Nonlinear Boussinesq Research Articles

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Repercussion of quadratic density variation on nonlinear mixed convective Darcy–Forchhimer Al2O3, Cu\\EG flow over an inclined plate with exponential heat source and Arrhenius equation

Generating and controlling energy at high temperatures is extremely challenging in solar energy applications with hybrid nanofluids (HNFs). This challenge is tackled with nonlinear changes in density with temperature and concentration. Hence, a nonlinear Boussinesq approximation is taken on Darcy – Forchhimer hybrid flow containing aluminium oxide ( A l 2 O 3 ) and copper ( Cu ) nanoparticles with base fluid Ethylene Glycol (EG) over an inclined plate at an angle of ξ = π 6 . The physical model is supported by a space-based exponential heat source (EHS) along with a temperature-based heat source (THS) in the energy equation and Arrhenius equation in diffusion relation with suction and blowing on the wall. The regulating equations are transformed into standard differential equations using the similarity conversions, and these equations are subsequently solved in MAPLE 2022. The thermal field is affected positively by EHS and THS parameters, which fulfil the purpose of including these parameters in the flow model. The rate of heat transport increment for Q E parameter by 1.04% for HNF whereas 1.06% for NF when Su > 0 . Also, Su < 0 heat transfer rate increases by 3.68% for HNF whereas 3.75% for NF. As A l 2 O 3 , Cu ∖EG HNF move from suction to injection, the Nusselt number reduces by more than 50%.

Significance of spatial variability in heat generation on nonlinear mixed convective Jeffrey nanofluid flows nearby an impermeable plate

The dynamics of Jeffrey nanofluids over an elongated surface heated by convection are addressed adequately under the nonlinear Boussinesq’s sense. Novel attributes of exponential heat source and thermal radiation are included. Buongiorno’s revised nanoliquid model is implemented to study the uneven dispersion and thermophoresis of nanoparticles subjected to passive control of nanoparticles on the surface. Wall suction or injection effects are also considered. Mathematical modeling is developed and treated numerically using the similarity approach. The results are presented considering numerous values of the physical parameters. Stimulus variables that regulate the resulting dimensionless physical quantities are examined graphically. Nonlinear convection has occurred to reinforce the nanofluid motion and reduce the thermal boundary layer thickness. The results of this study are useful in polymer industries.

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Optical soliton solutions of nonlinear differential Boussinesq water wave equation via two analytical techniques

In this article, we examine the optical soliton solutions of the 4th-order nonlinear Boussinesq water wave equation using the modified (G′/G2)− expansion and F-expansion methods. We utilize the similarity transformation to transform the partial differential form of the equation into an ordinary differential form. These techniques present a variety of solutions, including bright, dark, singular, dark-periodic, and singular-periodic soliton solutions. We plot the derived solutions in several profiles, such as 2D, 3D, and contour, to illustrate their physical appearance. The determined findings might be useful and have an enormous effect on the research of nonlinear phenomena in a variety of physical areas of science, like shallow water waves, acoustics, fluid dynamics, laser optics, communication systems, and heat transfer. The achieved results demonstrate the validity, applicability, and effectiveness of the presented method. We plot and validate the found solutions using the computational program MATHEMATICA.

Predicting heat transfer rate and system entropy based on combining artificial neural network with numerical simulation

PurposeThe purpose of this paper is to present numerical simulations for magnetohydrodynamics natural convection of a nanofluid flow inside a cavity with an H-shaped obstacle based on combining artificial neural network (ANN) with the finite element method (FEM), and predict the heat transfer rate and system entropy.Design/methodology/approachThe enclosure is assumed to be inclined. Changing the inclination angle results in a different obstacle shape, which affects the buoyancy force. Hence, different configurations of the contours of the fluid flow, isotherms and the entropy of the system are obtained. The outer walls of the cavity as well as the central part of the obstacle are kept adiabatic. The left vertical portion of the hindrance is cooled, whereas the right vertical part of the obstacle is a heated wall. Using dimensionless variables allows obtaining a dimensionless version of the governing system of equations that is solved via the consistency FEM. The coupled problem of pressure and velocity is overcome via the Increment Pressure Correction Scheme, which is known for its accuracy and stability for similar simple problems. A numerical computation is performed across a broad range of the governing parameters. A total of 304 data sets were used in the development of an ANN model. That data set was conducted from the numerical simulations. The data set underwent optimization, with 70% sets used for training the model, 15% for validation and another 15% for the testing phase. The training of the network model used the Levenberg–Marquardt training algorithm.FindingsFrom the numerical simulations, it is concluded that the H-shaped obstacle boosts heat transfer rate in comparison with the I-shaped case. Also, raising the value of the inclination angle improves the entropy of the system presented by the Bejen number. Furthermore, strength heat transfer rate is obtained via decreasing the Hartmann number while this decrease decays the values of the Bejen number for both positive and negative amounts of the nonlinear Boussinesq parameter. Slower velocity and a better heat transfer rate characterize nanofluid compared with pure fluid. Leveraging the capabilities of the ANN, the developed model adeptly forecasts the values of both the average Nusselt and Bejen numbers with a high degree of accuracy.Originality/valueA novel fusion of FEM and ANN has been tailored to forecast the heat transfer rate and system entropy of MHD natural convective flow within an inclined cavity containing an H-shaped obstacle, amid various physical influences.

