The “shooting and bouncing rays” (SBR) technique is used to analyze the electromagnetic scattering characters of ocean rough surfaces varying with time. Some numerical results are presented and compared with the met...The “shooting and bouncing rays” (SBR) technique is used to analyze the electromagnetic scattering characters of ocean rough surfaces varying with time. Some numerical results are presented and compared with the method of moments, and some factors, such as the incident angle, polarization and frequency are investigated which influence on electromagnetic scattering characters of ocean rough surfaces.展开更多
The significance of the thermophysical properties of Tetra hybrid nanofluid in enhancing heat transmission in various applications like heat exchangers, automobiles, and solar storage cannot be overstated. These featu...The significance of the thermophysical properties of Tetra hybrid nanofluid in enhancing heat transmission in various applications like heat exchangers, automobiles, and solar storage cannot be overstated. These features can be tampered with when nanoparticles are been introduced into the base fluid to produce an improved heat carrier fluid for the system. This study investigates the impact of temperature-dependent properties on the movement of TiO2-SiO2-ZnO-Fe2O3/PAO Tetra hybrid nanofluid along a vertical porous surface with suction. The system of governing Partial Differential Equations (PDEs) was formulated and transformed into the system of coupled nonlinear third-order Ordinary Differential Equations (ODEs) by similarity techniques. The resulting ODEs were solved numerically using the shooting method and fourth order Runge-Kutta method with the aid of Maple 18.0 software. Using numerical and statistical methods, the study analyzes velocity, temperature profiles, skin friction coefficient, and Nusselt number. It was found that as the variable thermal conductivity parameter upsurges both the skin friction coefficient and Nusselt number intensify at the rate of 0.011697519 and 8.043581616 respectively. This study underscores the vital role of Tetra hybrid nanofluid’s thermophysical properties in improving heat transmission for diverse appli cations. By manipulating nanoparticles within the base fluid, the heat carrier fluid’s efficiency can be enhanced, critical for industries like automotive and enewable energy. These insights inform the design of more efficient heat exchange systems, advancing sustainability and performance in real-world scenarios.展开更多
Stationary even single bump periodic solutions of the Swift Hohenberg equation are analyzed. The coefficient k in the equation is found to be a critical parameter. It is proved if 0<k<1 , there exist pe...Stationary even single bump periodic solutions of the Swift Hohenberg equation are analyzed. The coefficient k in the equation is found to be a critical parameter. It is proved if 0<k<1 , there exist periodic solutions having the same energy as the constant solution u=0; if 1<k<32 , there exist periodic solutions having the same energy as the stable states u=±k-1. The proof of the above results is based on a shooting technique, together with a linearization method and a scaling argument.展开更多
Stationary even periodic solutions of the Swift-Hohenberg equation areanalyzed for the critical parameter k = 1, and it is proved that there exist periodic solutionshaving the same energy as the constant solution u = ...Stationary even periodic solutions of the Swift-Hohenberg equation areanalyzed for the critical parameter k = 1, and it is proved that there exist periodic solutionshaving the same energy as the constant solution u = 0. For k ≤ 0, some qualitative properties ofthe solutions are also proved.展开更多
This paper presents an efficient numerical scheme for calculating the periodic motion of a harmonically forced piecewise linear oscillator very accurately. The scheme is based on the shooting technique with the tradi...This paper presents an efficient numerical scheme for calculating the periodic motion of a harmonically forced piecewise linear oscillator very accurately. The scheme is based on the shooting technique with the traditional numerical Poincare mapping and its Jacobian replaced by the piecewise analytic ones. Thus, the scheme gets rid of the requirement of the current schemes for an assumed order of the oscillator trajectory passing through different linear regions. The numerical examples in the paper demonstrate that the new scheme, compared with the current schemes, enables one to cope with more complicated dynamics of harmonically forced piecewise linear oscillators.展开更多
A suitable similarity transformation is introduced to reduce the laminarboundary layer equations of power law fluids to a class of singular nonlinear two-point boundaryvalue problems. The kin friction and shear stress...A suitable similarity transformation is introduced to reduce the laminarboundary layer equations of power law fluids to a class of singular nonlinear two-point boundaryvalue problems. The kin friction and shear stress distributions for boundary layer flow over amoving flat plate are investigated by utilizing the shooting technique. Results indicate that foreach fixed value of the power law exponent n or the velocity ratio parameter xi, the skin frictionand shear stress decrease with the increasing of n or xi respectively.展开更多
The transport behavior of free boundary value problems for a class ofgeneralized diffusion equations was studied. Suitable similarity transformations were used toconvert the problems into a class of singular nonlinear...The transport behavior of free boundary value problems for a class ofgeneralized diffusion equations was studied. Suitable similarity transformations were used toconvert the problems into a class of singular nonlinear two-point boundary value problems andsimilarity solutions were numerical presented for different representations of heat conductionfunction, convection function, heat flux function, and power law parameters by utilizing theshooting technique. The results revealed the flux transfer mechanism and the character as well asthe effects of parameters on the solutions.展开更多
