Entropy generation in electroosmotically aided peristaltic pumping of MoS₂ Rabinowitsch nanofluid

Here we examined the two-dimensional (2D) incompressible peristaltic pumping of nanofluids through a 2D symmetric microchannel. Two types of nanofluids are prepared by dispersing the spherical shaped molybdenum disulfide nanoparticles of average diameter 30 nm in two different types of base fluids i.e. kerosene oil, and CMC + water which is the solution of 0.3 wt% of CMC in water. Kerosene oil represents the Newtonian characteristics whereas CMC + water solution depicts the non-Newtonian shear thinning characteristics. The progressive sinusoidal wave of wavelength λ is propagated with constant speed c along the microchannel walls to engender the peristaltic propulsion. The peristaltic pumping is further mobilized by electroosmotic forces which are generated by applying the external electric source across the EDL. Cartesian coordinate system ($\tilde x$, $\widetilde {y,}\ \tilde t$) is adopted to formulate the problem mathematically and the physical sketch of the symmetric microchannel is shown in figure 1 with $\tilde x - {\text{axis}}$ being along the centerline and $\tilde y - {\text{axis}}$ perpendicular to it. The mathematical form of the deforming microchannel walls is considered as:

$$\begin{align}\tilde y = \pm \tilde H\left( {\tilde x,\tilde t} \right) = \pm d \pm a{\text{sin}}\left( {\frac{{2\pi }}{\lambda }\left( {\tilde x - c\tilde t} \right)} \right),\end{align} \tag{ 1 }$$

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where d is the half-width of the channel, $\tilde t$ is the time and $a$ is the amplitude of the wave. Moreover, $\tilde y\,$ = $\,\tilde H\left( {\tilde x,\tilde t} \right)$ and $\widetilde {y\,}$ = −$\tilde H\left( {\tilde x,\tilde t} \right)\,$ show the position of the upper and the lower walls of the channel.

In this investigation, Rabinowitsch fluid model is incorporated to specify the non-Newtonian aspects of CMC + water/MoS₂ nanofluid as well as Newtonian properties of kerosene oil/MoS₂ nanofluid by choosing a suitable value pseudo-plasticity parameter. The impact of the physical properties of particles and the base fluid is included through the Corcione model for thermal conductivity and viscosity of nanofluids, and the Buongiorno model for nanofluids is adopted to incorporate the Brownian and thermophoretic diffusion of nanoparticles in the fluid medium. It has been experimentally found that the solution of CMC + water with up to 6 wt% of CMC depicts the shear-thinning properties having the same thermophysical attributes as that of water. In this investigation, the solution of 0.3 wt% of CMC in water is considered. The electrostatic potential emerging due to applying an external electric field is modeled through linearized Poisson–Boltzmann ionic distribution. The additive aspect of Joule heating and viscous dissipation is also taken into consideration. The no-slip boundary conditions for velocity, convective boundary conditions for temperature, and condition of zero net mass flux at boundaries are enforced.

Subject to the above inferences, the momentum, energy, and concentration equations are derived as Akram et al (2020a), Shafey et al (2021):

$$\begin{align}\frac{{\partial \tilde u}}{{\partial \tilde x}} + \frac{{\partial \tilde v}}{{\partial \tilde y}} = 0,\end{align} \tag{ 2 }$$

$$\begin{align}{\rho _{{\text{nf}}}}\left( {\frac{{\partial \tilde u}}{{\partial \tilde t}} + \tilde u\frac{{\partial \tilde u}}{{\partial \tilde x}} + \tilde v\frac{{\partial \tilde u}}{{\partial \tilde y}}} \right) = - \frac{{\partial \tilde p}}{{\partial \tilde x}} + \frac{{\partial {{\tilde \tau }_{\tilde x\tilde x}}}}{{\partial \tilde x}} + \frac{{\partial {{\tilde \tau }_{\tilde x\tilde y}}}}{{\partial \tilde y}} + {\rho _{\text{e}}}{U_{{{\text{E}}_{\tilde x}}}},\end{align} \tag{ 3 }$$

$$\begin{align}{\rho _{{\text{nf}}}}\left( {\frac{{\partial \tilde v}}{{\partial \tilde t}} + \tilde u\frac{{\partial \tilde v}}{{\partial \tilde x}} + \tilde v\frac{{\partial \tilde v}}{{\partial \tilde y}}} \right) = - \frac{{\partial \tilde p}}{{\partial \tilde y}} + \frac{{\partial {{\tilde \tau }_{\tilde x\tilde y}}}}{{\partial \tilde x}} + \frac{{\partial {{\tilde \tau }_{\tilde y\tilde y}}}}{{\partial \tilde y}},\end{align} \tag{ 4 }$$

$$\begin{align} {\left( {\rho \zeta } \right)_{{\text{nf}}}}\left( {\frac{{\partial \tilde T}}{{\partial \tilde t}} + \tilde u\frac{{\partial \tilde T}}{{\partial \tilde x}} + \tilde v\frac{{\partial \tilde T}}{{\partial \tilde y}}} \right)& = {k_{{\text{nf}}}}\left( {\frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde x}^2}}} + \frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde y}^2}}}} \right) + \tau .{L^t} \nonumber \\ & \quad + {\left( {\rho \zeta } \right)_{\text{s}}}{D_{\text{B}}}\left( {\frac{{\partial \tilde \varPhi }}{{\partial \tilde x}}\frac{{\partial \tilde T}}{{\partial \tilde x}} + \frac{{\partial \tilde \varPhi }}{{\partial \tilde y}}\frac{{\partial \tilde T}}{{\partial \tilde y}}} \right)\nonumber \\ & \quad + \frac{{{D_{\text{T}}}}}{{{{\tilde T}_0}}}{\left( {\rho \zeta } \right)_{\text{s}}}\left( {{{\left( {\frac{{\partial \tilde T}}{{\partial \tilde x}}} \right)}^2} + {{\left( {\frac{{\partial \tilde T}}{{\partial \tilde y}}} \right)}^2}} \right) + {\sigma _{{\text{nf}}}}{U_{{{\text{E}}_{\tilde x}}}}^2,\nonumber\\ \end{align} \tag{ 5 }$$

$$\begin{align}\left( {\frac{{\partial \widetilde \Phi }}{{\partial \tilde t}} + \tilde u\frac{{\partial \widetilde \Phi }}{{\partial \tilde x}} + \tilde v\frac{{\partial \widetilde \Phi }}{{\partial \tilde y}}} \right) = {D_{\text{B}}}\left( {\frac{{{\partial ^2}\widetilde \Phi }}{{\partial {{\tilde x}^2}}} + \frac{{{\partial ^2}\widetilde \Phi }}{{\partial {{\tilde y}^2}}}} \right) + \frac{{{D_{\text{T}}}}}{{{T_0}}}\left( {\frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde x}^2}}} + \frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde y}^2}}}} \right).\end{align} \tag{ 6 }$$

where

$$\begin{equation*}L = {\text{grad}}\left( {\vec V} \right).\end{equation*}$$

Here $\vec V$ represents the velocity vector. The expression for the Rabinowitsch fluid model is given by:

$$\begin{align}{\tilde \tau _{\tilde x\tilde y}} + {\alpha^*}{\tilde \tau _{\tilde x\tilde y}}^3 = {\mu _{{\text{nf}}}}\frac{{\partial \tilde u}}{{\partial \tilde y}}.\end{align} \tag{ 7 }$$

