Exact solutions of non-singularized MHD Casson fluid with ramped conditions: A comparative study

Abstract

The study of magnetohydrodynamic (MHD) fluids has significant implications across various scientific disciplines, particularly in understanding complex phenomena in astrophysics, engineering, and geophysics. The Casson fluid model stands as a crucial tool for describing non-Newtonian behaviors observed in certain fluid systems. The current work shows an analytical analysis to determine the ramped effect on the fractionalized MHD Casson fluid over an vertical plate. Fractional partial differential equations are used to formulate the problem along with initial and boundary conditions. The governing equations are transformed into the dimensionless form and developed fractional models like Caputo-Fabrizio and Atangana-Baleanu Derivative. We used the Laplace transform technique to find the closed form solution of the dimensionless governing equation analytically. MATHCAD software is being used for numerical computations and the physical attributes of material and fractional parameters are discussed. To analyze their behavior clearly, two-dimensional graphical results are plotted for velocity profile and temperature as well. It has been concluded that the fluid’s velocity are reduced for larger values of the fractional parameter and Prandtl number and is maximum for small values of both parameters.

Introduction

Heat and mass transfer processes hold immense significance from an industrial perspective, captivating the attention of numerous researchers and scientists. In the realm of modern technologies and diverse industrial applications, the theory of non-Newtonian fluids exerts a profound influence due to the limitations of Newtonian fluid models in capturing a wide range of flow characteristics. Non-Newtonian fluids exhibit complex relationships involving shear strain rate and stress, transcending the simplistic assumptions of Newtonian fluid models. The theory of non-Newtonian fluids finds significant application in contemporary engineering, namely, within the petroleum sector, where it plays a crucial role in extracting crude oil from various petroleum reservoirs. Researchers have investigated a diverse range of non-Newtonian fluid models, each with its own unique computational properties. For instance, while the power-law model effectively captures viscosity characteristics, it fails to account for the effects of elasticity. This motivates researchers and mathematicians to delve deeper into the study of these complex fluids. For theoretical research as well as real world applications in contemporary engineering, a methodical examination of these fluid flow models is crucial. Due to this reason, non-Newtonian fluids give birth to abundant rheological mathematical models of fluids. Such fluids models are classified by us in terms of century wise: 18th century from 1867 to 1893 (Barus and Maxwell model) and 19th century from 1922 to 1995 (Blatter model, Ellis model, Giesekus model, Phan-Thien-Tanner model, Johnson-Tevaarwerk model, Carreau-Yasuda model, Carreau model, Cross model, Rivlin-Ericksen model, Oldroyd-8 constants model, Oldroyd-B model, Rivlin model, Generalized Burgers, Eyring and Williamson fluid model) and few others. Amongst these fluid models that is most accurate and treated fluid model in bio-field so called Casson fluid model (1959). The main significance of this model is to characterize yield stress pseudoplastic properties. The common useful examples of Casson model lie as concentrated fruit juices, jelly, tomato sauce, honey, and few others.^1–5 Asifa et al. ⁶ investigate the radiative MHD Casson fluid flow with heat source through a porous medium with generalized boundary conditions. Unsteady MHD natural convection flow of Casson fluid incorporating heat injection and suction mechanism under variable wall conditions studied by Anwar et al. ⁷ in existence of thermal radiative flux. Hussain et al. ⁸ investigates the mixed convection boundary-layer flow of Casson fluid with an internal heat source on an exponentially stretched sheet. The Buongiorno model, incorporating thermophoresis and Brownian motion, describes fluid temperature. Saeed et al. ⁹ analyzed the exact symmetric solutions of MHD Casson fluid using chemically reactive flow with generalized boundary conditions. Casson fluid over a persuaded steady spinning plane based on the three-dimensional nanofluid thin-film flow has been examined by Saeed et al. ¹⁰ They imposed homotopy analysis method on the governing equation for the numerical solutions of velocity, temperature, and concentration gradient.

Fractional calculus is the generalization classical calculus. In fractional calculus the derivative of real order are consider instead of integer order derivative and have been applied efficiently to explain the several real life problem. Several investigation are reported regarding nanofluid and ordinary fluid flows with different definitions of fractional derivatives. Imran et al. ¹¹ presented a comparative study of viscous fluid flow with two definitions of fractional derivatives. Sheikh et al. ¹² carried out the results for free convection flow of Casson fluid with Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. Rehman et al. ¹³ investigated the fractional models of generalized Casson fluid with oscillatory flow. Ahmad et al., ¹⁴ Sheikh et al., ¹⁵ Sadia et al., ¹⁶ and Sehra et al. ¹⁷ have successfully apply the fractional derivative approaches to different real life physical phenomenons. Qayyum et al. ¹⁸ developed the fractional modeling of Casson fluid between two parallel plates. Furthermore, author ¹⁹ investigated the fractional modeling and analysis of unsteady squeezing flow of Casson nanofluid via extended He-Laplace algorithm in Liouville-Caputo sense. Rehman et al. ²⁰ developed the application of novel hybrid fractional derivative operator on Casson fluid. Some more interesting studies with application of fractional derivatives are investigated in References 21–30.

