Mathematical and Computer Modeling of Electroosmotic Peristaltic Transport of a Biofluid with Double-Diffusive Convection and Thermal Radiation

1  Introduction

Peristaltic flow refers to a process in which fluids are transported through a tube or channel due to rhythmic squeezing of the tube’s walls. This type of flow is most commonly found in biological systems, where coordinated muscle contractions propel fluids forward. However, the principle is also widely applied in various engineering contexts. The mechanism’s efficiency and controllability make it vital in several fields, including biology, medicine, industry, and engineering. In the human body, peristalsis plays a crucial role in the digestive system, where muscle contractions help move food through the esophagus, stomach, and intestines, aiding in digestion and absorption. Similarly, the ureters depend on peristaltic motion to transport urine from the kidneys to the bladder, even working against gravity. Within the reproductive system, peristalsis assists in moving sperm through the male reproductive tract and in transporting the ovum through the female fallopian tubes. Additionally, peristaltic pumping occurs in certain blood vessels, such as small arteries, which help to sustain blood flow, especially under low-pressure conditions. In the medical field, this phenomenon is utilized in devices like renal replacement machines, intravenous infusion systems, and cardiopulmonary bypass devices. Industrial applications also benefit from peristaltic flow, particularly in the transport of sensitive, sanitary, or corrosive fluids. Researchers are increasingly turning their attention to non-Newtonian fluids due to their unique properties, which present significant potential in industrial, medicine, and technical fields. Unlike Newtonian fluids, which adhere to Newton’s law of viscosity, non-Newtonian fluids exhibit variable viscosity depending on the level of stress or strain rate. This leads to a more intricate relationship between stress and strain, resulting in different flow performances under various conditions. These fluids are vital in industries such as medical technology. Beauty products, food manufacturing, cosmetics, and manufacturing operations, as their ability to adjust viscosity in response to mechanical forces offer distinct advantages. However, their complex behavior also poses challenges in terms of processing, handling, and prediction. Raju and Devanathan [1,2] first investigated the peristalsis-driven motion of Generalized Newtonian fluids by applying a power-law fluid for simulating the peristaltic flow of blood. Recently, Elogail [3] used the shooting approach to numerically examine peristaltic permeation of a tangent hyperbolic fluid in a vertical asymmetric channel and concluded that a rise in temperature is subject to decrease in fluid’s viscosity. In order to investigate the homogeneous-heterogeneous reactions in peristaltic motion, Sheikh et al. [4] used a tangent hyperbolic fluid in an asymmetric channel under the influence of a magnetic field.

Nano-fluids are specially designed colloidal mixtures where nanoparticles, typically smaller than 100 nm, are suspended within a base fluid. These foundational fluids are often common liquids like biological liquids, water, lubricants and ethylene glycol. The nanoparticles are composed of substances like oxides compounds as aluminum oxide or silicon oxide, metallic elements e.g., copper and aluminum, or carbon-based materials including graphene and nanotubes. The addition of nanoparticles to fluids significantly enhances their thermal, rheological, and electrical properties, positioning nanofluids as a groundbreaking topic in the fields of heat transfer, biomedical applications, and fluid mechanics. The term “nanofluids” was first used in Argonne National Laboratory by Choi [5], where it was observed that dispersing nanoparticles in a liquid could considerably increase its thermal conductivity. This discovery paved the way for extensive research into nanofluids, using it in heat transfer systems like cooling, HVAC units, and heat exchangers. One of the pioneering studies in this area, conducted by Eastman et al. [6], demonstrated that adding copper nanoparticles to ethylene glycol could boost thermal conductivity by as much as 40%. Later, Das et al. [7] examined the nanofluid’s temperature dependency of thermal conductivity and concluded that the enhancement of thermal conductivity is subjective to temperature change where higher temperatures improve the heat transfer rates. Tripathi and Bég [8] studied the behavior of peristaltic flows in nanofluids for drug delivery applications, showing that by adjusting the concentration of nanoparticles and wave amplitude, it is possible to precisely control both the flow rate and drug diffusion. The peristalsis of a porous medium filled with a copper-water Colloidal fluid was investigated by Abbasi et al. [9]. According to their research, the copper–water nanofluid’s axial velocity falls as the volume percentage of nanoparticles rises.

Microfluidic and biomedical devices function through electroosmotic effects which operate together with peristaltic motion to achieve accurate management of flows that have low Reynolds numbers. Medical devices which use lab-on-a-chip systems require this particular coupling system to deliver drugs and perform biochemical tests and move samples without needing mechanical pumps. Electroosmosis creates an improved fluid flow system which operates through charged biological passages that include blood vessels and intestinal tracts in natural physiological flow systems. The electroosmotic–peristaltic system functions in micropumps and inkjet printers and micro-electronic cooling systems to achieve controlled fluid movement which operates at high efficiency. Mekheimer et al. [10] investigated the electron-osmosis force-induced peristaltic flow of Jeffery fluid across the tiny annulus. The computational analysis proves that the electric potential has a direct correlation with the Electric Debye Layer (EDL) thickness. Secondly, the excess axial flow is increased in the presence of an electropositive field while it is curtail under a negative electric field. Nadeem et al. [11] recognized the significance of the electro-osmosis force on micro vascular blood flow. Tripathi et al. [12] analyzed the contribution of the peristaltic motion in the presence of magnetic and electric fields of physiological fluids. In their conclusion, authors claimed that increased Hartmann numbers decrease bolus formation and change flow dynamics, whereas higher electro-osmotic parameters improve flow rates. The results have implications for the development of sophisticated fluid control systems and medical micro pumps in the nuclear and aerospace industries. Narla and Tripathi [13] assessed the electroosmotic induced blood circulation via a curved micro channel. Current research is being done on electro-osmotic transfers [14–17].

