Fluid dynamic research on rectangular and trapezoidal fins is aimed at increasing heat transfer by means of large surfaces.The trapezoidal cavity form is compared with its thermal and flow performance,and it is reveal...Fluid dynamic research on rectangular and trapezoidal fins is aimed at increasing heat transfer by means of large surfaces.The trapezoidal cavity form is compared with its thermal and flow performance,and it is revealed that trapezoidal fins tend to be more efficient,particularly when material optimization is critical.Motivated by the increasing need for sustainable energy management,this work analyses the thermal performance of inclined trapezoidal and rectangular porous fins utilising a unique hybrid nanofluid.The effectiveness of nanoparticles in a working fluid is primarily determined by their thermophysical properties;hence,optimising these properties can significantly improve overall performance.This study considers the dispersion of Graphene Oxide(GO)and Molybdenum Disulfide in the base fluid,engine oil.Temperature profiles are analysed by altering the radiative,porosity,wet porous,and angle of inclination parameters.Surface and contour plots are constructed by using the Lobatto IIIa Collocation Method with BVP5C solver in MATLAB and Gradient Descent Optimisation to predict the combined heat transfer rate.According to the study,fluid temperature consistently decreases when the angle of inclination,wet porous parameter,porosity parameter,and radiative parameter increase,suggesting significantly improved heat dissipation.The trapezoidal fin consistently exhibits a superior heat transfer mechanism than a rectangular fin.It is found that the trapezoidal fin transmits heat at a rate that is 0.05%higher than that of the rectangular fin.Validation of the present study is done through the comparison of previous studies.This research provides useful design insights for sophisticated engineering uses,including electrical cooling devices,heat exchangers,radiators,and solar heaters.展开更多
The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.I...The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.In addition to this,the impacts of thermal radiation and viscous dissipation are taken into account.With the use of various computing strategies,numerical results have been produced.Similarity transformation was utilized in order to convert the partial differential equations(PDEs)that regulated energy,rotational momentum,concentration,and momentum into ordinary differential equations(ODEs).As compared to earlier published research,MATLAB inbuilt solver solution shows an extremely good correlation in exceptional instances.In exceptional instances,the present MATLAB inbuilt solver solution has a very excellent connection with the findings of the previously published investigations.A variety of flow field factors impact the Nusselt number,the wall couple shear stress,the friction factor,Sherwood numbers the dimensionless distributions discussed in detail.When the Eckert number rises,the temperature rises,and the Schmidt number falls,the concentration falls.Velocity increases with increases in the material factor but drops with increases in the magnetic parameter and the surface condition factor.展开更多
Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized...Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized Fourier coefficients,respectively.In this paper,we investigate the correlations of triple sums associated to these Fourier coefficientsλ_(f)(n),λ_(g)(n),λ_(h)(n)over certain polynomials,and obtain some power-saving asymptotic estimates which beat the trivial bounds.展开更多
In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as ...In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.展开更多
We have examined an isotropic and homogeneous cosmological model in f(R,T^(φ))gravity,where R represents the Ricci scalar and T^(φ)exhibits the energy momentum tensor's trace.We examine the stability criteria by...We have examined an isotropic and homogeneous cosmological model in f(R,T^(φ))gravity,where R represents the Ricci scalar and T^(φ)exhibits the energy momentum tensor's trace.We examine the stability criteria by performing the dynamical system analysis for our model f(R,T^(φ))=R+2(a T^(φ)+b),where a and b are the constants.We derive a set of autonomous equations and find their solutions by assuming a flat potential V0.We assess the equilibrium points from these equations and find the eigenvalues.We analyze the physical interpretation of the phase space for this system.We obtain three stable equilibrium points.We also examine the interaction between the scalar field and dark energy,represented by Q=ψH_(ρde)and determine the equilibrium points for this interaction.We identify four stable equilibrium points for this interaction.We calculate the values of the physical parameters for both scenarios at each equilibrium point,indicating the Universe's accelerated expansion.展开更多
In this paper,Isogeometric analysis(IGA)is effectively integrated with machine learning(ML)to investigate the bearing capacity of strip footings in layered soil profiles,with a focus on a sand-over-clay configuration....In this paper,Isogeometric analysis(IGA)is effectively integrated with machine learning(ML)to investigate the bearing capacity of strip footings in layered soil profiles,with a focus on a sand-over-clay configuration.The study begins with the generation of a comprehensive dataset of 10,000 samples from IGA upper bound(UB)limit analyses,facilitating an in-depth examination of various material and geometric conditions.A hybrid deep neural network,specifically the Whale Optimization Algorithm-Deep Neural Network(WOA-DNN),is then employed to utilize these 10,000 outputs for precise bearing capacity predictions.Notably,the WOA-DNN model outperforms conventional ML techniques,offering a robust and accurate prediction tool.This innovative approach explores a broad range of design parameters,including sand layer depth,load-to-soil unit weight ratio,internal friction angle,cohesion,and footing roughness.A detailed analysis of the dataset reveals the significant influence of these parameters on bearing capacity,providing valuable insights for practical foundation design.This research demonstrates the usefulness of data-driven techniques in optimizing the design of shallow foundations within layered soil profiles,marking a significant stride in geotechnical engineering advancements.展开更多