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

Derivation and well-posedness for asymptotic models of cold plasmas

In this paper we derive three new asymptotic models for a hyperbolic–hyperbolic–elliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field. The first of these models takes the form of a non-linear and non-local Boussinesq system (for the ionic density and velocity) while the second is a non-local wave equation (for the ionic density). Moreover, we derive a unidirectional asymptotic model of the latter which is closely related to the well-known Fornberg–Whitham equation. We also provide the well-posedness of these asymptotic models in Sobolev spaces. To conclude, we demonstrate the existence of a class of initial data which exhibit wave breaking for the unidirectional model.

Fuzzy Finite Elements Solution Describing Recession Flow in Unconfined Aquifers

In this work, a novel fuzzy FEM (Finite Elements Method) numerical solution describing the recession flow in unconfined aquifers is proposed. In general, recession flow and drainage problems can be described by the nonlinear Boussinesq equation, while the introduced hydraulic parameters (Conductivity K and Porosity S) present significant uncertainties for various reasons (e.g., spatial distribution, human errors, etc.). Considering the general lack of in situ measurements for these parameters as well as the certain spatial variability that they present in field scales, a fuzzy approach was adopted to include the problem uncertainties and cover the disadvantage of ground truth missing data. The overall problem is encountered with a new approximate fuzzy FEM numerical solution, leading to a system of crisp boundary value problems. To prove the validity and efficiency of the new fuzzy FEM, a comparative analysis between the proposed approach and other well-known and tested approximations was carried out. According to the results, the proposed FEM numerical solution agrees with Karadinumerical method for the crisp case and is in close agreement with the original analytical solution proposed by Boussinesq in 1904 with the absolute reduced error to be 4.6‰. Additionally, the possibility theory is applied, enabling the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of hydraulic properties (e.g., water level, outflow volume) and make the right decisions for rational and productive engineering studies.

Estimate of Wave Overtopping Rate on Armoured Slope Structures Using FUNWAVE-TVD Model

In this study, the program was modified by adding the empirical formula of EurOtop (2018) to enable calculation of wave overtopping on armoured slope structures in the FUNWAVE-TVD model using the fully nonlinear Boussinesq equation. The validity of the modified numerical model was verified by comparing it with CLASH data and experiment data for the rubble mound structure. This model accurately reproduced the change in wave overtopping rate according to the difference in the roughness factor of the armoured block, and well reproduced the rate of decrease in wave overtopping rate due to the increase in relative freeboard. The overtopping rate of the armoured slope structures showed significant differences depending on the positioning condition of the armoured blocks. When Tetrapods were placed with regular positioning, the overtopping rate increased significantly compared to when they were placed with random positioning, and it was consistent with when they were placed with Rocks. Meanwhile, when rocks were placed in one row, the wave overtopping rate was greater than when rocks were placed in two rows, which is believed to be due to the influence of the roughness and permeability of the structure’s surface.

A set of fully nonlinear mild slope equations

A set of time-dependent fully nonlinear mild slope equations is established in terms of velocity vector at still water level and free surface elevation, which is derived from the Euler equations without approximation in nonlinearity and only with the potential flow assumption adopted for bottom slope terms. Apart from the fully nonlinear property, the model is also different from previous nonlinear mild slope equations by adopting the velocity expression accurate to the first-order in bottom slope, which allows the new expressions of higher-order bottom slope terms to be established and the fully nonlinear terms related to bottom slope to be included in the equations. The model can be reduced to the classical mild slope equation for linear waves and the fully nonlinear Boussinesq equations for nonlinear waves. The advantages of the present model are validated against the related laboratory experimental results and demonstrated by comparison with the numerical results of other relevant models.