The focal attention of this research is to extend the work ¹³ for ramped conditions on temperature and concentration and apply the Caputo-Fabrizio ¹¹ and Atangana-Baleanu ¹² fractional operator and also compare the result of both operators. Laplace transform and numerical inverse algorithms are utilized to acquired the solution of velocity, temperature, and concentration with ramped conditions.

Model of flow problem

In this problem, the fluid flow of Casson model along with heat and mass transfer over an infinite vertical plate has been considered. The direction of the fluid’s flow is x-axis, while y-axis represent the normal to the plate. A uniform magnetic field of strength $B_o$ is assumed to be applied along y-axis. The temperature and concentration levels of the plate increased linearly to $T_w$ and $C_w$ with time $t$ . The flow representation of generalized Casson fluid using geometry analyzed in Figure 1. The governing equation for generalized Casson fluid model with appropriate conditions is defined below ¹³ :

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial u(y,t)}{\partial t}=\nu (1+\frac{1}{\beta })\frac{{\partial }^{2}u(y,t)}{\partial y^2}+g{\beta }_C(C-C_{\infty })}\\ \multicolumn{1}{c}{+g{\beta }_T(T-T_{\infty })-\nu (1+\frac{1}{\beta })\frac{\zeta }{k_p}u(y,t)}\\ \multicolumn{1}{c}{-\frac{\sigma B_o^2}{\rho }u(y,t),}\end{array}$$

(1)

$$\rho C_p\frac{\partial T(y,t)}{\partial t}=k\frac{{\partial }^{2}T(y,t)}{\partial y^2}-\frac{\partial Q_r}{\partial y}-Q_o(T-T_{\infty }),$$

(2)

$$\frac{\partial C(y,t)}{\partial t}=D_m\frac{{\partial }^{2}C(y,t)}{\partial y^2}-K_c(C-C_{\infty }).$$

(3)

Figure 1.

Geometrical presentation for Casson Model.

The appropriate conditions are given below:

$$\begin{array}{c}u(y,0)=0,T(y,0)=T_{\infty },C(y,0)=C_{\infty },y\ge 0,\hfill \\ T(0,t)=\{\begin{array}{cc}{T}_{\infty }+(T_w-T_{\infty })\frac{t}{t_0},\hfill & 0<t\le t_0;\hfill \\ \multicolumn{1}{c}{T_w,}& t>t_0\hfill \end{array},\end{array}$$

(4)

$$u(0,t)=\{\begin{array}{cc}{U}_0\frac{t}{t_0},\hfill & 0<t\le t_0;\hfill \\ \multicolumn{1}{c}{U_0,}& t>t_0\hfill \end{array},$$

(5)

$$C(0,t)=\{\begin{array}{cc}{C}_{\infty }+(C_w-C_{\infty })\frac{t}{t_0},\hfill & 0<t\le t_0;\hfill \\ \multicolumn{1}{c}{C_w,}& t>t_0\hfill \end{array},$$

(6)

$$u(y,t)\to 0,T(y,t)\to T_{\infty },C(y,t)\to C_{\infty },y\to \infty .$$

(7)

By inserting the dimensionless variables:

$$\mathbb{R}=\frac{u}{U_0},\mathbb{Y}=\frac{y{U}_0}{\upsilon },\mathbb{T}=\frac{{\mathrm{tU}}_0^2}{\upsilon },t_0=\frac{\upsilon }{U_0^2},\theta =\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi =\frac{C-C_{\infty }}{C_w-C_{\infty }},\mathbb{M}=\frac{\sigma B_o^2\mu }{\rho U_o^2},$$

(8)

$${\mathbb{G}}_r=\frac{g{\beta }_T(T_w-T_{\infty })}{U_0^3},{\mathbb{G}}_m=\frac{g{\beta }_C(C_w-C_{\infty })}{U_0^3},{\mathbb{P}}_r=\frac{\mu C_p}{k},{\mathbb{S}}_c=\frac{\mu }{D_m},\frac{1}{\mathbb{K}}=\frac{{\nu }^2\zeta }{k_p{U}_o^2},\mathbb{Q}=\frac{Q_o\nu }{\rho C_p{U}_o^2}.$$

(9)

We have required set of dimensionless governing equations and corresponding conditions are given below:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathbb{R}(\mathbb{Y},\mathbb{T})}{\partial \mathbb{T}}=\mathbb{B}\frac{{\partial }^2\mathbb{R}(\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}+{\mathbb{G}}_r\theta (\mathbb{Y},\mathbb{T})}\\ \multicolumn{1}{c}{+{\mathbb{G}}_m\varphi (\mathbb{Y},\mathbb{T})-{\mathbb{M}}_1\mathbb{R}(\mathbb{Y},\mathbb{T}),}\end{array}$$