In fluid mechanics, thermal radiation becomes important when there are large temperature variations either within the fluid itself or between the fluid and its environment. This form of radiation involves the emission of electromagnetic waves, mainly in the infrared spectrum, due to an object’s temperature. It serves as a mode of energy transfer resulting from the random motion of charged particles within a material. Any object warmer than absolute zero emits this type of radiation. Thermal radiation is the term used to describe the phenomenon of heat transfer by electromagnetic waves. The large temperature differential between the two media is the cause of it. At this temperature, many technical processes take place. In the fields of aerospace, nuclear reactors, engineering and physics, power plants, glass manufacturing, furnace design, and other fields, thermal radiation plays a significant role in flow and heat propagation. Thermal heat transfer through radiation is known to have a major effect on a range of technological devices and industrial operations at high temperatures. These include fire propagation, materials processing, and combustion systems, propulsion of rockets [18], plume dynamics [19], and solar collector performance [20]. The thermal consequences of radiation on the boundary layer of viscous fluid motion that conducts electricity was investigated by Seddeek [21] and Raptis et al. [22]. The sources list mentions only a few current works on the subject [23–26]. Thermal radiation significantly influences heat transfer in peristaltic flows, affecting both the movement of the fluid and the distribution of temperature within the system. In the domains requiring accurate temperature control and effective fluid movement, like in medical technologies, refrigeration units, and manufacturing operations, comprehending how heat radiation interacts with wave-like fluid motion is vital for enhancing efficiency and functionality. Thermal radiation significantly influences nanofluid peristaltic flows by enhancing heat transfer and altering flow characteristics. The combination of nanoparticles, thermal radiation, and peristaltic motion is especially important in applications like healthcare equipment, thermal regulation technologies, and energy optimization, where efficient control of heat and fluid dynamics is vital for overall system performance. Numerous studies have investigated the effects of thermal radiation on nanofluidic peristaltic flows under various geometries [27–30].

Double diffusive convection (DDC) describes a fluid flow process where two distinct diffusion mechanisms—commonly mass transfer and heat transfer—take place simultaneously within a fluid system. These processes influence one another, generating convective movements because of the differences in their diffusion rates. This phenomenon is especially important in environments with both temperature and concentration gradients, as their interaction leads to intricate flow patterns. Examples of DDC include salt fingering and thermohaline circulation in marine environments. Double diffusive convection (DDC) is present in magma chambers and the Earth’s mantle, where temperature differences and variations in chemical composition generate convective currents that influence tectonic movements. DDC also plays a crucial role in systems like thermal transfer system, structural thermal barriers, and chemical reactors. In biological contexts, DDC may occur when both concentration gradients such as salt levels and temperature gradients drive the movement of fluids and nutrients across membranes. Due to its significance across various fields, this phenomenon has been extensively studied through multiple models, particularly in relation to peristaltic flows and nanofluid behavior [31–35].

Regardless of advancements in the research of electroosmotic and peristaltic flows, the combined impact of electroosmosis, double-diffusive convection, and non-Newtonian behavior specifically for tangent hyperbolic nanofluid is still too inadequately investigated. All the current models either ignore the rheological richness of shear-thinning fluids or exclude the role of solutal transport and electric field interactions in narrow spaces. To fill this gap, this work examines tangent hyperbolic nanofluid peristaltic flow under the concurrent action of double-diffusion convection and electroosmotic forces based on the application of the Debye–Hückel approximation to describe the electrokinetic potential. Formulation includes slip effects in terms of velocity, solutal, and nanoparticle, more accurately representing conditions in real microfluidics. This work is very much applicable to contemporary lab-on-a-chip devices, electrokinetic drug delivery, biosensors, and bio-micropumps, where fluid flow control and species transport in non-Newtonian fluids under electric fields are paramount. The findings shed more light on how electroosmotic transport can be optimized in non-Newtonian fluid environments, particularly in bioengineering and microfluidic applications.

2  Mathematical Modeling

The primary objective of this research focuses on conducting a detailed analysis of electrical impacts which occur when an incompressible tangent hyperbolic nanofluid moves through a two-dimensional channel with a total width of b3+b2 (see Fig. 1). The Cartesian coordinate system serves as the basis for studying fluid movement because the channel’s primary direction follows the x-axis while the perpendicular direction runs at right angles to the x-axis. A sinusoidal wave creates the fluid movement because it travels at a fixed velocity through the channel walls which simulate peristaltic motion. The fluid characteristics become established through the implementation of slip boundary conditions which use designated parameters.

Figure 1: Asymmetric peristaltic channel with nanoparticles and electroosmosis.

The functions of the asymmetric channel walls are defined as follows:

Right wall:H1(x,t)=b3+b4cos⁡[2πλ(X−ct)],Left Wall:H2(x,t)=−b2−b1cos⁡[2πλ(X−ct)+φ],(1)

where, c is the wave speed, (b4,b1) are the wave amplitudes, t gives the time, and λ provides the wavelength. The phase difference φ ranges in the interval 0≤φ≤π. For φ=0, the channel becomes symmetric having zero phase difference and φ=π means a channel influenced by a phase-shifted wave. Furthermore, the parameters b1,b2,b3,b4 and φ are interrelated through the inequality b42+b12+2b4b1cos⁡φ≤(b2+b3)2, thereby imposing a mathematical constraint on the system’s structural configuration. The constitutive stress tensor governing the tangent hyperbolic fluid [36] model is articulated as follows:

