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Analysis of the magnetohydrodynamic effects on non-Newtonian fluid flow in an inclined non-uniform channel under long-wavelength, low-Reynolds number conditions

  • Manjunatha Gudekote , Rajashekhar Choudhari , Prathiksha Sanil , Hanumesh Vaidya , Dharmendra Tripathi , Kerehalli Vinayaka Prasad and Kottakkaran Sooppy Nisar EMAIL logo
Published/Copyright: October 25, 2024
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This study utilises mathematical modelling and computations to analyse the magnetohydrodynamic (MHD) effects on non-Newtonian Eyring–Powell fluid flow in an inclined non-uniform channel under long-wavelength, low Reynolds number conditions. The governing equations are solved by applying slip boundary conditions to determine the velocity, temperature, concentration, and streamline profiles. The key findings show that the magnetic parameter dampens the flow rate. The relationship between the variable viscosity, velocity, and temperature is nonlinear. The wall rigidity parameter and axial velocity are directly proportional until a threshold. Increasing inclination angles distorts streamlines. The magnetic field alters concentration contours and thermal transport. MATLAB parametric analysis explores MHD effects. This study enhances the understanding of inclined channel fluid dynamics, offering insights into variable viscosity, magnetic fields, wall properties, and impacts of inclination angles on non-Newtonian flow characteristics. This knowledge can optimise industrial MHD conduit/channel transport applications.

Keywords: magnetohydrodynamic; non-Newtonian fluid; long-wavelength; variable viscosity; Eyring Powell fluid

1 Introduction

Peristalsis is a vital physiological process involving coordinated contraction and relaxation of muscles to propel fluids through tubular organs. It underlies essential functions, including swallowing, gastrointestinal motility, sperm transport, ovum movement in the fallopian tubes, and oesophageal transport. Technological advances have created sophisticated peristaltic devices for internal mobile liquid pumps. The ubiquity of peristalsis in human physiology, medicine, and bioengineering has made it an intriguing research area. Investigating the hydrodynamics of fluid flow through contracting/expanding chambers or channels is crucial with many practical applications. Peristaltic pumping is a standard method of fluid flow that propels fluids using progressive compression/decompression waves in flexible tubes. It is essential to understand this mechanism to comprehend natural fluid flows in the human body. The study of peristaltic motion was initiated by Latham [1], who first investigated the movement of urine through the ureter. After this, various researchers across the globe have carried out investigations aimed at comprehending the circulatory system in ducts and vessels, the process of spermatozoa development in the male reproductive system, and the movement of chyme in the gastrointestinal tract. The initial focus of peristaltic flow research was exclusively on fluids following Newton’s viscosity law. Analysis of non-Newtonian fluid flows to evaluate the peristaltic mechanism has recently become an important area of research. Non-Newtonian models with different geometries and assumptions have been documented in the literature [2,3,4,5,6].

Recent research has highlighted the importance of heat and mass transfer applications in understanding various cycles, such as inversion assimilation, propagation of synthetic pollutants, film division, and refining. Heat and mass transfer applications are critical in many areas and industries. Typically, temperature fluctuations result in a flow of heat that starts at one source and continues to the next. The rule of an assortment of temperatures can, similarly, be concentrated by the human body. Heat is typically delivered into the human stomach-related system because of its reliable methodology. The impact of heat transfer on peristalsis in non-Newtonian fluid flows is a crucial factor in biological sciences and bioengineering technology. Examining heat flow is essential for gaining insights into their properties, including conduction, convection, and radiation. The convection process is critical in medical procedures such as oxygenation and haemodialysis. Srinivas et al. [7] investigated the convection process in a peristaltic mechanism over a porous channel. Subsequently, Akbar and other researchers considered this strategy on the peristaltic transport of fluid [8,9,10,11,12]. The study of mass transfer and its methods on peristalsis is essential for blood to flow through arteries and their neighbouring tissues. Various researchers have mulled over the influence of heat and mass transfer by contemplating these limits with different mathematical models [13,14,15,16]. Heat and mass transfer applications improve energy efficiency, streamline operations, improve product quality, and develop technology across sectors. Understanding these principles enables innovative ideas and sustainable practices. The influence of heat and mass transport on these phenomena has been examined using numerical models.