We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of ...We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L^2(R^3), i.e., finite Dirichlet integral.展开更多
In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithm...In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.展开更多
An fKdV equation of two-layer how and an averaged fKdV equation (AfKdV equation) with respect to phase are derived to determine the theoretical amplitude and period of the precursor solitons in the present paper. In t...An fKdV equation of two-layer how and an averaged fKdV equation (AfKdV equation) with respect to phase are derived to determine the theoretical amplitude and period of the precursor solitons in the present paper. In terms of the AfKdV equation derived by the authors, a new theory on the precursor soliton generation based on Lee et al.'s concept is presented. Concepts of asymptotic mean hydraulic fall and level are introduced in our analysis, and the theoretical amplitude and period both depend on the asymptotic mi-an levels and stratified parameters. From the present theoretical results, it is obtained that when the moving velocity of the topography is at the resonant points, there exist two general relations: (1) amplitude relation (A) over circle = 2F, (2) period relation <(tau)over circle> = -8m(1)m(3)(-1)root 6m(4)m(3)(-1)F, in which (A) over circle and <(tau)over circle> are the amplitude and period of the precursor solitons at the resonant points respectively, m(1), m(3) and m(4) are coefficients of the fKdV equation, and F is asymptotic mean half-hydraulic fall at subcritical cutoff points. The theoretical results of this paper are compared with experiments and numerical calculations of two-layer flow over a semicircular topography and all these results are in good agreement. Due to the canonical character of the coefficients of fKdV equations, this theory also holds for any two-dimensional system, which can be reduced to fKdV equations.展开更多
We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the...We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.展开更多
The problem of the steady magnetohydrodynamic （MHD） stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The non...The problem of the steady magnetohydrodynamic （MHD） stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.展开更多
In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetso...In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.展开更多
Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investi...Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace＇s equation by a method based on Green＇s integral theorem with the introduction of appropriate Green＇s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.展开更多
The problem of oblique wave （internal wave） propagation over a small deformation in a channel flow consisting of two layers was considered. The upper fluid was assumed to be bounded above by a rigid lid, which is an...The problem of oblique wave （internal wave） propagation over a small deformation in a channel flow consisting of two layers was considered. The upper fluid was assumed to be bounded above by a rigid lid, which is an approximation for the free surface, and the lower one was bounded below by an impermeable bottom surface having a small deformation; the channel was unbounded in the horizontal directions. Assuming irrotational motion, the perturbation technique was employed to calculate the first-order corrections of the velocity potential in the two fluids by using Green＇s integral theorem suitably with the introduction of appropriate Green＇s functions. Those functions help in calculating the reflection and transmission coefficients in terms of integrals involving the shape ftmction c（x） representing the bottom deformation. Three-dimensional linear water wave theory was utilized for formulating the relevant boundary value problem. Two special examples of bottom deformation were considered to validate the results. Consideration of a patch of sinusoidal ripples （having the same wave number） shows that the reflection coefficient is an oscillatory function of the ratio of twice the x-component of the wave number to the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large. Similar results were observed for a patch of sinusoidal ripples having different wave numbers. It was also observed that for small angles of incidence, the reflected energy is greater compared to other angles of incidence up to π/ 4. These theoretical observations are supported by graphical results.展开更多
The study of energy transfer between coupled subsystems in a hybrid system is very important for applications. This paper presents an analytical analysis of energy transfer between plates of a visco-elastically connec...The study of energy transfer between coupled subsystems in a hybrid system is very important for applications. This paper presents an analytical analysis of energy transfer between plates of a visco-elastically connected double-plate system in free transversal vibrations. The analytical analysis shows that the visco-elastic connection between plates is responsible for the appearance of two-frequency regime in the time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are uncoupled, but energy transfer between plates in one eigen mode appears. It was shown for each shape of vibrations. Series of the two Lyapunov exponents corresponding to the one eigen amplitude mode are expressed by using the energy of the corresponding eigen amplitude time component.展开更多
This paper obtains the Cauchy-Pompeiu formula on certain distinguished boundary for functions with values in a universal Clifford algebra. This formula is just an extension of the Cauchy's integral formula obtaine...This paper obtains the Cauchy-Pompeiu formula on certain distinguished boundary for functions with values in a universal Clifford algebra. This formula is just an extension of the Cauchy's integral formula obtained in [11].展开更多
We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other app...We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.展开更多
基金supported by the“Regional Innovation System&Education(RISE)”through the Seoul RISE Center,funded by the Ministry of Education(MOE)and the Seoul Metropolitan Government(2025-RISE-01-027-04).
