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