found. In addition, we conducted a comprehensive parametric study to analyze the trends in the solutions obtained.

The study reveals that the parameters

have about 20% stronger impact on the nonlinear

Chebyshev Collocation Approach

cooling is an example of heat transfer fluid. Heat

transfer fluid has a lot of industrial applications and

are distinct ranging from the less complex static

design of advanced multi-loop systems. The process

by which heat is transferred to a moving fluid via a

heated surface is known as convective heat transfer

which can be viewed in two ways. The first is the

internal free convection at low flow rates. However,

when external devices such as stirrers or pump

ensure the flow rate, forced convection is

predominant. As described by, [1], combined

convection exists if the impact of buoyancy in non-

free convection and the impact of forced flow in

free convection are prevalent. They play a

significant role in the atmospheric boundary layer

flow, solar collectors, electronic equipment, and

nuclear reactors.

Jefferey fluid with a moving surface along with the

impact of temperature and velocity was considered,

[5]. In their work, [6], carried out a study on

generalized Mittag-Leffler via the analytical

solution of natural convection flow of Prabhakal

fractional Maxwell fluid with Newton heating. The

impact of various fluid behaviors of combined

convective heat transport with a nonlinear moving

sheet was explored by, [7]. The influence of Dufour

and Soret on convective viscoelastic fluid flow over

nanofluid along a non-linear radiation numerically.

A review of thermal boundary conditions of

Magnetohydrodynamics (MHD) free convective

nanofluid in a square enclosure is reported by, [11].

Various other works on convective heat transfer can

be found in, [12], [13], [14], [15], [16], [17], and

related literature.

The numerous advantages of convective

problems of electrically conducting fluid on a

magnetic field have attracted significant attention

considered water-based nanofluid MHD squeezing

flow between parallel disks through a saturated

medium. Ion slip influence on unsteady

magnetohydrodynamics rotating convective flow

has been explored by, [20]. [21], examined

magnetohydrodynamics micropolar fluid flow over

a curved surface. The work of MHD flow with

convection over a vertically oscillating porous wall

with constant heat flux is explored by, [22]. With

exponential permeable stretching porous surface,

slip-flow, researchers in their numbers have

developed an interest in it. In this period of modern

industrialization, technology, and science, the

significance of the slip-flow regime can never be

over-emphasized. In practical terms, the velocity of

the particle to the boundary no longer has the same

values as the surface. The surface particle takes a

definite velocity and slips on the plate. This kind of

flow is known as slip-flow regime and its impact

cannot be neglected. The phenomenon of this flow

introduces a novel aspect of incorporating nonlinear

mixed convection into the model proposed by [39].

The model assumes a first-order slip, with

temperature and concentration jumps. To solve the

resulting model, the Chebyshev Collocation

Approach (CCA) will be employed to obtain the

numerical approximations. The combination of a

nonlinear mixed convection, the varying conditions,

and concentration represents a new contribution to

the existing literature.

Considering 2D unsteady, laminar boundary layer

fluid flow in a horizontal rectangular wall with the

horizontal wall at

and at a variable distance

. The first wall is assumed slippery and

stretches with velocity

, temperature

of

,

and

, at

, (1)

following similarity solutions will be used to

non-linear ODEs:

The critical quantities are the coefficient of skin

friction, Nusselt number and Sherwood number

obtained as:

The reduced nonlinear coupled ODEs corresponding

residue. Thereby, utilizing a collocation method to

generate the error to almost zero. The accuracy of

the Colocation technique is found to be high and no

ambiguity to implement, [41], [42], [43], [44].

The functions to be determined

,

and

order to obtain the values of the unknown constants,

the Chebyshev base functions are substituted into

the boundary conditions in (10) corresponding to:

(11)

Newton’s method is employed to obtain the

solution of the system and Mathematical symbolic

package (MATHEMATICA 11.3) is used to carry

out all the computations.

The solution to the highly nonlinear coupled

expressions in (7-9) with their conditions in (10) are

numerically approximated with the Chebyshev

than conventional approaches. To check the

accuracy of this method, a comparative analysis is

carried out with the literature. An excellent

agreement is observed as presented in Table 2.

Figure 2a-2c describe the features of

magnetization on

respectively for both

linear

momentum. Fluid energy

and concentration

are escalated in Figure 2b and Figure 2c

respectively. The fluid velocities drop faster for

linear convection than nonlinear convection and the

temperature and concentration enhance faster with

linear convection. This phenomenon implies that the

Lorentz force has a greater impact on the linear

(c)

Fig. 2: The profiles of

values of

The variation of the mixed convection parameter

it is examined that

is increasing due to the

presence of the buoyancy effect while

decay for larger values of

has

a greater influence on

which signifies that the magnitude of buoyancy

forces is more prevalent in the nonlinear case. The

fluid horizontal velocity has the tendencies of

(c)

Fig. 3: The Profiles of

value of

respectively. The fluid velocity appreciates due to

convection

has a significant effect on

Additionally,

has a greater impact on the fluid

variables to

value of

The significance of Pr on

characterized in Figure 5 (a and b). The usefulness

of Prandtl number in industrial activities to attain

in industrial heating/cooling. It is explored here that

larger values of Prandtl number cause

decrease of the thermal boundary layer and solutal

concentration. Thus, Pr causes the fluid and solutal

concentration to drop quicker in the nonlinear sense.

Figure 6 (a and b) portrays the rheological

illustration of the profiles of

prevalent nature of the Brownian diffusion presence.

The weak coefficient of Brownian diffusion

signifies large values of Schmidt number which in

turns lower the temperature profiles and solutal

concentration. The Brownian diffusion’s less

prevalence in the presence of non-linear convection

resulted in the fluid energy and concentration to

drop faster compared to linear case.

(a)

Fig. 7: The profiles of