The fluid properties of the human body, especially in the blood, are necessary for physical processes. Heart function is greatly affected by blood’s non-Newtonian behaviour, which is influenced by variables like heart rate and temperature. In the anatomical architecture of the human body, it is imperative to contemplate the diversity in viscosity to elucidate blood circulation. This contemplation reveals that the viscosity of blood in regions proximate to the periphery is notably reduced when contrasted with that in the central areas. Along with the thickness factor, the changing thermal conductivity based on temperature is also considered in the ongoing research, as the fluctuation in viscosity and thermal conductivity affects the temperature. Models of non-Newtonian fluids are used to explore varying boundaries. They are developed by experts using the Jeffrey model [17], Bingham [18,19,20], Rabinowitsch [21], Carreau fluid [22], Williamson fluid [23], and third-grade fluid [24] liquid models. Understanding these differences is crucial for assessing cardiovascular health and conditions such as osteoarthritis. Studying different fluid properties in the body can lead to targeted medical interventions and diagnoses for various health problems.

Magnetohydrodynamics (MHD) peristaltic flow combines fluid mechanics with electromagnetism and studies the behaviour of electrically conducting fluids under the influence of both fluid motion and magnetic fields. Peristaltic waves, characterised by rhythmic contractions and relaxations, propel fluid through conduits. At the same time, magnetic fields influence and lead to complex phenomena such as flow instabilities and magnetic field-induced reductions in flow resistance. This interdisciplinary field has a wide range of applications in different areas. In biomedical engineering, MHD peristaltic flow helps with drug delivery systems by providing precise fluid control with external magnetic fields for targeted drug delivery. Microfluidics enables precise transport, mixing and pumping of liquids at the microscale, driving advances in lab-on-a-chip technologies. Moreover, MHD peristaltic flow holds promise for energy generation, particularly in MHD generators, enabling power generation without moving mechanical parts. As a result, MHD peristaltic flows are advantageous for industrial processes, as they are used in chemical reactors, metallurgical processes, and waste treatment, where precise liquid handling and better mixing performance are crucial. Ongoing research in MHD peristaltic flow promises further innovation in various engineering and biomedical applications. Current research investigates the dynamics of MHD peristaltic transport of non-Newtonian fluids. Eldabe et al. [25] investigated the influence of wall properties and Joule heating on Bingham non-Newtonian nanofluid flow. Magesh et al. [26] investigated non-Newtonian fluid flow in curved channels under peristalsis and induced magnetic fields. Li et al. [27] analysed the peristaltic transport of Ree-Eyring fluids in non-uniform compliant channels under varying conditions. Shaheen et al. [28] provided analytical solutions for non-Newtonian Williamson fluids influenced by MHD. These studies collectively advance our comprehension of intricate fluid dynamics in MHD peristaltic systems.

