Convective flow is a self-sustained flow with the effect of the temperature gradient.The density is non-uniform due to the variation of temperature.The effect of the magnetic flux plays a major role in convective flow...Convective flow is a self-sustained flow with the effect of the temperature gradient.The density is non-uniform due to the variation of temperature.The effect of the magnetic flux plays a major role in convective flow.The process of heat transfer is accompanied by a mass transfer process;for instance,condensation,evaporation,and chemical process.Due to the applications of the heat and mass transfer combined effects in a different field,the main aim of this paper is to do a comprehensive analysis of heat and mass transfer of MHD unsteady second-grade fluid in the presence of ramped boundary conditions near a porous surface.The dynamical analysis of heat transfer is based on classical differentiation with no memory effects.The non-dimensional form of the governing equations of the model is developed.These are solved by the classical integral(Laplace)transform technique/method with the convolution theorem and closed-form solutions are attained for temperature,concentration,and velocity.The physical aspects of distinct parameters are discussed via graph to see the influence on the fluid concentration,velocity,and temperature.Our results suggest that the velocity profile decrease by increasing the Prandtl number.The existence of a Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.Furthermore,to validate our results,some results are recovered from the literature.展开更多
This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom.The ice cover is modeled ...This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom.The ice cover is modeled as a thin elastic plate,and the bottom porosity is described by a real parameter.Using linear theory,the problem is formulated as an initial value problem for the velocity potential describing the motion.In the mathematical analysis,the Laplace and Hankel transform techniques have been used to obtain the depression of the ice-covered surface in the form of a multiple infnite integral.This integral is evaluated asymptotically by the method of stationary phase twice for a long time and a large distance from the origin.Simple numerical computations are performed to illustrate the efect of the ice-covered surface and bottom porosity on the surface elevation,phase velocity,and group velocity of the surface gravity waves.Mainly the far-feld behavior of the progressive waves is observed in two diferent cases,namely initial depression and an impulse concentrated at the origin.From graphical representations,it is clearly visible that the presence of ice cover and a porous bottom decreases the wave amplitude.Due to the porous bottom,the amplitude of phase velocity decreases,whereas the amplitude of group velocity increases.展开更多
The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded ...The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.展开更多
文摘Convective flow is a self-sustained flow with the effect of the temperature gradient.The density is non-uniform due to the variation of temperature.The effect of the magnetic flux plays a major role in convective flow.The process of heat transfer is accompanied by a mass transfer process;for instance,condensation,evaporation,and chemical process.Due to the applications of the heat and mass transfer combined effects in a different field,the main aim of this paper is to do a comprehensive analysis of heat and mass transfer of MHD unsteady second-grade fluid in the presence of ramped boundary conditions near a porous surface.The dynamical analysis of heat transfer is based on classical differentiation with no memory effects.The non-dimensional form of the governing equations of the model is developed.These are solved by the classical integral(Laplace)transform technique/method with the convolution theorem and closed-form solutions are attained for temperature,concentration,and velocity.The physical aspects of distinct parameters are discussed via graph to see the influence on the fluid concentration,velocity,and temperature.Our results suggest that the velocity profile decrease by increasing the Prandtl number.The existence of a Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.Furthermore,to validate our results,some results are recovered from the literature.
文摘This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom.The ice cover is modeled as a thin elastic plate,and the bottom porosity is described by a real parameter.Using linear theory,the problem is formulated as an initial value problem for the velocity potential describing the motion.In the mathematical analysis,the Laplace and Hankel transform techniques have been used to obtain the depression of the ice-covered surface in the form of a multiple infnite integral.This integral is evaluated asymptotically by the method of stationary phase twice for a long time and a large distance from the origin.Simple numerical computations are performed to illustrate the efect of the ice-covered surface and bottom porosity on the surface elevation,phase velocity,and group velocity of the surface gravity waves.Mainly the far-feld behavior of the progressive waves is observed in two diferent cases,namely initial depression and an impulse concentrated at the origin.From graphical representations,it is clearly visible that the presence of ice cover and a porous bottom decreases the wave amplitude.Due to the porous bottom,the amplitude of phase velocity decreases,whereas the amplitude of group velocity increases.
文摘The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.