(10)

$$\frac{\partial \theta (\mathbb{Y},\mathbb{T})}{\partial \mathbb{T}}=\frac{1}{P_{\mathrm{reff}}}\frac{{\partial }^2\theta (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-\mathbb{Q}\theta (\mathbb{Y},\mathbb{T}),$$

(11)

$${\mathbb{S}}_c\frac{\partial \varphi (\mathbb{Y},\mathbb{T})}{\partial \mathbb{T}}=\frac{{\partial }^2\varphi (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-{\mathbb{S}}_c{\mathbb{K}}_r\varphi (\mathbb{Y},\mathbb{T}).$$

(12)

At $\mathbb{T}=0$ ,

$$\mathbb{R}(\mathbb{Y},\mathbb{T})=0,\theta (\mathbb{Y},\mathbb{T})=0,\varphi (\mathbb{Y},\mathbb{T})=0,\mathrm{for}\mathbb{Y}\ge 0.$$

(13)

At $\mathbb{Y}=0$ ,

$$\varphi (\mathbb{Y},\mathbb{T})=\theta (\mathbb{Y},\mathbb{T})=\mathbb{R}(\mathbb{Y},\mathbb{T})\{\begin{array}{cc}\mathbb{T}\hfill & 0<\mathbb{T}\le {\mathbb{T}}_0;\hfill \\ \multicolumn{1}{c}{1}& \mathbb{T}>{\mathbb{T}}_0\hfill \end{array}.$$

(14)

At $\mathbb{Y}=\infty$ ,

$$\mathbb{R}(\mathbb{Y},\mathbb{T})=0,\theta (\mathbb{Y},\mathbb{T})=0,\varphi (\mathbb{Y},\mathbb{T})=0,$$

(15)

where $\mathbb{B}=1+\frac{1}{\beta }$ and ${\mathbb{M}}_1=\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}}$ .

Solution of fractional model with Caputo-Fabrizio (CF) differential operator

Concentration profile

Fractional operators offer remarkable flexibility in depicting the dynamics of mass transfer in Casson fluids through the formulation of governing equations. By employing the CF-fractional operator (17), facilitates the transformation of the concentration governing equation (12) into a fractionalized form, taking $\mathfrak{u}$ as a fractional order:

$${{}^{\mathrm{CF}}D}_t^{\mathfrak{u}}\varphi (\mathbb{Y},\mathbb{T})=\frac{1}{{\mathbb{S}}_c}\frac{{\partial }^2\varphi (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-{\mathbb{K}}_r\varphi (\mathbb{Y},\mathbb{T}),$$

(16)

where, ${{}^{\mathrm{CF}}D}_t^{\mathfrak{u}}$ is known as CF differential operator subject to the normalization function $M(\mathfrak{u})=M(1)=M(0)=1$ defined as:

$${{}^{\mathrm{CF}}D}_t^{\mathfrak{u}}f(y,t)=\frac{1}{1-\mathfrak{A}}{\int }_0^t\exp (-\frac{\mathfrak{u}(t-\tau )}{1-\mathfrak{u}})f^{/}(\tau )d\tau ,0<\mathfrak{A}<1.$$

(17)

Using the Laplace transform method, we solve equation (16). This requires utilizing a fundamental property of the CF operator as defined in equation (18).

$$\mathcal{L}({{}^{\mathrm{CF}}D}_t^{\mathfrak{A}}f(y,t))=\frac{\mathcal{J}\mathcal{L}(f(y,t))-f(y,0)}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}.$$

(18)

By utilizing the Laplace transform on equation (16) and resolving it with the assistance of equation (18), we investigated a second-order partial differential equation (PDE).

$${\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)\overline{\varphi }(\mathbb{Y},\mathcal{J})=\frac{{\partial }^2\overline{\varphi }(\mathbb{Y},\mathcal{J})}{\partial {\mathbb{Y}}^2},$$

(19)

invoking the elementary concept of differential equation for homogeneity, we get simplified expression as

$$\overline{\varphi }(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}}+c_2{e}^{\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}},$$

(20)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (20). We arrived at

$$\overline{\varphi }(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}}.$$

(21)

The convenient form of above function to implement the Laplace inversion is

$$\overline{\varphi }(\mathbb{Y},\mathcal{J})={\varphi }_1(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}{\varphi }_1(\mathbb{Y},\mathcal{J}).$$

(22)

The exact solution can be written as

$$\varphi (\mathbb{Y},\mathbb{T})={\varphi }_1(\mathbb{Y},\mathbb{T})-{\varphi }_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1).$$

(23)

In the above expression, $\mathbb{U}(\mathbb{T}-1)$ represents a Heaviside function, and

$$\varphi (\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}(\overline{\varphi }(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}}\right\}.$$

(24)

The current form of equation (24) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}{({\mathbb{S}}_c)}^{\frac{a_1}{2}}{({\mathbb{K}}_r)}^{\frac{a_1}{2}-a_2}\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{a_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(a_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(m-1)!}{a_3!(m-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(25)

