Effects of different shapes of nanoparticles on peristalsis of nanofluid possessing non-isothermal properties

Abstract

The exploration of various nanoparticles has become a focal point for researchers, driven by its potential applications in both industry and medicine. Nanoparticles can be incorporated in several shapes and sizes, hence this paper examines the nanofluid’s performance in a peristaltic channel under the impact of combined external applied electric and magnetic fields, in addition to thermal radiation and Joule heating effects. The present study also examines the non-isothermal characteristics with no slip convection for four distinct nanoparticles, namely Iron oxide ( $F{e}_3{O}_4$ ), Copper (Cu), Gold (Au), and Silver (Ag). Shape effects of nanomaterial are also considered. Arising nonlinear system is tackled numerically employing the integrated package NDSolve in Mathematica by the implementation of weaker Reynolds number and long wavelength assumptions. Current findings expose that radiation has a beneficial impact on controlling the velocity and temperature within the channel. Increase in radiation parameter leads to reduce both velocity and temperature by approximately $2.39\%$ and $18.18\%$ respectively. An improvement of up to $2.28\%$ in heat transfer rate can be achieved by utilizing spherical iron oxide nanoparticles for thermal conductivity parameter of range $0.00-0.10$ . Furthermore, silver nanoparticles provide better thermal conductivity compared to other nanoparticles.

Introduction

Nanofluids, which are suspensions of nanoparticles in a base fluid, have attracted considerable attention because of their advanced thermophysical properties, including thermal conductivity and convective heat transfer coefficients. These unique characteristics make nanofluids favorable for various applications in industries such as electronics cooling, solar thermal systems, biomedical engineering, and heat exchangers. Recent literature in the field of mathematics has focused on modeling the behavior of nanofluids to understand their flow and heat transfer mechanisms better under different flow conditions and nanoparticle concentrations. Firstly, the notion of nanofluids was offered by Choi ¹ who presented an improvement in thermal conductivity of fluids by adding small amount of nanoparticles. Additionally, mathematical analyses have contributed to understanding the fundamental interactions between distinct nanoparticles and the base fluid, thereby facilitating the modification of nanofluids with enhanced properties modified to specific applications. Dykman and Khlebtsov ² studied the applications of gold nanoparticles in biology and medicines. Sayed et al. ³ explored the slippery aspects of non-Newtonian material flow with peristaltic motion. Kothandapani and Prakash ⁴ evaluated the magnetized Jeffrey nanomaterial flow by considering the peristaltic motion. Nadeem et al. ⁵ conducted a study to analyze hybrid nanofluid over a curved surface. Results showed improved heat transfer rate for the hybrid nanofluid compared to the conventional nanofluid. Liu et al. ⁶ studied silver nanoparticles for treating breast cancer cells. This work highlighted the benefits of AgNPs for the generation of heat in breast cancer cells by targeting intracellular infections. Eshgarf et al. ⁷ studied single-phase and two-phase models of nanofluid to investigate the heat transfer rate. Furthermore, heat transfer analysis of nanofluid flowing modulated inside a double-pipe heat exchanger that includes spindle-shaped turbulators is presented by Izadi et al. ⁸ Results revealed that incorporating such turbulators enhances thermal efficiency when compared to smooth pipe. Additionally, Khan et al. ⁹ addressed the thermal applications of couple stress nanofluid subjected to triple diffusion effects and bioconvection. The investigation further incorporated variable viscosity and thermal conductivity, along with linear thermal radiation. The study highlighted the importance of triple diffusion in improving the heat and mass transfer rates. Some inquiries pertaining to the nanofluids can be observed via.^6,10–15

Peristalsis, the contraction and relaxation of muscles, serves as a fundamental mechanism in various physiological processes, including digestion, blood circulation, and urinary system function. This wave-like motion enables efficient movement of substances such as food, blood, and waste through tubes or channels. Experimental study of peristaltic flow was presented by Latham. ¹⁶ Shapiro et al. ¹⁷ used ‘long wavelength and low Reynolds number’ assumptions to investigate the peristaltic viscous fluid. Rao and Mishra ¹⁸ investigated the peristalsis of power law fluid via asymmetric porous tube. Their findings highlighted the influence of wave shape and porous wall properties of wavy channel. Further, Pandey and Tripathi ¹⁹ explored peristaltic flow of Casson fluid via finite channel. The findings showed insights into the influence of Casson fluid rheology on peristaltic motion, offering a better understanding of flow dynamics in the esophagus. Also, Rachid and Ouazzani ²⁰ examined the MHD peristalsis of Jeffrey fluid flow confined between a pair of deformable coaxial tubes having unequal wavelengths. Shahzad and Awan ²¹ conducted a study on the flow of heated Rabinowitsch fluid through an elliptic vertical peristaltic duct. The study provided insights into peristaltic flow in non-Newtonian systems. In addition, peristalsis together with nanofluids reveals new techniques for developing microfluidic technologies and refining their performances in various applications that is, biomedical, analytical, and industry. Akbar et al. ²² presented a model to explore nanofluid flow through peristaltic channel, considering slip boundary conditions. Abbasi et al. ²³ analytically studied peristaltically induced flow of nanomaterial through porous medium. Also, temperature-dependent characteristics of distinct nanoparticles, influencing peristaltic flow behavior in nanofluids is explored by Abbasi et al. ²⁴ Their study contributed to understanding the role of nanoparticle features in boosting heat transfer efficiency. Ebaid et al. ²⁵ adopted Homotopy Perturbation method to investigate flow of blood-based gold nanoparticles with a heat source via peristaltic conduit. The findings suggested potential applications of utilizing gold nanoparticles in cancer treatment. Numerous efforts have been recorded by researchers.^26–30 Moreover, Li et al. ³¹ analyzed turbulent Reynolds number at the regular, converging, and diverging outlets: Dynamics of air, water, and kerosene through y-shaped cylindrical copper ducts. Also, Animasaun et al. ³² explored half-cycle length of converging circular wavy duct with diverging-outlet.

Electro-magneto hydrodynamics (EMHD) and radiation properties display two interesting factors that can considerably affect the peristaltic performance of nanofluids. Researchers can involve the motion and delivery of nanoparticles within the nanofluid under external electric or magnetic fields, so varying the peristaltic pumping features. Besides, radiation effects, can affect the thermophysical properties of nanofluids, leading to deviations in the peristaltic flow behavior. Understanding the relationship between EMHD and radiation effects on nanofluid peristalsis is beneficial for various biomedical applications, including drug delivery systems and hyperthermia treatment. Prakash et al. ³³ investigated the combined effects thermal radiation and thermal wall slip on MHD electroosmotic flow of nanofluid in asymmetric peristaltic conduit. Hasona et al. ³⁴ explored the influence of thermal radiation and MHD effects on the flow of nanofluid in peristaltic channel, with application to radiotherapy and thermotherapy in cancer treatment. Abbasi et al. ³⁵ examined the peristaltic flow of Prandtl nanofluid modulated by electroosmosis with radiation and joule heating effects via tapered channel. In addition, Akbar et al. ³⁶ investigated the MHD peristaltic transport of radiative MWCNT-Ag/C₂H₆O₂ hybrid nanofluid with fluctuating characteristics. Besides, Saba et al. ³⁷ presented a study that considered combined effects of electric and magnetic fields on two-phase nanofluid flow through a curved peristaltic channel. Further related work can be seen in References 38–42. Incorporating different nanoparticles such as Gold (Au), Silver (Ag), Iron oxide ( $F{e}_3{O}_4$ ), and Copper (Cu) with diverse shapes into the study of electroosmotic peristaltic flow introduces several aspects that significantly enhance the understanding of nanofluid dynamics. Each nanoparticle exhibits distinct thermophysical and electromagnetic attributes, contributing uniquely to heat and mass transfer, flow behavior, and application-specific adaptability. Gold nanoparticles are ideal for biomedical imaging and targeted drug delivery due to their biocompatibility and moderate thermal conductivity. Fe₃O₄ nanoparticles are known for its superparamagnetic behavior and hence are considered beneficial in MRI, hyperthermia, catalytic degradation of pollutants and catalysis. Silver nanoparticles possess high thermal and electrical conductivity, making them suitable for sensor devices, solar cells, etc. Copper nanoparticles, a cost-effective alternative, provide better thermal conductivity and hence are highly effective for enhancing thermal transport, especially in industrial heat exchange systems. The integration of shape factor effects further adds importance to the study. Since each shape affects the thermophysical characteristics of the fluid differently due to variations in surface area and shape factor, influencing flow properties and impacting the thermal efficiency. Most prior work assumes constant fluid properties or focuses on single nanoparticles. Fluid’s viscosity reduces with increasing temperature, varying resistance to the flow and improving volumetric output, whereas thermal conductivity typically rises, enhancing heat transfer efficiency. Since these dependencies are vital for precise predictions in applications like biofluid heating, drug delivery, and microreactor cooling, where even minor temperature gradients can significantly affect system performance. Hence accounting for temperature-dependent properties enables more accurate and reliable prediction of fluid performance and ignoring these variations leads to errors in predicting fluid flow. The combination of electromagnetohydrodynamic and radiation effects in the study of peristaltic nanofluid flow is essential in evolving technologies such as microfluidics, energy converter devices, biomedical transport, and thermal management systems, where more precise heat and mass control at the nanoscale is critical. Despite their importance, few studies have addressed the coupled effects of peristalsis, electroosmosis, and temperature-dependent properties across multiple nanoparticle types and shapes in a unified framework under the influence of radiation. The results show that nanoparticle geometry significantly affects heat transfer and flow properties. These findings deliver new design strategies for enhancing thermal systems in biomedical engineering, microfluidics, and cooling application. Also, these results highlight the importance of particle morphology in the effective procedure of non-isothermal nanofluidic systems. This research fills that gap by offering a comprehensive analysis, focusing on how temperature fluctuations control nanofluid dynamics and identifying optimal conditions for thermal and flow performance. Additionally, this study addresses the following research questions, such as: (a) How do different shapes of Au, Fe₃O₄, Ag, and Cu nanoparticles influence heat transfer efficiency and the performance of peristaltic flow in non-isothermal and thermal transport of nanofluids subjected to electro-magneto-hydrodynamic (EMHD) impacts? (b) What is the comparative impact of distinct nanoparticles, and which nanoparticle offers the most effective thermal performance under combined electromagnetic and radiation effects? (c) How does thermal radiation interact with EMHD mechanisms to influence fluid motion in nanofluids containing various metallic nanoparticles? (d) In what ways temperature-induced changes in viscosity and thermal conductivity impact the transport mechanism in fluid flow situations?

In existing study, the authors have selected the nanoparticles of five different shapes that is, cylinders, spherical, blades, tetrahedrons, and platelets. Peristaltic phenomenon of Four distinct nanoparticles, namely Gold (Au), Silver (Ag), Iron oxide ( $F{e}_3{O}_4$ ), and Copper (Cu) having variable viscosity and thermal conductivity under electric, magnetic, and radiation effects are analyzed here. The Resultant nonlinear systems are solved numerically via NDSolve in Mathematica, with physical analysis facilitated through the presentation of graphs and tables.

