In a wide variety of mechanical and industrial applications,e.g.,space cooling,nuclear reactor cooling,medicinal utilizations(magnetic drug targeting),energy generation,and heat conduction in tissues,the heat transfer...In a wide variety of mechanical and industrial applications,e.g.,space cooling,nuclear reactor cooling,medicinal utilizations(magnetic drug targeting),energy generation,and heat conduction in tissues,the heat transfer phenomenon is involved.Fourier’s law of heat conduction has been used as the foundation for predicting the heat transfer behavior in a variety of real-world contexts.This model’s production of a parabolic energy expression,which means that an initial disturbance would immediately affect the system under investigation,is one of its main drawbacks.Therefore,numerous researchers worked on such problem to resolve this issue.At last,this problem was resolved by Cattaneo by adding relaxation time for heat flux in Fourier’s law,which was defined as the time required to establish steady heat conduction once a temperature gradient is imposed.Christov offered a material invariant version of Cattaneo’s model by taking into account the upper-connected derivative of the Oldroyd model.Nowadays,both models are combinedly known as the Cattaneo-Christov(CC)model.In this attempt,the mixed convective MHD Falkner-Skan Sutterby nanofluid flow is addressed towards a wedge surface in the presence of the variable external magnetic field.The CC model is incorporated instead of Fourier’s law for the examination of heat transfer features in the energy expression.A two-phase nanofluid model is utilized for the implementation of nano-concept.The nonlinear system of equations is tackled through the bvp4c technique in the MATLAB software 2016.The influence of pertinent flow parameters is discussed and displayed through different sketches.Major and important results are summarized in the conclusion section.Furthermore,in both cases of wall-through flow(i.e.,suction and injection effects),the porosity parameters increase the flow speed,and decrease the heat transport and the influence of drag forces.展开更多
The magnetohydrodynamic(MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined.The approximate solutions and skin friction coeffcients of the MHD boundar...The magnetohydrodynamic(MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined.The approximate solutions and skin friction coeffcients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method(DTM) with the Pad'e approximation called DTM-Pad'e.The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure.The approximate solutions are tabulated,plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique.It is found that the approximate solution agrees very well with the numerical solution,showing the reliability and validity of the present work.Moreover,the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.展开更多
The magnetohydrodynamics (MHD) Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite i...The magnetohydrodynamics (MHD) Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.展开更多
The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new ratio...The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients.To find the unknown coefficients and the auxiliary parameter contained in the polynomials,the collocation method with Chebyshev-Gauss points is used.The numerical examples show the efficiency of this method.展开更多
Falkner-Skan aspects are revealed numerically for a non-homogeneous hybrid mixture of 50%ethylene glycol-50%water,silver nanomaterials Ag,and molybdenum disul-fide nanoparticles MoS2 during its motion over a static we...Falkner-Skan aspects are revealed numerically for a non-homogeneous hybrid mixture of 50%ethylene glycol-50%water,silver nanomaterials Ag,and molybdenum disul-fide nanoparticles MoS2 during its motion over a static wedge surface in a DarcyForchheimer porous medium by employing the modified Buongiorno model.The Brownian and thermophoresis mechanisms are included implicitly along with the thermophysical properties of each phase via the mixture theory and some efficient phenomenological laws.The present simulation also accounts for the impacts of nonlinear radiative heat flux,magnetic forces,and Joule heating.Technically,the generalized differential quadrature method and Newton-Raphson technique are applied successfully for solving the resulting nonlinear boundary layer equations.In a limiting case,the obtained findings are validated accurately with the existing literature outcomes.The behaviors of velocity,temperature,and nanoparticles volume fraction are discussed comprehensively against various governing parameters.As crucial results,it is revealed that the temperature is enhanced due to magnetic field,linear porosity,radiative heat flux,Brownian motion,thermophoresis,and Joule heating effects.Also,it is depicted that the hybrid nanoliquids present a higher heat flux rate than the monotype nanoliquids and liquids cases.Moreover,the surface frictional impact is minimized via the linear porosity factor.Furthermore,the surface heat transfer rate receives a prominent improvement due to the radiative heat flux inclusion.展开更多