A second order heat equation with convection in an infinite medium isstudied. Suitable similarity transformations are used to reduce the parabolic heat equation to aClass of singular nonlinear boundary value problems....A second order heat equation with convection in an infinite medium isstudied. Suitable similarity transformations are used to reduce the parabolic heat equation to aClass of singular nonlinear boundary value problems. Numerical solutions are presented for differentrepresehtations of heat conduction, heat convection, heat flux, and power law parameters byutilizing the shooting technique. The results reveal the heat transfer characteristic and the effectof parameters on the solutions.展开更多
Analytical and numerical solutions are presented for the momentum and energy laminar boundary layer equations in power law fluids utilizing a similarity transformation and the shooting technique. The results indica...Analytical and numerical solutions are presented for the momentum and energy laminar boundary layer equations in power law fluids utilizing a similarity transformation and the shooting technique. The results indicated that for power law exponents 0< n ≤1, the skin friction σ decreases with increasing n , and the dimensionless shear force decreases with increasing dimensionless velocity t . When Pr =1, the velocity distribution in the viscous boundary layer is the same as the temperature distribution in the thermal boundary layer and δ=δ T. For Pr >1, the increase of the viscous diffusion exceeds that of thermal diffusion with increasing Pr , i.e. , δ T (t)<δ(t) . The thermal diffusion ratio increases with increasing n(0<n≤1) .展开更多
The current article investigates the impact of the bioconvection in an unsteady flow of magnetized Cross nanofluid with gyrotactic microorganisms and activation energy over a linearly stretched configuration.The analy...The current article investigates the impact of the bioconvection in an unsteady flow of magnetized Cross nanofluid with gyrotactic microorganisms and activation energy over a linearly stretched configuration.The analysis has been performed by utilizing the realistic Wu's slip boundary and zero mass flux conditions.The effects of nonlinear thermal radiation and the activation energy are also addressed.The governing flow equations are deduced to a dimensionless form by considering suitable transformations which are numerically targeted via a shooting algorithm.The physical visualization of each physical parameter governing the flow problem has been displayed graphically for distribution of velocity,temperature,concentration and motile microorganisms.The numerical treatment for the variation of skin friction coefficient,local Nusselt number,local Sherwood number and motile density number is performed in tabular forms.展开更多
A steady boundary layer flow over a porous flat plate has been considered in the present study.Mass transfer analysis with first order chemical reaction is also considered instead of heat transfer.The plate concentrat...A steady boundary layer flow over a porous flat plate has been considered in the present study.Mass transfer analysis with first order chemical reaction is also considered instead of heat transfer.The plate concentration is considered in the form of power law instead of taking constant.The goveming PDEs are transformed into ordinary differential equations using similarity transfomation and then these ODEs are solved by employing Runge-Kutta fourth order method associated with shooting technique.A parametric study of all involving parameters is obtained by the help of graphs.The major findings are:(i)the concentration of the fluid in its boundary layer decrease with increase in heavier species,the reaction rate parameter and the power law exponent;(ji)the rate of mass transfer increases with an increase in reaction rafe parameter and power-law exponent.展开更多
文摘The “shooting and bouncing rays” (SBR) technique is used to analyze the electromagnetic scattering characters of ocean rough surfaces varying with time. Some numerical results are presented and compared with the method of moments, and some factors, such as the incident angle, polarization and frequency are investigated which influence on electromagnetic scattering characters of ocean rough surfaces.
文摘The significance of the thermophysical properties of Tetra hybrid nanofluid in enhancing heat transmission in various applications like heat exchangers, automobiles, and solar storage cannot be overstated. These features can be tampered with when nanoparticles are been introduced into the base fluid to produce an improved heat carrier fluid for the system. This study investigates the impact of temperature-dependent properties on the movement of TiO2-SiO2-ZnO-Fe2O3/PAO Tetra hybrid nanofluid along a vertical porous surface with suction. The system of governing Partial Differential Equations (PDEs) was formulated and transformed into the system of coupled nonlinear third-order Ordinary Differential Equations (ODEs) by similarity techniques. The resulting ODEs were solved numerically using the shooting method and fourth order Runge-Kutta method with the aid of Maple 18.0 software. Using numerical and statistical methods, the study analyzes velocity, temperature profiles, skin friction coefficient, and Nusselt number. It was found that as the variable thermal conductivity parameter upsurges both the skin friction coefficient and Nusselt number intensify at the rate of 0.011697519 and 8.043581616 respectively. This study underscores the vital role of Tetra hybrid nanofluid’s thermophysical properties in improving heat transmission for diverse appli cations. By manipulating nanoparticles within the base fluid, the heat carrier fluid’s efficiency can be enhanced, critical for industries like automotive and enewable energy. These insights inform the design of more efficient heat exchange systems, advancing sustainability and performance in real-world scenarios.
