Analytical approach to parabolic flows along with inclined plate with uniform diffusion of mass

1. Introduction

Heat and mass flow often occur simultaneously in many engineering applications. An often-observed phenomenon is the process of lake water evaporating into the wind streaming across it. Water vapor diffuses into the air due to the partial pressure difference between the water’s surface and atmosphere. As the surface water evaporates, it absorbs heat from the remaining water, lowering the temperature of the lake. This cooling effect leads to convection, through which air is heated and then transferred to the water. As the lake water temperature drops, the saturation partial pressure of the vapor at the water’s surface also decreases, further slowing the mass transfer rate. Equilibrium is reached when heat flowing from the air to the liquid is sufficient for providing the latent heat required for evaporation, allowing liquid to disperse into the air.

Significant attention has been given to coupled mass and heat transfer issues due to their importance in various processes. Combined buoyancy mechanisms are involved in many applications, such as the curing of plastics and the production of bulb-insulated cables. For instance, if a colored substance like copper sulfate is placed at the bottom of a bottle filled with water and left undisturbed, the color will slowly spread from the bottom to the top, eventually resulting in uniformly colored water. This process, known as diffusion, describes the transfer of the colored substance from regions of higher concentration to regions of lower concentration. The flow phenomenon is characterized by various parameters, including velocity, concentration, and temperature distributions, Prandtl number, Schmidt number, time progression, and angle of inclination. Visual representations of the concentration, temperature, and velocity profiles have also been provided.

Impulsively started vertical plate flow, free convection currents, and the consequences of mass transfer were studied (Soundelgekar, 1979). Soundelgekar et al. (1979) further evaluated a mass transfer that started abruptly on a boundless perpendicular plate with varying temperatures. Das et al. (1994) impulsively started implications of mass transfer over a vertical infinite plate with a continuous mass flux in the flow. Xiao-Jun Yang (2017) analyzed a novel integral transform operator designed to address the heat-diffusion problem. This operator provides a new method for addressing heat-diffusion equations, contributing to more effective solutions in thermal analysis. Das et al. (1991) obtained an accurate solution for the dynamics of a viscous, incompressible fluid flowing past an impulsively initiated, infinitely thick vertical plate, considering uniform mass diffusion and a first-order chemical reaction. Meanwhile, scientists Xiao-Jun Yang et al. (2023) explored an odd entire-function solution to the one-dimensional diffusion equation within the context of the modular form theory. A numerical study by Muthucumaraswamy et al. (2016) investigated how flow through an abruptly commencing mass diffusion in a vertical plate might be affected by a first-order chemical reaction. Raptis et al. (1981) initiated the movement of a hydromagnetic convection past a vertical, accelerating plate with varying suction and flux. Mathematical methods were employed to solve the governing equations. Scientists also (Xiao-Jun Yang et al., 2023) examined special solutions for the Laplace and diffusion equations within the framework of algebraic number fields. The effects of continuous laminar free convection airflow with thermal radiation and chemical species concentration across an infinite vertical plate were studied (Chamkha et al., 2001). Sivakumar et al. (2014) examined radiative heat transfer’s impacts across an endless isothermal vertical plate. Xiao-Jun Yang et al. (2023) gave solutions for the diffusion equation in relation to L-functions associated with cusp forms. A further analysis, (Muthucumaraswamy et al., 2016) studied magnetohydrodynamics flow through a parabolic and an unbound plate in the existence of heat radiation. Scientists (Rajput et al., 2016) investigated the unsteady MHD flow past an impulsively inclined plate, considering the effects of variable temperature, mass diffusion, and Hall currents. Thenmozhi et al. (2024) examined the MHD Darcy-Forchheimer flow of a micropolar fluid using a predictor-corrector finite difference method, incorporating the effects of viscous dissipation and heterogeneous-homogeneous reactions. Xiao-Jun Yang (2024) examined a new approach to entire functions in the field of number theory and a further analysis, explored new conjectures concerning entire functions linked to fractional calculus. Siva Sankari et al. (2023) and Thenmozhi et al. (2024) analyzed the double stratification of Casson nanofluid flow over an exponentially stretching sheet through analytical methods.

2. Formulation of the problem

The vertical axis is taken in an upward trajectory, and the y-axis is oriented perpendicular to the plate, while the x-axis runs along the plate at an angle of inclination. At time $t^{\text{'}}\le 0,$ both the fluid and the plate are at a uniform temperature. ${\theta }_{\infty }.$ At time t’> 0, the plate is started with a velocity of $v=v_0.{t^{\prime }}^2$ against the gravitational field in its own plane, and the ambient temperature of the plate is increased to ${\theta }_w$ and the concentration levels adjacent to the plate are also elevated. to $C_w^{*}$. Following Boussinesq’s approximation, there are three equations that regulate the unstable motion: the momentum equation, the mass diffusion equation along with chemical reactions and the energy equation that includes radiation.

These are the starting and limiting conditions:

$v=0,\theta ={\theta }_{\infty },C*=C_{\infty }^{*}\text{for\ all}Y,t\le 0.$

$v\to 0,\theta \to {\theta }_{\infty },C*\to C_{\infty }^{*}\text{as}y\to \infty .$

To express equations (1), (2), and (3) in non-dimensional form, we will now introduce the non-dimensional variables.

These Non-dimensional form parameters are missing in Equation (5).

