of a gravity current (GC) in a horizontal channel along the horizontal coordinate x, with gravity g acting in the $$-z$$ direction, and y the horizontal–lateral coordinate. The bottom and top of the channel are at $$z=0,H$$ . The “standard” problem is concerned with 2D flow in a channel with rectangular (or laterally unbounded) cross-section area (CSA). Recent investigations have successfully extended the standard knowledge to the channels of CSA given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . This includes the practical $$\bigvee $$ -valley, triangle, circle/semi-circle and trapezoid; these geometries may be in “up” or “down” setting with respect to gravity, e.g., $$\bigtriangleup $$ and $$\bigtriangledown $$ . The major objective of the extended theory is to predict the height of the interface $$z=h(x,t)$$ and the velocity (averaged over the CSA) u(x, t), where t is time; the prediction includes the speed and position of the nose $$u_N(t), x_N(t)$$ . We show that the motion is governed by a set of simplified equations, called “model,” that provides versatile and insightful solutions and trends. The emphasis in on a high-Reynolds-number current whose motion is dominated by buoyancy–inertia balance; in particular a GC released from a lock, which also contains general effects such as front and internal jumps (shocks), and reflected bore. We discuss two-layer, one-layer, and box models; Boussinesq and non-Boussinesq systems; compositional and particle-driven cases; and the effect of stratification of the ambient fluid. The models are self-contained, and admit realistic initial and boundary conditions. The governing equations are amenable to analytical solutions in some special circumstances. Some salient features of the buoyancy-viscous regime, and the estimate for the length at which transition to this regime takes place, are also presented. Some experimental support to the theory, and open questions for further investigations, are also mentioned. The major conclusions are (1) The CSA geometry has significant influence on the motion of the GC; and (2) The new theory is a useful, very significant, extension of the standard two-dimensional GC problem. The standard current is just a particular case, $$f_{1,2} =$$ constants, among many other covered by the new theory .