In which ${\alpha ^*}$ symbolizes the parameter of pseudo-plasticity and ${\mu _{{\text{nf}}}}$ the effective dynamic viscosity of nanofluid. In the above relations, $\tilde p$ stands for pressure force, ${\tilde \tau _{\tilde x\tilde x}}$, $\,{\tilde \tau _{\tilde x\tilde y}}$ and ${\tilde \tau _{\tilde y\tilde y}}$ are the extra stress components, ${\rho _{{\text{nf}}}}$ is the effective density of nanofluid, ${k_{{\text{nf}}}}$ is the thermal conductivity of nanofluid, ${\left( {\rho \zeta } \right)_{{\text{nf}}}}$ and ${\left( {\rho \zeta } \right)_{\text{s}}}$ are the specific heat capacity for nanofluids and solid particles respectively, ${D_{\text{T}}}$ and ${D_{\text{B}}}$ are the parameters for thermophoresis and Brownian diffusion, ${\sigma _{{\text{nf}}}}$ is the effective electrical conductivity of nanofluids, ${U_{{E_{\tilde x}}}}$ is the electric body force, $\tilde T$ is the temperature and $\widetilde \Phi$ is the nanoparticle volume fraction.

The effective density and effective specific heat capacity defined by the general mixture rule are expressed as:

$$\begin{align}{\rho _{{\text{nf}}}} = \left( {1 - {\Phi _0}} \right){\rho _{{\text{bf}}}} + {\Phi _0}{\rho _{\text{s}}},\end{align} \tag{ 8 }$$

$$\begin{align}{\left( {\rho \zeta } \right)_{{\text{nf}}}} = \left( {1 - {\Phi _0}} \right){\left( {\rho \zeta } \right)_{{\text{bf}}}} + {\Phi _0}{\left( {\rho \zeta } \right)_{\text{s}}},{ }\end{align} \tag{ 9 }$$

here, the subscript 'bf' stands for the base fluids and the subscript 's' designates the properties of solid nanoparticles.

The ratio for effective thermal conductivity and viscosity of nanofluids as predicted by Corcione is given as:

$$\begin{align}\frac{{{k_{{\text{nf}}}}}}{{{k_{{\text{bf}}}}}} = \left( {1 + 4.4{R^{0.4}}{\text{P}}{{\text{r}}^{0.66}}{{\left( {\frac{{{T_0}}}{{{T_{{\text{fr}}}}}}} \right)}^{10}}{{\left( {\frac{{{k_{\text{s}}}}}{{{k_{{\text{bf}}}}}}} \right)}^{0.03}}{\Phi _0}^{0.66}} \right) = kr,\end{align} \tag{ 10 }$$

$$\begin{align}\frac{{{ }{\mu _{{\text{nf}}}}}}{{{\mu _{{\text{bf}}}}}} = \frac{1}{{\left( {1 - 34.87{{\left( {\frac{{{d_{\text{s}}}}}{{{d_{{\text{bf}}}}}}} \right)}^{ - 0.3}}{\Phi _0}^{1.03}} \right)}} = {\mu _{\text{r}}},\end{align} \tag{ 11 }$$

where ${\Phi _0}$ represents the nanoparticle volume fraction molybdenum disulfide nanoparticles, ${T_{{\text{fr}}}}$ is the freezing temperature for base fluid, ${d_{\text{s}}}$ and ${d_{\text{f}}}$ are the average diameter for nanoparticles and the molecule of the base fluid and $R$ is the Reynolds number for nanoparticles defined as:

$$\begin{align}R = \frac{{2{\rho _{{\text{bf}}}}{k_{\text{B}}}{T_0}}}{{\pi {\mu _{{\text{bf}}}}^2{d_{\text{s}}}}}\end{align} \tag{ 12 }$$

in which ${k_{\text{B}}}$ designates the Boltzmann constant. The electrical conductivity for nanofluid is calculated by Maxwell–Garnett model as:

$$\begin{align}{\sigma _{{\text{nf}}}} = {\sigma _{{\text{bf}}}}\left( {1 + \frac{{\left( {\frac{{{\sigma _{\text{s}}}}}{{{\sigma _{{\text{bf}}}}}} - 1} \right){\Phi _0}}}{{\left( {\frac{{{\sigma _{\text{s}}}}}{{{\sigma _{{\text{bf}}}}}} + 2} \right) - \left( {\frac{{{\sigma _{\text{s}}}}}{{{\sigma _{{\text{bf}}}}}} - 1} \right){\Phi _0}}}} \right).\end{align} \tag{ 13 }$$

The well-known Poisson equation is used to characterize the electric potential generated due to the mobility of excess of counterions in the vicinity of the EDL when an external source of electricity is applied as:

$$\begin{align}{\nabla ^2}{\tilde U_{\text{E}}} = - \frac{{{\rho _{\text{e}}}}}{{{\varepsilon _0}{\varepsilon _{\text{m}}}}},\end{align} \tag{ 14 }$$

where ${\varepsilon _{\text{m}}}$ and $\,{\varepsilon _0}$ designate the relative permittivity for the base fluid and the permittivity of vacuum respectively and ${\rho _{\text{e}}} = ez\left( {{n^ + } - {n^ - }} \right)$ is the net electric charge number density with e being the charge of the electron, z is the net valency of ionic species and ${n^ + }{\text{ and }}{n^ - }$ are the local ionic density of cations and anions respectively having bulk ionic concentration n₀.

