Roll of partial slip on Ellis nanofluid in the proximity of double diffusion convection and tilted magnetic field: Application of Chyme movement

Abstract

The current study aims to investigate the effect of partial slip conditions over double diffusion convection. The phenomenon is applied with the inclined magnetic flux of peristaltic flow on Ellis nanofluid model in an asymmetric channel. The fundamental differential equations are constructed and solved through lubrication approximation that gives the coupled system of ordinary differential equations. This resultant system is further solved numerically, and graphic representation are used to physically comprehend the flow quantity data. The whole procedure is carried out under the trapping mechanism by drawing contour streamlines. The knowledge gained from this work will be useful in the creation of intelligent magneto-peristaltic pumps for specific heat and medication delivery phenomena.

1. Introduction

Nanofluids containing nanoparticles of size 1–50 nm have a peculiar feature to enhance the thermal efficiency of liquids. Normal liquids like water, ethylene, oil etc. Have poor heat transfer qualities; hence, their heat efficiency can be enhanced by adding nanofluids. The utilization of nanoparticles in biology opens up a new trend in medicine. Nowadays, nanoparticles are used to transmit substances like drugs, heat, light etc. To the targeted areas in the human body, especially cancerous cells. For instance, iron-coated nanoparticles act as a tool to deliver the drug or radiation to the targeted areas; moreover, such material is safe for the rest of the body parts [1]. Drug transfer, and gene supply, examination of DNA formation, radiant biological labels, protein diagnosis, pathogen identification in the body etc. Are some of the areas of nanoparticle applications [[2], [3], [4], [5], [6], [7], [8], [9]]. Further, various aspects of nanofluid flow were investigated by researchers [[10], [11], [12], [13], [14], [15]].

Diversified application and utility of the phenomenon in the field of biomedicine reflect that natural biological procedures like peristaltic pumping used nanofluid transmission. Peristalsis is a phenomenon that pumps a fluid in a channel by contraction and expansion due to pressure change. This phenomenon is common in biological domains. The most common example is the fluid transfer in the human body. The natural excretion of wastes from human body like feces motion in intestine, urine excretion through kidneys; and circulation of fluids like blood flow within veins, arteries, and capillaries; movement of bile, spermatozoa, eggs; secretion and circulation of hormones; food movement throughout the digestive tract are all through peristalsis. Besides, peristalsis is involved in biomedical equipment such as heart pumps, heart & ortho machines, dialysis apparatus, lung machines, toxic and waste transport instruments etc. The phenomenon is also beneficial for nuclear industry such as toxic-wastes disposal instruments are made on the same principle.

Latham [16] started the research on the peristalsis, and many investigations on the same principle have been conducted to explore various parameters [[17], [18], [19], [20], [21], [22], [23]]. Another field of study, Magnetohydrodynamics (MHD), has many applications in medicine, manufacturing, and geophysics. In biomedicine, Magnetohydrodynamics (MHD) is beneficial to treat cancer cells, in magnetic cell segregation, rise in magnetic resonance, drug induction etc. Khan et al. [24] explored the magnetic field impact when externally applied to nanofluid flow within the stretching sheet and deliberated on an uneven solution to the problem. Sheikholeslami et al. [25] examined the magnetic flux impact on the flow of Cu-water. Furthermore, Sheikholeslami [26] and his co-authors applied Buongiorno's model to explore the uneven flow of nanofluid under magnetic force. Enforced convection of Ag–MgO nanofluid motion within a chamber under heat transport is explained by Ma et al. [27]. Other related works on magnetic influence are mentioned in studies [[28], [29], [30], [31], [32]].

The simultaneous mass and heat transmission along the fluid flow is called double diffusion. It has widespread application and utilization in mechanical & chemical engineering, biomedicine, oceanography, and astrophysics [33] along with the manufacturing field such as gas containers, crystallization process, solidification of liquid metals, solar plates etc. Currently, the research trend has seen the shift toward peristaltic flow under double-diffusive convection like Noreen et al. [34] has examined the same. Owing to its popularity, the phenomenon is further explored in nanofluid framework, and peristaltic thrust as given in [[35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]].

In the study of the characteristics of viscous fluid in tube walls, one can find out the limitations of no-slip boundary conditions. This shortcoming is obvious in the melting process of polymer which demonstrates the microscopic wall slip. Numerous applications of fluid flow mechanisms further strengthen the significance of boundary conditions in fluid flow. Because the fluids with the boundary conditions have many technological applications like in artificial heart polish and other inner cavities. Eldesoky [46] examined the peristaltic flow of a non-Newtonian Maxwellian fluid under slip conditions. While Chu and Fang [47] further studied the peristaltic flow under slip conditions for compressible Maxwell fluid in a tube under a porous medium. Other research findings related to the concept of slip boundary conditions have been mentioned in Refs. [[48], [49], [50], [51], [52], [53]].