The aim of this paper is to give a presentation of two new iterative methods forsolving non-linear differential equations, they are successive linearisation method and spectralhomotopy perturbation method. We applied ...The aim of this paper is to give a presentation of two new iterative methods forsolving non-linear differential equations, they are successive linearisation method and spectralhomotopy perturbation method. We applied these techniques on the non-linear boundary valueproblems of Falkner-Skan type. The methods used to find a recursive former for higher orderequations that are solved using the Chebyshev spectral method to find solutions that areaccurate and converge rapidly to the full numerical solution. The methods are illustrated byprogressively applying the technique to the Blasius boundary layer equation, the Falkner-Skanequation and finally, the magnetohydrodynamic (MHD) Falkner-Skan equation. The solutionsare compared to other methods in the literature such as the homotopy analysis method and thespectral-homotopy analysis method with focus on the accuracy and convergence of this newtechniques.展开更多
An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of -1. The solution is given in a clo...An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of -1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding ofthis equation.展开更多
This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial diffe...This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial differential equations describing laminar flow are converted to a self-simi- lar type third order ordinary differential equation known as the Falkner-Skan equation. For the purposes of a numerical solution, the Falkner-Skan equation is converted to a system of first order ordinary differential equations. These are numerically addressed by the conventional shooting and bisection methods coupled with the Runge-Kutta technique. However the accompanying energy equation lends itself to a hybrid numerical finite element-boundary integral application. An appropriate complementary differential equation as well as the Green second identity paves the way for the integral representation of the energy equation. This is followed by a finite element-type discretization of the problem domain. Based on the quality of the results obtained herein, a strong case is made for a hybrid numerical scheme as a useful approach for the numerical resolution of boundary layer flows and species transport. Thanks to the sparsity of the resulting coefficient matrix, the solution profiles not only agree with those of similar problems in literature but also are in consonance with the physics they represent.展开更多
基金Deanship of Scientific Research at King Khalid University for funding this work through Large Group Research Project(No.RGP2/19/44)。
文摘In a wide variety of mechanical and industrial applications,e.g.,space cooling,nuclear reactor cooling,medicinal utilizations(magnetic drug targeting),energy generation,and heat conduction in tissues,the heat transfer phenomenon is involved.Fourier’s law of heat conduction has been used as the foundation for predicting the heat transfer behavior in a variety of real-world contexts.This model’s production of a parabolic energy expression,which means that an initial disturbance would immediately affect the system under investigation,is one of its main drawbacks.Therefore,numerous researchers worked on such problem to resolve this issue.At last,this problem was resolved by Cattaneo by adding relaxation time for heat flux in Fourier’s law,which was defined as the time required to establish steady heat conduction once a temperature gradient is imposed.Christov offered a material invariant version of Cattaneo’s model by taking into account the upper-connected derivative of the Oldroyd model.Nowadays,both models are combinedly known as the Cattaneo-Christov(CC)model.In this attempt,the mixed convective MHD Falkner-Skan Sutterby nanofluid flow is addressed towards a wedge surface in the presence of the variable external magnetic field.The CC model is incorporated instead of Fourier’s law for the examination of heat transfer features in the energy expression.A two-phase nanofluid model is utilized for the implementation of nano-concept.The nonlinear system of equations is tackled through the bvp4c technique in the MATLAB software 2016.The influence of pertinent flow parameters is discussed and displayed through different sketches.Major and important results are summarized in the conclusion section.Furthermore,in both cases of wall-through flow(i.e.,suction and injection effects),the porosity parameters increase the flow speed,and decrease the heat transport and the influence of drag forces.
