**2. Mathematical Modeling**

The problem is to contemplate the effects of double diffusion on the peristaltic transport of an impermeable third-grade fluid in a compliant curved channel having radius *R* and uniform width 2*d* bent in the form around the curve with central point having the corresponding components *u* and *v* in above-mentioned sides (see Figure 1a). The walls have been structured to become wavy along the flow and have the mathematical expression as described below. The operating equations for the obstacle are [18]

$$\nabla \cdot V = 0,\tag{1}$$

$$
\rho\_f(\frac{\partial \mathbb{Z}}{\partial t} + V \cdot \nabla V) = -\nabla p + \mu \nabla \cdot \mathbb{S} \tag{2}
$$

$$\begin{array}{c} \stackrel{\text{r}}{+} (\stackrel{\text{out}}{\text{-}} + \stackrel{\text{r}}{(1-\text{q})} \stackrel{\text{r}}{\text{-}} (\stackrel{\text{r}}{-} - \stackrel{\text{r}}{\text{q}} (\stackrel{\text{r}}{\text{-}} - \stackrel{\text{r}}{\text{-}} (\stackrel{\text{r}}{\text{-}} - \stackrel{\text{r}}{\text{-}} (\text{C} - \text{C}\_{1}))) \text{g}, \end{array} \tag{2}$$

$$\begin{aligned} \left(\rho c\right)\_f \left[\frac{\partial T}{\partial t} + V \cdot \nabla T\right] &= K\nabla^2 T + \left(\rho c\right)\_p \left(D\_b \nabla \rho \cdot \nabla T + \frac{D\_t}{T\_1} (\nabla T \cdot \nabla T)\right) \\ &+ \left(\rho c\right)\_f D\_{lc} \nabla^2 C,\end{aligned} \tag{3}$$

$$\frac{\partial \mathcal{C}}{\partial t} + V \cdot \nabla \mathcal{C} = D\_s \nabla^2 \mathcal{C} + D\_{\mathcal{C}} \nabla^2 T\_{\prime} \tag{4}$$

$$\frac{\partial \rho}{\partial t} + V \cdot \nabla ]\varphi = D\_b \nabla^2 \varphi + (\frac{D\_t}{T\_1}) \nabla^2 T,\tag{5}$$

where ρ*<sup>f</sup>* and ρ*<sup>p</sup>* suggest the fluid and particles density in order; *c* stands for volumetric coefficient; *V* implies the velocity column; *f* gives the forcing factor; *P* delivers the pressure term; *e* represents the nanoparticles strength; *T*0, *C*0, and ϕ<sup>0</sup> describe the contextual representatives of *T*, *C*, and ϕ at lower wall, respectively; and *T*1, *C*1, and ϕ<sup>1</sup> are the correspondent at the upper wall; *Db* depicts the Brownian diffusion factor; *Dt* the thermophoretic diffusion coefficient; β*<sup>t</sup>* shows the volumetric volume expansion coefficient for the liquid; β*<sup>c</sup>* is the cognate solutal coefficient; *Dct* represents the soret diffusivity; *Ds* reveals the solutal diffusivity; *Dtc* directs the Dufer diffusivity; and *S* sweeps the fluid model tensor. We use the following dimensional quantities

$$\begin{array}{ll} \mathbf{x}^\* = \frac{\mathbf{x}}{\lambda}, \ r\_1^\* = \frac{r\_1}{d\_1}, \ t^\* = \frac{\mathbf{c}}{\lambda}, w\_1^\* = \frac{w\_1}{d\_1}, \ k^\* = \frac{R\_1^\*}{d\_1},\\ p^\* = \frac{d\_1^2 p}{c \lambda \mu'}, S\_{ij}^\* = \frac{d\_1 S\_{ij}}{c \mu}, \ \theta = \frac{T - T\_0}{T\_1 - T\_0}, \ \phi = \frac{\mathbf{C} - \mathbf{C}\_0}{\mathbf{C}\_1 - \mathbf{C}\_0},\\ \gamma = \frac{q - \varphi\_0}{\overline{\varphi\_1} - \varphi\_0}, \ N\_c = \frac{\beta\_c \mathbf{C}\_0}{\overline{\rho\_1} T\_0}, \ N\_{r\_1} = \frac{\rho\_r - \rho\_f}{(1 - \varphi\_0)\rho\_f \beta\_7 T\_0}, \ \tau = \frac{(\rho c)\_p}{(\rho c)\_f} \end{array} \tag{6}$$