基金National Natural Science Foundation of China (1 0 0 71 0 67)
文摘Stationary even single bump periodic solutions of the Swift Hohenberg equation are analyzed. The coefficient k in the equation is found to be a critical parameter. It is proved if 0<k<1 , there exist periodic solutions having the same energy as the constant solution u=0; if 1<k<32 , there exist periodic solutions having the same energy as the stable states u=±k-1. The proof of the above results is based on a shooting technique, together with a linearization method and a scaling argument.
文摘Stationary even periodic solutions of the Swift-Hohenberg equation areanalyzed for the critical parameter k = 1, and it is proved that there exist periodic solutionshaving the same energy as the constant solution u = 0. For k ≤ 0, some qualitative properties ofthe solutions are also proved.
文摘This paper presents an efficient numerical scheme for calculating the periodic motion of a harmonically forced piecewise linear oscillator very accurately. The scheme is based on the shooting technique with the traditional numerical Poincare mapping and its Jacobian replaced by the piecewise analytic ones. Thus, the scheme gets rid of the requirement of the current schemes for an assumed order of the oscillator trajectory passing through different linear regions. The numerical examples in the paper demonstrate that the new scheme, compared with the current schemes, enables one to cope with more complicated dynamics of harmonically forced piecewise linear oscillators.
基金This work was financially supported by the Cross-Century Talents Projects of Educational Ministry, China and the "973" Key Item(
文摘A suitable similarity transformation is introduced to reduce the laminarboundary layer equations of power law fluids to a class of singular nonlinear two-point boundaryvalue problems. The kin friction and shear stress distributions for boundary layer flow over amoving flat plate are investigated by utilizing the shooting technique. Results indicate that foreach fixed value of the power law exponent n or the velocity ratio parameter xi, the skin frictionand shear stress decrease with the increasing of n or xi respectively.
基金This work was financially supported by the Cross-Century Talents Projects of Educational Ministry of China and the 973 Key Item (No. G1998061510).]
文摘The transport behavior of free boundary value problems for a class ofgeneralized diffusion equations was studied. Suitable similarity transformations were used toconvert the problems into a class of singular nonlinear two-point boundary value problems andsimilarity solutions were numerical presented for different representations of heat conductionfunction, convection function, heat flux function, and power law parameters by utilizing theshooting technique. The results revealed the flux transfer mechanism and the character as well asthe effects of parameters on the solutions.
基金This work was supported by Cross-Century Talents Projects of Educational Ministry of China the "973" Key Foundation under the contract No.G1998061510.]
文摘A second order heat equation with convection in an infinite medium isstudied. Suitable similarity transformations are used to reduce the parabolic heat equation to aClass of singular nonlinear boundary value problems. Numerical solutions are presented for differentrepresehtations of heat conduction, heat convection, heat flux, and power law parameters byutilizing the shooting technique. The results reveal the heat transfer characteristic and the effectof parameters on the solutions.
文摘Analytical and numerical solutions are presented for the momentum and energy laminar boundary layer equations in power law fluids utilizing a similarity transformation and the shooting technique. The results indicated that for power law exponents 0< n ≤1, the skin friction σ decreases with increasing n , and the dimensionless shear force decreases with increasing dimensionless velocity t . When Pr =1, the velocity distribution in the viscous boundary layer is the same as the temperature distribution in the thermal boundary layer and δ=δ T. For Pr >1, the increase of the viscous diffusion exceeds that of thermal diffusion with increasing Pr , i.e. , δ T (t)<δ(t) . The thermal diffusion ratio increases with increasing n(0<n≤1) .
基金the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program under grant number(R.G.P.2/66/40)。
文摘The current article investigates the impact of the bioconvection in an unsteady flow of magnetized Cross nanofluid with gyrotactic microorganisms and activation energy over a linearly stretched configuration.The analysis has been performed by utilizing the realistic Wu's slip boundary and zero mass flux conditions.The effects of nonlinear thermal radiation and the activation energy are also addressed.The governing flow equations are deduced to a dimensionless form by considering suitable transformations which are numerically targeted via a shooting algorithm.The physical visualization of each physical parameter governing the flow problem has been displayed graphically for distribution of velocity,temperature,concentration and motile microorganisms.The numerical treatment for the variation of skin friction coefficient,local Nusselt number,local Sherwood number and motile density number is performed in tabular forms.
文摘A steady boundary layer flow over a porous flat plate has been considered in the present study.Mass transfer analysis with first order chemical reaction is also considered instead of heat transfer.The plate concentration is considered in the form of power law instead of taking constant.The goveming PDEs are transformed into ordinary differential equations using similarity transfomation and then these ODEs are solved by employing Runge-Kutta fourth order method associated with shooting technique.A parametric study of all involving parameters is obtained by the help of graphs.The major findings are:(i)the concentration of the fluid in its boundary layer decrease with increase in heavier species,the reaction rate parameter and the power law exponent;(ji)the rate of mass transfer increases with an increase in reaction rafe parameter and power-law exponent.