τ=−pI+S,S=μ∞+(μ+μ∞)tanh(Γγ˙)mA1.(2)

Herein, S designates the supplementary stress tensor, Γγ˙ assumes the material constant, μ signifies the zero-deformation viscosity, and m encapsulates the rheological flow rate index. Additionally, A1 corresponds to the inaugural Rivlin-Ericksen tensor, which characterizes the rate of deformation within the continuum framework. Given that Eq. (2) is scrutinized under the restrictive conditions wherein μ∞=0 and |Γγ˙| < 1, the constitutive formulation of the stress tensor undergoes a subsequent definition as follows:

γ˙=12ΣiΣjγ˙ijγ˙ji=12Π,(3)

hence

S=μ[1+m(Γγ˙−1)]A1.(4)

Herein, the invariant second strain tensor, denoted as Π, is mathematically expressed as Π=12trace(∇V+(∇V)∗)2, where the superscript (*) signifies the transpose operation. The velocity vector can be written as:

V=(U(X,Y,t),V(X,Y,t)).(5)

The general forms of continuity equation, momentum equation, Thermal distribution, nanoparticles and solute Concentration are given as:

∇.V=0,(6)

ρfdVdt=∇.S+J×B+g{(1−Θ0)ρf0{βT(T−T0)+βC(C−C0)}−(ρp−ρf0)(Θ−Θ0)}+ρeE,(7)

(ρc)fdTdt=k(∇2T)+S.L−∇⋅qr+(ρc)p((DB(∇Θ⋅∇T))+(DTT0)(∇T)2)+DTC∇2C,(8)

dCdt=Ds∇2C+DTC∇2T,(9)

dΘdt=DB∇2Θ+(DTT0)∇2T.(10)

The tangential component of Eq. (4) is defined as

SXY=μ0[1+m(Γγ˙−1)](∂U∂Y+∂V∂X),with,γ˙=[2(∂U∂X)2+(∂U∂Y+∂V∂X)2+2(∂V∂Y)2](11)

The components form of the Eqs. (6)–(10) are found as

∂U∂X+∂V∂Y=0,(12)

ρf(∂∂t~+U∂∂X+V∂∂Y)U=−∂P∂X+∂SXX∂X+∂SXY∂Y−σB02(Ucosω−Vsinω)cos⁡ω+g{(1−Θ0)ρf0{βT(T−T0)+βC(C−C0)}−(ρP−ρf0)(Θ−Θ0)}−(∂2ϑ∂X2+∂2ϑ∂Y2)εeffEx,(13)

ρf(∂∂t~+U∂∂X+V∂∂Y)V=−∂P∂Y+∂SYX∂X+∂SYY∂Y+σB02(Ucosω−Vsinω)sinω,(14)

(ρc)f(∂∂t~+U∂∂X+V∂∂Y)T=k(∂2T∂X2+∂2T∂Y2)(SXX∂U∂X+SXY(∂U∂Y+∂V∂X)+SYY∂V∂Y)−∂∂y(16σ∗T13∂T3k∗∂y)+(ρc)p(DB(∂Θ∂X∂T∂X+∂Θ∂Y∂T∂Y)+(DTT0)((∂T∂X)2+(∂T∂Y)2))+DTC(∂2C∂X2+∂2C∂Y2),(15)

(∂∂t~+U∂∂X+V∂∂Y)C=Ds(∂2C∂X2+∂2C∂Y2)+DTC(∂2T∂X2+∂2T∂Y2),(16)

(∂∂t~+U∂∂X+V∂∂Y)Θ=DB(∂2Θ∂X2+∂2Θ∂Y2)+(DTT0)(∂2T∂X2+∂2T∂Y2).(17)

Eq. (15) encompasses the term (16σ∗T13∂T3k∗∂y), which embodies the thermal radiative effects as articulated through the Rosseland approximation. In this context, the parameters k∗ and σ∗ signify the Rosseland mean absorption coefficient and Stefan-Boltzmann constant, respectively. A set of slip conditions for all the profiles are stated as [25]

U+β1SXY=0,atY=H1,U−β1SXY=0,atY=H2,(18)

T+β2∂T∂Y=T0atY=H1,T−β2∂T∂Y=T1atY=H2,(19)

C+β3∂C∂Y=C0,atY=H1,C−β3∂C∂Y=C1,atY=H2,(20)

Θ+β4∂Θ∂Y=Θ0,atY=H1,Θ−β4∂Θ∂Y=Θ1,atY=H2,(21)

where β1,β2,β3,β4 are slip parameters of velocity, temperature, concentration, and nanoparticle volume fraction, respectively.

2.1 Wave Frame Analysis

Here we introduce a set of wave transformations as follows [1–4]:

p(x,y)=P(X,Y,t),x=X−ct,u=U−c,y=Y,v=V.(22)

Putting the above values in the Eqs. (12)–(17), we have

∂u∂x+∂v∂y=0,(23)

ρf(u∂∂x+v∂∂y)u=−∂P∂x+∂Sxx∂x+∂Sxy∂y−σB02((u+c)cos⁡ω−vsin⁡ω)cos⁡ω+g{(1−Θ0)ρf0{βT(T−T0)+βC(C−C0)}−(ρP−ρf0)(Θ−Θ0)}−(∂2ϑ∂x2+∂2ϑ∂y2)εeffEx,(24)

ρf(u∂∂x+v∂∂y)v=−∂P∂y+∂Syx∂x+∂Syy∂y+σB02((u+c)cosω−vsinω)sinω,(25)

(ρc)f(u∂∂x+v∂∂y)T=k(∂2T∂x2+∂2T∂y2)+Sxx∂u∂x+Sxy(∂u∂y+∂v∂x)+Syy∂v∂y−∂∂y(16σ∗T13∂T3k∗∂y)+DTC(∂2C∂x2+∂2C∂y2)+(ρc)p(DB(∂Θ∂x∂T∂x+∂Θ∂y∂T∂y)+(DTT0)((∂T∂x)2+(∂T∂y)2)),(26)

(u∂∂x+v∂∂y)C=Ds(∂2C∂x2+∂2C∂y)+DTC(∂2T∂x2+∂2T∂y2),(27)

(u∂∂x+v∂∂y)Θ=DB(∂2Θ∂x2+∂2Θ∂y)+(DTT0)(∂2T∂x2+∂2T∂y2).(28)

Eq. (11) will become

Sxy=μ0[1+m(Γγ˙−1)](∂u∂y+∂v∂x),where γ˙=2(∂u∂x)2+(∂u∂y+∂v∂x)2+(∂v∂y)2.(29)

2.2 Scaling of Variables

Defining the dimensionless variables as follows [1–10]:

x¯=xλ,a=b4b3,y¯=yb3,δb3λ,u¯=uc,d=b2b3,t¯=ctλ,h2=H2b3,h1=H1b3,b=b1b3,p¯=b32pμ0cλ,Pr=(ρc)fυk,v¯=vc,Re=ρfcb3μ0,υ=μρf,Le=υDS,Ln=υDB,θ=T−T0T1−T0,γ=C−C0C1−C0,Ω=Θ−Θ0Θ1−Θ0,We=cΓb3,S¯xy=Sxyb3μ0c,S¯yy=Syyb3μ0c,S¯xx=Sxxλμ0c,Br=μ0c2kT,Rd=16σ∗T133k∗μ0cf,Nb=(ρc)pDB(Θ1−Θ0)(ρc)fυ,Grc=g(1−Θ0)ρfβC(C1−C0)b32μ0c,k=μ0cf,Grt=gb32(1−Θ0)(T1−T0)ρfβTμ0c,GrF=g(Θ1−Θ0)(ρp−ρf)b32μ0c,NCT=DCT(T1−T0)DS(C1−C0),NTC=DTC(C1−C0)(ρc)fυ(T1−T0),Uhs=−ζεeffExμ0c,Ec=c2(T1−T0)cf,Nt=(ρc)pDT(T1−T0)T0(ρc)fυ,eo=eza2n0εeffTvKB.(30)

After putting the above non-dimensional values in the Eqs. (23)–(29), we get

Re(δu¯∂∂x¯+v¯∂∂y¯)u¯=−∂p¯∂x¯+δ2∂S¯xx∂x¯+∂S¯xy∂y¯−M2((u¯+1)cosω−v¯sinω)cosω+Grtθ+Grcγ−GrFΩ−(δ2∂2ϑ¯∂x¯2+∂2ϑ¯∂y¯2)ζεeffExμ0c,(31)

Reδ(δu¯∂∂x¯+v¯∂∂y¯)v¯=−∂p¯∂y¯+δ2∂S¯yx∂x¯+δ∂S¯yy∂y¯+b32μ0δσB02((u¯+1)cosω−v¯sinω)sinω,(32)

RePr(δu¯∂θ∂x¯+v¯∂θ∂y¯)=(δ2∂2θ∂x¯2+∂2θ∂y¯2)+Nb(δ2∂Ω∂x¯∂θ∂x¯+∂Ω∂y¯∂θ∂y¯)+Nt(δ2(∂θ∂x¯)2+(∂θ∂y¯)2)+NTC(δ2∂2γ∂x¯2+∂2γ∂y¯2)+Rdk∂2θ∂y¯2+Br(δ2S¯xx∂u¯∂x¯+S¯xy(∂u¯∂y¯+δ∂v¯∂x¯)−S¯yy∂v¯∂y¯),(33)

ReLe(δu¯∂γ∂x¯+v¯∂γ∂y¯)=(δ2∂2γ∂x¯2+∂2γ∂y¯2)+NCT(δ2∂2θ∂x¯2+∂2θ∂y¯2),(34)

ReLn(δu¯∂Ω∂x¯+v¯∂Ω∂y¯)=(δ2∂2Ω∂x¯2+∂2Ω∂y¯2)+NtNb(δ2∂2θ∂x¯2+∂2θ∂y¯2).(35)

The stress components in dimensionless form are given as

S¯xy=[1+m(Weγ˙∗−1)](∂u¯∂y¯+δ∂v¯∂x¯), where γ˙∗=∂u¯∂y¯.

Through the utilization of the stream function formulation, the momentum equations are modified and expressed in terms of a single dependent variable ψ, thereby reducing the system’s complexity. Let us define, the stream function as below:

u¯=∂ψ∂y¯,v¯=−δ∂ψ∂x¯.(36)

By imposing the physical constraints of an extended wavelength approximation (δ≪1), and a significantly reduced Reynolds number (Re≈0), the governing Eqs. (31)–(35) are transformed into the following simplified forms

∂p¯∂x¯=∂∂y¯[[1+m(We∂2ψ∂y¯2−1)](∂2ψ∂y¯2)]−M2cos2ω(∂ψ∂y¯+1)+Grtθ+Grcγ−GrFΩ−eo2ϑ¯Uhs,(37)

∂p¯∂y¯=0,(38)

∂2θ∂y¯2+NTCPr∂2γ∂y¯2+NbPr∂θ∂y∂Ω∂y+NtPr(∂θ∂y)2+PrRd∂2θ∂y¯2+Br([1+m(We∂2Ψ∂y2¯−1)](∂2Ψ∂y¯2))∂2Ψ∂y2¯=0,(39)

∂2γ∂y¯2+NCT∂2θ∂y¯2=0,(40)

∂2Ω∂y¯2+NtNb∂2θ∂y¯2=0.(41)

Differentiating Eq. (37) with respect to the variable y and using Eq. (38), the pressure term is eliminated and we have the following momentum equation in terms of stream function [3–10]:

0=∂2∂y¯2[[1+m(We∂2ψ∂y¯2−1)](∂2ψ∂y¯2)]−M2cos2ω∂2ψ∂y¯2+Grt∂θ∂y¯+Grc∂γ∂y¯−GrF∂Ω∂y¯−eo2∂ϑ¯∂yUhs.(42)

2.3 Electroosmosis

The Poisson equation is defined as [10,11,13–15]

∇2ϑ=−ρeεeff,(43)

where εeff, ρe represents the electrolyte’s net charge density and dielectric constant, and it may then take the form ρe=ze(n+−n−), where n−, z, e, n+, symbolizes −ve ions, charge balance, charge, +ve ions sequentially n+=n0e−ezϑKBTv, n−=n0eezϑKBTv, where n0 and KB is Boltzmann constant and Bulk constant, respectively. After using these values, we get

ρe=−2zen0sinh⁡(zeϑTvKB).(44)

By Debye-Huckel linearization that is,

sinh⁡(zeϑTvKB)≈(zeϑTvKB).(45)

Now Eq. (43) becomes:

ρe=−2zen0(zeϑTvKB).(46)

Comparing Eqs. (43) and (46), we found

∇2ϑ=2(ze)2n0εeffTvKBϑ.(47)

Now using dimensionless parameters in above equation, we obtain

∂2ϑ¯∂y¯2=eo2ϑ¯,(48)

where eo is the electro osmotic parameter defined in Eq. (30). The boundary conditions of Eq. (48) are defined as [10,11,13,14]

ϑ(y¯)=1aty¯=h1,ϑ(y¯)=0aty¯=h2.(49)

The solution of the above system is found numerically using NDSolve tool on Mathematica.

The non-dimensional form of the mean flow (Q) is as follows:

Q=1+F+d,(50)

where

F=∫h2(x¯)h1(x¯)∂ψ∂y¯dy¯=ψ(h1)−ψ(h2).(51)

Here

h1=1+acos⁡2πx¯,h2=−d−bcos⁡(2πx¯+φ).(52)

In dimensionless form, the boundary constraints are expressed as:

Ψ=F2,∂Ψ∂y¯=−λ1S¯xy−1ony=h1,Ψ=−F2,∂Ψ∂y¯=λ1S¯xy−1ony=h2,(53)

θ+λ2∂θ∂y¯=0,ony=h1,θ−λ2∂θ∂y¯=1,ony=h2,(54)

γ+λ3∂γ∂y¯=0,ony=h1,γ−λ3∂γ∂y¯=1,ony=h2,(55)

Ω+λ4∂Ω∂y¯=0,ony=h1,Ω−λ4∂Ω∂y¯=1,ony=h2.(56)

The Sherwood number (mass transfer rate) and Nusselt number (heat transfer rate) are described as follows:

Sh=−∂Ω∂y¯,Nu=−∂θ∂y¯.(57)

The expression for pressure rise is given as Δp=∫01dp¯dx¯dx¯, where, the pressure gradient dp¯dx¯ can be found from Eq. (37). For simplicity, the bar symbols have been ignored in the graphs.

3  Results and Discussion

In the current portion of the article, the effects of electro osmosis-induced transport, radiative heat transfer, and the presence of a porous media on the peristaltic movement of a hyperbolic tangent model have been thoroughly investigated using comprehensive graphical diagrams. The numerical solutions corresponding to the thermal field, velocity distribution, rate of heat transfer, and the frictional coefficient associated with nanoparticles have been meticulously derived and computationally resolved employing the advanced algorithmic source of Mathematica software through NDSolve tool. These findings have been thoroughly illustrated and deeply discussed using a variety of graphical representations that elucidate the complex relationships and consequent impacts of the aforementioned physical phenomena on the fluid system’s motion and thermal behavior.

3.1 Velocity Profile

The graphical depictions in Fig. 2a–h elucidate the relation between flow speed and key governing parameters, including the velocity slip parameter λ1 the index of the power law for Tangent Hyperbolic model m, the solutal Grashof number Grc, the electroosmotic parameter eo and the hartman number M, both in 2D and 3D frames. Fig. 1a distinctly reveals a reduction in fluid velocity magnitude within the central channel region, y∈[−0.4,0.1] due to an increased influence of λ1. In contrast, the peripheral regions, y∈[−0.6,−0.4] and y∈[0.1,0.3], exhibit an opposing trend, with fluid velocity escalating as λ1 increases. A rise in the slip factor lessens the no-slip necessity at the boundary, enabling the fluid to slide against the wall. The decline in wall friction reduces viscous resistance and shear stress at the boundary. As a result, the fluid has less resistance to motion, resulting in increased axial velocity and a fuller (less steep) velocity profile. The effect dominates along the walls, with the core region remaining relatively less affected. Fig. 2e,g carefully shows the influence of the solute Grashof number Grc and the electro osmotic parameter eo, on fluid velocity. With the increase in Grc and eo fluid velocity decreases when moving towards the left wall, whereas the opposite trend is seen towards the right wall. This is due to the competing effects of solutal convection due to buoyancy and electro osmotic forces that create opposing flow regimes depending on directional forces. Fig. 1b shows divergent behavior for the power law index m with respect to Grc and eo. Along the left wall at y∈[−0.6], fluid velocity is constant but increases in y∈[−0.6,−0.1], while reducing towards the right wall in y∈[−0.1,0.3] as a result of the strengthening effects of eo. This highlights the non-Newtonian character of the fluid, where shear-thinning or shear-thickening characteristics asymmetrically regulate the velocity profile. Additionally, Fig. 2b,d,f,h presents a detailed three-dimensional analysis of velocity profile deviations influenced by λ1,m,Grc and eo. These numbers reveal deep insights into the complex dependencies and consequential changes of velocity profiles. The left panel of Fig. 2i demonstrates how the Hartmann number M creates different velocity profile patterns which do not follow a mirror image between the two sides of the channel. The axial velocity shows opposite trends against M, it decreases on the channel’s left side but it rises on the right side. The system displays this behavior because the magnetic field which has been applied works together with the uneven peristaltic wave shape to create a Lorentz force pattern which lacks symmetry. The magnetic damping force which opposes fluid flow becomes more powerful when the Hartmann number rises. The magnetic force does not stop fluid flow but it shifts how momentum moves throughout the system because of how peristaltic waves with uneven shapes interact with the flow. The fluid slows down near one side of the channel while its momentum increases toward the other wall which causes the velocity peak to move. The three-dimensional velocity surface shown in Fig. 2j demonstrates the redistribution process because the magnetic field moves the peak velocity position instead of creating a consistent velocity decrease throughout the entire channel.

Figure 2: Variations of velocity for (a,b) λ1, (c,d) m, (e,f) Grc, (g,h) eo and (i,j) M.

3.2 Temperature Profile

In an effort to look into thermal effects on the Hartmann number M, thermal radiation parameter Rd, Prandtl number Pr, and Helmholtz-Smoluchowski velocity Uhs, Fig. 3a–h is analyzed. The figures show decreasing heat transfer efficiency with rising Rd, Pr, and M due to magnetic damping, radiative dissipation, and reduced thermal diffusivity, respectively. Fig. 3g has the opposite trend close to the right wall at y>0.5, where heat transfer increases due to increased Helmholtz-Smoluchowski velocity Uhs This is caused by increased electro kinetic forces, which enhance boundary convective heat transfer, opposing the inhibitive effects of M, Rd, and Pr. Also, Fig. 3b,d,f,h illustrates the three-dimensional divergence of M, Pr, Uhs, and Rd, explaining their combined effect on thermal and flow behavior. The interaction of these parameters emphasizes the intricate balance among Lorentz forces, radiative heat dissipation, viscous dissipation, and electro osmotic flow, controlling the thermal response of the system. Physically, increasing M intensifies magnetic suppression, while higher Rd enhances energy loss. Conversely, Uhs drives electroosmotic flow, boosting heat transfer near boundaries. These findings are pivotal for optimizing thermal systems in magnetohydrodynamics, radiative cooling, and microfluidics, where precise thermal and flow control is paramount.

Figure 3: Variations of temperature for (a,b) M (c,d) Rd (e,f) Pr, and (g,h) Uhs.

3.3 Concentration Profile

To analyze the mastery of solvent concentration on the Soret parameter NCT, Brinkman number Br, power law index of Tangent Hyperbolic Fluid m, and Brownian motion Nb, Fig. 4a–h is examined. Fig. 4a,c,e reveals a decline in concentration with increasing NCT, Br, and m, attributed to thermophoretic effects, viscous dissipation, and shear-thinning behavior, respectively. However, Fig. 4c demonstrates a complex response for Nb, where concentration rises with enhanced Brownian motion toward the left wall, while remains uniform toward the right wall, y∈[0.4,0.6]. This asymmetry stems from the competing effects of particle diffusion and thermophoretic forces, which dominate in the opposite regions. Additionally, Fig. 4b,d,f,h illustrates the three-dimensional divergence of NCT, Br, m, and Nb, highlighting their collective impact on concentration profiles. Physically, increasing NCT enhances solute migration due to thermal gradients, while higher Br intensifies viscous heating, reducing concentration. The power law index m alters fluid rheology, influencing concentration distribution, and Nb drives nanoparticle diffusion, creating non-uniform concentration fields. These insights are crucial for applications in nanofluidics, thermal energy systems, and chemical processes, where precise control over solute distribution and heat transfer is essential.

Figure 4: Variations of concentrations for (a,b) NCT, (c,d) Br, (e,f) m and (g,h) Nb.

3.4 Nanoparticle Fraction Profile

To explore the stimulus of nanoparticle fraction on the solutal Grashof number Grc thermophoresis parameter Nt, thermal radiation effect Rd, and nanoparticle slip parameter λ4, Fig. 5a–h are meticulously analyzed. Fig. 5e demonstrates an augmentation in nanoparticle fraction due to the escalating thermal radiation effect Rd, as radiative heat transfer enhances particle mobility and distribution. Conversely, Fig. 5a,c,g reveals a reduction in nanoparticle fraction with increasing nanoparticle slip parameter (λ4), solutal Grashof number (Grc), and thermophoresis parameter (Nt). This decline is attributed to enhanced slip-induced particle migration, buoyancy-driven flow, and thermophoretic forces, which collectively redistribute nanoparticles away from concentrated regions. Furthermore, Fig. 5b,d,f,h depicts the three-dimensional divergence of λ4, Rd, Grc and Nt, elucidating their combined impact on nanoparticle distribution. Bodily, higher Rd intensifies radiative energy transfer, promoting nanoparticle dispersion, while increased λ4 facilitates particle slippage, reducing local concentration. Elevated Grc enhances solutal convection, and Nt drives thermophoretic migration, both contributing to non-uniform nanoparticle profiles. These findings are pivotal for optimizing nanofluid applications in thermal management, biomedical systems, and advanced material synthesis, where exact control over heat transmission and nanoparticle distribution is essential.

