2. Constitutive equations

The behaviour of an extra stress tensor for the linear PTT fluid model is (as seen in [Reference Hayat, Noreen, Ali and Abbasbanday16, Reference Vajravelu, Sreenadh, Dhananjaya and Lakshminarayana39])

$$ \begin{align*} T&=-pI+\tau,\\f(tr({\tau}))\tau+K\tau^{\nabla}&=2\mu \mathbb{D},\\\tau^{\nabla}&=\frac{d\tau}{dt}-\tau.\mathbb{L}^\prime-\mathbb{L}.\tau,\\f(tr(\tau))&=1+\frac{\epsilon \kappa}{\mu}tr(\tau),\end{align*} $$

where $\mathbb {L}=\text {grad }V$ .

In the above equations, p is the pressure, I is the identity tensor, $\mu $ is the dynamic viscosity, $\tau $ is the extra stress tensor, $\mathbb {D}$ is the deformation rate tensor, K is the relaxation time, $\tau ^{\nabla }$ denotes Oldroyds’s upper convected derivative, ${d}/{dt}$ is the material time derivative, $tr$ is the trace, V is the velocity vector and prime indicates the transpose. It is significant to note that when the extensional parameter is zero, the PTT model simplifies to an upper convicted Maxwell model.

3. Mathematical formulation

The geometric model of the electro-osmotic peristaltic flow of incompressible viscoelastic fluid in the current study is illustrated in Figure 1. We choose the electro-osmotic flow of a PTT (aqueous ionic solution) fluid through a two-dimensional channel of finite length (L) and uniform thickness of ( $2a$ ), with wavelength $\lambda $ and wave speed C. The Cartesian coordinate system ( $x,y$ ) is used in such a way that propagation of the wave is along the x-axis, and the y-axis is considered perpendicular to the direction of the flow. Motion is generated due to propagation of an infinite wave train travelling along the channel wall with speed C. The electro-osmotic flow occurs by the applied external electric field $E_x$ that flows parallel to the channel walls. As a result, the channel experiences electro-osmotic flow as the positive (cations) and negative (anions) ions form the EDL.

Figure 1 Geometry of the problem.

The simulation involves the geometry of deforming channel walls:

(3.1) $$ \begin{align} h(\bar{x},\bar{t})=a+b\sin\bigg(\frac{2\pi}{\lambda}(\bar{X}-C\bar{t})\bigg). \end{align} $$

Here, b, C, a and t stand for wave amplitude, wave velocity, half-width of the channel and time, respectively.

The motion is unsteady because of the moving boundary in the stationary frame ( $\bar {X},\bar {Y}$ ). However, it can be viewed as steady in the moving frame ( $\bar {x}, \bar {y}$ ). As a result, we can represent the transformations between the fixed and wave frames:

$$ \begin{align*} \bar{x}=\bar{X}-C\bar{t},\quad \bar{y}=\bar{Y}, \quad \bar{u}(\bar{x},\bar{y})=\bar{U}-C, \quad \bar{v}(\bar{x},\bar{y})=\bar{v}, \quad \bar{p}(\bar{x})=\bar{P}(\bar{X},\bar{t}). \end{align*} $$

The fundamental equations of continuity, momentum equations with temperature and thermal radiation may be constructed in accordance with the aforementioned [Reference Hayat, Noreen, Ali and Abbasbanday16, Reference Vajravelu, Sreenadh, Dhananjaya and Lakshminarayana39].

The equations, which can govern the present flow circumstances, are as follows:

(3.2) $$ \begin{align} \frac{\partial \bar{u}}{\partial \bar{x}}+\frac{\partial \bar{v}}{\partial \bar{y}}=0, \end{align} $$

(3.3) $$ \begin{align} \rho\bigg(\bar{u}\frac{\partial \bar{u}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{u}}{\partial \bar{y}}\bigg)=-\frac{\partial \bar{p}}{\partial \bar{x}}+\frac{\partial \bar{\tau}_{xx}}{\partial \bar{x}}+\frac{\partial \bar{\tau}_{xy}}{\partial \bar{y}}+\bar{\rho}_{e}\bar{E}_x, \end{align} $$

(3.4) $$ \begin{align} \rho\bigg(\bar{u}\frac{\partial \bar{v}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{v}}{\partial \bar{y}}\bigg)=-\frac{\partial \bar{p}}{\partial \bar{y}}+\frac{\partial \bar{\tau}_{yx}}{\partial \bar{x}}+\frac{\partial \bar{\tau}_{yy}}{\partial \bar{y}}+\bar{\rho}_{e}\bar{E}_y, \end{align} $$