Numerous researches have been conducted on MHD peristaltic flow of Ellis fluid model that have practical implications for technical and medical sciences. The common generalized Newtonian fluid (GNF) model are power law, Carreau fluid model, and Herschel Bulkley model. The Ellis model is a subcategory of GNF. The significant advantage of the Ellis equation is that it predicts the Newtonian tendency at minimum shear stresses and the power law inclination on higher stress.

Another notable advantage of Ellis fluid model is that it draws on Newtonian tendency upon lower sheer stress and also predicts power law on larger sheer stress. The said benefit of Ellis Model can be utilized to assess the mass and heat transfer for peristaltic transport along chemical reaction within asymmetrical channel [[54], [55], [56], [57], [58]]. Such implications of double diffusion phenomenon in nanofluids, as discussed above, the current examination explores the phenomenon of double-diffusion convection having temperature and concentration ingredients in irregular biological geometric conventions through mathematical computation to deal with the intricacy of biological or peristaltic flow scheme. Ellis nanofluid model with mathematical computation has been applied to get the dataset and develop the physical quantitative through graphical illustration. These bio-mathematical issues can be assumed to translate into the advanced diagnostic techniques by the process of peristalsis. The practical aspects include the utility in the domains of systems with pressure air, thermoelectric cooling, heat reservoirs, pipes and much more.

1.1. Formulation

The assumptions are based on dense electrically confined peristaltic movement for magneto-Ellis nanofluid in 2-dimensional conduit. Its width is taken as ${\alpha }_3+{\alpha }_4$. The model of the Cartesian coordinates system is applied to have the channel centre at X-axis while the Y-axis lies on the cross-sectional area. The speed track of sinusoidal wave remains constant on the channel walls. While temperatures, solvent concentrations, and nanoparticle concentrations are $T=T_0\text{,}$ $C=C_0\text{,}$ $\Theta ={\Theta }_0\text{,}$ (at $y=H_1$) and $T=T_1\text{,}$ $C=C_1\text{,}$ $\Theta ={\Theta }_1$ (at $y=H_2$), respectively on the right and left walls. At an angle, the magnetic field is uniform while Reynolds number and electric field are fixed to a low value to zero, respectively. Thus, we get the insignificantly induced magnetic field opposite to applied magnetic field. The geometry of the wall is defined in Fig. 1

Fig. 1.

Geometry of the problem.

The mathematical form of geometric shape of wall (is defined in Eq. (1)) is as follows [19,20]:

$$Y=H_1={\alpha }_3+{\alpha }_1\mathit{cos}\left(\frac{2\pi }{\lambda }\left(X-ct\right)\right),Y=H_2=-{\alpha }_4-{\alpha }_2\mathit{cos}\left(\frac{2\pi }{\lambda }\left(X-ct\right)+\omega \right)\text{,}$$ (1)

Where $c\text{,}$ $t\text{,}$ $({\alpha }_1\text{,}$ ${\alpha }_2)\text{,}$ $\left({\alpha }_3,{\alpha }_4\right)\text{,}$ $\lambda \text{,}$ denotes wave speed, time, channel width, and wave amplitude respectively. The range of phase difference $\omega$ is $0\le \omega \le \pi \text{,}$ with $\omega =0$ reflects a symmetric channel with no phase wave, and $\omega =\pi$ implies a channel having a phase wave. Also $\omega \text{,}$ ${\alpha }_1\text{,}$ ${\alpha }_2\text{,}$ ${\alpha }_3\text{,}$ ${\alpha }_4\text{,}$ obey the criteria ${\alpha }_1^2+{\alpha }_2^2+2{\alpha }_1{\alpha }_2\mathit{cos}\omega \le {\left({\alpha }_3+{\alpha }_4\right)}^2\text{.}$ In unsteady two-dimensional flow, lets consider velocity profile as $V=[U($ X $,Y,t),V($ X $,Y,t),0]$.

Here $(U\text{,}$ $V)$ are velocity components in fixed frame $(X\text{,}$ $Y)$. Now use Galilean transformation (Eq. (2)) to transfer equations from fixed $\left(X,Y\right)$ to moveable $\left(x,y\right)$ frame as

$$x=X-ct,p\left(x\right)=P\left(X,Y,t\right),y=Y,\mathrm{v}=V,u=U-c\text{,}$$ (2)

For Ellis fluid the stress tensor $S$ (Eq. (3)) is interpreted as [23]

$$S=\frac{\mu }{1+{\left(\frac{{\Pi }_s}{{\tau }_0^2}\right)}^{\beta -1}}{A}_1\text{,}$$ (3)

where $\mu \text{,}$ ${\Pi }_s\text{,}$ $A_1\text{,}$ $\left({\tau }_0,\beta \right)$ stands for dynamic viscosity, invariant stress tensor of second order, Rivilin Ericksen tensor, and material constants respectively. The constant ${\tau }_0$ generally defined as shear stress associated with half dynamic viscosity. The model (3) corresponds to Newtonian Model for $\beta =1$ and $1/{\tau }_0^2\to 0\text{.}$.

