| General note |
This study examines the heat and mass transfer in a steady Casson fluid over a<br/>stretching sheet. It considers the behavior of Casson fluid with and without slip velocity<br/>and variable heat flux boundary conditions under the influence of various control<br/>parameters. The mathematical model describing continuity, x-momentum, concentration,<br/>and energy transfer in the fluid is formulated along with necessary boundary conditions.<br/>To simplify the PDEs of this model, a new set of generalized transformations is derived<br/>using the Lie similarity method. A general vector field Lie symmetry generator is extended<br/>twice and applied to the fluid model and subjected boundary conditions, resulting in the<br/>invariance criteria in the form of linear PDEs. This invariance criterion, when applied to<br/>PDEs of the fluid model, yields the invariants, which, when applied to the model, reduce<br/>the number of independent variables, turning the complex set of PDEs into simpler ODEs<br/>while retaining the key physical features of the flow. These transformations, while<br/>satisfying the continuity equation, further reduce one dependent variable, decreasing the<br/>complexity of the system even further.<br/>This system of ODEs is then solved using the homotopy perturbation method. It is<br/>a semi-analytical technique that combines homotopy and perturbation methods to handle<br/>nonlinear problems. A higher-order perturbation series, written in the terms of the<br/>homotopy parameter and dependent variables is inserted in the system, which, during<br/>integration, helps refine the solution. This resulting system is then integrated with modified<br/>set of initial conditions having arbitrary constants. The equations resulting from<br/>integration, when subjected to final conditions, evaluate these arbitrary constant, which<br/>convert the boundary value problem into an initial values problem, which is then solved to<br/>get the solution of model.<br/>Different boundary condition sets are imposed on the considered flow model: one<br/>with slip velocity and variable heat flux, and the other without these conditions. The<br/>response of the velocity and temperature towards these conditions is observed and it is<br/>reported that for boundary conditions without slip velocity and variable heat flux, velocity<br/>increases with permeability and decreases with Casson fluid and magnetic field parameters;<br/>xix<br/>temperature increases with permeability, Prandtl number, radiation parameter and ratio of<br/>Lie control parameters and decreases with Casson fluid and magnetic field parameters; and<br/>concentration increases with permeability, Casson fluid parameter, ratio of Lie control<br/>parameters, and decreases with magnetic field parameter and Schmidt number. For slip<br/>velocity and variable heat flux boundary conditions, velocity increases with permeability<br/>and decreases with Casson fluid, slip velocity and magnetic field parameters; temperature<br/>decreases with permeability, Prandtl number, radiation parameter and ratio of Lie control<br/>parameters and increases with Casson fluid, slip velocity, heat flux and magnetic field<br/>parameters; and concentration increases with permeability, Schmidt number, ratio of Lie<br/>control parameters, and decreases with magnetic field, slip velocity and Casson fluid<br/>parameters. The use of Lie symmetry transformations and homotopy perturbation method<br/>proves to be a practical approach for solving complex fluid problems modeled using nonlinear PDEs, offering valuable insights for optimizing industrial processes like polymer<br/>extrusion, metal coating, and thermal management. |