Introducing the dimensionless numbers to facilitate the non-dimensional analysis:

$$\begin{equation*}x = \frac{{\tilde x}}{\lambda },{} y = \frac{{\tilde y}}{d},{} p = \frac{{\tilde p{d^2}}}{{{\mu _{{\text{bf}}}}c}},{} u = \frac{{\tilde u}}{c},v = \frac{{\tilde v}}{c},{} n = \frac{{\tilde n}}{{{n_0}}},{\tau ^*} = \frac{{{{\left( {\rho \zeta } \right)}_{\text{s}}}}}{{{{\left( {\rho \zeta } \right)}_{{\text{bf}}}}}},\Gamma = \frac{{R{D_{\text{B}}}{\Phi _0}}}{{{k_{{\text{bf}}}}}},\end{equation*}$$

$$\begin{equation*}h = \frac{{\tilde H}}{d} = 1 + \varepsilon \sin \left( {2\pi (x-t)} \right),{} {\text{Pr}} = \frac{{{\mu _{{\text{bf}}}}{\zeta _{{\text{bf}}}}}}{{{k_{{\text{bf}}}}}}{} ,{} \delta = \frac{d}{\lambda },{} \theta = \frac{{\tilde T - {T_0}}}{{\overline {\left( {{T_1} - {T_0}} \right)} }},\varPsi = \frac{{\tilde \varPsi }}{{cd}},\end{equation*}$$

$$\begin{equation*}\Phi = \frac{{\tilde \Phi - {\Phi _0}}}{{{\Phi _0}}},{U_{\text{E}}} = \frac{{ez{{\tilde U}_{\text{E}}}}}{{{k_{\text{B}}}{{\hat T}_{{\text{avg}}}}}},{\text{Re}} = \frac{{{\rho _{{\text{bf}}}}cd}}{{{\mu _{{\text{bf}}}}}},{\text{S}} = \frac{{{\sigma _{{\text{bf}}}}{U_{{{\text{E}}_x}}}^2{d^2}}}{{{k_{{\text{bf}}}}\left( {{T_1} - {T_0}} \right)}},\Omega = \frac{{\left( {{T_1} - {T_0}} \right)}}{{{T_0}}},\end{equation*}$$

$$\begin{equation*}{u_{{\text{hs}}}} = - \frac{{{\varepsilon _0}{\varepsilon _{\text{m}}}{k_{\text{B}}}{{\hat T}_{{\text{avg}}}}{U_{{{\text{E}}_{ x}}}}}}{{ez{\mu _{{\text{bf}}}}c}},\,m = \sqrt {\frac{{2{n_0}{e^2}{z^2}{d^2}}}{{{\varepsilon _0}{\varepsilon_m} {k_{\text{B}}}{{\hat T}_{{\text{avg}}}}}}} = \frac{d}{{{\lambda _{\text{d}}}}},\,{{N_{\text{t}}}} = \frac{{{D_{{{\overline{\text{T}}}}}}{\tau ^*}{\rho _{{\text{bf}}}}\left( {{T_1} - {T_0}} \right)}}{{{T_0}{\mu _{{\text{bf}}}}}}\,,\end{equation*}$$

$$\begin{equation*}{{N_{\text{b}}}} = \frac{{{D_{\text{B}}}{\tau ^*}{\rho _{{\text{bf}}}}{\Phi _0}}}{{{\mu _{{\text{bf}}}}}},{\text{ Ec}} = \frac{{{c^2}}}{{{\zeta _{{\text{bf}}}}\left( {{T_1} - {T_0}} \right)}},{\text{ Br }} = {\text{ Ec Pr}},u = {\varPsi _y},{ }v = - \delta {\varPsi _x}\end{equation*}$$

$$\begin{align}\alpha = \frac{{{\mu _{{\text{bf}}}}^2{c^2}{\alpha ^*}}}{{{d^2}}},{E_1} = \frac{{ - \tau {d^3}}}{{{\mu _{{\text{bf}}}}{\lambda ^3}c}},{ }{E_2} = \frac{{mc{d^3}}}{{{\mu _{{\text{bf}}}}{\lambda ^3}}},{E_3} = \frac{{{d^3}{d_1}}}{{{\mu _{{\text{bf}}}}{\lambda ^2}}}\end{align} \tag{ 15 }$$

where Re, m, u_hs, N_t, Ec, Br, S, N_b, $\theta$, E₁, E₂, E₃, Pr, $\Phi$, and α portend the Reynolds number, Debye length parameter, electroosmotic velocity or Helmholtz–Smoluchowski velocity, thermophoretic parameter, the Eckert number, Brinkmann number, Joule heating parameter, the Brownian diffusion parameter, the dimensionless temperature, the elastic tension parameter, the wall mass parameter, the wall damping parameter, Prandtl number, the dimensionless nanoparticle concentration parameter, and the dimensionless pseudo-plasticity parameter respectively. For suitable values of the pseudo-plasticity parameter, Rabinowitsch fluid model represents dilatant (α < 0), Newtonian (α = 0), and pseudoplastic fluid (α > 0).

Using equation (15) into equations (2)–(7) and (14) and considering the assumption of creeping flow, i.e. the long wavelength and low Reynolds number approximations, the continuity equation is trivially justified and equations (3)–(7) and (14) are reduced to:

$$\begin{align}\frac{{\partial p}}{{\partial x}} = \frac{{\partial {\tau _{xy}}}}{{\partial y}} + {u_{{\text{hs}}}}\frac{{{\partial ^2}{U_{\text{E}}}}}{{\partial {y^2}}},\end{align} \tag{ 16 }$$

$$\begin{align}\frac{{\partial p}}{{\partial y}} = 0,\end{align} \tag{ 17 }$$

$$\begin{align}{k_{\text{r}}}\frac{{{\partial ^2}\theta }}{{\partial {y^2}}} + {\text{PrNb}}\frac{{\partial \theta }}{{\partial y}}\frac{{\partial \Phi }}{{\partial y}} + {\text{PrNt}}{\left( {\frac{{\partial \theta }}{{\partial y}}} \right)^2} + {\text{Br }}{\tau _{xy}}\frac{{\partial u}}{{\partial y}} + {\sigma _{\text{r}}}S = 0,\end{align} \tag{ 18 }$$

$$\begin{align}\frac{{{\partial ^2}\Phi }}{{\partial {y^2}}} + \frac{{{\text{Nt}}}}{{{\text{Nb}}}}\frac{{{\partial ^2}\theta }}{{\partial {y^2}}} = 0,\end{align} \tag{ 19 }$$

$$\begin{align}{\tau _{xy}} + \alpha {\tau _{xy}}^3 = {\mu _{{\text{r }}}}\frac{{\partial u}}{{\partial y}},\end{align} \tag{ 20 }$$

$$\begin{align}\frac{{{\partial ^2}{U_{\text{E}}}}}{{\partial {y^2}}} = {m^2}\left( {\frac{{{n^ - } - {n^ + }}}{2}} \right).\end{align} \tag{ 21 }$$

