The current investigations are carried out to study the influence of the Darcy–Forchheimer relation on third-grade fluid flow and heat transfer over an angled exponentially stretching sheet embedded in a porous medium. In the current study, the applied magnetic field, Joule heating, thermaldiffusion, viscous dissipation, and diffusion-thermo effects are incorporated. The proposed model in terms of partial differential equations is transformed into ordinary differential equations using suitable similarity transformation. The reduced model is then solved numerically with the help of MATLAB built-in function bvp4c.The numerical solutions for velocity profile, temperature profile, and mass concentration under the effects of pertinent parameters involved in the model are determined and portrayed in graphical form. The graphical effects of the skin friction coefficient, the Nusselt number, and the Sherwood number are also shown. From the displayed results, we conclude that when the Joule heating parameter is enlarged, the velocity and the temperature of the fluid are increased. We observed that while enhancing the viscous dissipation parameter (Eckert number) the fluid’s velocity and temperature increase but decreases the mass concentration. By increasing the values of the thermal-diffusion parameter, the velocity distribution, the temperature field, and the mass concentration increase. When the diffusion–thermo parameter rises, the velocity field and the temperature distribution increase, and the reverse scenario is seen in the mass concentration. The results of the current study are compared with already published results, and a good agreement is noted to validate the current study.

The combined impact of a linear chemical reaction and Lorentz force on heat and mass transfer in a third-grade fluid with the Darcy–Forchheimer relation over an inclined, exponentially stretching surface embedded in a porous medium is investigated. The proposed process is mathematically expressed in terms of nonlinear and coupled partial differential equations, with the symmetry of the conditions normal to the surface. To solve the mathematical model of the proposed phenomenon, the partial differential equations are first reduced to ordinary differential equations; then, MATLAB built-in Numerical Solver bvp4c is used to obtain the numerical results of these equations. The influence of all the pertinent parameters that appeared in the flow model on the unknown material properties of interest is depicted in the forms of tables and graphs. The physical attitude of the unknown variables is discussed with physical reasoning. From the numerical solutions, it is inferred that, as Lorentz force parameter $M$ is increased, the velocity of the fluid decreases, but fluid temperature and mass concentration increase. This is due to the fact that Lorentz force retards the motion of fluid, and the increasing resistive force causes the rise in the temperature of the fluid. It is also noted that, owing to an increase in the magnitude of chemical reaction parameter $R$, the velocity profile and the mass concentration decline as well, but the fluid temperature increases in a reasonable manner. It is noted that, by augmenting the values of the local inertial coefficient $Fr$ and the permeability parameter $K^{*}$, the velocity field decreases, the temperature field increases, and mass concentration also increases with reasonable difference. Increasing values of Prandtl number $Pr$ results in a decrease in the profiles of velocity and temperature. All the numerical results are computed at the angle of inclination $\alpha =\pi /6$. The current results are compared with the available results in the existing literature for this special case, and there is good agreement between them that shows the validation of the present study. All the numerical results show asymptotic behavior by satisfying the given boundary conditions.

The current study aims to investigate the thermal-diffusion and diffusion-thermo effects on heat and mass transfer in third-grade fluid with Darcy–Forchheimer relation impact over an exponentially inclined stretching sheet embedded in a porous medium. The proposed mechanism in terms non-linear and coupled partial differential equations is reduced to set of ordinary differential equations by employing an appropriate similarity variable formulation. The reduced form of equations is solved by using the MATLAB built-in numerical solver bvp4c. The numerical results for unknown physical properties such as velocity profile, temperature field, and mass concentration along with their gradients such as the skin friction, the rate of heat transfer, and the rate of mass transfer at angle of inclination $\alpha =\pi /6$ are obtained under the impact of material parameters that appear in the flow model. The solutions are displayed in forms of graphs as well as tables and are discussed with physical reasoning. From the demonstration of the graphical results, it is inferred that thermal-diffusion parameter $Sr$ velocity, temperature, and concentration profiles are augmented. For the increasing magnitude of the diffusion-thermo parameter $D_f$ the fluid velocity and fluid temperature rise but the opposite trend in mass concentration is noted. The current results are compared with the available results in the existing literature, and there is good agreement between them that shows the validation of the present study.

The major aim of the current investigations is to study the magnetohydrodynamic effects on heat and mass transfer phenomena in third-grade fluid past an inclined exponentially stretching sheet fixed in a porous medium with Darcy–Forchheimer law influence. The constitutive equations compatible for heat and mass transportation in third-grade fluid in terms of partial differential equations are modeled. These partial differential equations are then converted to ordinary differential equations by using suitable similarity variables formulation. The transformed flow model is solved by using MATLAB built-in numerical solver bvp4c. Effects of pertinent parameters on physical properties that are velocity field, temperature field and mass concentration along with skin friction coefficient, Nusselt number and Sherwood number are demonstrated in graphs and tables. The impact of dimensionless numbers on the physical properties is analyzed and discussed with a physical view point at angle $\alpha =\pi /6$ (inclined sheet). It is seen that as the third-grade fluid parameter $\left(0.1\le \beta \le 11\right)$ is increased, the velocity profile increases, but the temperature field and mass concentration are decreased. It is observed that as the permeability parameter $(1\le K^{*}\le 11)$ is raised, the velocity distribution decreases and mass concentration increases. It is concluded from the results that owing to an increase in the local inertial coefficient $\left(0.1\le Fr\le 5\right)$, the velocity profile reduces but an increment in mass concentration is noted. It is concluded that by increasing values of magnetic field parameter $\left(0.1\le M\le 10\right)$ the velocity field is delineated and temperature field is elevated exactly according to the physics of magnetic field parameters. The present results are compared with already published results and it is observed that there is good agreement between them. This good agreement ensures the validation of accuracy of the results.

This paper concerns with the existence and uniqueness of fuzzy fractional evolution equation with uncertainty involves function of form ${}^c{D}^{\alpha }x\left(t\right)=f(t,x\left(t\right),D^{\beta }x\left(t\right)),I^{\alpha }x\left(0\right)=x_0,x^{\prime }\left(0\right)=x_1,$ where $1<\alpha <2,0<\beta <1$. After determining the equivalent integral form of solution we establish existence and uniqueness by using Rogers conditions, Kooi type conditions and Krasnoselskii-Krein type conditions. In addition, various numerical solutions have been presented to ensure that the main result is true and effective. Finally, a few examples which express fuzzy fractional evolution equations are shown.

We analyze a Dirichlet $\left(p\right(x),\mu q(x\left)\right)$-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter $\mu$ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates.