Entropy generation analysis for hybrid nanofluid mobilized by peristalsis with an inclined magnetic field

Abstract

The purpose of the present study is to analyse the entropy generation for the hybrid nanofluid mobilised by peristalsis. The hybrid nanoliquid is suspension of copper $\left(\mathrm{Cu}\right)$ and iron-oxide $\left(F{e}_3{O}_4\right)$ nanoparticles in water. Impacts of magnetic field, Joule heating, mixed convection, heat source/sink and viscous dissipation are reckoned. Governing set of equations are simplified by using lubrication approach. Obtained system of differential equations are solved numerically. Special attention is paid to analyse the effects of hybrid nanomaterial, Hartman and Grashoff numbers on entropy generation, Bejan number, axial velocity, temperature, heat transmission rate at walls, pressure gradient, skin friction, Nusselt number. Flow behaviour is visualised through streamlines. The study reveals that velocity and temperature decrease on increasing the volume fraction of solid nanomaterials. Higher Grashoff and Hartman numbers augment both velocity and temperature. Better heat transfer performance is recorded for strong Hartman number. $M$ and $\mathrm{Gr}$ improve Entropy generation and Bejan number. Higher Hartman number causes decrement in pressure gradient. Addition of nanoparticles concentration reduces skin friction. High flow rate increases trapping phenomenon.

Introduction

Nanofluids are best heat carriers and are colloidal suspensions of solid nanoparticles in base fluids. Nanomaterials are composed of metals (like gold, copper, silver, etc.), metallic oxides (like $F{e}_x{O}_y$ , $\mathrm{CuO},A{l}_2{O}_3$ , etc.), carbides/nitrides (like AIN, TiC, SiC, SiN, etc.). Ethylene-glycol, oil and water work as a base fluids. Initially Choi ¹ has studied the nanoliquids for heat transfer and concluded that the addition of solid nanomaterials improve thermal attributes of working fluid. Two-phase model was proposed by Tiwari and Das ² to contemplate the nanoliquids by making use of density, effective viscosity and thermal conductivity. Later, Buongiorno ³ recommended a model to scrutinise the nanofluid flow by employing the Brownian motion and thermophoresis effects. Turkyilmazoglu ⁴ utilised dual/single phase nanofluids models and got accurate results for the slippage flow in a concentric ring.

Peristalsis is a phenomena that occurs owing to the contraction and expansion of flexible walls. Its applications are found in engineering and biomedicine. For example food absorption in oesophagus, movement of urine from kidneys to the bladder, vasomotion in capillaries and arteries, movement of chyme in gastrointestinal track, finger pumping and many others. Initially, Latham ⁵ performed the experimental study for motion of the fluid mobilised by peristalsis. Later Shapiro et al. ⁶ extended it theoretically and explained the peristaltic flow by incorporating ‘small Reynolds number and large wave length’ approximations. Mishra and Rao ⁷ reported the transportation of Newtonian fluid through peristalsis in an asymmetric channel.

Magnetohydrodynamics is a securitisation of liquid motion under the influence of applied magnetic field. The investigation of MHD peristaltic flow has become attractive due to its ample implementation in biological and industrial fields. For example in MRI, tumour detection, petroleum industries and metal working process. Joule heating is among one of the salient features of the MHD. Reddy and and Venugopal Reddy ⁸ monitored the MHD peristalsis phenomenon of nanofluid in the presence of Joule heating with complaint walls. Hayat et al. ⁹ analysed the consequences of joule heating and slip on mixed convective peristaltic movement of nanofluid with Dufour and Soret impacts. Abbasi et al. ¹⁰ analysed the joule heating and radiation impacts on electroosmosis-modulated peristaltic flow of Prandtl nanofluid through tapered channel. Recently, Akram et al. ¹¹ analysed the electroosmotic peristaltic flow of nanofluid by using two different techniques.

The performance of thermodynamical system can be visualised through its entropy generation. Entropy generation overcomes energy disorder. Entropy generation analysis has key role in an industry and bio engineering. Initial work in this regard is done by Bejan^12,13 in which author illustrates the major steps to minimise the entropy generation at component level. Later on, Farooq et al. ¹⁴ computed the entropy generation and also reported the effects of thermal radiation over peristaltic movement of carbon nanotubes for the case of mixed convection. Abbasi et al. ¹⁵ reported entropy generation analysis of nanofluid activated by peristalsis with Hall and temperature dependent viscosity effects.