Now employing Eq. (2) the flow equations are defined in wave frame $\left(x,y\right)$ as [35] (Eq. (4) to Eq. (9))

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\text{,}$$ (4)

$${\rho }_f\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)u=-\frac{\partial p}{\partial x}+\frac{\partial }{\partial x}\left(S_{xx}\right)+\frac{\partial }{\partial y}\left(S_{xy}\right)$$

$$-\sigma B_0^2\mathit{cos}\Gamma \left(\mathit{cos}\Gamma \left(u+c\right)-v\mathit{sin}\Gamma \right)+g\{{\rho }_{f0}\left(1-{\Theta }_0\right)$$

$$\left\{{\beta }_T\left(T-T_0\right)+{\beta }_C\left(C-C_0\right)\right\}-\left({\rho }_p-{\rho }_{f0}\right)\left(\Theta -{\Theta }_0\right)\}\text{,}$$ (5)

$${\rho }_f\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)v=-\frac{\partial p}{\partial y}+\frac{\partial }{\partial x}\left(S_{yx}\right)+\frac{\partial }{\partial y}\left(S_{yy}\right)$$

$$+\sigma B_0^2\mathit{sin}\Gamma \left(\left(u+c\right)\mathit{cos}\Gamma -v\mathit{sin}\Gamma \right)\text{,}$$ (6)

$${\left(\rho c\right)}_f\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)T=k\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+{\left(\rho c\right)}_p\{D_B\left(\frac{\partial \Theta }{\partial x}\frac{\partial T}{\partial x}+\frac{\partial \Theta }{\partial y}\frac{\partial T}{\partial y}\right)$$

$$\left(\frac{D_T}{T_0}\right)\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]\}+D_{TC}\left(\frac{{\partial }^{2}C}{\partial x^2}+\frac{{\partial }^{2}C}{\partial y^2}\right)\text{,}$$ (7)

$$replace{D}_{TC}with{D}_{CT}$$ (8)

$$\left(u\frac{\partial }{\partial x}+v\frac{\partial }{\partial y}\right)\Theta =D_B\left(\frac{{\partial }^2\Theta }{\partial x^2}+\frac{{\partial }^2\Theta }{\partial y^2}\right)+\left(\frac{D_T}{T_0}\right)\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)\text{,}$$ (9)

where $S$ illustrates stress tensor, $g$ indicates acceleration, ${\rho }_p$ describes particles density, $k$ specifies thermal conductivity, ${\rho }_{f_0}$ represents fluid density at $T_0$ , ${\rho }_f$ indicates fluid density of base fluid, ${\left(\rho c\right)}_p$ describes nanoparticles heat capacity, $D_B$ indicates Brownian diffusion coefficient, ${\left(\rho c\right)}_f$ specifies heat capacity of fluid, $D_T$ illustrates thermophoretic diffusion coefficient, ${\beta }_C$ stands for fluid's volumetric solutal expansion coefficient, $D_{TC}$ represents Dufour diffusively, ${\beta }_T$ describes fluid's volumetrically thermal expansion coefficient, $D_{CT}$ indicates Soret diffusively, and $D_s$ illustrates solutal diffusively.

The nondimensional variables are characterized as (Eq. (10))

$$\bar{y}=\frac{y}{{\alpha }_3},\bar{x}=\frac{x}{\lambda },\bar{\mathrm{v}}=\frac{v}{c},\bar{u}=\frac{u}{c},\delta =\frac{{\alpha }_3}{\lambda },\bar{t}=\frac{ct}{\lambda },d=\frac{{\alpha }_4}{{\alpha }_3},{\mathrm{h}}_2=\frac{H_2}{{\alpha }_4},{\mathrm{h}}_1=\frac{H_1}{{\alpha }_3},b=\frac{{\alpha }_2}{{\alpha }_3},\bar{p}=\frac{{\alpha }_3^{2}p}{\mu c\lambda }\text{,}$$

$$a=\frac{{\alpha }_1}{{\alpha }_3},\mathit{Pr}=\frac{{\left(\rho c\right)}_f\upsilon }{k},Re=\frac{{\rho }_{f}c{\alpha }_3}{\mu },\upsilon =\frac{\mu }{{\rho }_f},Le=\frac{\upsilon }{D_s},\mathit{Ln}=\frac{\upsilon }{D_B},u=\frac{\partial \psi }{\partial y},v=-\delta \frac{\partial \psi }{\partial x}\text{,}$$