Introducing the Boltzmann equation for describing the local ionic density of each ionic species after inserting the expression for work done by the electric force on the ions,

$$\begin{align}{n^ \pm } = {{\text{e}}^{ \mp {U_{\text{E}}}}} = {{\text{e}}^{ \mp \frac{{ez{U_{\text{E}}}}}{{{k_{\text{B}}}\hat T}}}},\end{align} \tag{ 22 }$$

and inserting it in equation (25) to obtain the Poisson–Boltzmann potential distribution, we get:

$$\begin{align}\frac{{{\partial ^2}{U_{\text{E}}}}}{{\partial {y^2}}} = {m^2}\sinh \left( {{U_{\text{E}}}} \right).\end{align} \tag{ 23 }$$

The Poisson–Boltzmann equation can be linearized for a small potential of up to 50–80 mV. In general, at room temperature and for a wide range of pH of the electrolyte solution, the standard value of generated potential is less than or equal to 25 mV. Therefore the approximation of low zeta potential is applicable for electroosmotic flow i.e. $\sinh \left( {{U_{\text{E}}}} \right) \approx {U_{\text{E}}}$ and equation (23) is reduced to:

$$\begin{align}\frac{{{\partial ^2}{U_{\text{E}}}}}{{\partial {y^2}}} = {m^2}{U_{\text{E}}}.\end{align} \tag{ 24 }$$

Enforcing the condition of the same potential generated at both walls as:

$$\begin{align}{U_{\text{E}}}{|_{y = h}} = \xi ,{ }{U_{\text{E}}}{|_{y = - h}} = \xi ,{ }\end{align} \tag{ 25 }$$

and integrating equation (24) for potential generated within the EDL which yields as:

$$\begin{align}{U_{\text{E}}} = \xi \frac{{{\text{cosh}}\left( {my} \right)}}{{{\text{cosh}}\left( {mh} \right)}}.\end{align} \tag{ 26 }$$

The corresponding boundary conditions imposed along the microchannel walls in the dimensionless form can be written as:

$$\begin{align}& u = 0,\frac{{\partial \theta }}{{\partial y}} + {\beta _i}\theta = {\beta _i},\frac{{\partial \Phi }}{{\partial y}} + \frac{{{\text{Nt}}}}{{{\text{Nb}}}}\frac{{\partial \theta }}{{\partial y}} = 0,{\text{ at }}y = h, \nonumber\\[4pt] & u = 0,\frac{{\partial \theta }}{{\partial y}} - {\beta _i}\theta = 0,{ }\frac{{\partial \Phi }}{{\partial y}} + \frac{{{\text{Nt}}}}{{{\text{Nb}}}}\frac{{\partial \theta }}{{\partial y}} = 0,{\text{ at }}y = - h,\end{align} \tag{ 27 }$$

$$\begin{align}\frac{{\partial p}}{{\partial x}} = \frac{{\partial {\tau _{xy}}}}{{\partial y}} + {u_{{\text{hs}}}}\frac{{{\partial ^2}{U_{\text{E}}}}}{{\partial {y^2}}} = {E_1}\frac{{{\partial ^3}h}}{{\partial {x^3}}} + {E_2}\frac{{{\partial ^3}h}}{{\partial x\partial {t^2}}} + {E_3}\frac{{{\partial ^3}h}}{{\partial x\partial t}} = P,{\text{ at }}y = h,\end{align} \tag{ 28 }$$

where ${\beta _i}\,$ represents the temperature Biot number.

The heat transfer rate at the wall is calculated as:

$$\begin{align}Z = \frac{{\partial h}}{{\partial x}}{\left( {\frac{{\partial \theta }}{{\partial y}}} \right)_{y \to h}}.\end{align} \tag{ 29 }$$

The rate of volumetric entropy generation in the nanofluid flow produced by heat transfer, nanoparticles, electroosmosis, and fluid friction is given by:

$$\begin{align}{\dot S_{\text{G}}}& = \frac{{{k_{{\text{nf}}}}}}{{{{\tilde T}_0}}}\left( {\frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde x}^2}}} + \frac{{{\partial ^2}\tilde T}}{{\partial {{\tilde y}^2}}}} \right) + \frac{1}{{{{\tilde T}_0}}}{\tau _{\tilde x\tilde y}}\frac{{\partial \tilde u}}{{\partial \tilde y}} + \frac{{{\sigma _{{\text{nf}}}}{U_{{{\text{E}}_{\tilde x}}}}^2}}{{{{\tilde T}_0}}} + \frac{{R{D_{\text{B}}}}}{{{\Phi _0}}}{\left( {\frac{{\partial {\text{ }}\tilde \Phi }}{{\partial \tilde y}}} \right)^2}\nonumber\\ & \quad + \frac{{R{D_{\text{B}}}}}{{{{\tilde T}_0}}}\left( {\frac{{\partial {\text{ }}\tilde \Phi }}{{\partial \tilde x}}\frac{{\partial \tilde T}}{{\partial \tilde x}} + \frac{{\partial \;\tilde \Phi }}{{\partial \tilde y}}\frac{{\partial \tilde T}}{{\partial \tilde y}}} \right).\end{align} \tag{ 30 }$$

The dimensionless entropy generation is the ratio between actual entropy and the characteristics rate of entropy generation ${\acute{S}_{\text{G}}}$ defined as:

$$\begin{align}{\acute{S}_G} = \frac{{{k_{{\text{bf}}}}{{\left( {{{\tilde T}_1} - {{\tilde T}_0}} \right)}^2}}}{{{{\tilde T}_0}^2{d^2}}},\end{align} \tag{ 31 }$$

$$\begin{align}{\acute{N}_G} = \frac{{{{\dot S}_{\text{G}}}}}{\acute{S}_G} = {k_{\text{r}}}\frac{{{\partial ^2}\theta }}{{\partial {y^2}}} + {\text{ Br }}{\tau _{xy}}\frac{{\partial u}}{{\partial y}} + \frac{\Gamma }{\Omega }\frac{{\partial \theta }}{{\partial y}}\frac{{\partial \Phi }}{{\partial y}} + \frac{\Gamma }{{{\Omega ^2}}}{\left( {\frac{{\partial \Phi }}{{\partial y}}} \right)^2} + \frac{{{\sigma _{\text{r}}}}}{\Omega }S,\end{align} \tag{ 32 }$$

where $\Gamma$ and $\Omega$ are the dimensionless parameters defined in equation (15).