文摘Fluid dynamic research on rectangular and trapezoidal fins is aimed at increasing heat transfer by means of large surfaces.The trapezoidal cavity form is compared with its thermal and flow performance,and it is revealed that trapezoidal fins tend to be more efficient,particularly when material optimization is critical.Motivated by the increasing need for sustainable energy management,this work analyses the thermal performance of inclined trapezoidal and rectangular porous fins utilising a unique hybrid nanofluid.The effectiveness of nanoparticles in a working fluid is primarily determined by their thermophysical properties;hence,optimising these properties can significantly improve overall performance.This study considers the dispersion of Graphene Oxide(GO)and Molybdenum Disulfide in the base fluid,engine oil.Temperature profiles are analysed by altering the radiative,porosity,wet porous,and angle of inclination parameters.Surface and contour plots are constructed by using the Lobatto IIIa Collocation Method with BVP5C solver in MATLAB and Gradient Descent Optimisation to predict the combined heat transfer rate.According to the study,fluid temperature consistently decreases when the angle of inclination,wet porous parameter,porosity parameter,and radiative parameter increase,suggesting significantly improved heat dissipation.The trapezoidal fin consistently exhibits a superior heat transfer mechanism than a rectangular fin.It is found that the trapezoidal fin transmits heat at a rate that is 0.05%higher than that of the rectangular fin.Validation of the present study is done through the comparison of previous studies.This research provides useful design insights for sophisticated engineering uses,including electrical cooling devices,heat exchangers,radiators,and solar heaters.
文摘The purpose of this research is to investigate the influence that slip boundary conditions have on the rate of heat and mass transfer by examining the behavior of micropolar MHD flow across a porous stretching sheet.In addition to this,the impacts of thermal radiation and viscous dissipation are taken into account.With the use of various computing strategies,numerical results have been produced.Similarity transformation was utilized in order to convert the partial differential equations(PDEs)that regulated energy,rotational momentum,concentration,and momentum into ordinary differential equations(ODEs).As compared to earlier published research,MATLAB inbuilt solver solution shows an extremely good correlation in exceptional instances.In exceptional instances,the present MATLAB inbuilt solver solution has a very excellent connection with the findings of the previously published investigations.A variety of flow field factors impact the Nusselt number,the wall couple shear stress,the friction factor,Sherwood numbers the dimensionless distributions discussed in detail.When the Eckert number rises,the temperature rises,and the Schmidt number falls,the concentration falls.Velocity increases with increases in the material factor but drops with increases in the magnetic parameter and the surface condition factor.
基金Supported in part by NSFC(Nos.12401011,12201214)National Key Research and Development Program of China(No.2021YFA1000700)+3 种基金Shaanxi Fundamental Science Research Project for Mathematics and Physics(No.23JSQ053)Science and Technology Program for Youth New Star of Shaanxi Province(No.2025ZC-KJXX-29)Natural Science Basic Research Program of Shaanxi Province(No.2025JC-YBQN-091)Scientific Research Foundation for Young Talents of WNU(No.2024XJ-QNRC-01)。
文摘Let f,g and h be three distinct primitive holomorphic cusp forms of even integral weights k_(1),k_(2)and k_(3)for the full modular groupΓ=SL(2,Z),and denote byλ_(f)(n),λ_(g)(n),λ_(h)(n)the corresponding normalized Fourier coefficients,respectively.In this paper,we investigate the correlations of triple sums associated to these Fourier coefficientsλ_(f)(n),λ_(g)(n),λ_(h)(n)over certain polynomials,and obtain some power-saving asymptotic estimates which beat the trivial bounds.