(3.5) $$ \begin{align} \rho c_p\bigg(\bar{u}\frac{\partial \bar{T}}{\partial \bar{x}}+\bar{v}\frac{\partial \bar{T}}{\partial \bar{y}}\bigg)=k\bigg(\frac{\partial^2 \bar{T}}{\partial \bar{x}^2}+\frac{\partial^2 \bar{T}}{\partial \bar{y}^2}\bigg)+\bar{\tau}_{xx}\frac{\partial \bar{u}}{\partial \bar{x}}+\bar{\tau}_{yy}\frac{\partial \bar{v}}{\partial \bar{y}}+\bar{\tau}_{xy}\bigg(\frac{\partial \bar{u}}{\partial \bar{y}}+\frac{\partial \bar{v}}{\partial \bar{x}}\bigg)+\frac{\partial}{\partial \bar{y}}(\bar{q}_r), \end{align} $$

(3.6) $$ \begin{align} f\bar{\tau}_{xx}+k\bigg(\bar{u}\frac{\partial\bar{\tau}_{xx}}{\partial \bar{x}}+\bar{v}\frac{\partial\bar{\tau}_{xx}}{\partial \bar{y}}-2\frac{\partial\bar{u}}{\partial\bar{x}}\bar{\tau}_{xx}- 2\frac{\partial\bar{u}}{\partial\bar{y}}\bar{\tau}_{xy}\bigg)=2\mu\frac{\partial\bar{u}}{\partial\bar{x}}, \end{align} $$

(3.7) $$ \begin{align} f\bar{\tau}_{yy}+k\bigg(\bar{u}\frac{\partial\bar{\tau}_{yy}}{\partial \bar{x}}+\bar{v}\frac{\partial\bar{\tau}_{yy}}{\partial \bar{y}}-2\frac{\partial\bar{v}}{\partial\bar{x}} \bar{\tau}_{yx}-2\frac{\partial\bar{v}}{\partial\bar{y}}\bar{\tau}_{yy}\bigg)=2\mu \frac{\partial\bar{u}}{\partial\bar{y}}, \end{align} $$

(3.8) $$ \begin{align} f\bar{\tau}_{zz}+k\bigg(\bar{u}\frac{\partial\bar{\tau}_{zz}}{\partial \bar{x}}+\bar{v}\frac{\partial\bar{\tau}_{zz}}{\partial \bar{y}}\bigg)=0, \end{align} $$

(3.9) $$ \begin{align} f\bar{\tau}_{xy}+k\bigg(\bar{u}\frac{\partial\bar{\tau}_{xy}}{\partial \bar{x}}+\bar{v}\frac{\partial\bar{\tau}_{xy}}{\partial \bar{y}}-\frac{\partial\bar{v}}{\partial\bar{x}}\bar{\tau}_{xx}-\frac{\partial\bar{u}}{\partial\bar{x}}\bar{\tau}_{xy}-\frac{\partial \bar{u}}{\partial\bar{y}}\bar{\tau}_{yy}\bigg)= \mu\bigg(\frac{\partial\bar{u}}{\partial\bar{y}}+\frac{\partial\bar{v}}{\partial\bar{x}}\bigg), \end{align} $$

(3.10) $$ \begin{align} f=1+\frac{\epsilon k}{\mu}(\bar{\tau}_{xx}+\bar{\tau}_{yy}+\bar{\tau}_{zz}). \end{align} $$

Here, $\bar {\rho }_{e}\bar {E}_x$ represents the electro-kinetic body force, $\bar {p}$ represents the pressure, $\epsilon $ is the PTT fluid parameter, $\bar {\tau }$ is the stress tensor, $\bar {T}$ is the temperature, k denotes the thermal conductivity, $\bar {u}, \bar {v}$ are the respective velocity components in the radial y- and axial x-directions, $c_p$ is the specific heat constant at pressure, $\mu $ is the viscosity, and $\rho $ is the density of the fluid.

Using Rosseland’s approximation [Reference Vajravelu, Sreenadh, Dhananjaya and Lakshminarayana39], the radiative heat flux ( $q_r$ ) is defined as

(3.11) $$ \begin{align} q_r=\frac{-4\bar{\sigma}}{3\bar{k}}\frac{\partial\bar{T}^\ast}{\partial\bar{y}}\quad\text{and}\quad q=-\kappa\nabla T. \end{align} $$

Here $\sigma $ , k and q stand for the Stefan–Boltzman constant, thermal conductivity and vector of local heat flux, respectively. We also suppose that the temperature of the fluid within the flow region is sufficiently small. By expanding $T^4$ about $T_0$ and ignoring the higher order term obtained,

(3.12) $$ \begin{align} T_0^4=4T_{0}^3 T-3T_0^4, \end{align} $$

and the above appearance and equation (3.11) now yield

(3.13) $$ \begin{align} q_r=-\frac{16\sigma \bar{T}^3}{3\kappa \mu c_f}\frac{\partial\bar{T}}{\partial\bar{y}}. \end{align} $$

From the Gauss law,

(3.14) $$ \begin{align} \nabla.\bar{E}=\frac{\bar{\rho}_e}{\epsilon}, \end{align} $$

where

(3.15) $$ \begin{align} \bar{E}=-\nabla\Phi. \end{align} $$

Using equation (3.14) in equation (3.15) gives the equation for electrical potential distribution as

(3.16) $$ \begin{align} \nabla^2 \Phi=-\frac{\bar{\rho}_e}{\epsilon}, \end{align} $$

in which $\bar {\rho }_e$ and $\epsilon $ are the density of the total ionic energy and electric permittivity, respectively. The ions and counter-ions have the equal charge valance $z_+=z_-=z$ for a symmetric electrolyte.

Thus, the total density of the ionic charge is given by

(3.17) $$ \begin{align} \bar{\rho}_e=ez(\bar{n}_+-\bar{n}_-), \end{align} $$

where e and z stands for elementary charge and charge balance, respectively. The cations (positive) $\bar {n}_+$ and anions (negative) $\bar {n}_-$ have bulk concentration (number density). The charge number density must be determined to establish the potential distribution. The Nernst–Plank equation is used to define the distribution of ions inside the fluid as given by Tripathi et al. [Reference Tripathi, Sharma, Bég and Tiwari38]:

(3.18) $$ \begin{align} \frac{\partial\bar{n}_\pm}{\partial\bar{t}}+ \bar{u}\frac{\partial\bar{n}_\pm}{\partial\bar{x}}+ \bar{v}\frac{\partial\bar{n}_\pm}{\partial\bar{y}}=\mathbf{D} \bigg(\frac{\partial^2\bar{n}_\pm}{\partial\bar{x}^2}+\bar{v}\frac{\partial^2\bar{n}_\pm}{\partial\bar{y}^2}\bigg) \pm\frac{\mathbf{D}ez}{T_eK_B} \bigg[\frac{\partial}{\partial\bar{x}}\bigg(\bar{n}_\pm \frac{\partial\bar{\Phi}}{\partial\bar{x}}\bigg) + \frac{\partial}{\partial\bar{y}}\bigg(\bar{n}_\pm\frac{\partial\bar{\Phi}}{\partial\bar{y}}\bigg)\bigg]. \end{align} $$

Here it is presumed that for both species, the ionic diffusion coefficients are equal and that the mobility of the species is determined by the Einstein formula where $\mathbf {D}$ is the diffusion coefficient, $K_B$ is the Boltzmann constant and $T_e$ is the average temperature of the electrolytic solution.

Equation (3.16) is determined as follows in this limit:

(3.19) $$ \begin{align} \frac{\partial^2\Phi}{\partial y^2}=-m^2\bigg(\frac{n_{+}-n_{-}}{2} \bigg), \end{align} $$

where $m=aez\sqrt {{2n_0}/{\epsilon K_B T_e}}={a}/{\lambda _d}$ is known as the Debye–Hückel parameter [Reference Hayat, Noreen, Ali and Abbasbanday16], $\lambda _d$ is the EDL or the Debye length. Additionally, the simplified Nernst–Planck equations are intended to provide the ionic distribution:

(3.20) $$ \begin{align} \frac{\partial^2n_{+}}{\partial y^2}\pm\frac{\partial}{\partial y}\bigg(n_{\pm}\frac{\partial\Phi}{\partial y} \bigg)=0. \end{align} $$

Equation (3.20) with bulk $n_{\pm }=1$ at $\Phi =0,\ {\partial n_{\pm }}/{\partial y}=0$ at ${\partial \Phi }/{\partial y}=0,$ yields the Boltzmann distribution for the ions:

(3.21) $$ \begin{align} n_{\pm}=e^{\pm\Phi}. \end{align} $$

Substituting equation (3.21) into equation (3.20), one gets the Poisson–Boltzmann paradigm for calculating the distribution of electrical potentials

(3.22) $$ \begin{align} \frac{\partial^2\Phi}{\partial y^2}=-m^2\sinh(\Phi). \end{align} $$

Equation (3.22) has to be simplified to proceed with the analysis further. Under Debye–Hückel’s linearization [Reference Tripathi, Bushan and Bég36] (that is, $\Phi <25$ mV over a large pH range), $\sinh (\Phi )\approx \Phi $ . Consequently, equation (3.22) may be reduced to

(3.23) $$ \begin{align} \frac{\partial^2\Phi}{\partial y^2}=-m^2\Phi. \end{align} $$

Employing the boundary conditions ${\partial \Phi }/{\partial y}=0$ at $y=0$ , and $\Phi =1$ at $y=h$ to solve equation (3.23),

(3.24) $$ \begin{align} \Phi=\frac{\cosh(my)}{\cosh(mh)}, \end{align} $$

where the electro-osmotic parameter is denoted by $m={a}/{\lambda _D}$ .

The condition in equation (3.1) can be written as

(3.25) $$ \begin{align} h=1+\phi\sin(2\pi x). \end{align} $$

We introduce some nondimensional parameters as follows:

$$ \begin{align*} \begin{aligned} W_e&=\frac{kc}{a},\, u=\frac{\bar{u}}{C},\, v=\frac{\bar{v}}{C\delta},\,x=\frac{\bar{x}}{\lambda},\, y=\frac{\bar{y}}{a},\, h=\frac{\bar{h}}{a},\,\delta=\frac{a}{\lambda},\, p=\frac{\bar{p}a^2}{\mu c\lambda},\, \theta=\frac{\bar{T}-\bar{T}_0}{\bar{T}_0},\\ n&=\frac{\bar{n}}{n_0},\, t=\frac{C\bar{t}}{\lambda}, \, pr=\frac{\mu c_p}{K}, \, Ec=\frac{C^2}{c_f(\bar{T}_0)},\, \phi=\frac{b}{C}, \, \tau_{ij}=\frac{a\bar{\tau}_{ij}}{C\mu}, \, U_{hs}=-\frac{\psi\epsilon\bar{E}_x}{\mu C}, \\ \, Br&=Ecpr,\,Re=\frac{\rho Ca}{\mu}, \, \Phi=\frac{\bar{Z_e}\Phi}{K_BT_e},\, Rd=-\frac{16\delta \bar{T}^3}{3\kappa \mu c_f}, \end{aligned} \end{align*} $$

where $\delta $ is the wave number, $W_e$ is the Weissenberg number, $Re$ is the Reynolds number, $Br$ is the Brinkmann number, $pr$ is the Prandtl number and $U_{hs}$ is the Helmholtz–Smoluchowsky velocity.

Using the aforementioned nondimensional parameters and in accordance with the assumptions of large wave length ( $\delta \ll 1$ ), low Reynolds number ( $Re\ll 1$ ) and the Debay–Hückel linearization, the fundamental equations (3.3)–(3.10) can be scaled in the form:

(3.26) $$ \begin{align} \frac{\partial p}{\partial x}&=\frac{\partial\tau_{xy}}{\partial y}-m^2 U_{hs}\frac{\partial^2\Phi}{\partial y^2}, \end{align} $$

(3.27) $$ \begin{align} \frac{\partial p}{\partial y}&=0, \end{align} $$

(3.28) $$ \begin{align} f\tau_{xx}&=2W_e \frac{\partial u}{\partial y}\tau_{xy}, \end{align} $$

(3.29) $$ \begin{align} f\tau_{yy}&=0=f\tau_{zz}, \end{align} $$

(3.30) $$ \begin{align} f\tau_{xy}&=W_e \frac{\partial u}{\partial y}\tau_{yy}+\frac{\partial u}{\partial y}, \end{align} $$

and

(3.31) $$ \begin{align} (1+prRd)\frac{\partial^2\theta}{\partial y^2}=-Br\tau_{xy}\frac{\partial u}{\partial y}. \end{align} $$

The appropriate nondimensional boundary conditions are

(3.32) $$ \begin{align} \left\{\!\! \begin{array}{lll} \dfrac{\partial u}{\partial y}=0, & v=0 &\text{at } y=0,\\ u=-1 ,&v=-\dfrac{dh}{dx}&\ \text{at } y=h, \end{array} \right. \end{align} $$

(3.33) $$ \begin{align} \frac{\partial\theta}{\partial y}=0\quad\text{at } y=0,\quad\text{and}\quad \theta=1\quad \text{at } y=h. \end{align} $$

3.1. Volume flow rate

In the laboratory frame, the following is the expression of the dimensional volume flow rate:

(3.34) $$ \begin{align} Q=\int_{0}^{\bar{h}}\bar{U}(\bar{X},\bar{Y},\bar{t})\,d\bar{Y}, \end{align} $$

where $\bar {h} =\bar {h}(\bar {X},\bar {t})$ . In the wave frame of reference, the above equation reduces to

(3.35) $$ \begin{align} q=\int_{0}^{\bar{h}}u(\bar{x},\bar{y})\, dy , \end{align} $$

in which $\bar {h}=\bar {h}(\bar {x})$ .

From equations (3.1), (3.34) and (3.35),

$$ \begin{align*} Q=q+C\bar{h}(x). \end{align*} $$

The time averaged value over a fixed frame $\bar {X}$ is

$$ \begin{align*} \bar{Q}=\frac{1}{T}\int_{0}^{T}Q \,dt , \end{align*} $$

which, after using equation (3.35) and performing integration, leads to

$$ \begin{align*} \varphi=F+1, \end{align*} $$

where

(3.36) $$ \begin{align} \begin{aligned} \varphi&=\frac{\bar{Q}}{aC}, \\ F&=\frac{q}{aC}=\int_{0}^{h}u\, dy. \end{aligned} \end{align} $$

Equation (3.29) reveals that $\tau _{yy}=\tau _{zz}=0$ , and the stress tensors’ trace changes to $\tau _{xx}$ . Substituting equations (3.23) and (3.24) into equation (3.26), and solving with the boundary conditions $\tau _{xy}=0$ at $y=0$ (the line of symmetry),

(3.37) $$ \begin{align} \tau_{xy}=y\frac{dp}{dx}-mU_{hs}\frac{\sinh(my)}{\cosh(hm)}. \end{align} $$

With the aid of equations (3.29) and (3.30), we can write

$$ \begin{align*} \tau_{xx}=2\epsilon W_e \tau_{xy}^3. \end{align*} $$

From equations (3.10), (3.29) and (3.30),

(3.38) $$ \begin{align} \frac{\partial u}{\partial y}=\tau_{xy}+2\epsilon W_e^2 \tau_{xy}^3. \end{align} $$

Substituting equation (3.37) into equation (3.38),

(3.39) $$ \begin{align} \frac{\partial u}{\partial y}=y\frac{dp}{dx}-mU_{hs}\frac{\sinh(my)}{\cosh(hm)} +2\epsilon W_e^2\bigg(y\frac{dp}{dx}-\frac{mU_{hs}\sinh(my)}{\cosh(hm)}\bigg)^3. \end{align} $$

4. Analytical solutions

Employing the boundary conditions in equation (3.32) to solve equation (3.39),

(4.1) $$ \begin{align} u=\frac{1}{12}\bigg[6\bigg\{-2-h^2\frac{dp}{dx}+ \frac{dp}{dx}y^2+\bigg(\frac{dp}{dx}\bigg)^3(y^4-h^4)\epsilon W_e\bigg\}+L_1+L_2+L_3\bigg]. \end{align} $$

Using equation (3.36) in equation (4.1),

(4.2) $$ \begin{align} F=-h-\frac{h^3}{3}\frac{dp}{dx}-\frac{2}{5}h^5\bigg(\frac{dp}{dx}\bigg)^3\epsilon W_e+\frac{U_{hs}}{36m^3}[36m^2(hm-\tanh(hm))+L_4+L_5+L_6]. \end{align} $$

Due to nonlinearity, it is difficult to obtain the analytical solution to equation (4.2). As a result, the solution is obtained using the standard perturbation approach. We expand ${dp}/{dx}$ in terms of the parameter $(|W_e|)$ to use the perturbation approach as follows:

(4.3) $$ \begin{align}\frac{dp}{dx}=p_0+W_ep_1. \end{align} $$

The solution of equation (4.2) using equation (4.3) is given by

(4.4) $$ \begin{align} \frac{dp}{dx}&=\frac{1}{30h^9m^5}[-90h^6m^5(F+h)+U_{hs}(\tanh(hm)-hm)] + \epsilon W_e(972h^2(F+h)^3m^5) \nonumber \\ &\quad +L_7-L_8+L_9+L_{10}+L_{11} \nonumber \\ &\quad +(29160+13608h^2m^2+405h^4m^4-130h^6m^6)\tanh(hm), \end{align} $$