Temperature profile

Fractional operators offer remarkable flexibility in depicting the dynamics of heat transfer in Casson fluids through the formulation of governing equations. By employing the CF-fractional operator (17), facilitates the transformation of the concentration governing equation (11) into a fractionalized form, taking $\mathfrak{A}$ as a fractional order:

$${{}^{\mathrm{CF}}D}_t^{\mathfrak{A}}\theta (\mathbb{Y},\mathbb{T})=\frac{1}{{\mathbb{P}}_{\mathrm{reff}}}\frac{{\partial }^2\theta (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-\mathbb{Q}\theta (\mathbb{Y},\mathbb{T}).$$

(26)

By utilizing the Laplace transform on equation (26) and resolving it with the assistance of equation (18), we investigated a second-order partial differential equation (PDE).

$${\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})\overline{\theta }(\mathbb{Y},\mathcal{J})=\frac{{\partial }^2\overline{\theta }(\mathbb{Y},\mathcal{J})}{\partial {\mathbb{Y}}^2},$$

(27)

invoking the elementary concept of differential equation for homogeneity, we get simplified expression as

$$\overline{\theta }(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}}+c_2{e}^{\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}},$$

(28)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (28). We arrived at

$$\overline{\theta }(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}}.$$

(29)

The convenient form of above function to implement the Laplace inversion is

$$\overline{\theta }(\mathbb{Y},\mathcal{J})={\theta }_1(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}{\theta }_1(\mathbb{Y},\mathcal{J}).$$

(30)

The exact solution can be written as

$$\theta (\mathbb{Y},\mathbb{T})={\theta }_1(\mathbb{Y},\mathbb{T})-{\theta }_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1).$$

(31)

In the above expression, $(\mathbb{T}-1)$ represents a Heaviside function, and

$$\theta (\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}(\overline{\theta }(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}}\right\}.$$

(32)

The current form of equation (32) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\theta }_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}{({\mathbb{P}}_{\mathrm{reff}})}^{\frac{a_1}{2}}{(\mathbb{Q})}^{\frac{a_1}{2}-a_2}\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{a_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(a_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(m-1)!}{a_3!(m-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(33)

Velocity profile

We transform the governing equation of velocity (10) into a fractionalized form. Solving this uncoupled fractionalized equation of velocity (10) is accomplished through the Laplace transform method, yielding the desired outcome.

$$\begin{array}{c}\multicolumn{1}{c}{{{}^{\mathrm{CF}}D}_t^{\mathfrak{A}}\mathbb{R}(\mathbb{Y},\mathbb{T})=\mathbb{B}\frac{{\partial }^2\mathbb{R}(\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}+{\mathbb{G}}_r\theta (\mathbb{Y},\mathbb{T})+{\mathbb{G}}_m\varphi (\mathbb{Y},\mathbb{T})}\\ \multicolumn{1}{c}{-(\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})\mathbb{R}(\mathbb{Y},\mathbb{T}),}\end{array}$$

(34)

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mathbb{R}}(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{+c_2{e}^{\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_r(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}})}{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}}}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_m(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}})}{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}},}\end{array}$$

(35)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (35). We arrived at

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mathbb{R}}(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{+\frac{{\mathbb{G}}_r(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}})}}\\ \multicolumn{1}{c}{[e^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{Q})}}-e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}]}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_m(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}})}}\\ \multicolumn{1}{c}{[e^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+{\mathbb{K}}_r)}}-e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}].}\end{array}$$

(36)

The convenient form of above function to implement the Laplace inversion is

$$\overline{\mathbb{R}}(\mathbb{Y},\mathcal{J})=\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})+{\mathbb{G}}_r\overline{\mathrm{\Psi }}(\mathbb{Y},\mathcal{J})[\overline{\theta }(\mathbb{Y},\mathcal{J})-\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})]+{\mathbb{G}}_m\overline{\mathrm{\Psi }}(\mathbb{Y},\mathcal{J})[{\overline{\varphi }}_1(\mathbb{Y},\mathcal{J})-\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})],$$

(37)

and

$$\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})=\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J}).$$

(38)

The inverse Laplace of above equation is

$$\mathrm{\Phi }(\mathbb{Y},\mathbb{T})={\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})-{\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1),$$

(39)

where

$${\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}({\overline{\mathrm{\Phi }}}_1(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}\right\}.$$

(40)

The current form of equation (40) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}({\mathbb{M}}_1^{\frac{a_1}{2}-n})\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!(\mathbb{B}{)}^{\frac{a_1}{2}}\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{a_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(a_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(m-1)!}{a_3!(m-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(41)

Similarly,

$${\mathrm{\Psi }}_1(\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}({\overline{\mathrm{\Psi }}}_1(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{\mathbb{A}+\frac{\mathbb{D}\mathcal{J}}{(1-\mathfrak{A})\mathcal{J}+\mathfrak{A}}}\right\}.$$

(42)