Figure 5: Variations of nanoparticle fraction for (a,b) Grc, (c,d) Nt, (e,f) Rd and (g,h) λ4.

3.5 Pressure Gradient Profile

The impact of the pressure gradient on Helmholtz-Smoluchowski velocity (Uhs), thermal radiation parameter (Rd), Prandtl number (Pr), and electroosmotic parameter (eo) is delineated in Fig. 6a–d. Fig. 6a,d illustrates that the pressure gradient intensifies with rising Uhs and eo, driven by the amplification of electrokinetic forces and electroosmotic flow, which bolster fluid momentum and pressure gradients. Conversely, Fig. 6b,c exhibits a weakening of the pressure gradient as Rd and Pr increase. This attenuation is attributed to the energy dissipation caused by radiative heat transfer and the enhanced viscous resistance linked to higher Pr, both of which suppress fluid motion and diminish pressure gradients. Physically, the increase in Uhs and eo enhances electroosmotic flow, creating stronger pressure differentials, while elevated Rd promotes thermal energy loss, reducing fluid kinetic energy. Similarly, higher Pr increases viscous damping, further impeding flow dynamics. These phenomena highlight the intricate interplay between electrokinetic, thermal, and viscous mechanisms, which are critical for optimizing systems such as microfluidic devices, thermal management, and energy-efficient fluid transport, where precise control over pressure-driven flow is paramount.

Figure 6: Variations of pressure gradient for (a) Uhs, (b) Rd, (c) Pr and (d) eo.

3.6 Pressure Rise Profile

The implications of pressure rise on λ1, Uhs, M, and Nt are depicted in Fig. 7a–d. To examine the feature of pressure rise, the pumping zones are separated toward the subsequent groups: There are four types of pumping regions as given in the following Table 1.

Figure 7: Variation of pressure rise for (a) λ1, (b) Uhs, (c) M and (d) Nt.

Fig. 7a demonstrates that while a relatively larger pressure is seen in the retrograde flow regime, raising the velocity slip parameter λ1 decreases the pressure rise in the peristaltic and augmented pumping regions. Wall slip physically lessens the shear stress at the boundaries, facilitating fluid movement and lowering the pressure needed to sustain a specific flow rate. This drop is especially noticeable in an asymmetric channel because of the uneven shear distribution along the walls, which results in decreased pumping efficiency as slip increases. In all pumping locations, Fig. 7b shows that the pressure rise increases with the Helmholtz–Smoluchowski velocity Uhs. This phenomenon is explained by increased electroosmotic force, which fortifies the fluid’s axial momentum and supports the peristaltic process. Consequently, a greater pressure differential is produced in order to maintain the forced flow rate. In microfluidic systems, where electric-field-driven transport is predominant, the electroosmotic contribution is especially efficient. The impact of the Hartmann number M is shown in Fig. 7c, which shows that a stronger magnetic field causes a greater rise in pressure, especially in the reverse and peristaltic pumping zones. A greater pressure gradient is needed to propel the fluid since the applied magnetic field creates Lorentz forces that prevent fluid motion. The resistive function of magnetohydrodynamics forces in electrically conductive biofluids is proven by the greater significance of this magnetic damping effect with increasing M. The impact of the thermophoresis component Nt on pressure rise is seen in Fig. 7d. Especially at greater flow rates, an increase in Nt amplifies the pressure rise in the peristaltic pumping area. Temperature gradients cause thermophoretic particle migration, which modifies the fluid’s effective viscosity and momentum transmission. As a result, more pressure is needed to keep the volumetric flow rate constant. The substantial correlation between heat transport and hydrodynamic behavior in biofluids containing nanoparticles is shown by the nonlinear trend seen at higher Q.

3.7 Nusselt Number

Fig. 8a,b is built to investigate the promotion of the Nusselt number (Nu) on the Brownian motion parameter (Nb) and the Brinkman number (Br). Fig. 8a shows that for the range [−0.3, 0.6], the Nusselt number closer to the right channel’s wall contracts owing to the growing effect of the Brownian motion parameter. The decrease results from increased nanoparticle diffusion, which breaks down thermal boundary layers and lowers the efficiency of heat transfer. On the other hand, in the interval [−0.6, −0.3], a reverse trend is seen, where the Nusselt number rises, presumably because of the localized enhancement of thermal Brownian motion-induced gradients. Fig. 8b illustrates a very inverse relationship between the Brinkman number Br and the Nusselt number. The enhancement of Nusselt number with an increase in Br in the range [−0.6, 0.0] is owing to viscous dissipation to produce more heat, thus increasing thermal transfer. In the range [0.0, 0.7], however, the Nusselt number reduces as Br increases due to over-heating caused by viscosity dominating the thermal boundary layer and lessening heat transfer effectiveness. These opposing behaviors highlight the opposing roles of viscous dissipation and nanoparticle diffusion in controlling heat transfer dynamics, which are pivotal for thermal optimization nanofluid and high-viscosity flow systems.

Figure 8: Variations of Nusselt number for (a) Nb, (b) Br.