Solving equation (16) directly subject to boundary condition given in equation (28) yields,

$$\begin{align}{\tau _{xy}} = Py - m{u_{{\text{hs}}}}{ }\xi \frac{{\sinh \left( {ky} \right)}}{{\cosh \left( {kh} \right)}}.\end{align} \tag{ 33 }$$

Inserting this relation in equation (20) and then solving it, by imposing boundary conditions described in equation (27), for the velocity field, we get the following expression for the velocity of the nanofluid

$$\begin{align} u( y )& = \frac{1}{{24{m^2}\mu r}}( - 6{m^2}P( {h - y} )( {h + y} )( {2 + {P^2}( {{h^2} + {y^2}} )\alpha } ) - 24{u_{{\text{hs}}}} \nonumber \\ & \qquad\qquad\; \times ( {{m^2} + 3( {2 + {h^2}{m^2}} ){P^2}\alpha } ) \xi + 18{m^2}P{u_{{\text{hs}}}}^2\alpha {\xi ^2} - 8{m^4}{u_{{\text{hs}}}}^3\alpha {\xi ^3}\nonumber \\ & \qquad\qquad\; + {u_{{\text{hs}}}}\xi {\text{Sech}}\left[ {hm} \right](4{\text{cosh}}\left[ {my} \right](6( {{m^2} + 3{P^2}( {2 + {m^2}{y^2}} )\alpha } )\nonumber \\ & \qquad\qquad\; + {m^4}{u_{{\text{hs}}}}^2\alpha {\xi ^2}( { - 5 + {\text{cosh}}\left[ {2my} \right]} ){\text{Sech}}{\left[ {hm} \right]^2})\nonumber \\ & \qquad\qquad\; + 3m\alpha (12P(h( {4P - {m^2}{u_{{\text{hs}}}}\xi } ){\text{sinh}}\left[ {hm} \right] - 4Py{\text{sinh}}\left[ {my} \right])\nonumber \\ & \qquad\qquad \; + m{u_{{\text{hs}}}}\xi {\text{Sech}}\left[ {hm} \right](P( { - 3 + 6{m^2}( {h - y} )( {h + y} )} ) + 8{m^2}{u_{{\text{hs}}}}\xi \nonumber \\ & \qquad\qquad\qquad\quad - 3P{\text{cosh}}\left[ {2my} \right] + 6mPy{\text{sinh}}\left[ {2my} \right])))). \end{align} \tag{ 34 }$$

Integrating equation (34) subject to boundary condition Ψ = q/2 as y → h, we get

$$\begin{align} {{\varPsi }}& = \frac{1}{{360{m^3}\mu r}}( 6m( {m^2}( P{{( {h - y} )}^2}( 10( {2h + y} ) \nonumber \\ & \qquad\qquad\quad\; + 3{P^2}( {4{h^3} + 3{h^2}y + 2h{y^2} + {y^3}} )\alpha ) + 30q\mu r ) \nonumber \\ & \qquad\qquad\quad\; + 60{u_{{\text{hs}}}}\,( {{m^2}( {h - y} ) + 3{P^2}( {h( {6 + h{m^2}( {h - y} )} ) - 2y} )\alpha } )\xi\nonumber \\ & \qquad\qquad\quad\; + 45{m^2}P{u_{{\text{hs}}}}{\,^2}( { - 2h + y} )\alpha {\xi ^2} { + 20{m^4}{u_{{\text{hs}}}}{\,^3}( {h - y} )\alpha {\xi ^3}} )\nonumber \\ & \qquad\qquad\quad\; + 5{u_{{\text{hs}}}}\,\xi {\text{Sech}}\left[ {hm} \right]( - 864m{P^2}y\alpha {\text{cosh}}\left[ {my} \right]\nonumber\\ & \qquad\qquad\quad\; - 2( {648{P^2}\alpha } + 2{m^4}{u_{{\text{hs}}}}\,\alpha \xi ( {27hP( { - h + y} ) + 2{u_{{\text{hs}}}}\,\xi } )\nonumber \\ & \qquad\qquad\quad\; + 9{m^2}( {4 + 3P\alpha ( {4hP( {3h - 2y} ) - {u_{{\text{hs}}}}\,\xi } )} ) ){\text{sinh}}\left[ {hm} \right] \nonumber \\ & \qquad\qquad\quad\; + 72( {{m^2} + 3{P^2}( {6 + {m^2}{y^2}} )\alpha } ){\text{sinh}}\left[ {my} \right] - 9{m^2}{u_{{\text{hs}}}}\,\alpha \xi {\text{Sech}}\left[ {hm} \right] \nonumber \\ & \qquad\qquad\quad\; \times ( {mP} ({ - 6h + 4{h^3}{m^2} + 3y - 6{h^2}{m^2}y + 2{m^2}{y^3}} )\nonumber \\ & \qquad\qquad\quad\; + 8{m^3}{u_{{\text{hs}}}}\,( {h - y} )\xi - 3mPy{\text{cosh}}\left[ {2my} \right]\nonumber \\ & \qquad\qquad\quad\; + 3P{\text{sinh}}\left[ {2my} \right] )2{m^4}{u_{{\text{hs}}}}{\,^2}\alpha {\xi ^2}{\text{Sech}}{{\left[ {hm} \right]}^2}\nonumber \\ & \qquad\qquad\quad\; \times ( {28{\text{sinh}}\left[ {hm} \right] - 27{\text{sinh}}\left[ {my} \right] + {\text{sinh}}\left[ {3my} \right]} ) ) ). \end{align} \tag{ 35 }$$

The nonlinear coupled equations (18) and (19) are executed by employing the homotopy perturbation technique subject to boundary conditions for temperature and nanoparticle volume fraction expressed in equation (28). The deformation relations for the considered problem about parameter qp are described as:

$$\begin{align}\mathcal{H}( {q,\theta } )& = ( {1 - q} )\left[ {\mathcal{H}( \theta ) - \mathcal{H}( {{{\hat \theta }_0}} )} \right]\nonumber\\ & \quad + q\left[ {\mathcal{H}( {{\theta }} ) + \frac{{{\text{Pr}}{N_{\text{b}}}}}{{{k_{\text{r}}}}}\frac{{\partial \theta }}{{\partial y}}\frac{{\partial \Phi }}{{\partial y}} + \frac{{{\text{Pr}}{N_{\text{t}}}}}{{{k_{\text{r}}}}}{{\left( {\frac{{\partial \theta }}{{\partial y}}} \right)}^2} + \frac{{{\text{Br}}}}{{{k_{\text{r}}}}}\,{\tau _{xy}}\frac{{\partial u}}{{\partial y}} + \frac{{{\sigma _{\text{r}}}}}{{{k_{\text{r}}}}}S} \right],\nonumber\\\end{align} \tag{ 36 }$$