We get

$${\mathrm{\Psi }}_1(\mathbb{Y},\mathbb{T})=\sum _{m=0}^{\infty }\frac{{(-1)}^m{\mathbb{D}}^m}{{\mathbb{A}}^{m+1}}\{\frac{{\mathbb{T}}^{m-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(m-1)!}\times \sum _{k=0}^{n-1}\frac{{(-1)}^k(n-1)!}{k!(n-1-k)!}(\mathbb{T}-{(1-\mathfrak{A})}^k)\}.$$

(43)

The required velocity solution for CF operator after employing the definition of inverse Laplace is

$$\mathbb{R}(\mathbb{Y},\mathbb{T})=\mathrm{\Phi }(\mathbb{Y},\mathbb{T})+{\mathbb{G}}_r\mathrm{\Psi }(\mathbb{Y},\mathbb{T})*[\theta (\mathbb{Y},\mathbb{T})-\mathrm{\Phi }(\mathbb{Y},\mathbb{T})]+{\mathbb{G}}_m\mathrm{\Psi }(\mathbb{Y},\mathbb{T})*[\varphi (\mathbb{Y},\mathbb{T})-\mathrm{\Phi }(\mathbb{Y},\mathbb{T})].$$

(44)

Solution of fractional model with Atangana Baleanu (AB) differential operator

Concentration profile

Fractional operators offer remarkable flexibility in depicting the dynamics of mass transfer in Casson fluids through the formulation of governing equations. By employing the AB-fractional operator (45), facilitates the transformation of the concentration governing equation (12) into a fractionalized form, taking $\mathfrak{A}$ as a fractional order:

$${{}^{\mathrm{AB}}D}_t^{\mathfrak{A}}\varphi (\mathbb{Y},\mathbb{T})=\frac{1}{{\mathbb{S}}_c}\frac{{\partial }^2\varphi (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-{\mathbb{K}}_r\varphi (\mathbb{Y},\mathbb{T}),$$

(45)

where, ${{}^{\mathrm{AB}}D}_t^{\mathfrak{A}}$ is known as AB differential operator subject to the normalization function $M(\mathfrak{A})=M(1)=M(0)=1$ defined as:

$${{}^{\mathrm{ABC}}D}_t^{\mathfrak{A}}l(\mathbb{Y},t)=\frac{1}{1-\mathfrak{A}}{\int }_0^t{\mathit{E}}_{\mathfrak{A}}(-\frac{\mathfrak{A}{(t-\tau )}^{\mathfrak{A}}}{1-\mathfrak{A}})l\prime (\tau )d\tau .$$

(46)

Using the Laplace transform method, we solve equation (12). This requires utilizing a fundamental property of the AB operator as defined in equation (47).

$$\mathcal{L}({{}^{\mathrm{ABC}}D}_t^{\mathfrak{A}}l(\mathbb{Y},t))=\frac{s^{\mathfrak{A}}\mathcal{L}(l(\mathbb{Y},t))-s^{\mathfrak{A}-1}l(\mathbb{Y},0)}{(1-\mathfrak{A})s^{\mathfrak{A}}+\mathfrak{A}}.$$

(47)

By utilizing the Laplace transform on equation (45) and resolving it with the assistance of equation (47), we investigated a second-order partial differential equation (PDE).

$${\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)\overline{\varphi }(\mathbb{Y},\mathcal{J})=\frac{{\partial }^2\overline{\varphi }(\mathbb{Y},\mathcal{J})}{\partial {\mathbb{Y}}^2},$$

(48)

invoking the elementary concept of differential equation for homogeneity, we get simplified expression as

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\varphi }(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}}}\\ \multicolumn{1}{c}{+c_2{e}^{\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}},}\end{array}$$

(49)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (49). We arrived at

$$\overline{\varphi }(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}}.$$

(50)

The convenient form of above function to implement the Laplace inversion is

$$\overline{\varphi }(\mathbb{Y},\mathcal{J})={\varphi }_1(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}{\varphi }_1(\mathbb{Y},\mathcal{J}).$$

(51)

The exact solution can be written as

$$\varphi (\mathbb{Y},\mathbb{T})={\varphi }_1(\mathbb{Y},\mathbb{T})-{\varphi }_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1).$$

(52)

In the above expression, $\mathbb{U}(\mathbb{T}-1)$ represents a Heaviside function, and

$$\varphi (\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}(\overline{\varphi }(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}}\right\}.$$

(53)

The current form of equation (53) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\varphi }_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}{({\mathbb{S}}_c)}^{\frac{a_1}{2}}{({\mathbb{K}}_r)}^{\frac{a_1}{2}-a_2}\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{\mathfrak{A}{a}_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(\mathfrak{A}{a}_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(\mathfrak{A}m-1)!}{a_3!(\mathfrak{A}m-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(54)

Temperature profile

Fractional operators offer remarkable flexibility in depicting the dynamics of heat transfer in Casson fluids through the formulation of governing equations. By employing the AB-fractional operator (45), facilitates the transformation of the concentration governing equation (11) into a fractionalized form, taking $\mathfrak{A}$ as a fractional order:

$${{}^{\mathrm{AB}}D}_t^{\mathfrak{A}}\theta (\mathbb{Y},\mathbb{T})=\frac{1}{{\mathbb{P}}_{\mathrm{reff}}}\frac{{\partial }^2\theta (\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}-\mathbb{Q}\theta (\mathbb{Y},\mathbb{T}).$$