Previous studies have attempted to characterise the shear-thinning behaviour of non-Newtonian fluids; among these, the MHD peristaltic flow of the Powell–Eyring fluid model is the most applicable. Gas kinetic theory is used to derive the constitutive equation of the Powell–Eyring fluid model. In particular, this model provides a more thorough representation of flow behaviour at both low and high shear rates. Noreen and Qasim [29] showed that the Powell–Eyring fluid model deviates significantly from the Newtonian flow behaviour at low and high shear rates. Taking heat and mass transmission into account, Hina [30] investigated Eyring–Powell fluid peristaltic transport. Hayat et al. [31] examined how chemical reactions and convective circumstances affect the peristaltic movement of the Eyring–Powell fluid. Riaz studied the peristaltic movement of an Eyring–Powell fluid in a rectangular pipe with mass transfer [32]. Thermal study of biological transport was the primary area of their research. Hadimani et al. [33] and Gudekote et al. [34] delved further into Eyring–Powell fluid peristalsis mechanics. Fluid behaviour in various geometries, liquid properties, and no-slip circumstances were the primary foci of the experiments. Researchers have shown that adjusting the viscosity coefficient enhances peristalsis velocity and temperature profiles. Boujelbene et al. [35] studied the peristaltic flow of an Eyring–Powell fluid in an inclined channel, focusing on different slip conditions and wall characteristics. An AI-based perturbation technique is used to examine the phenomenon. To better understand how peristaltic flow behaves in this setup, this study delves into how these parameters impact flow characteristics. Gudekote et al. [36] examined the impact of MHD peristaltic transport of non-Newtonian Eyring–Powell fluid within an inclined uniform channel. The investigation elucidates the intricate interplay among fluid dynamics, MHD phenomena, and thermal transport, offering valuable insights into the behaviour of such systems.

Despite existing research on the peristaltic mechanism of an Eyring–Powell fluid, there needs to be more literature regarding the investigation of MHD peristalsis with variable fluid properties. To the best of the author’s knowledge, minimal or no research has been conducted in this area. The current model aims to address the abovementioned issue by integrating the impacts of fluctuating liquid properties on the movement of fluids. The model employs standard assumptions to simplify the governing non-linear equations. Furthermore, a perturbation technique is implemented for efficient problem-solving. The model considers varying liquid properties of the Eyring–Powell fluids and heat transport. It is noteworthy that these factors are of significance. A graphical representation has been created to facilitate the comprehension of the model’s pertinent parameters. The graphical depiction facilitates understanding the model's performance and attributes in varying circumstances. The current model is well-suited for analysing blood flow in constricted arteries, offering significant insights into this physiological scenario.

The key highlights captured and addressed in this article are:

  • How does variable thermal conductivity affect peristaltic blood flow?

  • How does fluctuating viscosity influence velocity, temperature, and flow dynamics during peristalsis?

  • How do the wall rigidity, stiffness, and damping parameters impact velocity, temperature, and flow characteristics?

  • What role does the magnetic parameter play in understanding concentration profiles in the system?

  • How do the Eyring–Powell parameters affect the size of the bolus in peristaltic flow?

  • What is the relationship between the bolus size and the non-uniform parameter?

  • How does the Eyring–Powell fluid model provide insights into peristaltic transport and hemodynamics in the human vascular system?

  • What factors must be incorporated to understand blood flow regulation and dysfunction comprehensively?

  • How can the knowledge gained from this study potentially contribute to diagnosing and treating vascular diseases?

2 Mathematical formulations

In the current study, an investigation is conducted on the peristaltic transport of an axisymmetric non-uniform channel with an incompressible, viscous Eyring–Powell fluid. The flow of fluid along the channel sides is caused by sinusoidal wave trains with a certain wavelength λ propagating at speed c (Figure 1).

Figure 1 Geometry of the non-uniform channel.
Figure 1

Geometry of the non-uniform channel.

The following expression gives the geometry of the channel:

(1) H ′ ( x ′ , t ′ ) = a ′ ( x ′ ) + b ′ sin 2 π λ ( x ′ − c t ′ ) ,

where H′ is the non-uniform wave, t′ is the time, a′(x′) is the non-uniform radius, and b′ is the wave amplitude.

For the fluid flow, the incompressibility conditions for the equation of continuity, motion, and energy are given as

(2) ∂ u ′ ∂ x ′ + ∂ v ′ ∂ y ′ = 0 ,

(3) ρ ∂ u ′ ∂ t ′ + u ′ ∂ u ′ ∂ x ′ + v ′ ∂ u ′ ∂ y ′ = − ∂ p ′ ∂ x ′ + ∂ τ x ′ x ′ ′ ∂ x ′ + ∂ τ x ′ y ′ ′ ∂ y ′ + ρ g sin γ − σ B 0 ( u ′ + 1 ) ,