where

$$ \begin{align*} L_{1} & = U_{hs}\bigg[12-12\frac{\cosh(my)}{\cosh(hm)} + \frac{\epsilon}{m^2} \bigg\{72\bigg(\frac{dp}{dx}\bigg)^2\bigg(2+h^2m^2 -\bigg\{(2+m^2y^2)\cosh(my)\\[2pt] &\quad +2m(h\sinh(hm)-y\sinh(my))\bigg\}\frac{1}{\cosh{hm}}\bigg)\bigg\}\bigg],\\[2pt] L_2 &=9m^2\frac{dp}{dx}[\cosh(2hm)+2m \{m(h^2-y^2) - h\sinh(2hm)+y\sinh(2my)\}]\frac{U_{hs}}{\cosh^2(hm)}, \\[2pt] L_3&=2m^4 [\cosh(3hm)-9\cosh(hm)+9\cosh(my)-\cosh(3my)] \frac{U_{hs}^2W_e}{\cosh(hm)},\\[2pt] L_4&=\epsilon W_e\bigg[9m^2\frac{dp}{dx}\{4h^3m^3+6hm\cosh(2hm) - 3(1+2h^2m^2) \sinh(2hm)\} \frac{U_{hs}}{\cosh^2(hm)}\bigg],\\[2pt] L_5&=2m^4[3hm\cosh(3hm)-27hm\cosh(hm)+27\sinh(hm)- \sinh(3hm)]\frac{U_{hs}^2}{\cosh^3(hm)},\\[2pt] L_6&=216\bigg(\frac{dp}{dx}\bigg)^2[hm(6+h^2m^2)-3(2+h^2m^2)\tanh(hm)],\\[2pt] L_7&=1944(F+h)^2m^2U_{hs}(hm(15+m^2h^2)-3(5+2h^2m^2)\tanh(hm),\\[2pt] L_8&=27(F+h)mU_{hs}^2\bigg[6h^2m^2(360+47m^2h^2)+(2h^2m^2-3)\frac{5h^4m^4}{\cosh^2(hm)}\\[2pt]&\quad -3hm(1440+413h^2m^2+10h^4m^4)\tanh(hm)\bigg],\\[2pt] L_9&=108(20+9h^2m^2)\tanh(hm)^2 ,\\[2pt] L_{10}&=U_{hs}^3 [87480hm+60669h^3m^3+5508h^5m^5+60h^7m^7-29160+101088h^2m^2\\[2pt] &\quad +22599h^4m^4+830h^6m^6]\tanh(hm),\\[2pt] L_{11}&=\frac{9hm (-9720-3501h^2m^2-135h^4m^4+10h^6m^6)}{\cosh^2(hm)}. \end{align*} $$

The results shown above are in line with the findings of Hayat et al. [Reference Hayat, Noreen, Ali and Abbasbanday16] when m and $U_{hs}$ approach zero.

The solution for temperature is obtained by solving equation (3.31) with equation (4.1) and boundary conditions in equation (3.33),

$$ \begin{align*} \theta &=\frac{1}{1080m^4(1+prRd)} \bigg[90m^4 \bigg(12(1+prRd)+Br\bigg(\frac{dp}{dx}\bigg)^2(h^4-y^4)\bigg) \nonumber \\ &\quad +Br\frac{dp}{dx}\bigg\{72m^4\bigg(\frac{dp}{dx}\bigg)^3(h^6-y^6)\epsilon W_e+135m^2\frac{dp}{dx}\epsilon\bigg((9+6h^2m^2)\frac{\cosh(2hm)}{\cosh^2(hm)} \nonumber \\ &\quad -3(3+2m^2y^2)\cosh(2my)\!+\!2m (m^3(y^4-h^4)\!-\!6h \sinh(2hm)+6y\sinh(2my))\bigg)U_{hs}^2W_e \nonumber \\ &\quad -20m^4\epsilon\bigg(162\cosh(hm)-2\cosh(3hm)-162\cosh(my)+2\cosh(3my) \nonumber \\ &\quad +3m\{-27h\sinh(hm)+h\sinh(3hm)+27y\sinh(my)-y\sinh(3my)\}\bigg)\frac{U_{hs}^3W_e}{\cosh^3(hm)} \nonumber \\ &\quad +1080U_{hs}\bigg(m^2\{2+(-2\cosh(my)-hm\sinh(hm))\}+ \frac{my\sinh(my)}{\cosh(hm)} \nonumber \\ &\quad+6\bigg(\frac{dp}{dx}\bigg)^2\epsilon\bigg\{6(4+h^2m^2)\cosh(hm)-6(4+m^2y^2)\cosh(my) \nonumber \\ &\quad -hm(18+h^2m^2)\sinh(hm)+my(18+m^2y^2)\sinh(my)\bigg\}\frac{W_e}{\cosh(hm)}\bigg)\bigg\}\bigg]. \end{align*} $$

From equation (4.4), the pressure difference over one wavelength is computed as

$$ \begin{align*} \Delta p=\int_{0}^{1}\frac{\partial p}{\partial x}\, dx. \end{align*} $$

We now introduce the stream function $\Psi $ as

$$ \begin{align*} u=\frac{\partial\Psi}{\partial y},\quad v=-\frac{\partial\Psi}{\partial x}. \end{align*} $$

5. Numerical results and discussion

In this section, we visually display the exact solutions that were computed in the previous section. This allows us to examine the effects of various influential parameters on the flow profile, such as Weissenberg number ( $W_e$ ), PTT parameter ( $\epsilon $ ), Helmholtz–Smoluchowski velocity ( $U_{hs}$ ), electro-osmotic parameter (m), thermal radiation parameter ( $Rd$ ) and Prandtl number ( $pr$ ). These parameters affect the temperature profile, velocity profile and pressure gradient.

The graphical plots in this section demonstrate the validity of the mathematical solutions and enable a more detailed analysis of the current work. Specifically, the velocity and temperature profiles clearly show that the boundary conditions we used in our situation were valid. The calculated mathematical solutions satisfy the accompanying equations and pertinent boundary conditions. It is worth noting that the axial velocity profiles exhibit a parabolic shape, which is a typical characteristic of no-slip flow. In this type of flow, the velocity is zero at the walls and reaches its maximum in the centre of the microchannel.

Figures 2(a)–(e) depict the analysis of velocity behaviour for different physical parameters in the obtained results. Figure 2(a) illustrates the velocity distributions for the Weissenberg number ( $W_e$ ). It is evident that higher values of $W_e$ lead to a substantial decrease in fluid velocity. This phenomenon occurs because it assesses the relaxation time of the fluid in relation to a specific process time. As a result, the Weissenberg number decreases the velocity of the fluid while simultaneously increasing its thickness. From Figure 2(b), it is evident that the velocity of the PTT liquid model is directly proportional to the mean flow rate, F. The graph clearly shows that as the intensity of the elastic force, F, increases, the velocity of the fluid also increases. Figure 2(c) is plotted to observe the effect of the parameter $\epsilon $ (which represents the extensibility of the fluid) on the velocity field. This graph indicates that the velocity, u, decreases as the value of $\epsilon $ increases. Figure 2(d) illustrates that an increase in the electro-osmotic parameter, m, reduces the amplitude of fluid velocity at the centre of the channel ( $y=0$ ). The electro-osmotic parameter demonstrates that the velocity profile acts as a declining function, although it has a greater influence on fluid velocity. This demonstrates that the electric field enforced along the length of the channel significantly influences controlled haemodynamics. It is evident that the electric field primarily controls the flow in this scenario, and by increasing the electro-kinetic parameter, m, we can regulate the speed of the fluid. This trend is caused by the presence of the electric double layer, which inhibits fluid flow. The argument against the Helmholtz–Smoluchowski parameter based on velocity is illustrated in Figure 2(e). It demonstrates that an increase in $U_{hs}$ velocity leads to a decrease. Therefore, the main purpose of the axial electric field is to control the flow. When $U_{hs}=0$ , there is no electro-osmosis happening during the peristaltic motion of the PTT fluid. The fluid velocity decreases as the EDL thickness increases due to the physical determination of $U_{hs}$ . Consequently, the presence of EDL results in a reduction in fluid flow.

Figure 2 Variations of the velocity concerning the y-axis for changed values for (a) Weissenberg number $W_e$ , (b) elastic force (F), (c) PTT parameter ( $\epsilon $ ), (d) electro-osmotic parameter (m) and (e) Helmholtz–Smoluchowski velocity $U_{hs}$ .

Figure 3 Variations of temperature $\theta $ concerning the y-axis for changed values for (a) Weissenberg number $W_e$ , (b) thermal radiation $Rd$ , (c) Prandtl number ( $pr$ ), (d) electroosmotic parameter (m), (e) Helmholtz–Smoluchowski velocity $U_{hs}$ and (f) Brinkman number ( $Br$ ).

Figure 4 Variations of pressure gradient ${dp}/{dx}$ concerning the x-axis for changed values for (a) electro-osmotic parameter (m), (b) Helmholtz–Smoluchowski velocity $U_{hs}$ , (c) Weissenberg number $W_e$ , (d) elastic force (F) and (e) PTT parameter ( $\epsilon $ ).

To gain insights into the presumed research, we discuss the influence of different constraints on temperature. Figures 3(a)–(f) illustrate the impact of various parameters on temperature distribution. Figure 3(a) depicts the variation in temperature profile with $W_e$ . According to Figure 3(a), it is evident that the temperature reaches its maximum value near the centre of the channel and its minimum value near the walls. This graph clearly shows that when the elastic forces are greater than the viscous forces ( $W_e>1$ ), the temperature of the liquid significantly decreases. Figure 3(b) demonstrates the effects of increasing the thermal radiation $Rd$ on temperature $\theta $ . It is observed that the temperature magnitude decreases as $Rd$ increases. This is due to the inverse relationship between heat radiation and thermal conduction (K). This indicates that when the system’s heat radiation is at its highest, the fluid’s ability to transmit heat is reduced. Figure 3(c) is plotted to reveal the impacts of Prandtl number $pr$ on heat transfer coefficient. Figure 3(c) indicates that increase in Prandtl number $pr$ slightly decreases the fluid temperature. In Figure 3(d), the electro-osmotic parameter m is plotted against the temperature distribution. The graph shows that as m increases, there is a decline in temperature. This indicates that the relationship between temperature and the electro-kinetic parameter is significant in regulating the heat transfer process. Therefore, it can be concluded that the electric field plays a primary role in controlling heat transmission. Figure 3(e) illustrates that temperature decreases as the values of $U_{hs}$ increase. Specifically, when $U_{hs}=0.5$ , the temperature reaches its highest point and gradually decreases with increasing $U_{hs}$ . This further emphasizes the dependence of temperature on the electric field and its role in regulating heat transport. Figure 3(f) shows the changes in temperature for various values of the Brinkman number $Br$ . It is demonstrated that the magnitude of the temperature $\theta $ increases with an increase in $Br$ . Temperature increases dramatically as a result of this behaviour because a greater magnitude of the Brinkman number means that less heat conduction is produced through viscous dissipation.