On the extraction of complex behavior of generalized higher-order nonlinear Boussinesq dynamical wave equation and (1+1)-dimensional Van der Waals gas system

&lt;p&gt;In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper.&lt;/p&gt;

Mixed convective magnetohydrodynamic and thermally radiative flow of reactive couple stress MWCNT–Ag/C2H6O2 hybrid nanofluid in a porous vertical channel: Entropy analysis

This paper is concerned with the analysis of the mixed convective magnetohydrodynamic (MHD) flow of a reactive couple stress multi-walled carbon nanotube −Ag/C2H6O2 hybrid nanofluid in a porous vertical channel subjected to quadratic thermal radiation along with a uniform inclined magnetic field applied to the channel walls. The flow is driven by the pressure gradient force and the buoyancy force, which is modeled based on the nonlinear Boussinesq approximation. The temperature-dependent reaction rate of the reactant molecule is derived using the Arrhenius law. The momentum and energy equations that govern the system are modeled in consideration of slip and convective conditions. The governing equations are non-dimensionalized by applying relevant dimensionless parameters and are solved using the homotopy analysis method (HAM). To analyze the irreversibilities in the system, the entropy generation number and the Bejan number are defined. Different important physical parameters developing in the system are considered for analysis, and their effects are scrutinized on the velocity and temperature profiles along with entropy generation. The emphasis is given to the concentration of nanoparticles along with the parameters arising due to the reactions of the fluid, buoyancy force, inclined magnetic field, thermal radiation, and porous material. The analysis reveals that the velocity and temperature of the fluid lowers with a higher concentration of nanoparticles, radiation parameter, and Hartmann number, whereas develops for the higher slip parameters and inclination of the magnetic field. The entropy generation rate increases with rising slip parameters and depletes for higher nanoparticle concentration, radiation parameter, Hartmann number, and inclination angle. The irreversibility in the system remains dominant due to heat transfer with higher Frank-Kameneskii and activation energy parameters, Hartmann number, and angle of inclination.

A wave-resolving modeling study of rip current variability, rip hazard, and swimmer escape strategies on an embayed beach

Abstract. Drownings due to rip currents are a major threat to beach safety. In this study a high-resolution Boussinesq model with a modified wave-resolving Lagrangian tracking module has been applied to a 2 km long embayed beach, Dadonghai of Sanya, Hainan Island, with the purpose of studying rip current variability, real-time rip hazard identification, and the optimal swimmer escape strategies. The beach stage evolves periodically at the study site and plays an important role in the long-term modulation of the occurrence and strength of rip currents according to the modeling. A series of tests are designed and confirm that rip current strength is closely related to wave properties and tidal levels. Spectral analysis of output time series at specific points shows that the modeled rip currents fluctuate on the orders of 1 and 10 min, which suggests the effects of wave-group-forced infragravity (IG) and very-low-frequency (VLF) motions. Rip hazard levels are defined by combining rip strength and its duration. An attempt to use the GPU-accelerated FUNWAVE-TVD (Total Variation Diminishing version of the Fully Nonlinear Boussinesq Wave Model) embedded with the spectral wave model WAM6-GPU (GPU version of the third-generation spectral wave model WAM Cycle 6) exhibits its capability to evaluate rip hazard levels in real time. One of the differences of the present study from previous works is that the random, wave-resolving tracking of virtual swimmers is performed with 1 m resolution to study beach safety strategies. The results demonstrate that multiple factors contribute to the survival of swimmers caught in the rip currents, including surf-zone bathymetry, rip strength, fine-scale flow patterns, the bather's position, and swimming ability. For weak-to-moderate rip currents and longshore currents, swim onshore consistently seems to be the most successful strategy across all the scenarios in this study. Higher surf-zone exit rates along Dadonghai beach are not favorable for stay afloat action, which puts swimmers at a higher risk of being expelled to deeper water. The effects of wave randomness of incoming wave trains and assignment of wave-following coefficients on Lagrangian tracking are also discussed.

Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative

Abstract A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time ( t ) \left(t) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.

Modeling of water table profile variations owing to stream-aquifer interaction.

Spatial and temporal variations of the water table could be explained by the one-dimensional Boussinesq equation by incorporating the variables of evapotranspiration and groundwater recharge with appropriate initial and boundary conditions. In this study, the stream-aquifer interaction has been investigated through a numerical example model with the implementations of Galerkin's method-based Finite Element Solution (FES), Hybrid Finite Analytic Solution (HFAS), Fully Implicit Finite Difference Solution (FIFDS) of one-dimensional nonlinear Boussinesq equation, and analytical solutions of the Boussinesq equation linearized by Baumann's transformation (AS I) as well as linearized by Werner's transformation (AS II). Considering HFAS as the benchmark solution, it was observed that in both recharging and discharging aquifers, water table profiles at 1 day and 5 days as obtained from FES followed by FIFDS were observed quite close to HFAS. Based on L2 and Tchebycheff norms, FES and FIFDS were ranked in first and second place, respectively. L2 and Tchebycheff norms could not consistently establish the performance ranking of analytical solutions but their performance ranking was certainly below the numerical solutions. The performance ranking of analytical solutions could not consistently be established using the L2 and Tchebycheff norms, but it was certainly below the numerical solutions.