Problem development

Peristalsis of nanoparticles that is, Copper (Cu), Iron oxide ( $F{e}_3{O}_4$ ), Silver (Ag), and Gold (Au) dispersed in water, conducted within asymmetric channel with a thickness of 2d. The viscosity of nanofluid is estimated using Brickman’s viscosity model ²⁴ :

$${\mu }_{\mathrm{nf}}=\frac{{\mu }_w}{{(1-\varphi )}^{2.5}},$$

(1)

where ${\mu }_{\mathrm{nf}},\varphi$ , and ${\mu }_w$ denote the nanofluid’s viscosity, the volume fraction of nanoparticles, and the water’s viscosity. It is assumed that the viscosity of water fluctuates with temperature using the following condition ²⁴ :

$${\mu }_w={\mu }_0(1-{\alpha }_0(T-T_0)).$$

(2)

Here α₀ represents the dimensional variable viscosity parameter, ${\mu }_0$ signifies the base fluid’s viscosity at constant temperature, T₀ denotes the ambient temperature, and T is nanofluid’s temperature. Upon substituting equation (2) into (1), the resultant relationship is provided as:

$${\mu }_{\mathrm{nf}}=\frac{{\mu }_0(1-{\alpha }_0(T-T_0))}{{(1-\varphi )}^{2.5}}.$$

(3)

The thermal conductivity that is, Hamilton-crosser’s (H-C) model of two-phase fluid is as follows ²⁴ :

$$\frac{K_{\mathrm{nf}}}{K_w}=\frac{K_{\mathrm{np}}+(n-1)K_w-(n-1)\varphi (K_w-K_{\mathrm{np}})}{K_{\mathrm{np}}+(n-1)K_w+\varphi (K_w-K_{\mathrm{np}})}.$$

(4)

Where, $K_{\mathrm{np}}$ , $K_{\mathrm{nf}}$ , and $K_w$ signify the thermal conductivity of nanomaterial, nanofluid, and water respectively. In Addition, here $n$ is a shape factor of nanoparticles and its values are reported via Table 1. The relationship between the temperature and water thermal conductivity is described as follows:

$$K_w=K_0(1+{\xi }_0(T-T_0)).$$

(5)

Table 1.

Nanoparticles shape with their shape factor. ³⁶

Shape of nanomaterials	Shape factors
Spherical	$n=3$
Tetrahedron	$n=4.0613$
Cylinders	$n=4.9$
Platelets	$n=5.7$
Blades	$n=8.6$

Here K₀ represents the base fluid’s thermal conductivity at constant temperature, while ${\xi }_0$ denotes the parameter of dimensional variable conductivity. Upon substitution of equation (5) into equation (4), the alterations to Hamilton-Crosser’s relations are delineated as^12,24:

$$\frac{K_{\mathrm{nf}}}{K_0}=(1+{\xi }_0(T-T_0))\frac{K_{\mathrm{np}}+K_0(n-1)(1+{\xi }_0(T-T_0))-(n-1)\varphi (K_0(1+{\xi }_0(T-T_0))-K_{\mathrm{np}})}{K_{\mathrm{np}}+K_0(n-1)(1+{\xi }_0(T-T_0))+\varphi (K_0(1+{\xi }_0(T-T_0))-K_{\mathrm{np}})}.$$

(6)

The expressions detailing the electric conductivity, effective heat capacitance, thermal expansion, and density of nanofluids are provided as ²⁴ :

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_w}=1+\frac{3(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}-1)\varphi }{(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}+2)-(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}-1)\varphi },}\\ \multicolumn{1}{c}{(\rho C{)}_{\mathrm{nf}}=(1-\varphi ){\left(\rho C\right)}_w+\varphi {\left(\rho C\right)}_{\mathrm{np},}}\\ \multicolumn{1}{c}{(\rho \beta {)}_{\mathrm{nf}}=(1-\varphi ){\left(\rho \beta \right)}_w+\varphi {\left(\rho \beta \right)}_{\mathrm{np}},}\\ \multicolumn{1}{c}{(\rho {)}_{\mathrm{nf}}=(1-\varphi ){\left(\rho \right)}_w+\varphi {\left(\rho \right)}_{\mathrm{np},}.}\end{array}$$

(7)

The Ohm’s law relationship is given as:

$$\mathit{J}={\sigma }_{\mathrm{nf}}[\mathit{V}\times \mathit{B}+\mathit{E}].$$

(8)

Here $\mathit{J}$ and ${\sigma }_{\mathrm{nf}}$ represent the nanofluid’s current density and electric conductivity. With regards to the specified electric and magnetic field, the components of Lorentz force for present problem $\mathit{V}=[\overline{U},\overline{V},0]$ are delineated as:

$$\mathit{J}\times \mathit{B}=[-{\sigma }_f\overline{U}{B}_0^2,-{\sigma }_f(B_0{E}_x+{\mathrm{VB}}_0^2)].$$

(9)

From Gauss’s law:

$$\nabla .\mathit{E}=\frac{{\rho }_e}{\epsilon },$$

(10)

where ε and ${\rho }_e$ are the electric permittivity and density. The formula of conservated electric field is:

$$\mathit{E}=-\nabla \overline{\Omega }.$$

(11)

Comparison of equations (10) and (11) leads to the Poisson equation as:

$${\nabla }^2\overline{\Omega }=-\frac{{\rho }_e}{\epsilon }.$$

(12)

The net charge density is defined by the Boltzmann distribution which is:

$${\rho }_e=\mathrm{ez}(n_{+}-n_{-}),$$

(13)

$n_{+}$ and $n_{-}$ from above equation are number density of cations and anions, stated as:

$$n_{\pm }=n_0{e}^{(\pm \frac{\mathrm{ez}}{T_{\mathrm{av}}{K}_B}\overline{\Omega })},$$

(14)

where $K_B$ the constant of Boltzmann and $T_{\mathrm{av}}$ illustrate the average temperature.

Using equations (13) and (14) in equation (12) and employing the Debye-Huckel assumption, Poisson’s equation takes form:

$${\nabla }^2\overline{\Omega }=k^2\overline{\Omega },$$

(15)

where $k=\frac{d}{{\lambda }_D}$ is an electroosmotic parameter and ${\lambda }_D=\frac{1}{\mathrm{ez}}{\left(\frac{\epsilon K_B{T}_{\mathrm{av}}}{2n_0}\right)}^{\frac{1}{2}}$ .