(4) ρ ∂ v ′ ∂ t ′ + u ′ ∂ v ′ ∂ x ′ + v ′ ∂ v ′ ∂ y ′ = − ∂ p ′ ∂ x ′ + ∂ τ x ′ y ′ ′ ∂ x ′ + ∂ τ y ′ y ′ ′ ∂ y ′ + ρ g cos γ ,

(5) ρ C P ∂ T ′ ∂ t ′ + u ′ ∂ T ′ ∂ x ′ + v ′ ∂ T ′ ∂ y ′ = k 1 ( T ′ ) ∂ 2 T ′ ∂ x ′ 2   + ∂ 2 T ′ ∂ y ′ 2 + τ x ′ x ′ ′ ∂ u ′ ∂ x ′   + τ y ′ y ′ ′ ∂ v ′ ∂ x ′ + τ x ′ y ′ ′ ∂ u ′ ∂ x ′ + ∂ v ′ ∂ x ′ ,

(6) ∂ C ′ ∂ t ′ + u ′ ∂ C ′ ∂ x ′ + v ′ ∂ C ′ ∂ y ′ = D ∂ 2 C ′ ∂ x 2 + ∂ 2 C ′ ∂ y 2 + D K T T m ∂ 2 T ′ ∂ x 2 + ∂ 2 T ′ ∂ y 2 ,

where u′ and v′ are the velocity components in axial and radial directions, respectively, p′ is the pressure, ρ is the fluid density, τ x ′ x ′ ′ , τ x ′ y ′ ′ , and τ y ′ y ′ ′ are extra stress components, k 1 is the mass diffusivity coefficient, T′ is the temperature, and C P is the specific heat at constant volume.

The dimensionless quantities of interest are as follows:

(7) x = x ′ λ , y = y ′ a ′ , u = u ′ c , v = λ v ′ c a ′ , τ x x = a ′ τ x ′ x ′ ′ c μ , τ x y = a ′ τ x ′ y ′ ′ c μ , τ y y = a ′ τ y ′ y ′ ′ c μ , t = c t ′ λ , Re = a ′ c ρ μ , ϑ = μ ρ , p = a ′ 2 p ′ c λ μ , θ = T ′ − T ′ 0 T 1 − T 0 , Pr = μ C P k 1 , Ec = c 2 δ T 0 , Br = Ec  Pr , δ = a ′ λ , ϵ = b ′ a ′ , ϕ = C ′ − C 0 ′ C 0 ′ , Sr = ρ D K T ( T ′ − T 0 ′ ) T m C 0 ′ , Sc = μ ρ D , F = μ c ρ g a ′ , M = σ μ B 0 a ′ , ψ = ψ ′ a ′ c , E 1 = − σ a ′ 3 λ 3 μ c , E 2 = m a ′ 3 c λ 3 μ , E 3 = a ′ 3 c λ 3 μ , μ 0 ′ = μ 0 μ , h = H ′ a ′ = 1 + m x + ϵ sin ( 2 π ( x − t ) ) .

On utilising the non-dimensional transformations from Eq. (7), by implementing long wavelength and small Reynolds number assumptions, Eqs (2)–(6) take the following form:

(8) ∂ τ x y ∂ y = ∂ p ∂ x − sin γ F + M 2 ∂ ψ ∂ y + 1 ,

(9) ∂ p ∂ y = 0 ,

(10) ∂ ∂ y k ( θ ) ∂ θ ∂ y + Br τ x y ∂ 2 ψ ∂ y 2 = 0 ,

(11) ∂ 2 ϕ ∂ y 2 + ScSr ∂ 2 θ ∂ y 2 = 0 ,

where γ is the angle of inclination, Br is the Brinkman number, and Sr and Sc are the Soret and Scmidtch numbers, respectively.

The constitutive equation τ x y of the Eyring Powell fluid is mathematically represented as follows:

(12) τ x y = ( μ ( y ) + B ) ∂ 2 ψ ∂ y 2 − A 3 ∂ 2 ψ ∂ y 2 3 .