3.8 Sherwood Number

The graphical plots in Fig. 9a,b clarify the influence of Sherwood number (Sh) on Brownian motion parameter (Nb) and Brinkman number (Br). Fig. 9a illustrates that in the range [−0.6, 0.1], the Sherwood number diminishes as Brownian motion parameter intensifies. This decrease is due to improved random motion of nanoparticles, which disturbs concentration boundary layers, thus decaying mass transfer efficiency. On the other hand, in the interval [0.1, 0.6], a reverse trend is seen, where the mass transfer rate, represented by Sherwood number improves with the increase in Nb, possibly because the increased diffusion of nanoparticles increases solute transport in this localized region. Fig. 9b shows that in the range [0.0, 0.6], the Sherwood number rises with the higher Brinkman number (Br). This is because viscous dissipation produces extra thermal energy, which enhances convective mass transfer. Yet, in the range [−0.6, 0.0], Sherwood number decreases with the rise in Br, as excessive viscous heating overpowers the concentration boundary layer and lowers mass transfer efficiency. Such divergent behaviors point to the double role of Brownian motion and viscous dissipation in controlling mass transfer dynamics, which are crucial for optimizing processes in nanofluid systems, chemical reactors, and thermal engineering applications.

Figure 9: Variations of Sherwood number for (a) Nb, (b) Br.

3.9 Stream Function

Trapping in peristaltic flow appears as the creation and transport of discrete masses of fluid or boluses through a conduit filled with fluid. Trapping is defined by the creation of distinctive streamline structures around the bolus, which is propelled by high pressure peristaltic waves. Trapping appears when significant occlusions capture and transport the bolus, a mechanism that enhance mixing and separation processes in chemical, biological, and microfluidic engineering applications. Figs. 10 and 11 illustrate the influence of streamlines for varying quantities of the thermal radiation parameter (Rd) and Helmholtz-Smoluchowski velocity (Uhs). These statistics show that the volume and number of boluses growing with the augmentation of the solutal Grashof number Grc and Uhs. Physically, increased Grc increases buoyancy-driven convection, driving the development of larger boluses, whereas augmenting Uhs enhances electroosmotic forces, further heightening fluid entrapment and transport. The above effects confirm the pivotal roles of thermal radiation solutal buoyancy, and electrokinetic forces in maximizing peristaltic flow dynamics and its necessity in medication administration systems, and next-generation fluid mixing technologies.

Figure 10: (a–d) Streamlines against Rd at 0.1, 0.6, 1.1, and 1.6.

Figure 11: (a–d) Streamlines against Uhs at 1, 2, 3, and 4.

3.10 Model Validation

Table 2 provides the comparison of current results with existing literature Naduvinamani et al. [36]. It shows that the current results are extremely valid and agreed with those of [36] through limiting case. It also shows that the slip effects consider for increasing the flow velocity at the peristaltic walls.

4  Conclusions

The quintessential goal of this research is to attempt a profound insight into the diverse hydrodynamic behavior of a tangent hyperbolic fluid, based on the postulates of a long wavelength and a substantially reduced Reynolds number in the context of an asymmetric channel. Numerical solutions, carefully computed, are meticulously determined for a list of significant physical variables, including velocity, temperature, concentration, pressure rise, volume fraction of nanoparticles, stream function, and pressure gradient. The solutions are aimed at defining The implications of several dimensionless parameters on the dynamics of flow are systematically examined and graphically outlined, with an extensive discussion of their resultant implications. The key findings and the general conclusions of this research effort are stated as below:

❖   The results show that velocity slip causes slower flow speeds at the center of the channel but faster flow speeds at the channel walls. The flow velocity decreases when solutal Grashof number and electroosmotic parameter values rise for leftward movement yet it increases for rightward flow direction. The power law index changes speed patterns in various areas of the flow which shows how these parameters interact with each other.

❖   A rise in Hartmann number (M), thermal radiation effect (Rd), and Prandtl number (Pr) results in a reduction in heat transmission. But at the right wall (y∈[0.5]), heat transmission increases with rising Helmholtz Smoluchowski velocity (Uhs).

❖   Concentration profile decreases with rising values of the Soret parameter (NCT), Brinkman number (Br), and power law index (m). For Brownian motion (Nb), on the other hand, concentration rises toward the left path while being almost constant close to the right wall.

❖   The volume fraction of nanoparticles rises as the thermal radiation effect (Rd) grows, but falls when the thermophoresis parameter (Nt), solutal Grashof number (Grc), and nanoparticle slip parameter (λ4) increase.

❖   As an Electroosmotic parameter(eo) and the Helmholtz-Smoluchowski velocity (Uhs) expand, the pressure gradient hike. But as the Prandtl number (Pr) and thermal radiation effect (Rd) rise, it somewhat declines.

❖   In the retrograde and free pumping zones, velocity slip (λ1) raises pressure, but in the augmented region, it decreases it. Prandtl number (Pr) and Brinkman number (Br) highlight the impacts of heat diffusion and viscous dissipation by influencing pressure in enhanced and retrograde regions while maintaining the free pumping zone.

❖   Streamlines graphs demonstrate that as the Solutal Grashof number (Grc) and Helmholtz-Smoluchowski velocity (Uhs) rise, so do the bolus quantity and volume.

❖   In future we can extend this work by using hybrid nanofluid models, various geometries and using certain modern methodologies.

Acknowledgement: The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number 0038-1446-S.

Funding Statement: This research work was supported by the Ministry of Education–Kingdom of Saudi Arabia through the project number 0038-1446-S.

Author Contributions: The authors confirm contribution to the paper as follows: Conceptualization, Yasir Khan; Methodology and supervision, Arshad Riaz; Writing—original draft and language, Iqra Batool; Software and resources, Safia Akram; Mathematical modeling, A. Alameer; Review and editing, geometry and validation, Ghaliah Alhamzi. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: All data generated or analyzed during this study are included in this article.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

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