$$\begin{align}\mathcal{H}( {q,\Phi } ) = ( {1 - q} )\left[ {\mathcal{H}( \Phi ) - \mathcal{H}( {{{\hat \Phi }_0}} )} \right] + q\left[ {\mathcal{H}( \Phi ) + \frac{{{N_{\text{t}}}}}{{{N_{\text{b}}}}}\frac{{{\partial ^2}\theta }}{{\partial {y^2}}}} \right],\end{align} \tag{ 37 }$$

where the linear operator $\mathcal{H}$ is defined as $\mathcal{H} = \frac{{{\partial ^2}}}{{\partial {y^2}}}$. Here ${\hat \theta _0}$ and ${\widehat \Phi _0}$ are the initial approximations for temperature and nanoparticle volume fraction respectively which are defined as:

$$\begin{align}{{{\hat \theta }}_0} = \frac{{{\beta _i}( {y + h} ) + 1}}{{2 + 2{\beta _i}h}},\end{align} \tag{ 38 }$$

$$\begin{align}{\hat \Phi _0} = \frac{{ - {\beta _i}{N_{\text{t}}}y}}{{2{N_{\text{b}}}( {{\beta _i}h + 1} )}}.\end{align} \tag{ 39 }$$

Now expanding $\theta$ and $\Phi$ in a series solution about small parameter q as:

$$\begin{align}\begin{array}{*{20}{l}} {\theta ( {y,q} ) = {\theta _0} + q{\theta _1} + \ldots } \\ {\Phi ( {y,q} ) = {\Phi _0} + q{\Phi _1} + \ldots } \end{array}.\end{align} \tag{ 40 }$$

Following the technique of HPM, the final solutions for temperature and nanoparticle volume fraction are obtained by setting qp→ 1 as:

$$\begin{align}\begin{array}{*{20}{l}} {\theta ( {y,1} ) = {\theta _0} + {\theta _1} + \ldots } \\ {\Phi ( {y,1} ) = {\Phi _0} + {\Phi _1} + \ldots } \end{array}.\end{align} \tag{ 41 }$$

And we get

$$\begin{align} \theta ( y )& = \frac{{1 + ( {h + y} ){\beta _i}}}{{1 + h{\beta _i}}} + \frac{1}{{17280{m^4}\mu r{k_{\text{r}}}{\beta _i}}}\nonumber\\ & \quad \times ( {{\beta _i}({\text{Br}}(8(36{m^4}{P^2}( {5{h^4} - 5{y^4} + 2{P^2}( {{h^6} - {y^6}} )\alpha } )} \nonumber \\ & \quad - 8640P{u_{{\text{hs}}}}( {{m^2} + 6( {4 + {h^2}{m^2}} ){P^2}\alpha } )\xi\nonumber \\ & \quad + 540{m^2}{u_{{\text{hs}}}}^2( {{m^2} + 3( {3 + 2{h^2}{m^2}} ){P^2}\alpha } ){\xi ^2} \nonumber \\ & \quad - 640{m^4}P{u_{{\text{hs}}}}^3\alpha {\xi ^3} + 135{m^6}{u_{{\text{hs}}}}^4\alpha {\xi ^4})\nonumber \\ & \quad + 5{u_{{\text{hs}}}}\xi {\text{Sech}}\left[ {hm} \right]( {27{m^6}{u_{{\text{hs}}}}^3\alpha {\xi ^3}(17 + 24{m^2}} \nonumber \\ & \quad \times {( {h - y} )( {h + y} ) + 16{\text{cosh}}\left[ {2my} \right] - {\text{cosh}}\left[ {4my} \right]} ){\text{Sech}}{\left[ {hm} \right]^3}\nonumber \\ & \quad + 384P( {36( {{m^2} + 6{P^2}( {4 + {m^2}{y^2}} )\alpha } )} \nonumber \\ & \quad \times {\text{cosh}}\left[ {my} \right] + hm( 648{P^2}\alpha + 4{m^4}{u_{{\text{hs}}}}^2\alpha {\xi ^2}\nonumber \\ & \quad + 9{m^2}( {2 + P\alpha ( {4{h^2}P - 3{u_{{\text{hs}}}}\xi } )} ) ){\text{sinh}}\left[ {hm} \right] \nonumber \\ & \quad - 18my( {{m^2} + 2{P^2}( {18 + {m^2}{y^2}} )\alpha } ){\text{sinh}}\left[ {my} \right] )\nonumber \\ & \quad - 24{m^2}{u_{{\text{hs}}}}\xi {\text{Sech}}\left[ {hm} \right]( {18({m^2} + 2{m^4}( {h - y} )} \nonumber \\ & \quad \times {( {h + y} ) + {P^2}( {9 + 6{h^2}{m^2} + 2{h^4}{m^4} - 2{m^4}{y^4}} )\alpha } )\nonumber \\ & \quad - 896{m^2}P{u_{{\text{hs}}}}\alpha \xi + 45{m^4}{u_{{\text{hs}}}}^2\alpha {\xi ^2}\nonumber \\ & \quad + {18( {{m^2} + 3{P^2}( {3 + 2{m^2}{y^2}} )\alpha } ){\text{cosh}}\left[ {2my} \right] - 216m{P^2}y\alpha {\text{sinh}}\left[ {2my} \right]} )\nonumber \\ & \quad - 128{m^4}P{u_{{\text{hs}}}}^2 \alpha {\xi ^2}{\text{Sech}}{\left[ {hm} \right]^2}( 162{\text{cosh}}\left[ {my} \right] - 2{\text{cosh}}\left[ {3my} \right]\nonumber \\ & \quad + 84hm{\text{sinh}}\left[ {hm} \right] - 81my{\text{sinh}}\left[ {my} \right] \nonumber \\ & \quad + 3my{\text{sinh}}\left[ {3my} \right]))) + 8640{m^4}S( {h - y} )( {h + y} )\mu r{\sigma _{\text{r}}})\nonumber \\ & \quad + 48m(360h{m^3}S\mu r{\sigma _{\text{r}}} + {\text{Br}}( {24hmP} \nonumber \\ & \quad \times ( 3{h^4}{m^2}{P^3}\alpha + 5{h^2}{m^2}P( {1 + 12P{u_{{\text{hs}}}}\alpha \xi } ) + 5{u_{{\text{hs}}}}\xi ( 72{P^2}\alpha\nonumber \\ & \quad + {m^2}( {6 + {u_{{\text{hs}}}}\alpha \xi ( { - 9P + 4{m^2}{u_{{\text{hs}}}}\xi } )} ) ) ) \nonumber \\ & \quad + 5{u_{{\text{hs}}}}\xi (27h{m^7}{u_{{\text{hs}}}}^3\alpha {\xi ^3}{\text{Sech}}{\left[ {hm} \right]^4}\nonumber \\ & \quad + 2( { - 864{P^3}\alpha + 9{m^6}{u_{{\text{hs}}}}^3\alpha {\xi ^3} + 2{m^4}{u_{{\text{hs}}}}\xi } \nonumber \\ & \quad \times {( {9 + 54{h^2}{P^2}\alpha - 8P{u_{{\text{hs}}}}\alpha \xi } ) - 18{m^2}P( {4 + 3P\alpha ( {8{h^2}P - {u_{{\text{hs}}}}\xi } )} )} )\nonumber \\ & \quad \times {\text{tanh}}\left[ {hm} \right] + {m^3}{u_{{\text{hs}}}}\xi {\text{Sech}}{\left[ {hm} \right]^2} \nonumber \\ & \quad \times ( - 36h( { - 3{P^2}\alpha + {m^2}( {1 + 2P\alpha ( {{h^2}P + 4{u_{{\text{hs}}}}\xi } )} )} )\nonumber \\ & \quad + m{u_{{\text{hs}}}}\alpha \xi ( {224P - 45{m^2}{u_{{\text{hs}}}}\xi } ){\text{tanh}}\left[ {hm} \right] ))))), \end{align} \tag{ 42 }$$