$$\theta =\frac{T-T_0}{T_1-T_0},\gamma =\frac{C-C_0}{C_1-C_0},\Omega =\frac{\Theta -{\Theta }_0}{{\Theta }_1-{\Theta }_0},\xi =\frac{c}{{\alpha }_3{\tau }_0^2},{\bar{S}}_{xx}=\frac{S_{xx}\lambda }{\mu c},M=\sqrt{\frac{\sigma }{\mu }}{B}_0{\alpha }_3,{\bar{S}}_{xy}=\frac{S_{xy}{\alpha }_3}{\mu c}\text{,}$$

$${\bar{S}}_{yy}=\frac{S_{yy}{\alpha }_3}{\mu c},G_{rt}=\frac{g{\alpha }_3^2\left(1-{\Theta }_0\right)\left(T_1-T_0\right){\rho }_{f0}{\beta }_T}{\mu c},G_{rc}=\frac{g\left(1-{\Theta }_0\right){\rho }_{f0}{\beta }_c\left(C_1-C_0\right){\alpha }_3^2}{\mu c}\text{,}$$

$$N_{CT}=\frac{D_{CT}\left(T_1-T_0\right)}{\left(C_1-C_0\right)D_s},N_b=\frac{{\left(\rho c\right)}_p{D}_B\left({\Theta }_1-{\Theta }_0\right)}{k},N_{TC}=\frac{D_{CT}\left(C_1-C_0\right)}{k\left(T_1-T_0\right)},N_t=\frac{{\left(\rho c\right)}_p{D}_T\left(T_1-T_0\right)}{T_{0}k}\text{,}$$

$$G_{rF}=\frac{g\left({\rho }_p-{\rho }_{f0}\right)\left({\Theta }_1-{\Theta }_0\right){\alpha }_3^2}{\mu c}\text{,}$$ (10)

where Re symbolizes Reynolds number, $\delta$ indicates wave number, $G_{rt}$ stands for thermal Grashof number, $M$ signifies Hartmann number, $G_{rF}$ indicates nanoparticle Grashof number, $N_t$ signifies thermophoresis parameter, $\mathit{Ln}$ illustrates nanofluid Lewis number, $N_{CT}$ denotes Soret parameter, $Le$ indicates Lewis number, $\Omega$ stands for nanoparticle fraction, $G_{rc}$ denotes solutal Grashof number, Pr represents Prandtl number, $\gamma$ indicates solutal concentration, $N_{TC}$ represents Dufour parameter, $\theta$ illustrates temperature, $p$ indicates pressure, and $N_b$ symbolizes Brownian motion.

Now using Eq. (10), Eq. (4) is identically satisfied and Eqs. (5), (6), (7), (8), (9) in terms of stream function (dropping bars) are as follows (Eqs. (11), (12), (13), (14), (15)):

$$Re\delta \left({\Psi }_y{\Psi }_{xy}-{\Psi }_x{\Psi }_{yy}\right)=-\frac{\partial p}{\partial x}+\delta \frac{\partial S_{xx}}{\partial x}+\frac{\partial S_{xy}}{\partial y}-M^2\mathit{cos}\Gamma \left(\mathit{cos}\Gamma \left({\Psi }_y+1\right)+\delta {\Psi }_x\mathit{sin}\Gamma \right)$$

$$+G_{rt}\theta +G_{rc}\gamma -G_{rF}\Omega \text{,}$$ (11)

$$Re{\delta }^3\left({\Psi }_x{\Psi }_{xy}-{\Psi }_y{\Psi }_{xx}\right)=-\frac{\partial p}{\partial y}+{\delta }^2\frac{\partial S_{yx}}{\partial x}+\delta \frac{\partial S_{yy}}{\partial y}$$

$$+\delta M^2\mathit{sin}\Gamma \left(\left({\Psi }_y+1\right)\mathit{cos}\Gamma +\delta {\Psi }_x\mathit{sin}\Gamma \right)\text{,}$$ (12)

$$Re\mathit{Pr}\delta \left({\Psi }_y{\theta }_x-{\Psi }_x{\theta }_y\right)=\left({\theta }_{yy}+{\delta }^2{\theta }_{xx}\right)+N_{TC}\left({\delta }^2{\gamma }_{xx}+{\gamma }_{yy}\right)+N_b\left({\delta }^2{\Omega }_x{\theta }_x+{\theta }_y{\Omega }_y\right)$$

$$+N_t\left({\delta }^2{\left({\theta }_x\right)}^2+{\left({\theta }_y\right)}^2\right)\text{,}$$ (13)