(55)

By utilizing the Laplace transform on equation (55) and resolving it with the assistance of equation (47), we investigated a second-order partial differential equation (PDE).

$${\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})\overline{\theta }(\mathbb{Y},\mathcal{J})=\frac{{\partial }^2\overline{\theta }(\mathbb{Y},\mathcal{J})}{\partial {\mathbb{Y}}^2},$$

(56)

invoking the elementary concept of differential equation for homogeneity, we get simplified expression as

$$\overline{\theta }(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}}+c_2{e}^{\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}},$$

(57)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (57). We arrived at

$$\overline{\theta }(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}}.$$

(58)

The convenient form of above function to implement the Laplace inversion is

$$\overline{\theta }(\mathbb{Y},\mathcal{J})={\theta }_1(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}{\theta }_1(\mathbb{Y},\mathcal{J}).$$

(59)

The exact solution can be written as

$$\theta (\mathbb{Y},\mathbb{T})={\theta }_1(\mathbb{Y},\mathbb{T})-{\theta }_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1).$$

(60)

In the above expression, $(\mathbb{T}-1)$ represents a Heaviside function, and

$$\theta (\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}(\overline{\theta }(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}}\right\}.$$

(61)

The current form of equation (32) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\theta }_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}{({\mathbb{P}}_{\mathrm{reff}})}^{\frac{a_1}{2}}{(\mathbb{Q})}^{\frac{a_1}{2}-a_2}\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{\mathfrak{A}{a}_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(\mathfrak{A}{a}_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(\mathfrak{A}m-1)!}{a_3!(\mathfrak{A}m-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(62)

Velocity profile

We transform the governing equation of velocity (10) into a fractionalized form. Solving this uncoupled fractionalized equation of velocity (10) is accomplished through the Laplace transform method, yielding the desired outcome.

$${{}^{\mathrm{AB}}D}_t^{\mathfrak{A}}\mathbb{R}(\mathbb{Y},\mathbb{T})=\mathbb{B}\frac{{\partial }^2\mathbb{R}(\mathbb{Y},\mathbb{T})}{\partial {\mathbb{Y}}^2}+{\mathbb{G}}_r\theta (\mathbb{Y},\mathbb{T})+{\mathbb{G}}_m\varphi (\mathbb{Y},\mathbb{T})-(\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})\mathbb{R}(\mathbb{Y},\mathbb{T}),$$

(63)

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mathbb{R}}(\mathbb{Y},\mathcal{J})=c_1{e}^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{+c_2{e}^{\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_r(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}})}{e}^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}}}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_m(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}})}{e}^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}},}\end{array}$$

(64)

applying the initial, boundary, and natural conditions as defined in equation (14), we eliminated the arbitrary constants $c_1$ and $c_2$ from equation (64). We arrived at

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mathbb{R}}(\mathbb{Y},\mathcal{J})=(\frac{1-e^{\mathcal{J}}}{{\mathcal{J}}^2})e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}}\\ \multicolumn{1}{c}{+\frac{{\mathbb{G}}_r(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}})}}\\ \multicolumn{1}{c}{[e^{-\mathbb{Y}\sqrt{{\mathbb{P}}_{\mathrm{reff}}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{Q})}}-e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}]}\\ \multicolumn{1}{c}{-\frac{{\mathbb{G}}_m(1-e^{\mathcal{J}})}{{\mathcal{J}}^2(\mathbb{A}+\frac{\mathbb{D}{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}})}}\\ \multicolumn{1}{c}{[e^{-\mathbb{Y}\sqrt{{\mathbb{S}}_c(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+{\mathbb{K}}_r)}}-e^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}].}\end{array}$$

(65)

The convenient form of above function to implement the Laplace inversion is

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{\mathbb{R}}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})={\overline{\mathrm{\Phi }}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})+{\mathbb{G}}_r{\overline{\mathrm{\Psi }}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})}\\ \multicolumn{1}{c}{[{\overline{\theta }}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})-{\overline{\mathrm{\Phi }}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})]}\\ \multicolumn{1}{c}{+{\mathbb{G}}_m{\overline{\mathrm{\Psi }}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})[{\overline{\varphi }}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})-{\overline{\mathrm{\Phi }}}_{\mathrm{AB}}(\mathbb{Y},\mathcal{J})],}\end{array}$$

(66)

and

$$\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})=\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J})-e^{-\mathcal{J}}\overline{\mathrm{\Phi }}(\mathbb{Y},\mathcal{J}).$$

(67)

The inverse Laplace of above equation is

$$\mathrm{\Phi }(\mathbb{Y},\mathbb{T})={\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})-{\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})\mathbb{U}(\mathbb{T}-1),$$