$$\begin{align} {{\Phi }}( y )& = \frac{1}{{1440}}{\text{Nt}}y\Bigg( - \frac{{720{\beta _i}}}{{{\text{Nb}} + h{\text{Nb}}{\beta _i}}} + 1/( {{m^3}{\text{Nb}}\mu r{k_{\text{r}}}} )4( 360h{m^3}S\mu r{\sigma _{\text{r}}}\nonumber\\ & \quad + {\text{Br}}( {24hmP( {3{h^4}{m^2}{P^3}\alpha } } + 5{h^2}{m^2}P( {1 + 12P{u_{{\text{hs}}}}\alpha \xi } )\nonumber \\ & \quad + 5{u_{{\text{hs}}}}\xi ( 72{P^2}\alpha + {m^2}( 6 + {u_{{\text{hs}}}}\alpha \xi ( { - 9P + 4{m^2}{u_{{\text{hs}}}}\xi } ) ) ) )\nonumber \\ & \quad + 5{u_{{\text{hs}}}}\xi ( 27h{m^7}{u_{{\text{hs}}}}^3\alpha {\xi ^3}{\text{Sech}}{{\left[ {hm} \right]}^4}\nonumber\\ & \quad + 2( - 864{P^3}\alpha + 9{m^6}{u_{{\text{hs}}}}^3\alpha {\xi ^3} + 2{m^4}{u_{{\text{hs}}}}\xi ( {9 + 54{h^2}{P^2}\alpha - 8P{u_{{\text{hs}}}}\alpha \xi } ) \nonumber \\ & \quad - 18{m^2}P( {4 + 3P\alpha ({8{h^2}P - {u_{{\text{hs}}}}\xi})})){\text{tanh}}\left[ {hm} \right] + {m^3}{u_{{\text{hs}}}}\xi {\text{Sech}}{\left[ {hm} \right]^2} \nonumber \\ & \quad \times ( { - 36h} ( { - 3{P^2}\alpha + {m^2}( {1 + 2P\alpha ( {{h^2}P + 4{u_{{\text{hs}}}}\xi } )} )} ){m^3}{u_{{\text{hs}}}}\xi {\text{Sech}}{\left[ {hm} \right]^2}\nonumber\\ & \quad \times ( - 36h( { - 3{P^2}\alpha + {m^2}} {( {1 + 2P\alpha ( {{h^2}P + 4{u_{{\text{hs}}}}\xi } )} )} ). \end{align} \tag{ 43 }$$

Figures 2–5 are prepared to clarify the alterations in velocity profiles subject to development is electroosmotic velocity parameter, Debye length parameter, zeta potential, and wall elasticity parameters. The velocity profiles for both of the nanofluids are compared and it is found that under the same physical conditions, the velocity of MoS₂/kerosene oil nanofluid is relatively lower than the velocity of MoS₂/CMC + water. It is due to the shear-thinning aspect of the CMC + water solution which facilitates the motion of the fluid. The change in velocity profile for an increase in electroosmotic velocity is clarified in figure 2. From the definition of the electroosmotic velocity parameter, it is obvious that velocity produced by the phenomenon of electroosmosis is directly related to the strength of the electric field and negative values of this parameter correspond to the mobility of ions in the direction of the peristaltic wave. An increase in u_hs in a negative direction physically implies the presence of a strong electric field which produces an enhancement in the velocity profile. It is evident from figure 3 that there is a substantial enhancement in the velocity for increment in the Debye length parameter. As there is an inverse relationship between the thickness of the EDL and the Debye length parameter m, a rise in m corresponds to a thinner EDL. The electric potential significantly enhances for a thin EDL which acts as the main driving force of the ions. Hence, electroosmotic velocity is regulated and an overall elevation in the velocity of the fluid is noticed. A similar enhancing response of velocity towards larger zeta potential is illustrated in figure 4. Figure 5 provides an insight into the extent of variation produced in velocity via larger wall elasticity parameters. As for the increase in the wall tension parameter, the speed of the peristaltic wave tends to grow therefore a substantial increment in velocity profile is observed. A rise in the wall mass parameter also boosts the fluid motion. However, an uplift in the wall damping parameter strengthens the viscous damping forces leading to a strong deceleration in the fluid velocity. Figure 6 reveals that a significant decline in the velocity of both nanofluids occurs when nanofluid are prepared with large fraction nanoparticles. Clearly, it is due to the higher viscosity of the inserted molybdenum dioxide nanoparticles.