$$Re\delta Le\left({\Psi }_y{\gamma }_x-{\Psi }_x{\gamma }_y\right)=\left({\delta }^2{\gamma }_{xx}+{\gamma }_{yy}\right)+N_{CT}\left({\delta }^2{\theta }_{xx}+{\theta }_{yy}\right)\text{,}$$ (14)

$$Re\delta \mathit{Ln}\left({\Psi }_y{\Omega }_x-{\Psi }_x{\Omega }_y\right)=\left({\delta }^2{\Omega }_{xx}+{\Omega }_{yy}\right)+\frac{N_t}{N_b}\left({\delta }^2{\theta }_{xx}+{\theta }_{yy}\right)\text{,}$$ (15)

The representation of stress tensor is computed in component form as (Eqs. (16), (17), (18), (19)):

$$S_{xx}=\frac{2\delta \frac{{\partial }^2\Psi }{\partial y^2}}{1+{\left(\xi \chi \right)}^{\beta -1}}\text{,}$$ (16)

$$S_{xy}=\frac{\left(\frac{{\partial }^2\Psi }{\partial y^2}-{\delta }^2\frac{{\partial }^2\Psi }{\partial x^2}\right)}{1+{\left(\xi \chi \right)}^{\beta -1}}\text{,}$$ (17)

$$S_{yy}=\frac{-\delta \frac{{\partial }^2\Psi }{\partial y\partial x}}{1+{\left(\xi \chi \right)}^{\beta -1}}\text{,}$$ (18)

$$\chi ={\left(\frac{1}{2}\left({\left(S_{xx}\right)}^2+2{\left(S_{xy}\right)}^2+{\left(S_{yy}\right)}^2\right)\right)}^{1/2}\text{,}$$ (19)

Using constraints of $Re\to 0$ (low number of Reynolds), and $\delta \ll 1$ (long wavelength), Eqs. (11), (12), (13), (14), (15), (16), (17), (18), (19) becomes (Eqs. (20), (21), (22), (23), (24), (25))

$$\frac{\partial p}{\partial x}=\frac{\partial S_{xy}}{\partial y}-M^2\left(\frac{\partial \psi }{\partial y}+1\right){\mathit{cos}}^2\Gamma +G_{rt}\theta +G_{rc}\gamma -G_{rF}\Omega \text{,}$$ (20)

$$\frac{\partial p}{\partial y}=0\text{,}$$ (21)

$$\frac{{\partial }^2\theta }{\partial y^2}+N_{TC}\frac{{\partial }^2\gamma }{\partial y^2}+N_b\left(\frac{\partial \theta }{\partial y}\frac{\partial \Omega }{\partial y}\right)+N_t{\left(\frac{\partial \theta }{\partial y}\right)}^2=0$$ (22)

$$\frac{{\partial }^2\gamma }{\partial y^2}+N_{CT}\frac{{\partial }^2\theta }{\partial y^2}=0$$ (23)

$$\frac{{\partial }^2\Omega }{\partial y^2}+\frac{N_t}{N_b}\frac{{\partial }^2\theta }{\partial y^2}=0$$ (24)

$$S_{xx}=0,S_{yy}=0,S_{xy}=\frac{\frac{{\partial }^2\psi }{\partial y^2}}{1+{\left(\xi S_{xy}\right)}^{\beta -1}}$$ (25)

$$\frac{{\partial }^2{S}_{xy}}{\partial y^2}-M^2{\mathit{cos}}^2\Gamma \frac{{\partial }^2\psi }{\partial y^2}+G_{rt}\frac{\partial \theta }{\partial y}+G_{rc}\frac{\partial \gamma }{\partial y}-G_{rF}\frac{\partial \Omega }{\partial y}=0$$ (26)

The mean flow $\left(Q\right)$ in dimensionless form is computed as (Eq. (27)):

$$Q=d+F+1$$ (27)

Where $F$ is defined in Eq. (28)

$$F=\underset{h_2\left(x\right)}{\overset{h_1\left(x\right)}{\int }}\frac{\partial \psi }{\partial y}dy=\psi \left({\mathrm{h}}_1\left(x\right)-{\mathrm{h}}_2\left(x\right)\right)$$ (28)

here ${\mathrm{h}}_1$ and ${\mathrm{h}}_2$ is defined in Eq. (29)

$${\mathrm{h}}_1\left(x\right)=1+a\mathit{cos}2\pi x,{\mathrm{h}}_2\left(x\right)=-d-b\mathit{cos}\left(2\pi x+\omega \right)$$ (29)