(68)

where

$${\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}({\overline{\mathrm{\Phi }}}_1(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{{\mathcal{J}}^2}{e}^{-\mathbb{Y}\sqrt{\mathbb{B}(\frac{{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}+\mathbb{M}+\frac{\mathbb{B}}{\mathbb{K}})}}\right\}.$$

(69)

The current form of equation (69) makes it challenging to compute the Laplace inverse. Therefore, it is necessary to convert it into a series form, which will yield the following expression

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Phi }}_1(\mathbb{Y},\mathbb{T})=\sum _{a_1=0}^{\infty }\sum _{a_2=0}^{\infty }\frac{{(-\mathbb{Y})}^{a_1}({\mathbb{M}}_1^{\frac{a_1}{2}-n})\mathrm{\Gamma }(\frac{a_1}{2}+1)}{a_1!a_2!(\mathbb{B}{)}^{\frac{a_1}{2}}\mathrm{\Gamma }(\frac{a_1}{2}-a_2+1)}\times }\\ \multicolumn{1}{c}{\{\frac{{\mathbb{T}}^{\mathfrak{A}{a}_2-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(\mathfrak{A}{a}_2-1)!}\times \sum _{a_3=0}^{m-1}\frac{{(-1)}^{a_3}(m^{\mathfrak{A}}-1)!}{a_3!(m^{\mathfrak{A}}-1-a_3)!}(\mathbb{T}-{(1-\mathfrak{A})}^{a_3})\}.}\end{array}$$

(70)

Similarly,

$${\mathrm{\Psi }}_1(\mathbb{Y},\mathbb{T})={\mathcal{L}}^{-1}({\overline{\mathrm{\Psi }}}_1(\mathbb{Y},\mathcal{J}))={\mathcal{L}}^{-1}\left\{\frac{1}{\mathbb{A}+\frac{\mathbb{D}{\mathcal{J}}^{\mathfrak{A}}}{(1-\mathfrak{A}){\mathcal{J}}^{\mathfrak{A}}+\mathfrak{A}}}\right\}.$$

(71)

We get

$${\mathrm{\Psi }}_1(\mathbb{Y},\mathbb{T})=\sum _{m=0}^{\infty }\frac{{(-1)}^m{\mathbb{D}}^m}{{\mathbb{A}}^{m+1}}\{\frac{{\mathbb{T}}^{\mathfrak{A}m-1}{e}^{(1-\mathfrak{A})\mathbb{T}}}{(\mathfrak{A}m-1)!}\times \sum _{k=0}^{n-1}\frac{{(-1)}^k(\mathfrak{A}n-1)!}{k!(\mathfrak{A}n-1-k)!}(\mathbb{T}-{(1-\mathfrak{A})}^k)\}.$$

(72)

The required velocity solution for AB operator after employing the definition of inverse Laplace is

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbb{R}}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})={\mathrm{\Phi }}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})+{\mathbb{G}}_r{\mathrm{\Psi }}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})*}\\ \multicolumn{1}{c}{[{\theta }_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})-{\mathrm{\Phi }}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})]}\\ \multicolumn{1}{c}{+{\mathbb{G}}_m{\mathrm{\Psi }}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})*[{\varphi }_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})-{\mathrm{\Phi }}_{\mathrm{AB}}(\mathbb{Y},\mathbb{T})].}\end{array}$$

(73)

Skin friction

The non dimensional skin friction is given in equation (74)

$$S_k=\frac{\partial \mathbb{R}(0,\mathbb{T})}{\partial \mathbb{Y}}$$

(74)

Nusselt number

The non dimensional Nusselt number, the rate of heat transport is given in equation (75)

$$N_u=-\frac{\partial \theta (0,\mathbb{T})}{\partial \mathbb{Y}}$$

(75)

Sherwood number

The non dimensional Sherwood number provides the rate of mass transfer is given in equation (76)

$$S_h=-\frac{\partial \varphi (0,\mathbb{T})}{\partial \mathbb{Y}}$$

(76)

Results and discussion

This study contributes to the understanding of complex fluid flow phenomena in MHD Casson fluids and provides valuable insights into their behavior under ramped conditions. This part focus on chemical effect in the MHD flow of fractional Casson fluid over an infinite plate with with consideration of heat and mass transfer through a porous media. This study presents analytical solutions for the velocity, temperature, and concentration using non-integer differential operators. The results are achieved in series form and also represent in term of special functions. Additionally, the physical impact of the relevant parameters is illustrated through graphical representation.

Figure 2 describes the influence of fractional parameter $\mathfrak{A}$ on the temperature profile. As the value of $\mathfrak{A}$ increases, the boundary layer thickens, leading to decrease in temperature. The validity of the obtained results can be easily confirmed by considering $\mathfrak{A}\to 1$ . Figure 3 discuss the temperature behavior for various values of $P_{\mathrm{reff}}$ . The effective Prandtl number is a dimensionless quantity that characterizes the relative importance of momentum diffusion to thermal diffusion in a fluid. It provides valuable information about the rate of heat transfer and the thermal boundary layer in various fluid flow systems. The prevalence of mass diffusivity in fluid flow leads to a decrease in the thermal boundary layer, consequently causing a reduction in temperature. It has been declared that when the value of $P_r$ increase, the temperature falling. The quantity of heat absorption and generation discuss in Figure 4. It can be observed that the energy profile decreases as the values of $\mathbb{Q}$ increase, indicating the considerable impact of heat extraction or generation in cooling and heating procedures.