**Figure 1.** (**a**) Geometry of the problem. (**b**) Comparison of current work with literature [25].

The new discovered parameters like *Ld*, *Nb*, *Nc*, *Nd*, *Nr*<sup>1</sup> and *Nt* take place for a Dufour Lewis number, a Brownian motion parameter, the regular double-diffusive buoyancy ratio, a modified Dufour parameter, the nanofluid buoyancy ratio, and the thermophoresis parameter, accordingly. According to the mechanism of flow, the velocity field is supposed as **V** = (*v*, *u*). After using above defined parameters and applying the conditions of low Reynolds number and long wavelength, the Equations (1)–(5) get the next coming form

$$-\frac{l}{r\_1 + k\_1} \frac{\partial p}{\partial \mathbf{x}} + \frac{1}{r\_1 + k\_1} \frac{\partial}{\partial r\_1} [(r\_1 + k\_1)^2 S\_{r\_1 \mathbf{x}}] + N\_\mathbf{c} \gamma + \theta - N\_{r\_1} \phi = 0,\tag{7}$$

$$
\left[ \frac{\partial^2 \theta}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial \theta}{\partial r\_1} + N\_b \frac{\partial \phi}{\partial r\_1} \frac{\partial \theta}{\partial r\_1} + N\_l (\frac{\partial \theta}{\partial r\_1})^2 + N\_d \left[ \frac{\partial \gamma}{\partial r\_1} + \frac{1}{r\_1 + k\_1} \frac{\partial \gamma}{\partial r\_1} \right] \right] = 0,\tag{8}$$

$$
\left[\frac{\partial^2}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial}{\partial r\_1}\right] \phi + \frac{N\_t}{N\_b} \left[\frac{\partial^2}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial}{\partial r\_1}\right] \theta = 0,\tag{9}
$$

$$\left[\frac{\partial^2 \gamma}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial \gamma}{\partial r\_1} + L\_d \left[\frac{\partial^2 \theta}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial \theta}{\partial r\_1}\right] = 0,\tag{10}$$

by using the no-slip boundary conditions and compliant walls phenomenon [22,29]

$$\begin{array}{l} \mathcal{U} = c \,\,\text{at } r\_1 = \pm \eta = \pm (d\_1 + a \sin\left(\frac{2\pi(X - ct)}{\lambda}\right)) \\\mathcal{T} = T\_0 \,\,\text{at } r\_1 = -\eta \,\,\text{and}\,\,\mathcal{T} = T\_1 \,\,\text{at } r\_1 = \eta \\\mathcal{C} = \mathcal{C}\_0 \,\,\text{at } r\_1 = -\eta \,\,\text{and}\,\,\mathcal{C} = \mathcal{C}\_1 \,\,\text{at } r\_1 = \eta \\\mathcal{\boldsymbol{\eta}} = \boldsymbol{\eta} \boldsymbol{\eta} \,\,\,\text{at } r\_1 = -\eta \,\,\text{and}\,\,\boldsymbol{\eta} = \boldsymbol{\eta} \boldsymbol{\eta} \,\,\,\text{at } r\_1 = \eta \end{array} \Big| \begin{array}{l} \\ \mathcal{U} = \mathcal{U} \,\,\text{at } r\_1 = \eta \\\mathcal{\boldsymbol{\eta}} = \boldsymbol{\eta} \,\,\,\text{and}\,\,\boldsymbol{\eta} = \boldsymbol{\eta} \boldsymbol{\eta} \,\,\,\text{at } r\_1 = \eta \end{array} \Big| \begin{array}{l} \\ \mathcal{U} = \mathcal{U} \,\,\,\text{at } r\_1 = \eta \\\mathcal{U} = \mathcal{U} \,\,\,\,\text{at } r\_1 = \eta \,\,\,\,\,\end{array} \Big| \begin{array}{l} \\ \mathcal{U} = \mathcal{U} \,\,\,\, \,\, \mathcal{U} = \mathcal{U} \,\,\, \,\, \mathcal{U} = \mathcal{U} \,\,\, \,\, \mathcal{U} = \mathcal{U} \,\, \,\, \, \mathcal{U} = \mathcal{U} \,\, \, \, \, \,$$