In the X-direction, the radiative heat flux is assumed insignificant when compared to the Y-direction. According to the Rosseland approximation perspective, the radiative heat flux ( $q_r$ ) satisfies the following equation. ³⁶

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial \overline{Y}}.$$

(16)

Here, ${\mathrm{\sigma }}^{*}$ represents the Stefan-Boltzmann constant and $k^{*}$ denotes the mean absorption coefficient. It is supposed that the change in temperature is adequately small within the flow. By expanding the term $T^4$ as a Taylor series around $T_0$ , the free stream temperature. Ignoring higher-order terms, we obtain the following expression:

$$T^4=T_0^4+4T_0^3(T-T_0)+6T_0^2{(T-T_0)}^2+\dots$$

(17)

After employing equation (17), equation (16) becomes:

$$q_r=-\frac{16{\sigma }^{*}{T}_0^3}{3k^{*}}\frac{\partial T}{\partial \overline{Y}}.$$

(18)

Table 2 represents the physical parameters values corresponding to water and multiple nanoparticles. Here, $\overline{X}$ -axis is aligned along its length and $\overline{Y}$ -axis perpendicular to it. Uniform magnetic and electric fields $B(=(0,0,B_0))$ and $\mathit{E}(=(E_x,0,0))$ are applied to the channel. Mathematical form of geometry is given as:

$$\begin{array}{c}{\overline{H}}_1(\overline{t},\overline{X})=d_1+a_1\mathrm{Cos}\left(\frac{2\pi }{\lambda }(-c\overline{t}+\overline{X})\right),\hfill \\ {\overline{H}}_2(\overline{t},\overline{X})=-d_2-a_2\mathrm{Cos}(ϵ+\frac{2\pi }{\lambda }(-c\overline{t}+\overline{X})),\hfill \end{array}$$

(19)

where $a_1$ , $a_2\lambda$ , and $\overline{t}$ represent the amplitude of the left and right walls, wavelength, and time of peristaltic wave. Such waves induce fluid movement in the channel. The governing equations can be delineated as ²⁴ :

$$\frac{\partial \overline{U}}{\partial \overline{X}}+\frac{\partial \overline{V}}{\partial \overline{Y}}=0,$$

(20)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f(\overline{U}\frac{\partial }{\partial \overline{X}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\frac{\partial }{\partial \overline{t}})\overline{U}=-\frac{\partial \overline{P}}{\partial \overline{X}}+2\frac{\partial }{\partial \overline{X}}}\\ \multicolumn{1}{c}{\left({\mu }_{\mathrm{nf}}\left(T\right)\frac{\partial \overline{U}}{\partial \overline{X}}\right)+\frac{\partial }{\partial \overline{Y}}\left[{\mu }_{\mathrm{nf}}\left(T\right)(\frac{\partial \overline{V}}{\partial \overline{X}}+\frac{\partial \overline{U}}{\partial \overline{Y}})\right]}\\ \multicolumn{1}{c}{-{\sigma }_{\mathrm{nf}}{B}_0^2\overline{U}+g{\left(\rho \beta \right)}_{\mathrm{nf}}(T-T_0)+{\rho }_e{E}_x,}\end{array}$$

(21)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_f(\overline{U}\frac{\partial }{\partial \overline{X}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\frac{\partial }{\partial \overline{t}})\overline{V}=-\frac{\partial \overline{P}}{\partial \overline{Y}}+2\frac{\partial }{\partial \overline{Y}}}\\ \multicolumn{1}{c}{\left({\mu }_{\mathrm{nf}}\left(T\right)\frac{\partial \overline{V}}{\partial \overline{Y}}\right)+\frac{\partial }{\partial \overline{X}}\left[{\mu }_{\mathrm{nf}}\left(T\right)(\frac{\partial \overline{V}}{\partial \overline{X}}+\frac{\partial \overline{U}}{\partial \overline{Y}})\right]}\\ \multicolumn{1}{c}{-{\sigma }_{\mathrm{nf}}(E_x{B}_0+\overline{V}{B}_0^2),}\end{array}$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{(\rho C{)}_f(\overline{U}\frac{\partial }{\partial \overline{X}}+\overline{V}\frac{\partial }{\partial \overline{Y}}+\frac{\partial }{\partial \overline{t}})T=\overline{\nabla }.\left[K_{\mathrm{nf}}\left(T\right)\left(\overline{\nabla }T\right)\right]}\\ \multicolumn{1}{c}{+{\mu }_{\mathrm{nf}}\left(T\right)[2\{{\left(\frac{\partial \overline{U}}{\partial \overline{X}}\right)}^2+{\left(\frac{\partial \overline{V}}{\partial \overline{Y}}\right)}^2\}+{(\frac{\partial \overline{V}}{\partial \overline{X}}+\frac{\partial \overline{U}}{\partial \overline{Y}})}^2]}\\ \multicolumn{1}{c}{+{\sigma }_{\mathrm{nf}}[{(E_x+\overline{V}{B}_0)}^2+{\overline{U}}^2{B}_0^2]-\frac{\partial q_r}{\partial \overline{Y}}.}\end{array}$$

(23)

Where ${\mu }_{\mathrm{nf}}\left(T\right)$ and $K_{\mathrm{nf}}\left(T\right)$ are viscosity and thermal conductivity respectively, which vary with temperature.

Table 2.