Latest development is to use hybrid nanofluid, which are manufactured by the suspension of the two or more nanoparticles into base fluid. Thus they have significant thermal conductivity and better heat transfer rate as compared to the traditional nanofluids. Saleem et al. ¹⁶ explained the heat transmission rate of blood-based hybrid nanofluid through a tube with ciliated walls. Ali et al. ¹⁷ investigated MHD peristalsis of hybrid nanofluid with slip conditions. Zahid et al. ¹⁸ and Abbasi et al. ¹⁹ carried out thermodynamic analysis of electroosmosis regulated by peristalsis of hybrid nanofluid. Tripathi et al. ²⁰ studied the peristaltic propulsion of electroosmosis induced hybrid nanofluids numerically through microchannel. Prakash et al. ²¹ and Afsar et al. ²² are some useful recent developments in peristaltic transport of hybrid nanofluid.

The novel features of this investigation are the analysis of entropy generation for the $(\mathrm{Cu}-F{e}_3{O}_4)/H_{2}O$ hybrid nanofluid triggered by peristaltic via asymmetric channel under the effect of an inclined magnetic field. Some more effects mixed convection, viscous dissipation, ohmic heating and heat generation/absorption are also accounted. Resulting system is evaluated numerically. Graphical study is performed for axial velocity, temperature, heat transmission rate at boundaries, Bejan number, entropy generation, pressure gradient, skin friction and Nusselt number for the different influential parameters. Flow behaviour is visualised through streamlines. Present study investigates the flow under the long wavelength and low Reynolds number assumptions. Moreover, there is a very limited work available in literature on peristaltic transport of hybrid nanomaterials. This study is devoted to pursue this development.

Problems formulation

Problems statement

Consider the motion of electrically conducting $F{e}_3{O}_4-\mathrm{Cu}/H_{2}O$ hybrid nanofluid driven by peristaltic waves flowing through an asymmetric $2D$ channel of width $d(=d_1+d_2)$ . Geometry of the problem is given through Figure 1. Let $T_1$ and $T_2$ be the temperatures of right $({\hat{H}}_1)$ and left $({\hat{H}}_2)$ walls respectively such that $(T_1<T_2)$ . An inclined magnetic field is applied on the channel making an inclination angle $\omega$ with $\hat{Y}$ -axis. Mathematical form of wavy channel walls as per Abbasi et al. ²³ is described as:

$${\hat{H}}_1(\hat{X},t)={\gamma }_1\cos \frac{2\zeta }{\lambda }(\hat{X}-\mathrm{ct})+d_1,$$

(1)

$${\hat{H}}_2(\hat{X},t)=-{\gamma }_2\cos \frac{2\zeta }{\lambda }((\hat{X}-\mathrm{ct})+\mathrm{\Omega })-d_2.$$

(2)

Where $\mathrm{\Omega }$ , $c$ , ${\gamma }_1$ , ${\gamma }_2$ , $\lambda$ and $t$ indicate phase difference, speed, amplitudes, wavelength of peristaltic waves and time respectively. These parameters satisfy the relations:

$${\gamma }_1^2+{\gamma }_2^2+2{\gamma }_1{\gamma }_2\cos \mathrm{\Omega }\le (d_1-d_2{)}^2.$$

(3)

Figure 1.

Schematic picture of the problem.

The considerations and assumptions for present study are narrated as:

• Asymmetric channel

• Hybrid nanofluid

• Inclined magnetic field

• Mixed convection

• Viscous dissipation

• Ohmic heating

• Heat generation/absorption

• Long wavelength and low Reynolds number assumptions.

The Ohms law in the absence of the applied electric field is ²⁴ :

$$\mathbf{J}={\sigma }_{\mathrm{hn}\Im }(\mathbf{V}\times \mathbf{B}),$$

(4)

where J denotes the current density. The components of inclined external magnetic field and velocity vector are $\mathbf{B}=B_0(\sin \mathbf{\omega },\cos \mathbf{\omega },0)$ and $\mathbf{V}=(\hat{U},\hat{V},0)$ respectively for the given problem. The Lorentz force becomes:

$$\begin{array}{c}\mathbf{J}\times \mathbf{B}={\sigma }_{\mathrm{hn}\Im }{B}_0^2\hfill \\ (\hat{V}\cos \omega \sin \omega -\hat{U}\mathrm{co}{s}^2\omega ,\hat{U}\cos \omega \sin \omega -\hat{V}\mathrm{si}{n}^2\omega ,0).\hfill \end{array}$$

(5)

Here ${\sigma }_{\mathrm{hn}\Im }$ stands for hybrid nanofluid’s electric conductivity. The expression for Joule heating is obtained as:

$$\frac{\mathbf{J}.\mathbf{J}}{{\sigma }_{\mathrm{hn}\Im }}={\sigma }_{\mathrm{hn}\Im }{B}_0^2({\hat{U}}^2\mathrm{co}{s}^2\omega +{\hat{V}}^2\mathrm{si}{n}^2\omega -2\hat{U}\hat{V}\cos \omega \sin \omega ).$$

(6)

Governing equations

The basic equations incorporating MHD (magnetohydrodynamics), viscous heating, mixed convection, heat generation/absorption and Joule heating as per Khazayinejad et al. ²⁴ are described:

$$\frac{\partial }{\partial \hat{X}}(\hat{U})+\frac{\partial }{\partial \hat{Y}}(\hat{V})=0,$$

(7)

$$\begin{array}{c}{\rho }_{\mathrm{hn}\Im }(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial t})\hat{U}=-\frac{\partial P}{\partial \hat{X}}\hfill \\ +{\mu }_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})\hat{U}+g(\rho \beta {)}_{\mathrm{hn}\Im }(T-T_m)\hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2(\hat{V}\cos \omega \sin \omega -\hat{U}\mathrm{co}{s}^2\omega ),\hfill \end{array}$$

(8)

$$\begin{array}{c}{\rho }_{\mathrm{hn}\Im }(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial t})\hat{V}=-\frac{\partial P}{\partial \hat{Y}}+{\mu }_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})\hat{V}\hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2(\hat{U}\cos \omega \sin \omega -\hat{V}\mathrm{si}{n}^2\omega ),\hfill \end{array}$$

(9)

$$\begin{array}{c}(\rho C{)}_{\mathrm{hn}\Im }(\hat{U}\frac{\partial }{\partial \hat{X}}+\hat{V}\frac{\partial }{\partial \hat{Y}}+\frac{\partial }{\partial t})T=k_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{X}}^2}+\frac{{\partial }^2}{\partial {\hat{Y}}^2})T\hfill \\ +{\mu }_{\mathrm{hn}\Im }\left[\begin{array}{c}2{\left(\frac{\partial \hat{U}}{\partial \hat{X}}\right)}^2+2{\left(\frac{\partial \hat{V}}{\partial \hat{Y}}\right)}^2\hfill \\ \multicolumn{1}{c}{+{(\frac{\partial \hat{U}}{\partial \hat{Y}}+\frac{\partial \hat{V}}{\partial \hat{X}})}^2}\end{array}\right]+\mathrm{\Phi }\hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2({\hat{U}}^2\mathrm{co}{s}^2\omega +{\hat{V}}^2\mathrm{si}{n}^2\omega -2\hat{U}\hat{V}\cos \omega \sin \omega ).\hfill \end{array}$$

(10)

In above mentioned equations, $P$ , $B_0$ , $g$ , $T$ and $T_m=\left(\frac{T_1+T_2}{2}\right)$ indicate the pressure, applied magnetic field, gravity, temperature and mean temperature, respectively. While ${\mu }_{\mathrm{hn}\Im }$ , ${\rho }_{\mathrm{hn}\Im }$ , ${\beta }_{\mathrm{hn}\Im }$ , $(\rho C{)}_{\mathrm{hn}\Im }$ , $\mathrm{\Phi }$ , $k_{\mathrm{hn}\Im }$ , ${\sigma }_{\mathrm{hn}\Im }$ represent dynamic viscosity, density, thermal expansion coefficient, heat capacity, heat generation/absorption parameter, thermal and electrical conductivity of hybrid nanofluid respectively.

The mathematical expression of skin friction and Nusselt number for hybrid nanofluids at right wall as per References 25–27 are:

$$C_f=-\left(\frac{{\mu }_{\mathrm{hn}\Im }}{{\mu }_{\Im }}\right)\frac{\partial \hat{U}}{\partial \hat{Y}},$$

(11)

$$\mathrm{Nu}=-\left(\frac{K_{\mathrm{hn}\Im }}{K_{\Im }}\right)\frac{\partial \hat{T}}{\partial \hat{Y}}.$$

The Bachelor’s effective viscosity model ²⁸ and Maxwell electric conductivity model ²⁹ for two-phase flow are:

$$\frac{{\mu }_{n\Im }}{{\mu }_{\Im }}=(1+2.5\phi +6.5{\phi }^2),$$

(12)

$$\frac{{\sigma }_{n\Im }}{{\sigma }_{\Im }}=\frac{(1+2\phi ){\sigma }_s+2(1-\phi ){\sigma }_{\Im }}{(1-\phi ){\sigma }_s+(2+\phi ){\sigma }_{\Im }}.$$

(13)

Where ${\sigma }_{\Im }$ and ${\sigma }_s$ denote electric conductivity of the fluid and solid nanoparticles respectively. The Hamilton-crosser’s (H-C) thermal conductivity model for two phase fluid ³⁰ is given as:

$$\frac{k_{n\Im }}{k_{\Im }}=\frac{k_s+(\wp -1)k_{\Im }-\phi (\wp -1)(k_{\Im }-k_s)}{k_s+(\wp -1)k_{\Im }+\phi (k_{\Im }-k_s)},$$

(14)

where $k_{n\Im }$ indicates the thermal conductivity of the nanofluid, $k_{\Im }$ the fluid and $k_s$ solid nanoparticle. Remember that, for $\wp =3$ it becomes Maxwell’s model and is reserved for spherical nanoparticles and for $\wp =6$ it is utilised for cylindrical nanoparticles. ³¹ The generalised form of above model’s and other thermophysical properties for hybrid nanoparticles are written according to Bibi and Xu ³² and Xu and Sun ³³ are provided through Table 1:

Table 1.

Hybrid nanofluid models.

Properties	Hybrid nanofluid models
Viscosity	$\frac{{\mu }_{\mathrm{hn}\Im }}{{\mu }_{\Im }}=(1+2.5{\phi }_a+6.5{\phi }_a^2)(1+2.5{\phi }_b+6.5{\phi }_b^2)$
Thermal conductivity	$\frac{k_{\mathrm{hn}\Im }}{k_{b\Im }}=\frac{k_{\mathrm{sb}}+(\wp -1)k_{b\Im }-\phi (\wp -1)(k_{b\Im }-k_{\mathrm{sb}})}{k_{\mathrm{sb}}+(\wp -1)k_{b\Im }+\phi (k_{b\Im }-k_{\mathrm{sb}})}$ , where $\frac{k_{b\Im }}{k_{\Im }}=\frac{k_{\mathrm{sa}}+(\wp -1)k_{\Im }-\phi (\wp -1)(k_{\Im }-k_{\mathrm{sa}})}{k_{\mathrm{sa}}+(\wp -1)k_{\Im }+\phi (k_{\Im }-k_{\mathrm{sa}})}$
Electric conductivity	$\frac{{\sigma }_{\mathrm{hn}\Im }}{{\sigma }_{b\Im }}=\frac{{\sigma }_{\mathrm{sb}}(1+2{\phi }_b)+2{\sigma }_{b\Im }(1-{\phi }_b)}{{\sigma }_{\mathrm{sb}}(1-{\phi }_b)+{\sigma }_{b\Im }(2+{\phi }_b)}$ , where $\frac{{\sigma }_{b\Im }}{{\sigma }_{\Im }}=\frac{{\sigma }_{\mathrm{sa}}(1+2{\phi }_a)+2{\sigma }_{\Im }(1-{\phi }_a)}{{\sigma }_{\mathrm{sa}}(1-{\phi }_a)+{\sigma }_{\Im }(2+{\phi }_a)}$ .
Density	${\rho }_{\mathrm{hn}\Im }=(1-{\phi }_a-{\phi }_b){\rho }_{\Im }+{\phi }_a{\rho }_{\mathrm{sa}}+{\phi }_b{\rho }_{\mathrm{sb}}$
Heat capacity	$(\rho C_p{)}_{\mathrm{hn}\Im }=(1-{\phi }_a-{\phi }_b)(\rho C_p{)}_{\Im }+{\phi }_a(\rho C_p{)}_{\mathrm{sa}}+{\phi }_b(\rho C_p{)}_{\mathrm{sb}}$
Thermal expansion	$(\rho \beta {)}_{\mathrm{hn}\Im }={\phi }_a(\rho \beta {)}_{\mathrm{sa}}+{\phi }_b(\rho \beta {)}_{\mathrm{sb}}+(1-{\phi }_a-{\phi }_b)(\rho \beta {)}_{\Im }$

Here subscripts ℑ stands for fluid, $b\Im$ for base fluid, $\mathrm{sa}$ for first and $\mathrm{sb}$ second phase of the solid nanoparticles. Numerical data of thermo-physical properties of copper $\left(\mathrm{Cu}\right)$ iron oxide $\left(F{e}_3{O}_4\right)$ and water $\left(H_{2}O\right)$ are provided in Table 2:

Table 2.

Numerical data for the thermophysical properties.

Phase	$\rho (\mathrm{kg}{m}^{-3})$	$K(\mathrm{Wm}{k}^{-1})$	$C(\mathrm{Jk}{g}^{-1}{K}^{-1})$	$\beta (1/k)\times {10}^{-6}$	$\sigma (S{m}^{-1})$
Water	997.1	0.613	4179	210	0.05
Copper	8933	401	385	16.65	5.96 × 107
IronOxide	5200	6	670	13	25,000

The transformation from fixed $(\hat{X},\hat{Y})$ into moving $(\hat{x},\hat{y})$ frames is determined as:

$$\underbrace{\hat{y}=\hat{Y},\hat{x}=\hat{X}-\mathrm{ct},p(\hat{x},\hat{y})=P(\hat{X},\hat{Y},t),\hat{u}=\hat{U}-c,\hat{v}=\hat{V}.}$$

(15)

Making use of the above transformation, the system of equations (7)–(10) yields:

$$\frac{\partial \hat{u}}{\partial \hat{x}}+\frac{\partial \hat{v}}{\partial \hat{y}}=0,$$

(16)

$$\begin{array}{c}{\rho }_{\mathrm{hn}\Im }((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})(\hat{u}+c)\hfill \\ =-\frac{\partial p}{\partial \hat{x}}+{\mu }_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})(\hat{u}+c)+g(\rho \beta {)}_{\mathrm{hn}\Im }(T-T_m)\hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2(\hat{v}\cos \omega \sin \omega -(\hat{u}+c)\mathrm{co}{s}^2\omega ),\hfill \end{array}$$

(17)

$$\begin{array}{c}{\rho }_{\mathrm{hn}\Im }((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})\hat{v}=-\frac{\partial p}{\partial \hat{y}}+{\mu }_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})\hat{v}\hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2((\hat{u}+c)\cos \omega \sin \omega -\hat{v}\mathrm{si}{n}^2\omega ),\hfill \end{array}$$

(18)

$$\begin{array}{c}(\rho C{)}_{\mathrm{hn}\Im }((\hat{u}+c)\frac{\partial }{\partial \hat{x}}+\hat{v}\frac{\partial }{\partial \hat{y}})T=k_{\mathrm{hn}\Im }(\frac{{\partial }^2}{\partial {\hat{x}}^2}+\frac{{\partial }^2}{\partial {\hat{y}}^2})T\hfill \\ +{\mu }_{\mathrm{hn}\Im }\left[\begin{array}{c}2{\left(\frac{\partial \hat{u}}{\partial \hat{x}}\right)}^2+2{\left(\frac{\partial \hat{v}}{\partial \hat{y}}\right)}^2\hfill \\ \multicolumn{1}{c}{+{(\frac{\partial \hat{u}}{\partial \hat{y}}+\frac{\partial \hat{v}}{\partial \hat{x}})}^2}\end{array}\right]+\epsilon \hfill \\ +{\sigma }_{\mathrm{hn}\Im }{B}_0^2((\hat{u}+c{)}^2\mathrm{co}{s}^2\omega +{\hat{v}}^2\mathrm{si}{n}^2\omega -2(\hat{u}+c)\hat{v}\cos \omega \sin \omega ).\hfill \end{array}$$

(19)

Non-dimensionalisation

Incorporating the dimensionless parameters and variables defined as per Khazayinejad et al. ²⁴ :

$$\begin{array}{c}x=\frac{\hat{x}}{\lambda },y=\frac{\hat{y}}{d_1},u=\frac{\hat{u}}{c},v=\frac{\hat{\nu }}{c\delta },\delta =\frac{d_1}{\lambda },\hfill \\ a=\frac{{\gamma }_1}{d_1},b=\frac{{\gamma }_2}{d_1},d=\frac{d_2}{d_1},h_1=\frac{{\hat{H}}_1}{d_1},h_2=\frac{{\hat{H}}_2}{d_1},\hfill \\ p=\frac{d_1^{2}p}{c\lambda {\mu }_{\Im }},\mathrm{Re}=\frac{{\rho }_{\Im }c{d}_1}{{\mu }_{\Im }},M=\sqrt{\frac{{\sigma }_{\Im }}{{\mu }_{\Im }}}{B}_0{d}_1,\hfill \\ \mathrm{Br}=\mathrm{Pr}.\mathrm{Ec},\mathrm{Gr}=\frac{{\rho }_{\Im }g{\beta }_{\Im }(T_2-T_1)d_1^2}{{\mu }_{\Im }c},\theta =\frac{T-T_m}{T_2-T_1},\hfill \\ \mathrm{Ec}=\frac{c^2}{C_{\Im }(T_2-T_1)},\mathrm{Pr}=\frac{{\mu }_{\Im }{C}_{\Im }}{k_{\Im }},\epsilon =\frac{d_1^2\mathrm{\Phi }}{(T_2-T_1)k_{\Im }},\hfill \\ u=\frac{\partial \psi }{\partial y},v=-\delta \frac{\partial \psi }{\partial x},\hfill \end{array}$$

(20)

and under the assumption of large wave length and small Reynolds number, equations (16)–(19) can be re-written as:

$$A_4\frac{{\partial }^4\psi }{\partial y^4}+A_1\mathrm{Gr}\frac{\partial \theta }{\partial y}-A_2{M}^2\mathrm{co}{s}^2\omega \frac{{\partial }^2\psi }{\partial y^2}=0,$$

(21)

$$\frac{\partial p}{\partial y}=0,$$

(22)

$$A_3\frac{{\partial }^2\theta }{\partial y^2}+\mathrm{Br}{A}_4{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2+A_2\mathrm{Br}{M}^2\mathrm{co}{s}^2\omega {(1+\frac{\partial \psi }{\partial y})}^2+\epsilon =0.$$

(23)

In the earlier mentioned equations $\mathrm{Gr}$ , $M$ , $\mathrm{Re}$ , $\mathrm{Br}$ and $\mathrm{Pr}$ are non-dimensional parameters representing the Grashoff, Hartmann, Reynolds, Brinkman and Prandtl numbers respectively. While ${\phi }_a$ , ${\phi }_b$ and $\epsilon$ stand for volume fraction of copper, iron oxide nanomaterials and heat absorption/generation parameter respectively. $p$ , $\theta$ , $\epsilon$ , $\psi$ are dimensionless temperature, pressure, heat generation/absorption parameter and stream function respectively. Further, a detailed nomenclature is also provided.

Moreover $(A_1-A_4)$ are given as:

$$\begin{array}{c}{A}_1=\frac{{(\rho \beta )}_{\mathrm{hn}\Im }}{{(\rho \beta )}_{\Im }}=(1-{\phi }_a-{\phi }_b)+\frac{1}{{(\rho \beta )}_{\Im }}\{{\phi }_a{(\rho \beta )}_{\mathrm{sa}}+{\phi }_b{(\rho \beta )}_{\mathrm{sb}}\},\hfill \\ A_2=\frac{{\sigma }_{\mathrm{hn}\Im }}{{\sigma }_{\Im }},A_3=\frac{K_{\mathrm{hn}\Im }}{K_{\Im }},A_4=(1+2.5{\phi }_a+6.5{\phi }_a^2)\hfill \\ (1+2.5{\phi }_b+6.5{\phi }_b^2).\hfill \end{array}$$

(24)

The dimensionless mean flow rates in fixed frame $\eta =\left(\frac{\hat{Q}}{c{d}_1}\right)$ and in moving frame $\mathrm{\Theta }=\left(\frac{\hat{q}}{c{d}_1}\right)$ satisfying the relation:

$$\eta =\mathrm{\Theta }+d+1.$$

(25)

Further $\mathrm{\Theta }$ is defined as:

$$\mathrm{\Theta }=\underset{h_1}{\overset{h_2}{\int }}{\psi }_y\mathrm{dy}.$$

(26)

The modified form of skin friction and Nusselt number are:

$$\begin{array}{c}{C}_f=-\left(\frac{{\mu }_{\mathrm{hn}\Im }}{{\mu }_{\Im }}\right){\frac{\partial u}{\partial y}|}_{h_2},\hfill \\ \mathrm{Nu}=-\left(\frac{K_{\mathrm{hn}\Im }}{K_{\Im }}\right){\frac{\partial \theta }{\partial y}|}_{h_2}.\hfill \end{array}$$

(27)

The stream function satisfies the equation (16) identically. To compute the temperature and stream function, the dimensionless boundary conditions are modified according to Zahid et al. ¹⁸ as:

$$\psi =\frac{\mathrm{\Theta }}{2},{\psi }_y=-1,\theta =-\frac{1}{2},\mathrm{at}y=h_1=a\cos (2\zeta x)+1,$$

$$\psi =-\frac{\mathrm{\Theta }}{2},{\psi }_y=-1,\theta =\frac{1}{2},\mathrm{at}y=h_2=-b\cos (2\zeta x+\mathrm{\Omega })-d.$$

(28)

Entropy generation and Bejan number

The dimensional formulation of entropy generation is:

$$\begin{array}{c}\multicolumn{1}{c}{S_G=\frac{K_{\mathrm{hn}\Im }}{{(T_2-T_1)}^2}[{\left(\frac{\partial \hat{T}}{\partial \hat{X}}\right)}^2+{\left(\frac{\partial \hat{T}}{\partial \hat{Y}}\right)}^2]}\\ +\frac{{\mu }_{\mathrm{hn}\Im }}{(T_2-T_1)}\left[\begin{array}{c}2{\left(\frac{\partial \hat{U}}{\partial \hat{X}}\right)}^2+2{\left(\frac{\partial \hat{V}}{\partial \hat{Y}}\right)}^2\hfill \\ \multicolumn{1}{c}{+{(\frac{\partial \hat{U}}{\partial \hat{Y}}+\frac{\partial \hat{V}}{\partial \hat{X}})}^2}\end{array}\right]+\frac{\mathrm{\Phi }}{(T_2-T_1)}\hfill \\ +\frac{{\sigma }_{\mathrm{hn}\Im }}{(T_2-T_1)}{B}_0^2({\hat{U}}^2\mathrm{co}{s}^2\omega +{\hat{V}}^2\mathrm{si}{n}^2\omega -2\hat{U}\hat{V}\cos \omega \sin \omega ).\hfill \end{array}$$

(29)