After adopting wave frame phenomeno and creeping characteristics of the current ransport, we have the following conclusive form of the above-defined boundary relations in dimensionless format

$$
\mu = 0 \text{ at } \tau \mathbf{l} = \pm \eta = \pm (1 + \varepsilon \sin 2\pi (\mathbf{x} - \mathbf{t})),
\tag{11}
$$

$$
\theta = 0, \,\phi = 0, \,\text{y} = 0 \,\text{at} \,\, r\_1 = -\eta, \tag{12}
$$

$$
\theta = 1, \,\phi = 1, \,\eta = 1 \,\text{at} \, r\_1 = \eta, \tag{13}
$$

$$k\left[E\_1\frac{\partial^3}{\partial \mathbf{x}^3} + E\_2\frac{\partial^3}{\partial \mathbf{x}\partial t^2} + E\_3\frac{\partial^2}{\partial t\partial \mathbf{x}}\right]\eta = \frac{\partial p}{\partial \mathbf{x}}\text{ at }r\_1 = \pm \eta\_\prime \tag{14}$$

$$S\_{r\_1x} = -\mu\_{r\_1} + \frac{1}{r\_1 + k\_1}u - 2\beta (\mu\_{r\_1} + \frac{1}{r\_1 + k\_1}u)^3. \tag{15}$$

where *E*1, *E*2, and *E*<sup>3</sup> are the representatives of the compliant wall properties [10].

#### **3. Solution of the Problem**

We utilize the method of series expansion to solve coupled differential equations which are given before. The deformation equations for *u*, θ, γ, and φ are defined as [30]

$$\mathbb{E}\left[\left(1-q\right)\mathbb{E}[\mu-\mu\_{0}]+q\right]\begin{bmatrix}-\frac{l}{r\_{1}+k\_{1}}\frac{\partial A}{\partial x}+\frac{1}{r\_{1}+k\_{1}}\frac{\partial}{\partial r\_{1}}\left[\left(r\_{1}+k\_{1}\right)^{2}S\_{r\_{1}x}\right] \\ +N\_{c}r\_{1}+\theta-N\_{r\_{1}}\phi\end{bmatrix}=0,\tag{16}$$

$$\begin{aligned} (1-q)\mathbb{E}[\theta-\theta\_0]+q\begin{bmatrix} \frac{\partial^2\theta}{\partial r\_1^2} + \frac{1}{r\_1+k\_1}\frac{\partial\Omega}{\partial r\_1} + \mathcal{N}\_b\frac{\partial\varphi}{\partial r\_1}\frac{\partial\Omega}{\partial r\_1} + \mathcal{N}\_l\left(\frac{\partial\Omega}{\partial r\_1}\right)^2\\ + \mathcal{N}\_d\left[\frac{\partial\mathcal{V}}{\partial r\_1} + \frac{1}{r\_1+k\_1}\frac{\partial\mathcal{V}}{\partial r\_1}\right] \end{bmatrix} = \mathbf{0},\tag{17}$$

$$(1 - q)\mathbb{E}[\varphi - \varphi\_0] + q \left[ \left| \frac{\partial^2}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial}{\partial r\_1} \right| \varphi + \frac{N\_t}{N\_b} \left| \frac{\partial^2}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial}{\partial r\_1} \right| \theta \right] = 0,\tag{18}$$

$$q(1-q)\mathbb{E}[\gamma - \gamma \nu] + q \left[ \frac{\partial^2 \gamma}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial \gamma}{\partial r\_1} + L\_d \left[ \frac{\partial^2 \theta}{\partial r\_1^2} + \frac{1}{r\_1 + k\_1} \frac{\partial \theta}{\partial r\_1} \right] \right] = 0. \tag{19}$$