The expression of slip constraints for current scenario in non-dimensional form is defined as (Eqs. (30), (31), (32), (33)):

$$\psi =\frac{F}{2},\frac{\partial \psi }{\partial y}=-L_1{S}_{xy}-1\text{on}y={\mathrm{h}}_1\left(x\right)$$

$$\psi =-\frac{F}{2},\frac{\partial \psi }{\partial y}=L_1{S}_{xy}-1\text{on}y={\mathrm{h}}_2\left(x\right)$$ (30)

$$\theta +L_2\frac{\partial \theta }{\partial y}=0,\text{on}y={\mathrm{h}}_1,\text{and}\theta -L_2\frac{\partial \theta }{\partial y}=1,\text{on}y={\mathrm{h}}_2$$ (31)

$$\gamma +L_3\frac{\partial \gamma }{\partial y}=0,\text{on}y={\mathrm{h}}_1,\text{and}\gamma -L_3\frac{\partial \gamma }{\partial y}=1,\text{on}y={\mathrm{h}}_2$$ (32)

$$\Omega +L_4\frac{\partial \Omega }{\partial y}=0,\text{on}y={\mathrm{h}}_1,\text{and}\mathrm{\Omega }-L_4\frac{\partial \Omega }{\partial y}=1,\text{on}y={\mathrm{h}}_2$$ (33)

If $L_1,L_2,L_3,L_4=0$ in Eqs. (30)-(33) the no slip boundary conditions exist in this case.

Special Case: In the absence of slip conditions ($L_1=L_2=L_3=L_4=0$), $M=0$, $\xi =0\text{,}$ or $\beta =1\text{,}$ $G_{rt}=0\text{,}$ $G_{rc}=0\text{,}$ $G_{rF}=0$, or the results of [19] can be obtained as a special case of the current problem. Table 1 shows a comparison of the current problem with the results of [19]

Table 1.

The comparison of the velocity profile to the current literature.

$y$	Current problem	Mishra and Rao [19]
0.676373	$-0.773118$	$-1$
0.568366	$-0.854148$	$-1.01004$
0.460359	$-0.921623$	$-1.01777$
0.352351	$-0.977005$	$-1.02323$
0.244344	$-1.02154$	$-1.02649$
0.136337	$-1.05627$	$-1.02757$
0.0283299	$-1.08209$	$-1.02649$
$-0.0796772$	$-1.09976$	$-1.02323$
$-0.187684$	$-1.10989$	$-1.01777$
$-0.295692$	$-1.11298$	$-1.01004$
$-0.403699$	$-1.10945$	$-1.0$

1.1.1. Numerical computation and graphical outcomes

The Eqs. $\left(26\right)\text{,}$ (20) and (22)-(24) are solved to get the numerical computation. In the current analysis, we have used MATHEMATICA, the mathematical software, and applied its build-in command NDSolve to get the results. This command uses Interpolating function objects to find solutions iteratively. Further, these numerical solutions are graphically depicted to demonstrate the impact of the model's various parameters.