基金supported by the NSFC(12331007)the National Key Research and Development Program of China(2020YFA0713803)。
文摘In this paper,for the 1-D semilinear wave equation∂_(t)^(2)u-∂_(x)^(2)u+μ/t∂_(t)u=|u|~p with scaling invariant damping,where t≥1,p>1 andμ∈(0,1)∪(1,4/3),we establish the global weighted space-time estimates as well as the global existence of small data weak solution u when the nonlinearity power p is larger than some critical power p_(crit)(μ).Our proof is based on a class of new weighted Strichartz estimates with the weight t^(θ)|(1-μ)^(2)t^(2/|1-μ|)-x^(2)|^(γ)(θ>0andγ>0 are appropriate constants)for the solution of linear generalized Tricomi equation∂_(t)^(2)φ-t^(m)∂_(x)^(2)φ=0 with m being any fixed positive number.
基金funded by Researchers Supporting Project No.RSPD2025R733,King Saud University,Riyadh,Saudi Arabia。
文摘We have examined an isotropic and homogeneous cosmological model in f(R,T^(φ))gravity,where R represents the Ricci scalar and T^(φ)exhibits the energy momentum tensor's trace.We examine the stability criteria by performing the dynamical system analysis for our model f(R,T^(φ))=R+2(a T^(φ)+b),where a and b are the constants.We derive a set of autonomous equations and find their solutions by assuming a flat potential V0.We assess the equilibrium points from these equations and find the eigenvalues.We analyze the physical interpretation of the phase space for this system.We obtain three stable equilibrium points.We also examine the interaction between the scalar field and dark energy,represented by Q=ψH_(ρde)and determine the equilibrium points for this interaction.We identify four stable equilibrium points for this interaction.We calculate the values of the physical parameters for both scenarios at each equilibrium point,indicating the Universe's accelerated expansion.
文摘In this paper,Isogeometric analysis(IGA)is effectively integrated with machine learning(ML)to investigate the bearing capacity of strip footings in layered soil profiles,with a focus on a sand-over-clay configuration.The study begins with the generation of a comprehensive dataset of 10,000 samples from IGA upper bound(UB)limit analyses,facilitating an in-depth examination of various material and geometric conditions.A hybrid deep neural network,specifically the Whale Optimization Algorithm-Deep Neural Network(WOA-DNN),is then employed to utilize these 10,000 outputs for precise bearing capacity predictions.Notably,the WOA-DNN model outperforms conventional ML techniques,offering a robust and accurate prediction tool.This innovative approach explores a broad range of design parameters,including sand layer depth,load-to-soil unit weight ratio,internal friction angle,cohesion,and footing roughness.A detailed analysis of the dataset reveals the significant influence of these parameters on bearing capacity,providing valuable insights for practical foundation design.This research demonstrates the usefulness of data-driven techniques in optimizing the design of shallow foundations within layered soil profiles,marking a significant stride in geotechnical engineering advancements.
基金supported by the Engineering and Physical Sciences Research Council [EP/L015811/1]
文摘We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically zero.This is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L^2(R^3), i.e., finite Dirichlet integral.
文摘In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.
基金The project supported by the foundation of The State Education Commission"The dynamics of upper ocean"the open grants of Physical Oceanography Laboratory
文摘An fKdV equation of two-layer how and an averaged fKdV equation (AfKdV equation) with respect to phase are derived to determine the theoretical amplitude and period of the precursor solitons in the present paper. In terms of the AfKdV equation derived by the authors, a new theory on the precursor soliton generation based on Lee et al.'s concept is presented. Concepts of asymptotic mean hydraulic fall and level are introduced in our analysis, and the theoretical amplitude and period both depend on the asymptotic mi-an levels and stratified parameters. From the present theoretical results, it is obtained that when the moving velocity of the topography is at the resonant points, there exist two general relations: (1) amplitude relation (A) over circle = 2F, (2) period relation <(tau)over circle> = -8m(1)m(3)(-1)root 6m(4)m(3)(-1)F, in which (A) over circle and <(tau)over circle> are the amplitude and period of the precursor solitons at the resonant points respectively, m(1), m(3) and m(4) are coefficients of the fKdV equation, and F is asymptotic mean half-hydraulic fall at subcritical cutoff points. The theoretical results of this paper are compared with experiments and numerical calculations of two-layer flow over a semicircular topography and all these results are in good agreement. Due to the canonical character of the coefficients of fKdV equations, this theory also holds for any two-dimensional system, which can be reduced to fKdV equations.
基金supported in part by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1 and EP/L015811/1
文摘We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem （initial value problem） in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.
基金supported by the Fundamental Research Grant Scheme (FRGS) of the Ministry of Higher Education (MOHE) of Malaysia (No. UKM-ST-07-FRGS0036-2009)
文摘The problem of the steady magnetohydrodynamic （MHD） stagnation-point flow of an incompressible viscous fluid over a stretching sheet is studied. The effect of an induced magnetic field is taken into account. The nonlinear partial differential equations are transformed into ordinary differential equations via the similarity transformation. The transformed boundary layer equations are solved numerically using the shooting method. Numerical results are obtained for various magnetic parameters and Prandtl numbers. The effects of the induced magnetic field on the skin friction coefficient, the local Nusselt number, the velocity, and the temperature profiles are presented graphically and discussed in detail.