where £ is the linear operator which is chosen as £ = <sup>∂</sup><sup>2</sup> ∂*r*<sup>2</sup> 1 . The initial guesses for *u*, θ, φ, and γ are defined as

$$\begin{split} \overline{u}\_{0} = \frac{1}{2w\_{1}} [-2(k\_{1} + r\_{1})w \ln \left(k\_{1} + r\_{1}\right) + (k\_{1} - w\_{1})(-r\_{1} + w\_{1}) \ln \left(k\_{1} - w\_{1}\right) + \\ (k\_{1} + w\_{1})(r\_{1} + w\_{1}) \ln \left(k\_{1} + w\_{1}\right)] \end{split} \tag{20}$$

$$\begin{aligned} \overline{\theta}\_0 &= \frac{1}{2w\_1} [-2(k\_1 + r\_1)w\_1 \ln\left(k\_1 + r\_1\right) + (k\_1 - w\_1)(r\_1 + w\_1)\ln\left(k\_1 - w\_1\right) + \\ &\quad (k\_1 + w\_1)(r\_1 + w\_1)\ln\left(k\_1 + w\_1\right)], \end{aligned} \tag{21}$$

$$\overline{\varphi}\_{0} = \frac{1}{2w\_{1}} [-2(k\_{1} + r\_{1})w\_{1} \ln\left(k\_{1} + r\_{1}\right) + (k\_{1} - w\_{1})(-r\_{1} + w\_{1})\ln\left(k\_{1} - w\_{1}\right) + \\\frac{1}{2}(k\_{1} + w\_{1})\ln\left(k\_{1} + w\_{1}\right)],\tag{22}$$

$$\begin{split} \overline{\gamma}\_{0} = \frac{1}{2w\_{1}} [-2(k\_{1} + r\_{1})w \ln \left(k\_{1} + r\_{1}\right) + (r\_{1} - w\_{1})(r\_{1} + w\_{1}) \ln \left(k\_{1} - w\_{1}\right) + \\ (k\_{1} + w\_{1})(r\_{1} + w\_{1}) \ln \left(k\_{1} + w\_{1}\right)]. \end{split} \tag{23}$$

Now we use the following perturbation series for *u*, θ, γ, and φ

$$\begin{aligned} \mu &= \mu\_0 + q\mu\_1 + \dots \\ \theta &= \theta \mathbf{o} + q\theta\_1 + \dots \\ \gamma &= \gamma\_0 + q\gamma\_1 + \dots \\ \rho &= \rho \mathbf{o}\_0 + q\rho \mathbf{o}\_1 + \dots \end{aligned} \tag{24}$$

After using the above series solutions in Equations (11) to (14) and comparing the coefficients of *q*, we get the same solutions for zeroth order terms and the first order systems found the following solutions

$$\mu\_1 = \mathbb{C}\_1 + r\mathbb{C}\_2 - \frac{1}{4\eta^3} (\frac{1}{9}r\eta^2 (6k(6 + k + k\mathbb{N}\_{\mathbb{C}} - k\mathbb{N}\_{\mathbb{T}1} + 6A)\eta + 3(3 + 5k)(1 + \mathbb{N}\_{\mathbb{C}} - \mathbb{N}\_{\mathbb{T}1})))$$