Figure 2.

Comparability of temperature profile to Caputo-Fabrizio and Atangana-Baleanu for fractional parameter $\mathfrak{u}$ .

Figure 3.

Comparability of temperature profile to Caputo-Fabrizio and Atangana-Baleanu for effective Prandtl number $P_{\mathrm{reff}}$ .

Figure 4.

Comparability of temperature profile to Caputo-Fabrizio and Atangana-Baleanu for $\mathbb{Q}$ .

The effect of Grashof number ${\mathbb{G}}_r$ illustrate in Figure 5. As we raise the value of ${\mathbb{G}}_r$ , velocity fluid rises. ${\mathbb{G}}_r$ represent the relationship between heat buoyant force to the viscous force. Grashof number controls the flow regime in natural convection. Figure 6 discuss the impact of ${\mathbb{G}}_m$ on the velocity profile. Velocity is understood to be quicker with higher ${\mathbb{G}}_m$ levels. The speed of the velocity increases because ${\mathbb{G}}_m$ is connected to buoyancy forces, which increase natural convection. In Figure 7, illustrate the impact of magnetic field $\mathbb{M}$ on the velocity profile. The results indicates that both the magnitude of the boundary layer thickness and velocity decrease when a strong magnetic field is applied. As we enhance the value of $\mathbb{M}$ , the dragging forces dominate, as a result velocity decreases.

Figure 5.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for ${\mathbb{G}}_r$ .

Figure 6.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for ${\mathbb{G}}_m$ .

Figure 7.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for $\mathbb{M}$ .

Figure 8 investigate the behavior of Prandtl number $P_r$ which is defined as the ratio of kinematic viscosity and thermal diffusivity. As $P_r$ increase, the resultant velocity reduce. The less $P_r$ helps to enlarge thermal conductivity. It plays a significant role in determining the behavior of velocity profile within boundary layer. Figure 9 illustrate the behavior of $S_c$ on velocity profile. It is deceptive that maximum value of $S_c$ , decay in velocity profile noticed. This is due to the fact that reduction of boundary layers of velocity happened corresponding to enlarge value of $S_c$ . $\mathrm{AB}$ and $\mathrm{CF}$ models describe the impact of $\mathfrak{A}$ for ramped and isothermal condition on velocity profile via graphs as shown in Figure 10. With large value of $\mathfrak{A}$ , the velocity profile reduce. But in case of isothermal condition, the velocity enhanced as compared to ramped condition. A Casson fluid with $\mathrm{AB}$ fractional derivative as compared to $\mathrm{CF}$ is the best option according to improve the fluid motion. The comparison of velocity distribution between current study and Rehman et al. ²¹ is shown in Figure 11. If we take $\mathfrak{A}=0.5$ and ${\mathbb{G}}_m=0$ in Rehman et al., ²¹ the velocity profile are identical. Figure 12 is a comparison among different fluids at room temperature for both the ramped and isothermal conditions. It is established that mercury has the least heat transfer rate than oxygen, air, water, and ethanol.

Figure 8.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for ${\mathbb{P}}_r$ .

Figure 9.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for ${\mathbb{S}}_c$ .

Figure 10.

Comparability of velocity profile to Caputo-Fabrizio and Atangana-Baleanu for $\mathfrak{A}$ .

Figure 11.

Comparison with published work.

Figure 12.

Comparison of the rate of heat transfer.

Conclusion

This work explore the results for Casson fluid by obtaining analytical solutions using Caputo-Fabrizio and Atangana Baleanu fractional derivative. The fluid flow of Casson model along with heat and mass transfer over an infinite vertical plate has been considered. The Laplace transform is used to analyze the solution of non-dimensional fractional governing equations. Special functions with particular characteristics that are frequently utilized to explain complicated phenomena are also used to display the derived solutions. More accurate and succinct results can be displayed by using series solution. The following parameter includes the Prandtl number, Grashof number, Schmidt number, mass Grashof number, and fractional parameter as well as their impacts on velocity and temperature.

• Temperature and velocity profiles in case of ramped conditions are less than in isothermal conditions.

• Applications of the magnetic field retards the flow to a remarkable extent.

• Flow is accelerated under mass and heat Grashof effect.

• Fluid velocity decrease as the magnetic force rises.

• Fluid temperature drops with the increase in radiation parameter.

• Ramped plate has the tendency to augment the heat transfer rate.

• Rate of heat transfer of mercury is less than oxygen, air, water, and ethanol at room temperature.

Developing efficient and accurate numerical methods for solving fractional differential equations and integral equations. This includes exploring new algorithms, software implementations, and optimization techniques tailored to the computational challenges posed by fractional calculus.