The non-dimensional form is:

$$\begin{array}{c}\multicolumn{1}{c}{N_s=\frac{S_G}{S_c}=A_3{\left(\frac{\partial \theta }{\partial y}\right)}^2+\mathrm{Br}{A}_4{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2}\\ \multicolumn{1}{c}{+A_2\mathrm{Br}{M}^2\mathrm{co}{s}^2\omega {(1+\frac{\partial \psi }{\partial y})}^2+\epsilon ,}\end{array}$$

(30)

$$B_e=\frac{A_3{\left(\frac{\partial \theta }{\partial y}\right)}^2}{A_3{\left(\frac{\partial \theta }{\partial y}\right)}^2+\mathrm{Br}{A}_4{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2+A_2\mathrm{Br}{M}^2\mathrm{co}{s}^2\omega {(1+\frac{\partial \psi }{\partial y})}^2+\epsilon }.$$

(31)

Where $S_c=\frac{k\Im }{d_1^2}$ is the characteristics entropy generation and range of $\mathrm{Be}$ is $(0\le B_e\le 1)$ .

Numerical procedure

The built-in numerical technique NDSolve is utilised to solve the dimensionless system of equations (21)–(23) with the aid of boundary conditions (28) in Mathematica. This method is very effective and efficient to solve boundary value problems especially highly non-linear system of equations. Both $x$ and $y$ are plotted simultaneously with small step size 0.01 and small error. Thus this technique has unconditional stability and accuracy.

Outcomes and discussions

This section analyses the consequences of influential parameters on axial velocity, temperature, heat transmission rate, entropy generation, Bejan number and axial pressure gradient. Graphs are obtained for the fixed parameters $a=0.7,{\phi }_a=0.01,{\phi }_b=0.01,\wp =6,b=0.6,x=0,d=0.8,\mathrm{\Omega }=\frac{\pi }{2},\eta =0.7,\mathrm{Gr}=3,\mathrm{Br}=0.3,M=1,$ $\omega =\frac{\pi }{4}$ and $\epsilon =2.5$ , unless mentioned alternate.

Velocity and temperature

Figure 2(a) to (h) illustrates the consequences of sundry parameters over axial velocity and temperature of hybrid nanoliquids. Velocity profiles are placed on left side and temperature profiles are placed on right side. Velocity of hybrid nanofluid declines for higher values of ${\phi }_a$ and ${\phi }_b$ . As large amount of nanomaterial develops confrontation to the flow and stops the fluid to attain its maximum velocity. Similarly velocity of hybrid nanofluid significantly slumps for strong value of $M$ . This is due to strong effect of magnetic field which produces drag force (Lorentz force). This Lorentz force leads resistance against the flow. On the contrary velocity hikes at large values of $\mathrm{Gr}$ . Higher value of $\mathrm{Gr}$ implies strong buoyancy driven force. As a result velocity of the hybrid nanofluid rises. Parts (b), (d), (f) and (h) depict the effects of sundry parameters on temperature profile. It is evident that temperature of the hybrid nanofluid declines on addition of nanoparticles. It happens owing to the strong thermal conductivity of hybrid nanofluid and as a result fluid will loose heat rapidly. An increasing behaviour is observed in $\theta$ at a large values of $M$ . Strong Hartman number depends upon Lorentz force (drag force) which creates resistance against the fluid motion and permits the hybrid nanoliquid to heat up more. Temperature of hybrid nanofluid grows with the enhancement of Grashoff number.

Figure 2.

Impact of pertinent parameters on velocity and temperature.

Axial pressure gradient

Figure 3(a) to (d) illustrates the change in pressure gradient for the mentioned parameters. Fluctuation and periodic behaviour in pressure gradient graphs are observed for all values of ${\phi }_a$ , ${\phi }_b$ , $M$ and $\mathrm{Gr}$ . Increase in amount of nanoparticles compress pressure gradients. Actually addition of nonmaterial creates resistance against flow. Similar behaviour is spotted on variation of Hartman number at the middle part of the channel. However, strong $\mathrm{Gr}$ increases pressure gradient.

Figure 3.

Impact of pertinent parameters on pressure gradient.

Heat transmission rate at walls

Figure 4(a) to (d) manifest the heat transmission rate at walls for sundry parameters. Parts (a) and (b) exhibit the increments in heat transmission rate by increasing the amount of hybrid nanomaterials because addition of solid nanoparticles improve the thermal conductivity of the hybrid nanofluid and helps the heat interchange process. An increasing trend of heat transfer rate is noticed on the enlargements of $M$ and $\mathrm{Gr}$ as well.

Figure 4.

Impact of pertinent parameters on heat transfer rate at wall.