Figs. 2(a)-2(d) are drawn to demonstrate the impression of slip velocity $L_1$, thermal slip $L_2$, solutal slip concentration $L_3\text{,}$ and Hartmann number $M$. Fig. 2(a)shows the rise in the velocity value for the region $\in \left[-\mathrm{0.2,0.5}\right]$ , whereas the opposite effects is observed for $y\in \left[-0.4,-0.2\right]$ and $y\in \left[\mathrm{0.5,0.65}\right]$ . Here the magnitude of the velocity declines with the rise in slip velocity. The fluid velocity decreases further, within the channel, due to the deviation of fluid particles under slip impact. The impact of velocity on thermal slip $L_2$ and solutal slip concentration $L_3$ are drawn in Fig. 2(b) and (c) respectively. Moreover, the Fig. 2(b) and (c) illustrate the decline in velocity profile with $y\in \left[-\mathrm{0.4,0.3}\right]$ whereas with $y\in \left[\mathrm{0.3,0.65}\right]$ velocity magnitude rises because of the increase of $L_2$ and $L_3$ values. The impact of velocity on $M$ is plotted in Fig. 2(d) which depicts the surge in velocity magnitude when $y\in \left[-\mathrm{0.4,0.3}\right]$ while with $y\in \left[\mathrm{0.3,0.65}\right]$ velocity magnitude decreases owing to the increase in the value of $M$. When the magnetic number increases, it enhances the Lorentz force that ultimately acted as a retarding energy for the fluid motion, so it declines. Fig. 3(a)-(e) are drawn to analysis the impact of temperature on thermal slip $L_2\text{,}$ Soret $N_{CT}\text{,}$ Dufour $N_{TC}\text{,}$ thermophoresis parameter $N_t\text{,}$ and Brownian motion $N_b\text{.}$ Fig. 3(a) shows that the rise in thermal slip parameter reduces the temperature profile in the region $y\in \left[-\mathrm{0.4,0.3}\right]$ , but the behavior is opposite for the area $y\in \left[-\mathrm{0.4,0.3}\right]$ . The temperature rises because of rise in thermal slip parameter. Here, the slipness causes the rise in the kinetic energy of the fluid particles which raises the fluid temperature. The impact of temperature profile on $N_{CT}\text{,}$ $N_{TC}\text{,}$ $N_t\text{,}$ and $N_b$ are shown in Fig. 3(b)-3(e) which further reveals that the rising behaviour of $N_{CT}\text{,}$ $N_{TC}\text{,}$ $N_t\text{,}$ and $N_b$ brings the rise in temperature profile. The phenomena of diffusion-thermo or Dufour effect is defined as phenomena when a chemical system undergoes a concentration gradient to produce heat flux. Thermal diffusion plays a vital role in the phenomena tough have a small impact that can be very vital when applied on the species with differing molecular weights. A concentration gradient is formed with the mass diffusion of uneven gradients having different molecular weight. Thermo-diffusion or Soret effect is a process when mass diffusion occurs due to the impact of temperature gradient. Hence, the higher is the temperature gradient, the larger would be the Soret effect. Furthermore, the increase in the intensity of the Brownian motion gears up the nanoparticle's movement along the fluid wall which causes the rise in temperature with the rise in $N_t\text{,}$ and $N_b\text{.}$ The detail analysis of solute concentration impact $\gamma$ on solute slip concentration $L_3$, Soret $N_{CT}$, Dufour $N_{TC}\text{,}$ thermophoresis parameter $N_t$, and Brownian motion $N_b$ are drawn in Fig. 4(a)-(e). From all these, we can see the drop with the rising effect of $L_3\text{,}$ $N_{CT}$, $N_{TC}\text{,}$ $N_t$ , and $N_b\text{.}$ Fig. 5(a)-(e) illustrate the impact of nanoparticle slip concentration $L_4$ , Soret $N_{CT}$ , Dufour $N_{TC}$ , thermophoresis parameter $N_t$ , and Brownian motion $N_b$ on nanoparticle concentration $\Omega$ . Fig. 5(a)-(d) show the drop in nanoparticle concentration profile with the rising behaviour of $L_4$, $N_{CT}$, $N_{TC}\text{,}$ and $N_t\text{.}$ Big values of $L_4$ drops the nanoparticle concentration because the channel walls create less disturbance for the particles under slip effect which causes decrease in mass transfer rate of nanoparticles. Greater $N_t$ value reduces the fluid velocity, and correspondingly reduces the volume fraction for less dense particles. Fig. 5(e)shows the increased profile of nanoparticle fraction with greater $N_b$ values. The influence of pressure gradient on velocity slip parameter $L_1$ , Brownian motion $N_b$ , Hartmann number $M$ , thermal Grashof number $G_{rt}$ , and nanoparticle slip parameter $L_4$ are drawn in Fig. 6(a)-(e). The rising values of $L_1$ and $N_b$ reduces the pressure gradient as shown in Fig. 6(a), (b). While increase in Hartmann number also increases the pressure gradient as depicted by Fig. 6(c). Strong magnetic flux is required for the flow field to pass the higher-pressure gradient. Hence, we can deduce the fact that the fluid pressure can be controlled by the appropriate intensity of the magnetic force. This is useful to regulate the bleeding during surgical procedures and in other critical conditions. Moreover, Fig. 6(d) and (e) reveal that increasing of $G_{rt}$ and $L_4$ cause the pressure gradients to rise. Now, to assess the results of pressure rise on velocity slip parameter $L_1$ , Hartmann number $M$ , Grashof number of nanoparticles $G_{rF}$, thermophoresis parameter $N_t$, and nanoparticle slip constraint $L_4$ Figs. 7(a)-(e) are plotted. Peristaltic pumping regions are further divided up into various regions to elaborate the pressure impact on it. (a) Region of peristaltic pumping $\left(\Delta p>0,Q>0\right)$. Here the wave of peristalsis controls the pressure and direct the fluid in the line of its propagation. (b) Region of co-pumping/augmented $\left(\Delta p<0,Q>0\right)$ where the pressure strengthens the flow due to peristaltic force. (c) Region of free pumping $\left(\Delta p=0\right)$. As the term implied, the only force acts on the flow is peristalsis. (d) Region of retrograde pumping $\left(\Delta p>0,Q<0\right)$. As the Fig. 7(a) depicts the outcome of velocity slip parameter $L_1$. We can see the decrease in pumping in peristaltic, free, retrograde area while in co-pumping zone increased velocity slip parameter $L_1$ raises the pumping thrust. The study of Fig. 7(b) demonstrates the contrary situation with increase in $M$ as compared to $L_1$. Here pumping increases in peristaltic, and retrograde, but decline with the enhanced effect of $M$ in free and co-pumping zones. Decline in the pumping rate in all the zones of peristalsis with the rise of nanoparticle Grashof number $G_{rF}$ (see Fig. 7(c)). Moreover, Fig. 7(d) and (e)illustrate the opposite behaviour of $N_t$ and $L_4$ as compared to $G_{rF}\text{.}$ Because it causes the rise in pumping rate for all the peristaltic regions.