*<sup>r</sup>*<sup>η</sup> + (<sup>1</sup> + Nc <sup>−</sup>Nr1)*r*2(<sup>3</sup> + <sup>5</sup>η)) + <sup>1</sup> 12(*k*+*r*) <sup>3</sup> (768(*k* + *r*) 3 βη3Log[*k* + *r*] 3 −3β(*k* − η) 3 (*k* + η)(17*k*<sup>2</sup> + 40*kr* + 24*r*<sup>2</sup> − 6*k*η − 8*r*η + η2)Log[*k* − η] 3 +288(*k* + *r*) 2 βη2Log[*k* + *r*] 2 (12(*k* + *r*)η + (*k* − η)(3*k* + 4*r* − η)Log[*k* − η] −(*k* + η)(3*k* + 4*r* + η)Log[*k* + η]) + β(*k* − η) 2 (*k* + η)Log[*k* − η] 2 (−8(*k* + *r*)(31*k* + 36*r* − 5η)η +3(3*k*(17*k*<sup>2</sup> + 40*kr* + 24*r*2)η + (5*k* + 8*r*)η<sup>2</sup> − 3η3)Log[*k* + η]) +(*k* + η)Log[*k* + η](4(*k* + *r*) 2 η2(*r*3(6 + *r* + Nc*r* − Nr1*r*) + 3(1 + (1 + Nc − Nr1)*r*<sup>3</sup> − 30β)η +*k*(−3 + 90β + *r*2(6 + *r* + Nc*r* − Nr1*r* + 3(1 + Nc − Nr1)η))) +β(*k* − η)(*k* + η)Log[*k* + η](−8(*k* + *r*)η(31*k* + 36*r* + 5η) +3(*k* + η)(17*k*<sup>2</sup> + 40*kr* + 24*r*<sup>2</sup> + 6*k*η + 8*r*η + η2)Log[*k* + η])) −(*k* − η)Log[*k* − η](4(*k* + *r*) 2 η2(*r*3(6 + *r* + Nc*r* − Nr1*r*) − 3(1 + (1 + Nc − Nr1)*r*<sup>3</sup> − 30β)η +*k*(−3 + 90β + *r*2(6 + *r* + Nc*r* − Nr1*r* − 3(1 + Nc − Nr1)η))) +β(*k* + η)Log[*k* + η](−16(*k* + *r*)η(31*k*2*kr* + 5η2) + 3(*k* + η)(3*k*(17*k*<sup>2</sup> + 40*kr* + 24*r*2) −(11*k*<sup>2</sup> + 32*kr* + 24*r*2)η + (5*k* + 8*r*)η<sup>2</sup> + 3η3)Log[*k* + η])) −8(*k* + *r*)ηLog[*k* + *r*]((*k* + *r*) 2 (6 + (*k* + *r*)(*k*2(1 + Nc − Nr1) +*r*(6 + *r* + Nc*r* − Nr1*r*) + 2*k*(3 + 3*A* + *r* + Nc*r* − Nr1*r*)) − 936β)η<sup>2</sup> +6β(−(*k* − η) 2 (7*k*<sup>2</sup> + 18*k* + 12*r*<sup>2</sup> − 4*k*η − 6*r*η + η2)Log[*k* − η] 2 +2(*k* − η)Log[*k* − η](−6(*k* + *r*)(5*k* + 6*r* − η)η +(*k* + η)(7*k*<sup>2</sup> + 18*kr* + 12*r*<sup>2</sup> − η2)Log[*k* + η]) +(*k* + η)Log[*k* + η](12(*k* + *r*)η(5*k* + 6*r* + η) −(*k* + η)(7*k*<sup>2</sup> + 18*kr* + 12*r*<sup>2</sup> + 4*k*η + 6*r*η + η2)Log[*k* + η]))))), (25) <sup>θ</sup><sup>1</sup> = *<sup>C</sup>*<sup>3</sup> + *rC*<sup>4</sup> <sup>−</sup> <sup>1</sup> <sup>4</sup>η<sup>2</sup> (2(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*)(−<sup>1</sup> <sup>−</sup> Nd + (Nb <sup>+</sup> Nt)(*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*))η2Log[*<sup>k</sup>* <sup>+</sup> *<sup>r</sup>*] 2 +2(*k* + *r*)η + Log[*k* + *r*](1 + Nd − (Nb + Nt)(*k* + *r*) − 2η −(2Nd + (Nb + Nt)(*k* + *r*))η + (−1 − Nd + (Nb + Nt)(*k* +*r*))((*<sup>k</sup>* <sup>−</sup> <sup>η</sup>)Log[*<sup>k</sup>* <sup>−</sup> <sup>η</sup>] <sup>−</sup> (*<sup>k</sup>* + <sup>η</sup>)Log[*<sup>k</sup>* + <sup>η</sup>])) + <sup>1</sup> <sup>2</sup> *r*((Nb +Nt)*r* + 2(−2(1 + Nd)+(Nb + Nt)(2*k* + *r*))η + 2(4 + 4Nd +2*k*(Nb + Nt) + <sup>3</sup>(Nb + Nt)*r*)η<sup>2</sup> + ((*<sup>k</sup>* <sup>−</sup> <sup>η</sup>)Log[*<sup>k</sup>* <sup>−</sup> <sup>η</sup>] <sup>−</sup> (*<sup>k</sup>* +η)Log[*k* + η])(−2(Nb + Nt)*r* − 2(−2(1 + Nd)+(Nb +Nt)(2*k* + *r*))η + (Nb + Nt)*r*((*k* − η)Log[*k* − η] − (*k* +η)Log[*k* + η])))), (26) φ<sup>1</sup> = *C*<sup>5</sup> + *rC*<sup>6</sup> + <sup>1</sup> <sup>2</sup>*Nb*<sup>η</sup> ((Nb + Nt)(*<sup>k</sup>* + *<sup>r</sup>*)(<sup>1</sup> <sup>−</sup> <sup>2</sup><sup>η</sup> + <sup>η</sup>Log[*<sup>k</sup>* + *<sup>r</sup>*] 2 +(−*k* + η)Log[*k* − η] + (*k* + η)Log[*k* + η] − Log[*k* + *r*](1 − 2η (27)