$V=v{\left(\frac{v_0}{{\gamma }^2}\right)}^{\frac{1}{3}},t={\left(\frac{v_0^2}{\gamma }\right)}^{\frac{1}{3}}t\text{'},Y=y{\left(\frac{v_0}{{\gamma }^2}\right)}^{\frac{1}{3}},\mathrm{Pr}=\frac{\mu C_p}{k},T=\frac{\theta -{\theta }_{\infty }}{{\theta }_w-{\theta }_{\infty }},$

When we insert equation (5) into equations (1), (2), and (3), the governing equations take on a non-dimensional form.

The preliminary and boundary circumstances based on equation (4), it is now expressed as

$V=0,T=0,\phi =0\text{for\ every}Y,t\le 0.$

$V\to 0,T\to 0,\phi \to 0\text{as}Y\to \infty .$

3. Solution of the problem

The dimensionless fundamental equations (6) to (8), along with the equation (9), are solved. The resulting solution is presented below.

$\begin{array}{c}T=L_2\\ \phi =L_3\\ V=2\left(\frac{t^2}{6}\left[L_1\left(L_{05}\right)-L_4\left(L_{00}\right)\left(L_{04}\right)\right]\right)-\left(a+b\right)\left[t\left[L_1\left(L_{01}\right)-2L_4\left(L_{00}\right)\right]\right]\\ +a\left(t\left[L_2\left(L_{02}\right)-2L_5\left(L_{00}\right)\sqrt{\mathrm{Pr}}\right]\right)+b\left(t\left[L_3\left(L_{03}\right)-2L_6\left(L_{00}\right)\sqrt{Sc}\right]\right)\end{array}$

Where, $a=-\frac{GrCos\alpha }{(\mathrm{Pr}-1)},b=-\frac{GcCos\alpha }{(Sc-1)}$ and $\eta =\frac{Y}{2\sqrt{t}}$.

4. Results and analysis of the problem

Fig. 1 shows the physical model and the coordinate system used to describe the problem. Fig. 2(a) and 2(b) velocity profiles are plotted respectively. It is very esvident that when Pr falls in the boundary layer and raises the velocity profile. In Fig. 3(a) and 3(b) represents the impact of velocity with respect to time. Increases in speed are consistently measured throughout time. The velocity curves for various plate angles are represented in Figs. 4(a) and 4(b). It has been seen that when the plate angle is raised, the velocity decreases. Fig. 5(a) and 5(b) velocity profiles are plotted respectively. It is very evident that when Sc falls in the boundary layer and raises the velocity profile. Fig. 6(a), 6(b), 7(a), and 7(b) present several values of the Gr and Gc. This is clear that the velocity magnifies when the Gr or Gc increases because of buoyancy force. Fig. 8(a) and 8(b) displays the result of Sc on the concentration profiles respectively. It is investigational that the concentration curves diminish as the (Sc) increases. The fluid’s temperature for Pr is proffered at Fig. 9(a) and 9(b), this is clearly demonstrating that temperature profile decreased as increase Pr.

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Fig. 1.

Governing system and physical model.

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Fig. 2.

(a) Shows a comparison of velocity for different Pr values at t = 0.4, (b) Shows a comparison of velocity for different Pr values at t = 0.6.

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Fig. 3.

(a) Shows a comparison of velocity for different t values, (b) Shows a comparison of velocity for different t values.

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Fig. 4.

(a) Presents a comparison of velocity for various α values, (b) Shows a comparison of velocity for different α values.

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Fig. 5.

(a) Shows a comparison of velocity for different Sc values at t = 0.6, (b) Shows a comparison of velocity for different Sc values at t = 0.2.

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Fig. 6.

(a) Displays a comparison of velocity for different Gr values at t = 0.2, (b) Shows a comparison of velocity for different Gr values at t = 0.6.

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Fig. 7.

(a) Shows a comparison of velocity for different Gc values at t = 0.4, (b) Shows a comparison of velocity for different Gc values at t = 0.6.

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Fig. 8.

(a) Shows a comparison of concentration for different Sc values, (b) Shows a comparison of concentration for different Sc values.

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Fig. 9.

(a) Shows a comparison of temperature for different Pr values, (b) Shows a comparison of temperature for different Pr values.

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Table 1 presents Pr’s velocity changes over time, with data shown in Fig. 2(a) and 2(b). Table 2 shows the velocity of t for different time values, linked to Fig. 3(a) and 3(b). Table 3 provides a comparison of α’s velocity over time, supported by Fig. 4(a) and 4(b). Table 4 displays how Sc’s velocity varies, with Fig. 5(a) and 5(b) illustrating the results. Tables 5 and 6 compare the velocities of Gr and Gc over time, using Figs. 6(a), 6(b), 7(a), and 7(b) for reference. Table 7 focuses on concentration variations due to Sc, supported by Fig. 8(a) and 8(b). Lastly, Table 8 highlights temperature changes based on different Pr values, shown in Fig. 9(a) and 9(b).

5. Conclusion

The intention of the complete review in closed form is to investigate the effect of an inclined plate with uniform mass diffusion, and the solutions are obtained through the Laplace procedure. The consequences of the problem are presented below.

• The velocity improves as well as a rise in (Gr), (Gc), and time (t). It exhibits the opposite effect when (Pr) and (Sc) increases.

• Temperature profile diminishes with an increase of (Pr).

• As the (Sc) increases, the concentration profile diminishes.