基金Supported by BRNS of Bhaba Atomic Research Centre,Mumbai under Department of Atomic Energy,Government of India vide under Grant No.2012/37P/54/BRNS/2382
文摘In the present paper, we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics; namely the space-time fractional Zakharov–Kuznetsov(ZK) and modified Zakharov–Kuznetsov(m ZK) equations by using fractional sub-equation method. As a result, new types of exact analytical solutions are obtained. The obtained results are shown graphically. Here the fractional derivative is described in the Jumarie's modified Riemann–Liouville sense.
文摘Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean of finite depth, where the free surface has an ice-cover being modelled as an elastic plate of very small thickness, is investigated within the framework of linearized water wave theory. The effect of surface tension at the surface below the ice-cover is neglected. There exists only one wave number propagating at just below the ice-cover. A perturbation analysis is employed to solve the boundary value problem governed by Laplace＇s equation by a method based on Green＇s integral theorem with the introduction of appropriate Green＇s function and thereby evaluating the reflection and transmission coefficients approximately up to first order. A patch of sinusoidal ripples is considered as an example and the related coefficients are determined.
文摘The problem of oblique wave （internal wave） propagation over a small deformation in a channel flow consisting of two layers was considered. The upper fluid was assumed to be bounded above by a rigid lid, which is an approximation for the free surface, and the lower one was bounded below by an impermeable bottom surface having a small deformation; the channel was unbounded in the horizontal directions. Assuming irrotational motion, the perturbation technique was employed to calculate the first-order corrections of the velocity potential in the two fluids by using Green＇s integral theorem suitably with the introduction of appropriate Green＇s functions. Those functions help in calculating the reflection and transmission coefficients in terms of integrals involving the shape ftmction c（x） representing the bottom deformation. Three-dimensional linear water wave theory was utilized for formulating the relevant boundary value problem. Two special examples of bottom deformation were considered to validate the results. Consideration of a patch of sinusoidal ripples （having the same wave number） shows that the reflection coefficient is an oscillatory function of the ratio of twice the x-component of the wave number to the ripple wave number. When this ratio approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of incident wave energy occurs if this number is large. Similar results were observed for a patch of sinusoidal ripples having different wave numbers. It was also observed that for small angles of incidence, the reflected energy is greater compared to other angles of incidence up to π/ 4. These theoretical observations are supported by graphical results.
基金the Ministry of Sciences and Enviromental Protection of Republic Serbia through Mathematical Institute SANU Belgrade Grants No.ON144002
文摘The study of energy transfer between coupled subsystems in a hybrid system is very important for applications. This paper presents an analytical analysis of energy transfer between plates of a visco-elastically connected double-plate system in free transversal vibrations. The analytical analysis shows that the visco-elastic connection between plates is responsible for the appearance of two-frequency regime in the time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are uncoupled, but energy transfer between plates in one eigen mode appears. It was shown for each shape of vibrations. Series of the two Lyapunov exponents corresponding to the one eigen amplitude mode are expressed by using the energy of the corresponding eigen amplitude time component.
基金Project supported by RFDP of Higher Education and NNSF of China, SF of Wuhan University.
文摘This paper obtains the Cauchy-Pompeiu formula on certain distinguished boundary for functions with values in a universal Clifford algebra. This formula is just an extension of the Cauchy's integral formula obtained in [11].
文摘We study separable and self-similar solutions to the Hunter-Saxton equation, a nonlinear wave equation which has been used to describe an instability in the director field of a nematic liquid crystal （among other applications）. Essentially, we study solutions which arise from a nonlinear inhomogeneous ordinary differential equation which is obtained by an exact similarity transform for the ttunter-Saxton equation. For each type of solution, we are able to obtain some simple exact solutions in closed-form, and more complicated solutions through an analytical approach. We find that there is a whole family of self-similar solutions, each of which depends on an arbitrary parameter. This parameteressentially controls the manner of self-similarity and can be chosen so that the self-similar solutions agree with given initial data. The simpler solutions found constitute exact soIutions to a nonlinear partial differential equation, and hence are also useful in a mathematical sense. Analytical solutions demonstrate the variety of behaviors possible within the wider family of similarity solutions. Both types of solutions cast light on self-similar phenomenon arising in the Hunter-Saxton equation.