Figure 5 Stream lines for $\Phi =0.3, m=1, W_e=0.02, \epsilon =0.1, F=0.2$ . For (a) $U_{hs}=0.5$ , (b) $U_{hs}=1.0$ and (c) $U_{hs}=1.5$ .

Figure 6 Stream lines for $\Phi =0.3, U_{hs}=1, W_e=0.02, \epsilon =0.1, F=0.2$ . For (a) $m=1$ , (b) $m=2$ and (c) $m=3$ .

Figures 4(a)–(e) demonstrate the impact of various parameters, namely the electro-osmotic parameter (m), Helmholtz–Smoluchowski velocity ( $U_{hs}$ ), Weissenberg number ( $W_e$ ), elastic force (F) and PTT parameter ( $\epsilon $ ), on the pressure gradient ( ${dp}/{dx}$ ). It is important to note that the distribution of ${dp}/{dx}$ is shown along the axial coordinate (x). The graph clearly shows that the pressure slope along the axial length follows a sinusoidal pattern, with high-pressure gradients during contraction and low-pressure gradients during relaxation. This pattern is crucial in facilitating the peristaltic pumping process. As a result, controlled volume of liquid can be transferred from one area to another without contamination. The pressure gradient for increasing the Debey–Hückel parameter, which is the inverse EDL thickness ( $m={a}/{\lambda _d}$ ), is shown in Figure 4(a). It has been observed that as the Debey–Hückel parameter increases, the pressure gradient also rises. This means that as the typical thickness of the EDL decreases, a negative pressure gradient may occur. Additionally, it is noted that the pressure gradient is insignificant for $x\in [0,0.6]$ and $x\in [0.9,1]$ , but a large pressure gradient occurs at $x=0.8$ . Figure 4(b) illustrates the variation of ${dp}/{dx}$ for different values of the Helmhotz–Smoluchowski velocity ( $U_{hs}$ ). The pressure gradient consistently increases with an increase in $U_{hs}$ . The electric field in the direction of fluid flow ( $U_{hs}>0$ ) exhibits the maximum axial pressure gradient compared with the reverse direction of fluid flow ( $U_{hs}<0$ ). The effect of the rheological parameter $W_e$ on the axial pressure gradient is shown in Figure 4(c). It demonstrates that the axial pressure gradient increases with an increase in the rheological parameter $W_e$ , as there is a direct relationship between ${dp}/{dx}$ and the Weissenberg number $W_e$ . The effects of F and $\epsilon $ on ${dp}/{dx}$ are shown in Figures 4(d) and 4(e). From these graphs, it can be observed that ${dp}/{dx}$ increases for large values of F and the PTT fluid parameter $\epsilon $ . Furthermore, it is noticed that the maximum pressure gradient occurs at $x=0.8$ .

Figure 7 Stream lines for $\Phi =0.3, U_{hs}=1, m=1, \epsilon =0.1, F=0.2$ . (a) $W_e=0.01$ , (b) $W_e=0.03$ and (c) $W_e=0.05$ .

Figure 8 Stream lines for $\Phi =0.3, U_{hs}=1, m=1, \epsilon =0.1, W_e=0.02$ . (a) $F=1.0$ , (b) $F=1.5$ and (c) $F=2.0$ .

Trapping is significantly important, because it creates a circulatory area that moves forward with the speed of the wave, providing continuous transportation. In the human gastrointestinal tract, smooth muscle tissues contract sequentially to create the peristaltic wave, which propels the food ball (referred to as a bolus in the oesophagus and upper gastrointestinal tract, and chyme in the stomach). This leads to the formation of trapped boluses in confined streamlines. The peristaltic motion of the fluid is shown in Figures 5–8. Figures 5(a)–(c) indicate that the magnitude of the bolus tends to decrease with an increase in the Helmhotz–Smoluchowski parameter $U_{hs}$ . Figures 6(a)–(c) reveal that the size of bolus increases as the Debey–Hückel parameter m increases while the number of boluses is decreasing. Figures 7(a)–(c) conclude that the extent of the trapped bolus is increased with a rise in the parameter $W_e$ . Figures 8(a)–(c) display the effect of different values of the PTT liquid parameter F on the streamlines. It shows that an increase in F leads to a decrease in the size of the trapped bolus, while only slightly increasing the number of streamlines in the top symmetric micro-channel.