+(−*k* + η)Log[*k* − η] + (*k* + η)Log[*k* + η]))),

$$\begin{array}{ll} \gamma\_1 = \mathsf{C}\_7 + r \mathsf{C}\_8 & + \frac{1}{2\eta} ((1 + \mathsf{L}\_d)(k + r)(1 - 2\eta + \eta \mathsf{Log}[k + r]^2 \\ & + (-k + \eta) \mathsf{Log}[k - \eta] + (k + \eta) \mathsf{Log}[k + \eta] - \mathsf{Log}[k + r](1 - 2\eta \\ & + (-k + \eta) \mathsf{Log}[k - \eta] + (k + \eta) \mathsf{Log}[k + \eta]))), \end{array} \tag{28}$$

where the constants *Ci*, *i* = 1, 2, 3, ... 8 can be found by using boundary conditions are described in the Appendix A and the quantity *A*(*x,t*) contains the subsequent expression

$$A(\mathbf{x},t) = -2\varepsilon\pi^3 k \left(\frac{E\_3}{2\pi}\sin\left(\mathbf{x}-t\right)2\pi - \left(E\_1 + E\_2\right)\cos\left(\mathbf{x}-t\right)2\pi\right). \tag{29}$$

Therefore, the final solutions can be composed by injecting above evaluated expressions of *u*0, θ0, φ0, γ<sup>0</sup> and *u*1, θ1, φ1, γ<sup>1</sup> into Equation (24).