Fig. 2.

(a)-(d): Roll of velocity profile on $L_1\text{,}$$L_2\text{,}$$L_3\text{,}$ and $M\text{.}$.

Fig. 3.

(a)-(e): Roll of temperature on $L_2\text{,}$$N_{CT}\text{,}$$N_{TC}\text{,}$$N_t\text{,}$ and $N_b$.

Fig. 4.

(a)-(e): Roll of solvent concentrations on $L_3\text{,}$$N_{CT}\text{,}$$N_{TC}\text{,}$$N_t\text{,}$ and $N_b$.

Fig. 5.

(a)-(e): Roll of nanoparticle concentrations on $L_4\text{,}$$N_{CT}\text{,}$$N_{TC}\text{,}$$N_t\text{,}$ and $N_b$.

Fig. 6.

(a)-(e): Roll of pressure gradient on $L_1\text{,}$$N_b\text{,}$$M\text{,}$$G_{rt}\text{,}$ and $L_4$.

Fig. 7.

(a)-(e): Roll of pressure rise on $L_1\text{,}$$M\text{,}$$G_{rF}\text{,}$$N_t$ and $L_4$ .

1.2. Trapping phenomena

Trapping phenomenon is described as the formation of Bolus fluid due to circulation. The bolus which is trapped passes on with the wave. Fig. 8 shows the actions of thermal slip parameter $L_2$ on streamlines. It reveals that the surge in thermal slip parameter $L_2$ , the trapped bolus decreases in top portion of channel. Whereas lower part of channel exhibits the contrary behaviour as the bolus volume increases in the lower part. Fig. 9 is drawn to show the streamline effect on solute Grashof number $G_{rc}$ . It is obvious that in the top portion of the channel, the bolus increases in both amount & size, but drop in the bottom of the channel owing to the increase in $G_{rc}$ . The streamlines action on nanoparticle slip parameters $L_4$ is illustrated in Fig. 10 . It shows the rise in trapped bolus with the rise in the effect of nanoparticle slip parameters $L_4$ in the top of the channel while decrease in the bottom of the channel. Fig. 11 explains the impact of streamlines on $N_t\text{.}$ It is observed that the behavior of streamlines is opposite to the nanoparticle slip parameters $L_4$. In upper part of channel, trapped bolus increases in size with rise in $N_t$ but decreases in lower half of channel.

Fig. 8.

Significance of streamlines on $L_2$.

Fig. 9.

Significance of streamlines on $G_{rc}$.

Fig. 10.

Significance of streamlines on $L_4\text{.}$.

Fig. 11.

Significance of streamlines on $N_t\text{.}$.

2. Conclusions

The main theme of this work is to look at the theoretical consequences of slip boundaries and double diffusion over peristaltic flow using Ellis nanofluid as base fluid in an asymmetric channel with slanted magnetic fields. The mathematical characteristics of double-diffusion convection on magneto Ellis nanofluid model for nanofluids are investigated thoroughly. Numerical solution of the current problem is computed, and graphical results are displayed for various flow parameters. It is found that increasing the intensity of Brownian motion speeds up the migration of nanoparticles along the fluid wall, causing temperature to rise as thermophoresis and Brownian motion parameters increase. It is also observed that slip generates a rise in the kinetic energy of the fluid particles, which raises the fluid temperature. Moreover, the deviation of fluid particles during slip impact reduces the fluid velocity. It is also noted that the trapped bolus rises with the rise in nanoparticle slip parameters in the top of channel while decreases in the bottom of channel.

Author contribution statement

Yasir Khan, Safia Akram, Alia Razia: Conceived and designed the analysis; Contributed analysis tools or data; Wrote the paper.

Maria Athar, A. Alameer: Conceived and designed the analysis; Analyzed and interpreted the data.

Khalid Saeed: Analyzed and interpreted the data; Contributed analysis tools or data; Wrote the paper.

Funding statement

This research work was funded by institutional fund projects under no. (IFP-A-2022-2-5-24) by ministry of education and University of Hafr Al Batin, Saudi Arabia.

Data availability statement

No data was used for the research described in the article.

Declaration of interest's statement

The authors declare no conflict of interest.