Numeric values of the thermo-physical characteristics. ²⁴

Phase	$\rho \left(\frac{\mathrm{kg}}{m^3}\right)$	$C_p\left(\frac{j}{\mathrm{kgK}}\right)$	$K\left(\frac{W}{\mathrm{mk}}\right)$	$\beta \left(\frac{1}{k}\right)$ × ${10}^{-6}$	$\sigma \left(\frac{S}{m}\right)$
$H_{2}O$	997.1	4179	0.613	210	0.05
Cu	$8933$	385	401	16.7	$5.96$ x ${10}^7$
Ag	$10,500$	$235$	$429$	$18.9$	$6.30$ x ${10}^7$
Au	$19,300$	$129.1$	$318$	$14$	$4.52$ x ${10}^7$
$F{e}_3{O}_4$	$5200$	$670$	$80.6$	$13$	$25,000$

The conversions from the laboratory ( $\overline{X}$ , $\overline{Y},\overline{t}$ ) to the wave ( $\overline{x},\overline{y}$ ) frame are outlined as:

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

After implementing the above aforementioned relationships, the modeled equations in the wave frame adopt the form:

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

(24)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\rho }}_{\mathrm{nf}}((\overline{u}+c)\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}})(\overline{u}+c)=-\frac{\partial \overline{p}}{\partial \overline{x}}+2\frac{\partial }{\partial \overline{x}}}\\ \multicolumn{1}{c}{\left({\mu }_{\mathrm{nf}}\left(T\right)\frac{\partial \overline{u}}{\partial \overline{x}}\right)+\frac{\partial }{\partial \overline{y}}\left[{\mu }_{\mathrm{nf}}\left(T\right)(\frac{\partial \overline{v}}{\partial \overline{x}}+\frac{\partial \overline{u}}{\partial \overline{y}})\right]}\\ \multicolumn{1}{c}{-{\sigma }_{\mathrm{nf}}{B}_0^2(\overline{u}+c)+g(\rho \beta {)}_{\mathrm{nf}}(T-T_0)+{\rho }_e{E}_x,}\end{array}$$

(25)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\rho }}_{\mathrm{nf}}((\overline{u}+c)\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}})\overline{v}=-\frac{\partial p}{\partial \overline{y}}+2\frac{\partial }{\partial \overline{y}}\left({\mu }_{\mathrm{nf}}\left(T\right)\frac{\partial \overline{v}}{\partial \overline{y}}\right)}\\ \multicolumn{1}{c}{+\frac{\partial }{\partial \overline{x}}\left[{\mu }_{\mathrm{nf}}\left(T\right)(\frac{\partial \overline{v}}{\partial \overline{x}}+\frac{\partial \overline{u}}{\partial \overline{y}})\right]-{\sigma }_{\mathrm{nf}}(E_x{B}_0+\overline{v}{B}_0^2),}\end{array}$$

(26)

$$\begin{array}{c}\multicolumn{1}{c}{(\mathrm{\rho }\mathrm{C}{)}_{\mathrm{nf}}((\overline{u}+c)\frac{\partial }{\partial \overline{x}}+\overline{v}\frac{\partial }{\partial \overline{y}})T=\overline{\nabla }.[K_{\mathrm{nf}}\left(T\right)(\overline{\nabla }T)]}\\ \multicolumn{1}{c}{+{\mu }_{\mathrm{nf}}\left(T\right)[2\{{\left(\frac{\partial \overline{u}}{\partial \overline{x}}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial \overline{y}}\right)}^2\}+{(\frac{\partial \overline{v}}{\partial \overline{x}}+\frac{\partial \overline{u}}{\partial \overline{y}})}^2]}\\ \multicolumn{1}{c}{+{\sigma }_{\mathrm{nf}}[{(E_x+\overline{v}{B}_0)}^2+{(\overline{u}+c)}^2{B}_0^2]-\frac{\partial q_r}{\partial \overline{y}}.}\end{array}$$

(27)

The dimensionless variables and parameters are characterized as:

$$\begin{array}{c}\multicolumn{1}{c}{y=\frac{\overline{y}}{d},x=\frac{\overline{x}}{\lambda },v=\frac{\overline{v}}{c\delta },u=\frac{\overline{u}}{c},\delta =\frac{d}{\lambda },h_1=\frac{{\overline{H}}_1}{d_1},}\\ \multicolumn{1}{c}{h_2=\frac{{\overline{H}}_2}{d_1},a=\frac{a_1}{d_1},b=\frac{a_2}{d_1},}\\ \multicolumn{1}{c}{p=\frac{d^2\overline{p}}{c\lambda {\mu }_o},\alpha ={\alpha }_o{T}_o,d=\frac{d_2}{d_1},\xi ={\xi }_o{T}_o,}\\ \multicolumn{1}{c}{M={\left(\frac{{\sigma }_w}{{\mu }_0}\right)}^{\frac{1}{2}}{B}_{0}d,t=\frac{c\overline{t}}{\lambda },S=\frac{d^2{\sigma }_w{E}_x^2}{k_0{T}_0},}\\ \multicolumn{1}{c}{U_{\mathrm{hs}}=-\frac{\epsilon E_x}{c{\mu }_0},\mathrm{Gr}=\frac{{\rho }_{w}g{\beta }_w{T}_0{d}^2}{{\mu }_{0}c},\mathrm{Re}=\frac{{\rho }_w\mathrm{cd}}{{\mu }_0},}\\ \multicolumn{1}{c}{\mathrm{Pr}=\frac{{\mu }_0{C}_w}{K_0},E=\frac{c^2}{C_w{T}_0},}\\ \multicolumn{1}{c}{\theta =\frac{(T-T_0)}{T_0},\mathrm{Br}=\mathrm{PrEc},N_r=\frac{16{\sigma }^{*}{T}_o^3}{3k^{*}{k}_0},}\\ \multicolumn{1}{c}{u={\psi }_y,v=-{\psi }_x.}\end{array}$$

(28)

Where $\mathrm{Pr}$ , $E$ , $\mathrm{Re}$ , $M$ , $\mathrm{Br}$ , $\delta$ , $S$ , $U_{\mathrm{hs}}$ mean the Prandtl, Eckert, Reynolds, Hartman, Brinkman, wave numbers, joule heating parameter, and Helmholtz-Smoluchowski (electroosmotic) velocity respectively. Furthermore, $\alpha$ , $\xi ,N_r$ , and $\theta$ show the dimensionless viscosity, thermal conductivity, and radiation parameter.

Applying the stated dimensionless constraints, considering the weaker Reynolds number and long wavelength approximation, equations (24)–(27) and the peristaltic wall expression simplify to:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial p}{\partial x}=\frac{1}{{(1-\varphi )}^{2.5}}\frac{\partial }{\partial y}\left[(1-\alpha \theta )\frac{{\partial }^2\psi }{\partial y^2}\right]-A_1{M}^2(\frac{\partial \psi }{\partial y}+1)}\\ \multicolumn{1}{c}{+\mathrm{Gr}\{1-\varphi +\varphi \left(\frac{{\left(\rho \beta \right)}_{\mathrm{np}}}{{\left(\rho \beta \right)}_w}\right)\}\theta +U_{\mathrm{hs}}\frac{d^2\Omega }{d{y}^2},}\end{array}$$

(29)

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

(30)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial }{\partial y}\left[\left\{(1+\xi \theta )\frac{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )-(n-1)\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )+\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}\right\}\frac{\partial \theta }{\partial y}\right]+\frac{\mathrm{Br}(1-\alpha \theta )}{{(1-\varphi )}^{2.5}}{\left(\frac{{\partial }^2\psi }{\partial y^2}\right)}^2}\\ \multicolumn{1}{c}{+\mathrm{Br}{A}_1{M}^2{(\frac{\partial \psi }{\partial y}+1)}^2+N_r\frac{{\partial }^2\theta }{\partial y^2}+A_{1}S=0,}\end{array}$$

(31)

$$\begin{array}{c}{h}_1\left(x\right)=1+\mathrm{acos}\left(2\pi x\right),\hfill \\ h_1\left(x\right)=-d-b\cos (2\pi x+\gamma ).\hfill \end{array}$$

Where

$$A_1=1+\frac{3(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}-1)\varphi }{(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}+2)-(\frac{{\sigma }_{\mathrm{np}}}{{\sigma }_w}-1)\varphi }.$$

(32)

The convective conditions are given as^3,4:

$$\begin{array}{c}\psi =\frac{F}{2},\frac{\partial \psi }{\partial y}+\frac{\beta (1-\alpha \theta )}{{(1-\varphi )}^{2.5}}\frac{{\partial }^2\psi }{\partial y^2}=-1,B_i(\theta +\frac{1}{2})\hfill \\ \multicolumn{1}{c}{+(1+\xi \theta )}\\ \multicolumn{1}{c}{\left(\frac{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )-(n-1)\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )+\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}\right)}\\ \multicolumn{1}{c}{\frac{\partial \theta }{\partial y}=0}\\ \mathrm{at}y=h_1,\hfill \\ \psi =-\frac{F}{2},\frac{\partial \psi }{\partial y}-\frac{\beta (1-\alpha \theta )}{{(1-\varphi )}^{2.5}}\frac{{\partial }^2\psi }{\partial y^2}=-1,-B_i(\theta -\frac{1}{2})\hfill \\ \multicolumn{1}{c}{+(1+\xi \theta )}\\ \multicolumn{1}{c}{\left(\frac{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )-(n-1)\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}{K_{\mathrm{np}}+k_0(n-1)(1+\xi \theta )+\varphi (k_0(1+\xi \theta )-K_{\mathrm{np}})}\right)}\\ \multicolumn{1}{c}{\frac{\partial \theta }{\partial y}=0}\\ \mathrm{at}y=h_2.\hfill \end{array}$$

(33)

Here, $B_i(=\frac{\mathrm{ld}}{k_o})$ and $\beta (=\frac{{\beta }_1{\mu }_0}{d})$ denote the Biot number and velocity slip constraint.

Dimensionless volume flow rates in fixed $\eta (=\frac{\overline{Q}}{\mathrm{cd}})$ and moving $F(=\frac{\overline{q}}{\mathrm{cd}})$ frames are given as:

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

(34)

where

$$F=\underset{0}{\overset{h}{\int }}\frac{\partial \psi }{\partial y}\mathrm{dy}.$$

(35)

The resulting nonlinear systems presented in equations (29)–(31), along with conditions (33), are complex. As analytical solution is not readily attainable and time-consuming, it is necessary to adopt a numerical scheme to tackle these equations. Hence build-in routine NDSolve in Mathematica which is based on highly effective and efficient shooting method is utilized. Some of the advantages of using NDSolve are as follows:

• Consistently adjust variables such as $\prime x\prime$ and $\prime y\prime$ with a step size of $0.01$ .

• Demonstrates accuracy and remains unconditionally stable.

Also, Figure 1 shows the flowchart of the solution procedure.

Figure 1.

Flow chart of solution methodology.

Discussion

In this section, acquired numerical data has been observed to explore the effects of evolving constraints on heat transfer rate, temperature and velocity of $\mathrm{Cu}-H_{2}O$ nanofluid. Additionally, the comparisons of four nanoparticles with different shapes are discussed. To enhance clarity, this section is subdivided into three subsections.

This portion focuses the effects of various parameters on the velocity, temperature, and heat transfer rate plots via graphs. Solutions for different flow constraints are discussed that is, M the Hartman number, $k$ the electroosmotic, $S$ the Joule heating, $U_{\mathrm{hs}}$ the Helmholtz-Smoluchowski velocity, Grashoff number $\mathrm{Gr},$ thermal conductivity $\xi$ parameter, viscosity parameter $\alpha$ , nanoparticles volume fraction $\varphi$ . radiation parameter $N_r$ and Biot number $\mathrm{Bi}$ Values of the parameters are taken to be $\alpha =0.05,\eta =0.7,\xi =0.05,$ $x=1.0,\varphi =0.05,a=0.3,b=0.4,\mathrm{Br}=0.2,\gamma =0.05,$ $M=1.0,\beta =0.05,Gr=2.0,Bi=10,Nr=1,Uhs=-1,$ $S=1,k=1$ , unless stated elsewise.

Velocity profile analysis

The variations in velocity are illustrated in Figures 2 to 9. Figure 2 demonstrated an increasing trend near channel’s center with an increase in the ‘ $\alpha$ ’ value, with a slight decay in velocity observed near the channel wall. Figure 3 addressed that an increase in ‘ $\varphi$ ’ leads to enhances resistance to flow, resulting in reduced velocity of $\mathrm{Cu}-H_{2}O$ nanomaterial near the channel’s center, while the reverse behavior is noticed near the walls of channel. Figure 4 indicated a significant impact of ‘ $\mathrm{Gr}$ ’ on velocity. It reduced the drag force which augment the velocity near the center of channel and decreasing it near channel walls with a rise in ‘ $\mathrm{Gr}$ ’. The influence of ‘ $M$ ’ is shown in Figure 5. It resulted a decrement in the velocity near the center of channel, but opposite performance is depicted near the walls of channel. This suggests that the higher $M$ corresponds to a stronger retarding effect, which is evident from current observations. Lower velocity is observed for higher ‘ $\beta$ ’ values near the channel’s center, however a slight increasing result of velocity is seen for ‘ $B_i$ ’ values (see Figures 6 and 7). In addition, it is noticed that the velocity for the $U_{\mathrm{hs}}$ has increasing trend near bottom wall although this performance is witnessed to be contrasting near higher wall (see Figure 8). Also, it is inspected that the velocity decreases with a growth in $k$ near bottom wall whereas the contrary trend is witnessed for the higher wall (see Figure 9). This can be attributed to the direct relationship between $U_{\mathrm{hs}}$ and the electric field, where positive values support in postponing force in the momentum equation, whereas the negative values support fluid flow. Figure 10 shows increase in $S$ leads to increment in velocity near channel’s center. While reduction is seen near channel walls. This is because Joule heating parameter allows fluid to move by increasing temperature and lowering viscosity. Further, reduced profile near center of the channel is seen for increased values of $N_r$ (see Figure 11) due to the heat loss, which in turns increase resistance to the flow and hence velocity decreases. In addition, Figure 12 is prepared to check the behavior of velocity for shape parameter. It is observed that copper-water nanofluid has greater velocity near center for spherical while smaller if they are of blades shape.

Figure 2.

Velocity profile for $\alpha$ .

Figure 3.

Velocity profile for $\varphi$ .

Figure 4.

Velocity profile for $\mathrm{Gr}$ .

Figure 5.

Velocity profile for $M$ .

Figure 6.

Velocity profile for $\beta$ .

Figure 7.

Velocity profile for $\mathrm{Bi}$ .

Figure 8.

Velocity profile for $U_{\mathrm{hs}.}$

Figure 9.

Velocity profile for $k$ .

Figure 10.

Velocity profile for $S$ .

Figure 11.

Velocity profile for $\mathrm{Nr}$ .

Figure 12.

Velocity profile for shape factor.

Temperature profile analysis

Temperature dispersals resulting from variations in parameters are showed through Figures 13 to 20. Figure 13 illustrates that a rise in ‘ $\xi$ ’ indications to a drop in temperature when the ambient temperature is less than the fluid’s temperature. It is noteworthy that small increase in ‘ $\varphi$ ’ enhances the temperature particularly near the center line Figure 14. As increasing the number of nanoparticles improves thermal conductivity, leading to higher temperature distribution. Further, increase in Hartman number $M$ and Joule parameter $S$ tend to increase temperature (see Figures 15 and 16). Since these both parameters cause resistance due to magnetic and electric fields, which generate heat and hence temperature raises. when $N_r$ and $B_i$ increase, they both enhance heat loss phenomena to the surrounding and at the walls respectively and hence a drop in temperature is witnessed through Figures 17 and 18. Further, Figure 19 shows a slight increasing behavior of temperature profile as $U_{\mathrm{hs}}$ increases. Also, temperature decreases as $\beta$ increases (see Figure 20). In addition, Figure 21 is prepared to check the behavior of temperature for shape parameter. It is observed that copper-water nanofluid has greater temperature for spherical while smaller if they are of blades shape.

Figure 13.

Temperature profile for $\xi$ .

Figure 14.

Temperature profile for $\varphi$ .

Figure 15.

Temperature profile for $M$ .

Figure 16.

Temperature profile for $S$ .

Figure 17.

Temperature profile for $\mathrm{Nr}$ .

Figure 18.

Temperature profile for $\mathrm{Bi}$ .

Figure 19.

Temperature profile for $\mathrm{Uhs}$ .

Figure 20.

Temperature profile for $\beta$ .

Figure 21.

Temperature profile for shape factor.

Rate of heat transmission

This section is comprised of two portions. In first portion bar charts shown through Figures 22 to 26 are studied to analyze the presentation of heat transfer rate of four nanoparticles Gold (Au), Silver (Ag), Iron oxide ( $F{e}_3{O}_4$ ), and Copper (Cu) for change in different parameters. Second part of this section consists of plots which are prepared to study heat transfer rate for different shapes of iron oxide-water nanofluid (Figures 27–33).

Figure 22.

Heat transfer rate for $\beta$ .

Figure 23.

Heat transfer rate for $\mathrm{Gr}$ .

Figure 24.

Heat transfer rate versus S for various nanoparticles.

Figure 25.

Heat transfer rate versus Uhs for various nanoparticles.

Figure 26.

Heat transfer rate versus for various nanoparticles.

Figure 27

Heat transfer rate versus for various nanoparticle shapes.

Figure 28.

Heat transfer rate versus S for various nanoparticle shapes.

Figure 29.

Heat transfer rate versus for various nanoparticle shapes.

Figure 30.

Heat transfer rate for $\mathrm{Bi}$ .

Figure 31.

Heat transfer rate for $M$ .

Figure 32.

Heat transfer rate for $\alpha$ .

Figure 33.

Heat transfer rate for $\xi$ .

Comparison for different nanoparticles

Alike behavior is seen for both $\beta$ , $\mathrm{Gr},\mathrm{Uhs}$ , and $\mathrm{Nr}$ that is, higher parametric values of these constraints resulted lower rate of heat transmission and vice versa (see Figures 22, 23 and 25, 26). Conversely, increasing outcome is seen on the rate of heat transmission for gradual rise in $S$ values (see Figure 24). It is observed that iron oxide $F{e}_3{O}_4$ nanoparticles provided better rate of heat transmission as comparative to other nanoparticles.

Comparison for different shapes

Heat transfer rate is seen to be decreased for enhancement in $\mathrm{Nr},\mathrm{Uhs}$ , and $\alpha$ (see Figures 27, 29, and 32). Also, Figure 28 depicts increasing behavior of heat transfer rate when $S$ increases. Additionally, the heat transfer rate for blades is greater than the other shapes whereas spherical nanoparticles have lower transfer rate near lower wall, but opposite trend is seen near upper wall. Increment in heat transfer rate is observed for $\mathrm{Bi},M$ , and $\xi$ (see Figures 30, 31, and 33). It is perceived that heat transfer rate is higher if nanoparticles are of spherical shape while it has low values for blade shape.

Comparison of nanoparticles shapes of thermal conductivity

In this portion evaluation of two nanoparticles of thermal conductivity for different shapes that is, are provided through Figure 34(a)–(e).

Figure 34.

(a–e) Thermal conductivity for different nanoparticles.

Figure 34(a) to (e) portrays that thermal conductivity inclines to grow when temperature rises. Likewise, thermal conductivity for $\mathrm{Ag}$ shows larger enhancements compared to other nanoparticles, while $F{e}_3{O}_4$ shows low thermal conductivity for all shaped nanoparticles. Also, these figures portray an increasing behavior of thermal conductivity which displays advanced enhancement for blades shaped nanoparticles and less enhancements if nanoparticles are of spherical shape.

Conclusions

This study develops a mathematical model to examine a nanofluid flow via peristaltic channel. It mainly investigates the impact of non-isothermal features, including temperature dependent thermal conductivity and viscosity, on performance of the fluid. A comparative analysis of various nanoparticle shapes is carried out to inspect their effects on velocity and temperature profiles. The study assesses how different parameters influence the thermal transport across distinct types of nanoparticles. The collective effects of thermal radiation and electro-magneto-hydrodynamic forces are also studied to offer a comprehensive understanding of the system’s thermal performance. This study shows potential applications across numerous sectors. In aerospace, automotive, microelectronics, and power generation systems, nanoparticle and EMHD effects can improve cooling systems for high-performance devices by boosting heat transfer. In biomedical applications that is, hyperthermia therapy and drug delivery systems, it can improve thermal treatment techniques, design nano-drug carrier and temperature regulation, especially using $F{e}_3{O}_4$ nanoparticles. Furthermore, the study’s relevance extends to energy systems, where improved thermal conductivity and flow control support applications in solar collectors, fuel cells, and waste heat recovery units. The findings are summarized as follows:

• Silver $\mathrm{Ag}$ nanoparticles respond to stronger thermal conductivity.

• Heat transfer rate of iron oxide-water nanofluid is improved by up to $2.28\%$ for spherical nanoparticles as the thermal conductivity parameter rises from $0.00-0.10$ . However, increase in the thermal conductivity parameter leads to reduce temperature.

• The outcomes for energy transmission rate, temperature, and velocity are stronger for spherical nanoparticles comparative to other shapes.

• Radiation effect helps to control the thermal by approximately $18.18\%$ and flow of the system by up to $2.39\%$ .

• Iron oxide $F{e}_3{O}_4$ nanoparticles provide better heat transfer rate, with an amplification of approximately 0.11% compared to other nanoparticles.

• The heat transfer rate for blades is greater than the other shapes whereas spherical nanoparticles have lower transfer rate near lower wall, but opposite trend is seen near upper wall.

• An increase in nanoparticles concentration $\varphi$ results in a reduction in velocity within the central region of the channel, whereas simultaneously enhancing the temperature profile.

• Joule heating parameter significantly enhances velocity and temperature of the nanofluid.

This study develops the theoretical understanding of nanofluid by representing the essential role of nanoparticle geometry, comparative thermal efficiency of different nanoparticles, interaction of thermal radiation with EMHD flow in non-isothermal situations and proposes practical guidance for the design of thermally effective systems in microfluidics.

Furthermore, the improved thermal conductivity performance for silver nanoparticles over others is consistent with results supported by prior work that is, Abbasi et al., 2022, yet this study diverges by showing that this advantage is shape-sensitive and context-dependent under radiation and electromagnetohydrodynamic effects. Hence these finding highlight a critical distinction such as performance of nanoparticles when properly shaped and thermally stabilized in a flow domain influenced by external and magnetic fields.

However, the assumption of uniform particle distribution may restrict applicability to complex biological or industrial systems where particle aggregation plays a vital role. While absence of entropy generation analysis also limits thermodynamic understanding into the system’s irreversibility.

Future studies could address these limitations by incorporating irregular and complex geometries using hybrid nanofluids and non-Newtonian rheology. Such extension could advance the generalizability and engineering relevance of the current finding.