基金supported by the National Natural Science Foundation of China (Nos. 50936003 and 51076012)the Open Project of State Key Laboratory for Advanced Metals and Materials (No. 2009Z-02)
文摘The magnetohydrodynamic(MHD) Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined.The approximate solutions and skin friction coeffcients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method(DTM) with the Pad'e approximation called DTM-Pad'e.The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure.The approximate solutions are tabulated,plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique.It is found that the approximate solution agrees very well with the numerical solution,showing the reliability and validity of the present work.Moreover,the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.
基金supported by the Imam Khomeini International University of Iran(No.751166-1392)the Deanship of Scientific Research(DSR)in King Abdulaziz University of Saudi Arabia
文摘The magnetohydrodynamics (MHD) Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined.
文摘The new rationalα-polynomials are used to solve the Falkner-Skan equation.These polynomials are equipped with an auxiliary parameter.The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients.To find the unknown coefficients and the auxiliary parameter contained in the polynomials,the collocation method with Chebyshev-Gauss points is used.The numerical examples show the efficiency of this method.
文摘Falkner-Skan aspects are revealed numerically for a non-homogeneous hybrid mixture of 50%ethylene glycol-50%water,silver nanomaterials Ag,and molybdenum disul-fide nanoparticles MoS2 during its motion over a static wedge surface in a DarcyForchheimer porous medium by employing the modified Buongiorno model.The Brownian and thermophoresis mechanisms are included implicitly along with the thermophysical properties of each phase via the mixture theory and some efficient phenomenological laws.The present simulation also accounts for the impacts of nonlinear radiative heat flux,magnetic forces,and Joule heating.Technically,the generalized differential quadrature method and Newton-Raphson technique are applied successfully for solving the resulting nonlinear boundary layer equations.In a limiting case,the obtained findings are validated accurately with the existing literature outcomes.The behaviors of velocity,temperature,and nanoparticles volume fraction are discussed comprehensively against various governing parameters.As crucial results,it is revealed that the temperature is enhanced due to magnetic field,linear porosity,radiative heat flux,Brownian motion,thermophoresis,and Joule heating effects.Also,it is depicted that the hybrid nanoliquids present a higher heat flux rate than the monotype nanoliquids and liquids cases.Moreover,the surface frictional impact is minimized via the linear porosity factor.Furthermore,the surface heat transfer rate receives a prominent improvement due to the radiative heat flux inclusion.
文摘The aim of this paper is to give a presentation of two new iterative methods forsolving non-linear differential equations, they are successive linearisation method and spectralhomotopy perturbation method. We applied these techniques on the non-linear boundary valueproblems of Falkner-Skan type. The methods used to find a recursive former for higher orderequations that are solved using the Chebyshev spectral method to find solutions that areaccurate and converge rapidly to the full numerical solution. The methods are illustrated byprogressively applying the technique to the Blasius boundary layer equation, the Falkner-Skanequation and finally, the magnetohydrodynamic (MHD) Falkner-Skan equation. The solutionsare compared to other methods in the literature such as the homotopy analysis method and thespectral-homotopy analysis method with focus on the accuracy and convergence of this newtechniques.
文摘An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of -1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding ofthis equation.
文摘This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial differential equations describing laminar flow are converted to a self-simi- lar type third order ordinary differential equation known as the Falkner-Skan equation. For the purposes of a numerical solution, the Falkner-Skan equation is converted to a system of first order ordinary differential equations. These are numerically addressed by the conventional shooting and bisection methods coupled with the Runge-Kutta technique. However the accompanying energy equation lends itself to a hybrid numerical finite element-boundary integral application. An appropriate complementary differential equation as well as the Green second identity paves the way for the integral representation of the energy equation. This is followed by a finite element-type discretization of the problem domain. Based on the quality of the results obtained herein, a strong case is made for a hybrid numerical scheme as a useful approach for the numerical resolution of boundary layer flows and species transport. Thanks to the sparsity of the resulting coefficient matrix, the solution profiles not only agree with those of similar problems in literature but also are in consonance with the physics they represent.