Therefore, the non-dimensional slip conditions are given by

(13) ∂ ψ ∂ y + β 1 τ x y = − 1 at y = h ,

(14) ∂ 2 ψ   ∂ y 2 = 0 at y = 0 ,

(15) θ + β 2 ∂ θ ∂ y = 1 at y = h ,

(16) ∂ θ ∂ y = 0 at y = 0 ,

(17) ϕ + β 3 ∂ ϕ ∂ y = 1 at y = h ,

(18) ∂ ϕ ∂ y = 0 at y = 0 .

The varying viscosity μ(y) and thermal conductivity k(θ) are given by

(19) μ ( y ) = 1 − α 1 y α 1 ≪ 1 ,  

(20) k ( θ ) = 1 + α 2 θ α 2 ≪ 1 ,

where α 1 is the coefficient of variable viscosity, and α 2 is the coefficient of variable thermal conductivity.

3 Solution methodology

Consider Eq. (8) and integrate P = ∂ p ∂ x and f = sin γ F . Compare the obtained solution with Eq. (12). Then,

(21) ( P − f + M 2 ) y + M 2 ψ = ( 1 − α 1 y + B ) ∂ 2 ψ ∂ y 2 − A 3 ∂ 2 ψ ∂ y 2 3 ,

(22) ∂ ∂ y k ( θ ) ∂ θ ∂ y + Br ( 1 − α 1 y + B ) ∂ 2 ψ ∂ y 2 − A 3 ∂ 2 ψ ∂ y 2 3 ∂ 2 ψ ∂ y 2 = 0 .

The non-linear character of Eqs. (21) and (22) makes analytical solutions unattainable. A series solution of the perturbation technique is introduced to obtain the solutions.

3.1 Perturbation technique

A series perturbation technique is used to derive the solution for the streamline function and temperature expressions, as follows:

(23a) ψ = ∑ A n ψ n ,

(23b) θ = ∑ A n θ n .

By ignoring O(A 2) terms in Eq. (23a), the expression for streamline is obtained as

(24) ψ = ψ 0 + A ψ 1 .

The equation for streamlines of zeroth order is given by

(25) ( P − f + M 2 ) y + M 2 ψ 0 = ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 ,

with boundary conditions

(26) ∂ 2 ψ 0 ∂ y 2 = 0 at y = 0 and ∂ ψ 0 ∂ y + β 1 ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 = − 1 at y = h .

The equation for streamlines of the first order is given by

(27) ( 1 − α 1 y + B ) ∂ 2 ψ 1 ∂ y 2 − M 2 ψ 1 − 1 3 ∂ 2 ψ 0 ∂ y 2 3 = 0 ,

with boundary conditions

(28) ∂ 2 ψ 1 ∂ y 2 = 0 at y = 0 and ∂ ψ 1 ∂ y + β 1 ( 1 − α 1 y + B ) ∂ 2 ψ 1 ∂ y 2 −   1 3 ∂ 2 ψ 0 ∂ y 2 3 = 0 at y = h .

On ignoring O(A 2) terms in Eq. (23b), the expression for temperature is considered as

(29) θ =   θ 0 + A   θ 1 .

The equation for the temperature of zeroth order is given by

(30) ∂ θ 0 ∂ y + α 2 θ 0 ∂ θ 0 ∂ y + ∫ Br ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 2 ∂ y = 0 ,

with boundary conditions

(31) ∂ θ 0 ∂ y = 0 at y = 0 and θ 0 + β 2 ∂ θ 0 ∂ y = 1 at y = h .

The equation for the temperature of the first order is given by

(32) ∂ θ 1 ∂ y + α 2 θ 0 ∂ θ 1 ∂ y + α 2 θ 1 ∂ θ 0 ∂ y + ∫ Br 2 ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 ∂ 2 ψ 1 ∂ y 2 − 1 3 ∂ 2 ψ 0 ∂ y 2 4 ∂ y = 0 ,

with boundary conditions

(33) ∂ θ 1 ∂ y = 0 at y = 0 and θ 1 + β 2 ∂ θ 1 ∂ y = 0 at y = h .

Since the aforementioned equations are non-linear, the double perturbation approach is employed to find the solutions.

(34) u i = ∑ α 1 j u i j , where 0 ≤ j ≤ n , i = { 0 , 1 } ,

(35) θ i = ∑ α 2 j θ i j , where 0 ≤ j ≤ n , i = { 0 , 1 } .

Higher-order terms are disregarded to simplify solutions for streamlines and temperature, i.e. O ( α 1 2 ) and O ( α 2 2 ) . The resulting streamline and temperature equations are as follows.

(36a) ψ 0 = ψ 00 + α 1 ψ 01 ,

(36b) ψ 1 = ψ 10 + α 1 ψ 11 ,

(37a) θ 0 = θ 00 + α 2 θ 01 ,

(37b) θ 1 = θ 10 + α 2 θ 11 .

Substituting Eq. (36a) in Eqs (25) and (26), the equation for zeroth-order stream function, with slip boundary condition, is derived as

(38) ∂ 2 ψ 00 ∂ y 2 − M 2 ( 1 + B ) ψ 00 = ( P − f + M 2 ) ( 1 + B ) y ∂ 2 ψ 00 ∂ y 2 = 0 at y = 0 and ∂ ψ 00 ∂ y + β 1 ( 1 + B ) ∂ 2 ψ 00 ∂ y 2 = − 1 at y = h .

Then, the solution is given by

(39) ψ 00 = − y − ( P − f ) M 2 y + 2 C 2 sinh ( C 1 y ) .

The equation for the first-order stream function, along with slip boundary conditions, is derived as

(40) ∂ 2 ψ 01 ∂ y 2 − M 2 ( 1 + B ) ψ 01 = y ∂ 2 ψ 00 ∂ y 2 ∂ 2 ψ 01 ∂ y 2 = 0 at y = 0 and ∂ ψ 01 ∂ y + β 1 ( 1 + B ) ∂ 2 ψ 01 ∂ y 2 + β 1 y ∂ 2 ψ 00 ∂ y 2 = 0 at y = h .

Then, the solution is given by

(41) ψ 01 = 2 C 4 sinh ( C 1 y ) − C 3 y sinh ( C 1 y ) + C 1 C 3 y 2 cosh ( C 1 y ) .

Substituting Eq. (36b) into Eqs (27) and (28), the equation for the zeroth-order stream function, with slip boundary condition, is derived as

(42) ∂ 2 ψ 10 ∂ y 2 − M 2 ( 1 + B ) ψ 10 = 1 3 ( 1 + B ) ∂ 2 ψ 00 ∂ y 2 3 ∂ 2 ψ 10 ∂ y 2 = 0 at y = 0 and ∂ ψ 10 ∂ y + β 1 ( 1 + B ) ∂ 2 ψ 10 ∂ y 2 − 1 3 ∂ 2 ψ 00 ∂ y 2 3 = 0 at y = h .

Finally, the solution is given by

(43) ψ 10 = 2 C 6 sinh ( C 1 y ) − C 1 5 C 5 y cosh ( C 1 y ) + 1 4 C 1 4 C 5 sinh ( 3 C 1 y ) .

The equation for the first-order stream function, along with slip boundary conditions, is derived as

(44) ∂ 2 ψ 11 ∂ y 2 − M 2 ( 1 + B ) ψ 11 = 1 ( 1 + B ) y ∂ 2 ψ 10 ∂ y 2 + ∂ 2 ψ 00 ∂ y 2 2 ∂ 2 ψ 01 ∂ y 2 ∂ 2 ψ 11 ∂ y 2 = 0 at y = 0 and ∂ ψ 11 ∂ y + β 1 ( 1 + B ) ∂ 2 ψ 11 ∂ y 2 − y ∂ 2 ψ 10 ∂ y 2 − ∂ 2 ψ 00 ∂ y 2 2 ∂ 2 ψ 01 ∂ y 2 = 0 at y = h .