Entropy generation and Bejan number (Be)

Figure 5(a) to (h) demonstrates the impacts of pertinent parameters on entropy generation and Bejan number. It is worth noting that results for entropy generation are directly related to the temperature. The augmentation in temperature creates disorder in the system and thus enhances entropy. It is evident from first two figure that nanoparticles concentration enhancement invites decrement in $N_s$ . As addition of nanoparticles drops the temperature of nanoliquids. Opposite behaviour of entropy generation is noticed on increasing $M$ , because joule heating augments temperature of hybrid nanofluid. Variation of Grashoff number also amplify the entropy generation. As $\mathrm{Gr}$ number greatly depends upon mixed convection which motivates the temperature. Figures on right panel reveal the influences of various parameters on $\mathrm{Be}$ . Figure (b) and (d) show the decreasing trend in $B_e$ on increasing the volume fraction of solid nanoparticles. As total entropy generation becomes dominant to entropy generation motivated by heat transportation. $B_e$ increases on increasing the Hartman and Grashoff numbers. These results are in qualitative agreement with the results available in the literature. Plot (i) depicts the impact of $\epsilon$ on $N_s$ . Entropy generation increases with the amplification of $\epsilon$ . Actually, increasing effects of heat source parameter generate more heat and lead to higher irreversibility. Similar increasing trend is noticed for Bejan number at large values of $\epsilon$ .

Figure 5.

Impact of pertinent parameters on entropy and Bejan number.

Skin friction and Nusselt number

Figure 6(a) to (h) are sketched to show variation in skin friction and Nusselt number against longitudinal distance $x$ . The impact of nanoparticles volume fraction on skin friction can be seen from (a) and (c). Skin friction decreases on addition of nanoparticles concentration. Hartman number also reduces $C_f$ . Strong $\mathrm{Gr}$ increases skin friction. Figure (b) and (d) exhibit the impacts of ${\phi }_a$ and ${\phi }_b$ on Nussel number. Enlargement of nanoparticle volume fraction increase Nusselt number. Meanwhile high $M$ and $\mathrm{Gr}$ reduce it.

Figure 6.

Impact of pertinent parameters on Skin friction and Nusselt number.

Trapping phenomenon

Bolus and Trapping phenomena are interesting features in peristaltic transport. Streamlines can be visualised through these features. Bolus identifies the rotational flow under the wave crests and Trapping indicates the flow circumstances near the central line. Figure 7 illustrates the impacts of Hartman number and flow rate on streamlines. The graphs of streamlines are obtained for symmetric channel. It is clear from graphs of left panel that bolus size become small on increasing $M$ . Actually magnetic field provides Lorentz force which acts against the flow. This battle of forces produce rotational flow and as a result trapped bolus will decrease. Moreover, flow rate $\eta$ is responsible for trapping near the boundary and central line. Graphs of right panel are sketched to support this argument. Note that at low value of $\eta$ , there is small trapping phenomenon and thus trapped bolus is small in size. The trapped bolus significantly increases on increasing $\eta$ .

Figure 7.

Impacts of Hartmann number and flow rate on streamlines.

Comparison with previous results

A comparison table is constructed to check the validity of the adapted technique for limiting case. Numerical values of $(-\frac{K_{\mathrm{hn}\Im }}{K_{\Im }}\theta \prime (h_1))$ received from present investigation are compared with the results calculated by Zahid et al. ¹⁸ (Shooting method) and Abbasi et al. ²³ (numerical technique) are provided in Table 3. Which depicts good agreement with previous studies.

Table 3.

Comparison table for limiting case of the present investigation with the results available in previous literature for $\begin{array}{l}{\phi }_b=0.1,\wp =3,b=0.6,x=0,d=0.8,\mathrm{\Omega }=\frac{\pi }{2},\eta =0.7,\\ Gr=3,Br=0.3,m=2,\omega =0\mathrm{and}\epsilon =2.5.\end{array}$

M	Zahid et al. 18	Abbasi et al. 23	Present
0	3.93751	3.937511	3.93937
1	3.95900	3.959001	3.96088
2	4.02357	4.023571	4.02551
3	4.13140	4.131391	4.13342

Conclusions

The entropy generation for electrically conducting hybrid nanofluid triggered by peristalsis via asymmetric channel is analysed. Impacts of MHD mixed convection, Ohmic heating, heat generation/absorption and viscous dissipation are reckoned. Upshots drawn from this study are summarised below:

• Both temperature and velocity slump on the enhancement of volume fraction of solid nanoparticles.

• Strong M improves temperature and heat transfer rate.

• Enhancement of the concentration of solid nanomaterials and Hartmann number increase pressure gradient.

• Higher Gr increases pressure gradient rapidly.

• M and Gr increase Entropy generation and Bejan number while concentration of nanoparticles acts oppositely.

• Addition of nanoparticles decrease skin friction.

• High flow rate increases trapping phenomenon.