TSUNAMI MODELLING IN A BUILT-IN COASTAL ENVIRONMENT WITH ADAPTIVE MESH RIFINEMENT AND VARYING BOTTOM FRICTION

The great Japan earthquake on March/11/2011provoked an extreme tsunami wave toward the northeastern Japanese coast. The tsunami inundation covered a wide range of territories stretching over 500 km2 and caused devastating influences that claimed more than 15,000 lives (Mori et al. (2011)). Miyagi Prefecture, the closest mainland to the epicentre, received massive tsunami waves that reached over 40 m in run-up height. Prasetyo et al. (2019) aimed to reproduce the tsunami wave inundation in Onagawa Town of Miyagi Prefecture by building a physical model using Hybrid Tsunami Open Flume in Ujigawa Open Laboratory (HyTOFU) at Kyoto University to study the inundation process by measuring the wave height and arrival time. For that purpose, two wave types were used, the (Pump-type) hydraulic bore, and the (Piston-type) solitary wave. Also, 2D and quasi-3D (Q3D) models were tested for their robustness and applied to emulate tsunami propagation in the complex coastal city model. In this study, we adopted adaptive mesh refinement (AMR) for its efficiency in tsunami modelling with Basilisk open-source flow solver for non-linear shallow water (Saint-Venant) equations and fully non-linear Boussinesq (Green-Naghdi) equations (Popinet (2015)). The numerical experiment was conducted to validate the model by comparing the wave height with the physical model experiment, the 2D, and Q3D model results by Prasetyo et al. (2019).

A Nonlinear Recession Model for Horizontal Aquifers

AbstractRecession analysis is a powerful tool for determining the hydraulic characteristics of a riparian aquifer. The basis of the method relies on two analytical solutions of the nonlinear Boussinesq equation: one applies to a uniform water table profile in a semi‐infinite aquifer while the other pertains to a hypothetical initial profile in a finite aquifer. Both solutions assume that the water depth in the adjoining stream is negligible. In the present work, a nonlinear solution of the one‐dimensional Boussinesq equation is derived using the traveling wave approximation. The solution better represents the field conditions as it pertains to a uniform water table profile in a finite aquifer that is adjoining a stream with a non‐zero water level. It allows the derivation of simple algebraic expressions for the recession of the water table, the flow rate, and the associated drainage volume. Approximations relating the discharge rate to the outflow volume were also obtained for practical implementation in the estimation of the hydraulic parameters of the aquifer. A comparison of the proposed methodology of parameter estimation with the standard method of recession analysis showed that the present approach is superior as it avoids the pitfalls associated with the classical method, especially when daily discharge values with inherent observational errors are used. Application of the recession model to real‐world cases demonstrates its effectiveness in the estimation of the catchment‐scale hydraulic conductivity and drainable porosity.

Propagation characteristics of tsunami waves in shear flow based on the Boussinesq model with constant vorticity

Propagation of tsunami wave in linear shear flow with constant vorticity is studied in this paper. The nonlinear and dispersive Boussinesq equations for water waves in shear flow are derived by introducing uniform vorticity into the Euler equations. Generally, the transoceanic propagation of tsunami can be well simulated by the weakly nonlinear Boussinesq model. Thus, the nonlinear and dispersive terms are retained to the first-order and second-order (ϵ,μ2) respectively in the Boussinesq equations with uniform vorticity in the present work, which satisfies the simulation accuracy especially the dispersion effects of the long-distance propagation of tsunamis and does not require huge computational cost for the simulation. The set of equations is solved with the total variation diminishing (TVD) Lax–Wendroff scheme which is second-order both in space and time. This TVD scheme is applicable to handle the dispersion effects of the N-waves and undular bores. Different patterns of tsunami wave in the water with presence of vorticity are simulated by the numerical model. Wave height and wavelength are affected by the current with vorticity. The pattern of solitary wave is significantly changed in upstream and downstream conditions. The wave deformation and propagation speed of N-waves and undular bores in the shear flow are also studied. The vorticity has impacts on the leading crest amplitude and the maximum wave appearance time for N-wave and changes the wave amplitude and the following water level for undular bores.