The expression for the heat transfer coefficient is described as

$$z = \eta\_{\mathbf{x}} \theta\_{\mathbf{f}}(\eta). \tag{30}$$

Hence, it is calculated as

$$\begin{array}{l} z = & -\frac{2}{2+2\epsilon\text{Li}[2\pi(-t+x)]}\pi\epsilon\mathbb{E}\cos[2\pi(-t+x)](\mathsf{5}(\text{Nb}+\text{Nt})-4\mathsf{C}\_{4}+(-1+k)^{2}(\text{Nb}+\text{Nt})) \\ & + \mathsf{Nt}\,\mathsf{I}\,\mathsf{Log}[-1+k+\epsilon\text{Si}[2\pi(t-t)]] \\ & + 2(-1+k)\mathsf{Log}[-1+k+\epsilon\text{Si}[2\pi(t-t)]](-\mathsf{Nb}-\mathsf{Nt} \\ & + (-1+\mathsf{Nt}+k\text{Nb}-\mathsf{Nt}+\mathsf{Nt}+k\text{Nt})\mathrm{Log}[1+k+\epsilon\text{Si}[2\pi(-t+x)]]) + \mathsf{Log}[1+k] \\ & + \epsilon\text{Si}[2\pi(-t+x)]((-2\langle\mathsf{3}+\mathsf{Nb}+k\mathsf{N}\mathsf{N}+\mathsf{N}\mathsf{N}+k\mathsf{N})+(\mathsf{Nb}+\mathsf{Nt}-\mathsf{N}^{2}\,\mathsf{(Nb} \\ & + \mathsf{Nt})-2\mathsf{k}(-1+\mathsf{Nt}-\mathsf{N}\mathsf{N}+\mathsf{N})\mathrm{Log}[1+k+\epsilon\mathsf{Si}[2\pi(t-t+x)]]) + \epsilon\text{Si}[2\pi(t-t+x)] \\ & + \mathsf{x}]((2\mathsf{4}(\mathsf{Nb}+\mathsf{Nt})-2\mathsf{C}[2]-(-1+k)(\mathsf{Nb}+\mathsf{Nt})\mathrm{Log}[-1+k+\epsilon\mathsf{Si}[2\pi(t-t)]])^{2} \\ & - (\mathsf{4}+\mathsf{Nb}+\mathsf{Nd}+\mathsf{Nt})\mathrm{Log}[1+k+\epsilon\mathsf{Si}[2\pi(t-t+x)]]-(-1+k)(\mathsf{Nb}+\mathsf{Nt})\mathrm{Log}[1+k+\mathsf{N}] \\ & + \epsilon\text{Si}[2\pi(t-t+x)]]^{2}+\mathsf{Log$$

#### **4. Graphical Results and Discussion**

The above analysis composes the effects of double diffusion on pumping flow of non-Newtonian (third order) fluid travelling through a curved channel and also described the wall properties. The formulation is carried out by introducing non-dimensional parameters and imposing the features of the lubrication approach. After achieving system of four nonlinear coupled differential equations, exact analytical solutions have been found by an appropriate analytical highly converging technique (HPM). In this segment of the study, we have included graphical treatment of various obtained quantities like comparison graph, velocity, temperature, solutal concentration, and nanoparticle phenomenon. Figure 1b is included just to validate the present results by comparing analytical solution with exact solution [25]. This graph contains the data of velocity obtained in the current study by neglecting the effects of double diffusion convection (*Nc* = *Nr*<sup>1</sup> = 0) and the data of [25]. One can find the reading that the current analytical solutions are very much in agreement with the exact solutions found by Hayat et al. [25]. In Figure 2, the velocity is displayed under the variation of the regular buoyancy ratio *Nc*. We conclude from this figure that the velocity of fluid is increasing with increasing quantity of *Nc* and become highest in the middle part of the channel. This result stresses that *Nc* being the ration of concentration variance to temperature gradient, when gets increased meant that concentration change is higher than the temperature difference which is actually causing the fluid to travel with greater intensity. From Figure 3, it is very clear that the velocity is showing totally opposite behavior against the buoyancy parameter *Nr*<sup>1</sup> as compared to *Nc* which is also prominent physically that when we increase the density of particles the fluid travels slowly. Figure 4 is portrayed to find the influence of complaint wall parameters *E*1, *E*2, *E*<sup>3</sup> and it can be concluded here that the velocity of the nanofluid is minimized with the complaint wall parameters. In Figure 5, the temperature profile

