Computational simulation for MHD peristaltic transport of Jeffrey fluid with density-dependent parameters

This study aimed to give a new theoretical recommendation for non-dimensional parameters depending on the fluid temperature and concentration. This suggestion came from the fact of fluid density may change with the fluid temperature (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}θ) and concentration (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}φ). So, a newly released mathematical form of Jeffrey fluid with peristalsis through the inclined channel is constructed. The problem model defines a mathematical fluid model which converts using non-dimensional values. A sequentially used technique called the Adaptive shooting method for finding the problem solutions. Axial velocity behavior has become a novel concern to Reynolds number. In contradiction to different values of parameters, the temperature and concentration profiles are designated/sketched. The results show that the high value of the Reynolds number acts as a fluid temperature damper, while it boosts the concentration of the fluid particle. The non-constant fluid density recommendation makes the Darcy number controls with a fluid velocity which is virtually significant in drug carries applications or blood circulation systems. To verify the obtained results, a numerical comparison for obtained results has been made with a trustful algorithm with aid of AST using wolfram Mathematica version 13.1.1.


Analysis of model
In a 2D asymmetric channel of thickness 2a, the bloodstream of a Jeffery fluid is premeditated. B 0 is the magnetic field strength with an unvarying mesmeric pasture. The Reynolds number and Froude number have a new related definition with the fluid density behavior. The fluid geometry is presented graphically in an inclined form as shown in Fig. 1.
Here, the wave (b) amplitude, the wave ( ) length, the inclination (γ ) angle of the channel, and the wave (c) speed. Now, the coordinates and velocities in the laboratory (X,Y ) frame and the wave frame (x,y) are linked by The fluid density is proposed as in Ref. 5 .
Here, the parameter of thermal α = − 1 ρ 0 ∂ρ ∂Ť expansion reflects the density β = 1 ρ 0 ∂ρ ∂Č contrast among the flush and the deferred particles. In this paper, αandβ are referred to as the parameters of temperaturedependent and concentration-dependent fluid density.
In a wave frame, the governing system of equations is Refs. 2,3,5 : Here, the Density (ρ) of fluid, the gravity ( g ), electrical (σ ) conductivity, the thermal (k) conductivity parameter, the strength of the magnetic (B 0 ) field, and the Hall (m) parameter. The Jeffrey fluid ( τ ) equation can be Refs. 2,3 .

Numerical treatments for a physical model
In the mathematics field known, a direct adaptive shooting technique (AST) is a numerical/semi-analytical method for solving BVP. The method establishes substantial progress in the nonlinear and numerical distribution over individual shoot techniques. In this algorithm/technique, we can attain the closest guess or solution to the systems of differential equations with highly nonlinear terms. In this paper, we let Consequently, the Eqs. (18), (19) and (20) with boundary conditions (21) and (22) will be transformed into the recurrence relation as follows:

Results and discussions
This section is subdivided into two subsections, the first of them is to approve the validity of proposed results and the second subsection is to present a sketch of physical pertinent parameters of interest against the fluid distributions. Note that the standard values of physical parameters q → −0.5, → 0.5, M → .5,

Validation of results.
The results of the proposed model by Eqs. (18), (19) and (20)  Discussions and analysis of results. Through this section, sketches of pressure dp dx gradient, velocity u y , temperature θ y , and concentration ϕ y profiles are obtained/discussed. All graphs are offered against pertinent physical parameters of interest for four and five different values. Eventually, sketches were obtained at the fact of variable non-dimensional parameters (Reynolds ( R e ) number, Soret (S r ) number, Schmidt ( S c ) number, and Prandtl ( P r ) number). Pressure dp dx gradient distribution is obtained versus values of non-constant parameters (α, β) of density, Jeffrey ( ) parameter, Froude (F r ) number, and inclination (γ ) angel channel through Figs. 3, 4, 5 and 6. At the edges of the channel, no sight effects are observed in Fig. 3 at high values of α and β on the pressure gradient distribution. Further, the different values of α and β cause a diminishment in the dp dx distribution. The values of non-constant fluid density ( α = 0.2 and β = 0.2 ) combined with the high values of Jeffrey ( ) parameter get more sight effects on the fluid pressure through Fig. 4. Consequently, impoverishment in pressure dp dx gradient (21)  Fig. 5 that the dp dx is considered as a decreasing function in a Froude (F r ) number, i.e. at the edges and the core parts of the channel the fluid pressure gradient is declined at high values of F r . In a physical cause, a diminishing/shrinking in the fluid dp dx is alike to decreasing in the fluid potential energy, which indicates a development in the kinetic energy. Furthermore, the inclination (γ ) angle of channel values is visualized versus dp dx through Fig. 6, growing on the pressure gradient behavior is noted through all parts of the channel, like puffins in a fluid at the core and edges of fluid movements.
• Pressure gradient has two opposite behavior at different values of Froude (Fr) number and inclination (γ ) angle of the channel along the channel wall. • AST is assured as one powerful technique for solving highly nonlinear systems equations.
• Soret ( S r ) number and Jeffrey ( ) parameter have an opposite behavior on the fluid concentration.
• Hartmann (M) number has more view impact on the fluid velocity in the case of non-constant fluid density.
• The supposition of variable fluid density is approved to be better for fluid distributions.

Data availability
The datasets generated and/or analyzed during the current study are not publicly available due [All the required data are only with the corresponding author] but are available from the corresponding author on reasonable request.