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The streamline contours for the peristaltic transport in a microchannel are displayed in figures 7–12. The trapping phenomenon is observed for both types of nanofluids and it has been observed that fewer streamlines are circulated in the case of MoS₂/kerosene oil nanofluid when compared with the streamline pattern of MoS₂/CMC + water nanofluid. The trapping procedure is strongly retarded for an increase in the strength of the axial electric field and for a further increase in electroosmotic velocity, there is a substantial suppression in the size of the trapping bolus and the decrease in the number of circulating streamlines (see figures 7(a)–(c)). The trapping phenomenon for MoS₂/kerosene oil nanofluid is similarly affected by the rise in the electroosmotic velocity parameter as shown in figures 8(a)–(c). The evolution in the circulatory flow pattern of the peristaltic flow regime for enhanced values of the Debye length parameter is inspected in figures 9(a)–(c) for MoS₂/CMC + water nanofluid and in figures 10(a)–(c) for MoS₂/kerosene oil nanofluid. A wider circular structure with a smaller inner bolus is observed for a smaller value of the Debye length parameter i.e. for a thick EDL for both nanofluids. However, for a gradual increase in values of m, there is an extension in the volume of the central circular structure and a decrease in the number of circulations is noticed. As for enhancement in both electroosmotic velocity and the Debye length parameter, fluid motion is accelerated and the fluid particles are forced to move rapidly along the straight paths producing a reduction in the rotatory flow pattern. Consequently, less number of streamline trapped bolus are observed. Streamline contours modification for variation in elastic wall properties are inspected through figures 11(a)–(c) and 12(a)–(c). It can further be observed that with growth in the wall tension parameter, the size of the central circulatory bolus shrinks. For increasing the wall mass parameter, an extension in the volume of the central zone is produced however straightening in the external streamlines also occurs. However, for modification in the wall damping parameter, a very minor alteration in streamline structure is noticed. Again for the larger wall tension parameter and wall mass parameter, the fluid velocity is augmented and fluid particles follow the laminar flow pattern. However, rising values of the wall damping parameter resist the motion of fluid therefore fluid particles are deflected to follow the rotatory path. Hence volume occupied by streamlines grows.

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The evolution in axial temperature profile for Biot number, electroosmotic velocity parameter Debye length parameter, Joule heating parameter, and nanoparticle volume fraction is visualized in figures 13–17. A keen observation of these plots manifests that the temperature of kerosene oil nanofluid is comparatively larger than that of the temperature of CMC + water nanofluid. Due to the shear-thinning nature of CMC + water solution and less viscosity as compared to the viscosity of the kerosene oil, the acceleration of CMC + water nanofluid is notably higher than the speed of kerosene oil nanofluid. As a result, more cooling effect within the flow regime is produced by CMC + water nanofluid depicting a lesser temperature profile than kerosene oil nanofluid. Figure 13 elaborates on the alteration in the temperature of nanofluids via larger values Joule heating parameter. The temperature of the nanofluid is augmented when more electric energy is transformed into heat energy. It can be is visualized from figure 14 that there is a significant reduction in the temperature of both nanofluids as the Biot number is augmented. Biot number is directly linked with convective heat transfer. Increasing Biot number means that the thermal convection is boosted and the process of cooling occurs very rapidly. Consequently temperature within fluid medium drops. A consistent increment in the temperature profile is observed for enhancement in the electroosmotic velocity parameter (see figure 15). As increasing negative values of the electroosmotic parameter correspond to a rise in the strength of the aligned electric field which energizes the nanofluid particle. These energized particles contribute to raising the temperature of nanofluid. Furthermore, applying a strong electric field also boosts the joule heating phenomenon which is also a reason behind this increase in temperature. A substantial elevation in the temperature of the fluid for growing values of the Debye length parameter is demonstrated in figure 16. It is due to the higher potential generated across EDL which elevates the drag force produced by the rapid motion of ions on the fluid particles. The variation in the thermal profile of the fluid for inserting a larger fraction of nanoparticles is examined in figure 17. As expected, due to the growth in the conductive heat transfer, there is a reduction in the temperature of the nanofluid for a larger nanoparticle volume fraction. Consequently, the cooling effect of the working fluid is boosted.

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The heat transfer characteristics for both nanofluids subject to alteration in diverse active parameters are analyzed in figures 18–21. The heat transfer tendency of both nanofluids i.e. MoS₂/CMC + water nanofluid and MoS₂/kerosene oil nanofluid is compared through graphical results. The heat transfer coefficient basically calculates the change in the temperature of the fluid on the flow regime. Evidently, the efficiency of MoS₂/CMC + water in transferring the heat is higher than that of the MoS₂/kerosene oil nanofluid. As mentioned earlier, the kerosene oil is more viscous than the CMC + water solution having a lower thermal conductivity than the thermal conductivity of CMC + water. Furthermore, under the same pumping power and applied forces, the motion of CMC + water is facilitated due to its shear-thinning aspects. All these properties of MoS₂/CMC + water nanofluid are advantageous in raising its thermal efficiency. The evolution in the magnitude of the heat transfer coefficient for the multiple values of the Biot number is elaborated through figure 18. As enhancement in Biot number corresponds to the consistent growth in convective heat transfer, the cooling process is encouraged. As a result, the magnitude of the heat transfer coefficient increases. The alteration in the heat transfer coefficient for different values Joule heating parameter is delineated in figure 19. As the Joule heating parameter is connected with the strength of the axial electric field raised to power 2, increasing S physically means that a stronger electric source is connected. As a result, the phenomenon of electroosmosis is accelerated and the heat transfer rate rises. Rapid transmission of heat in response to a rise in the electroosmotic velocity parameter and the Debye length parameter is recorded through figures 20 and 21. Clearly, the pumping rate is increased by energizing the electroosmotic procedure which tends to raise the heat transfer due to convection. Consequently, the magnitude of the heat transfer coefficient rises.

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Computing the amount of entropy generation is very important to get an insight into the efficiency of a working system. Higher the rate of generation of volumetric entropy, lower the efficiency of the system. Observation manifest that more entropy is generated by the flow of kerosene oil nanofluid however very less amount of energy is dissipated for CMC + water nanofluid. Figures 22–26 are prepared to visualize the alteration in the process of entropy generation for multiple values of the involved parameters. Decay in the amount of entropy generation is noticed for increasing values of the Biot number as observed through figure 22. As mentioned earlier Biot number is responsible for facilitating the transfer of heat due to convection producing a decline in the amount of entropy generation. An augmentation in entropy generation is noticed via increasing the Joule heating parameter through figure 23. As Joule heating is the amount of electric energy of ions utilized to overcome the resistance offered by the fluid medium to the passage of the electrons. A rise in the Joule heating parameter means that more resistance is experienced by ions which results in the enhancement of entropy generation. Insertion of more nanoparticles in both types of nanofluid increases their efficiency by enhancing their thermal conductance ability as evaluated through figure 24. A very minute rise in the entropy generation by MoS₂/CMC + water nanofluid is observed at the center of the channel for larger values of both the electroosmotic velocity parameter and the Debye length parameter as seen through figures 25 and 26. However, a very significant elevation in entropy is produced in the case of MoS₂/kerosene oil nanofluid.

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