θ is portrayed to measure the influence of the regular buoyancy parameter *Nb*. According to this graph, it is noticed that the temperature is increasing with the increasing value of *Nb* and the highest temperature is observed at *r* = 0.2, which is near the centerline of channel. Figure 6 is showing the effect of a modified Dufour parameter *Nd* on temperature profile θ. This graph is emphasized that *Nd* is lowering the temperature throughout the geometry which represents the cooling effects due to change in nanoparticles concentration. The temperature profile θ for various values of thermophoresis parameter *Nt* is plotted in Figure 7. According to this figure, we can analyze that as we increase the magnitude of *Nt*, the temperature θ is also increased and gets higher magnitude in the central region of the channel. Figures 8 and 9 highlights the variation of nanoparticle concentration φ when there is an increase the values of Brownian motion parameter *Nb* and thermophoresis parameter *Nt*. It can be supposed from these figures that nanoparticle concentration is increasing with *Nb* but decreasing with *Nt*. It is also observed that nanoparticles are less in numbers in the central part and minimum quantity is at the position *r* = *h*. Figures 10 and 11 are drawn to manage the behavior of curvature parameter *k* and Defour-Solutal Lewis number *Ld* on solutal concentration. Figure 10 depicts that γ is increasing with the increasing values of *k*. It means that as we use the curved channel with large curvature, the solutal concentration will get increased. On the other hand, Figure 11 emphasizes that γ is decreasing with *Ld* and quite opposite behaviour is observed in this figure as we have seen from Figure 10. Figures 12–14 are captured to visualize the effects the *Nb*, *Nd*, and *Nt* respectively on the heat transfer coefficient *z*. It is found from these figures that heat transfer is decreasing with *Nb* and *Nt* on the left and right sides but increasing in the centre. It is depicted here that *Nd* reflects the opposite behaviour on heat transfer. It is also noted from Figures 12–14 that amount of heat transfer is maximum at the center of the channel.

**Figure 2.** Alteration of *Nc* on u when *x* = 0.2; *t* = 0; *Q* = 10; β = 0.1; = 0.6; *Nr*<sup>1</sup> = 2; *k* = 2; *E*<sup>1</sup> = 0.1; *E*<sup>2</sup> = 0.1; *E*<sup>3</sup> = 0.9.

**Figure 3.** Alteration of *Nr*<sup>1</sup> on u when *x* = 0.2; *t* = 0; *Q* = 10; β = 0.1; = 0.6; *Nc* = 2; *k* = 2; *E*<sup>1</sup> = 0.1; *E*<sup>2</sup> = 0.1; *E*<sup>3</sup> = 0.9.

**Figure 4.** Variation of complaint wall parameters on u when = 0.6; β = 0.01; *k* = 2; *x* = 0.2; *t* = 0; *Nc* = 2; *Nr*<sup>1</sup> = 1; *Q* = 10.

**Figure 5.** Alteration of *Nb* on θ when *x* = 0.5; *t* = 0; *Q* = 10; β = 0.2; = 0.6; *Nt* = 5; *Nd* = 2; *k* = 2.

**Figure 6.** Alteration of *Nd* on θ when *x* = 0.5; *t* = 0; *Q* = 10; β = 0.2; = 0.6; *Nt* = 5; *Nb* = 2; *k* = 2.

**Figure 7.** Alteration of *Nt* on θ when *x* = 0.5; *t* = 0; *Q* = 10; β = 0.2; = 0.2; *Nb* = 5; *Nd* = 2; *k* = 2.

**Figure 8.** Alteration of *Nb* on φ when *x* = 0.1; *t* = 0; *Q* = 10; β = 0.2; = 0.1; *Nt* = 10; *Nd* = 2; *k* = 2.

**Figure 9.** Alteration of *Nt* on φ when *x* = 0.1; *t* = 0; *Q* = 10; β = 0.2; = 0.1; *Nb* = 10; *Nd* = 2; *k* = 2.

**Figure 10.** Alteration of k on γ when *x* = 0.1; *t* = 0; *Q* = 10; β = 0.2; = 0.1; *Nb* = 2; *Nd* = 1; *Ld* = 0.1.

**Figure 11.** Alteration of *Ld* on γ when *x* = 0.1; *t* = 0; *Q* = 10; β = 0.2; = 0.1; *Nb* = 2; *Nd* = 1; *k* = 1.5.

**Figure 12.** Alteration of *Nb* on *z* when *t* = 0; *Q* = 10; β = 0.2; = 0.2; *Nt* = 15; *Nd* = 2; *k* = 2.

**Figure 13.** Alteration of *Nd* on *z* when *t* = 0; *Q* = 10; β = 0.2; = 0.2; *Nt* = 15; *Nb* = 20; *k* = 2.

**Figure 14.** Alteration of *Nt* on *z* when *t* = 0; *Q* = 10; β = 0.2; = 0.2; *Nb* = 5; *Nd* = 2; *k* = 2.