The final solution is given by

(45) ψ 11 = 2 C 10 sinh ( C 1 y ) − 3 C 1 5 C 2 2 C 4 y cosh ( C 1 y ) + 1 4 C 1 4 C 2 2 C 4 cosh ( 3 C 1 y ) + 1 2 C 6 y cosh ( C 1 y ) sinh ( C 1 y ) + 1 2 C 6 C 1 cosh ( C 1 y ) + 1 2 C 6 C 1 y 2 cosh ( C 1 y ) − C 6 y sinh ( C 1 y ) + 1 144 C 5 C 1 3 ( ( 9 C 1 2 y 2 + 2 ) cosh ( 3 C 1 y ) − 6 C 1 y sinh ( 3 C 1 y ) ) − 11 144 C 5 C 1 3 ( cosh ( 3 C 1 y ) − 3 C 1 y sinh ( 3 C 1 y ) ) − 5 4 C 1 3 C 5 ( C 1 y sinh ( C 1 y ) − cosh ( C 1 y ) ) − 1 32 3 C 5 C 1 3 cosh ( 3 C 1 y ) + 1 4 C 1 3 C 5 ( 3 ( C 1 2 y 2 + 2 ) cosh ( C 1 y ) − C 1 y ( C 1 2 y 2 + 6 ) sinh ( C 1 y ) ) − 2 C 1 3 C 5 ( ( C 1 2 y 2 + 2 ) cosh ( C 1 y ) − 2 y sinh ( C 1 y ) ) + 5 4 C 1 3 C 5 cosh ( C 1 y ) + 5 32 C 1 3 C 5 e − ( C 1 y ) .

Combining Eqs (39), (41), (43), and (45), the expression for stream function is obtained as

ψ = ψ 00 + α 1 ψ 01 + A   ψ 10 + A   α 1 ψ 11 .

Utilising the equation u = ∂ ψ ∂ y , the analytical solution for velocity can be determined.

Similarly, using Eq. (37a) in Eqs (30) and (31), the equation for the zeroth-order temperature expression with slip boundary conditions is derived as

(46) ∂ θ 00 ∂ y + ∫ Br  ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 2   ∂ y   = 0 ∂ θ 00 ∂ y = 0 at y = 0 and θ 00 + β 2 ∂ θ 00 ∂ y = 1 at y = h .  

The equation for the first-order temperature expression with slip boundary conditions is derived as

(47) ∂ θ 01 ∂ y + θ 00 ∂ θ 00 ∂ y = 0 ∂ θ 01 ∂ y = 0 at y = 0 and θ 01 + β 2 ∂ θ 01 ∂ y = 0 at y = h .

Substituting Eq. (37b) in Eqs (32) and (33), the equation for the zeroth-order temperature expression with slip boundary conditions is derived as

(48) ∂ θ 10 ∂ y + ∫ Br 2 ( 1 − α 1 y + B ) ∂ 2 ψ 0 ∂ y 2 ∂ 2 ψ 1 ∂ y 2 − 1 3 ∂ 2 ψ 0 ∂ y 2 4 ∂ y = 0 ∂ θ 10 ∂ y = 0 at y = 0 and θ 10 + β 2 ∂ θ 10 ∂ y = 0 at y = h .

The equation for the first-order temperature expression with slip boundary conditions is derived as

(49) ∂ θ 11 ∂ y + θ 00 ∂ θ 10 ∂ y + θ 10 ∂ θ 00 ∂ y = 0 ∂ θ 11 ∂ y = 0 at y = 0 and θ 11 + β 2 ∂ θ 11 ∂ y = 0 at y = h   .

By solving Eqs (46)–(49) analytically and substituting into Eq. (29), the expression for temperature function is obtained as

θ = θ 00 + α 2 θ 01 + A   θ 10 + A   α 2 θ 11 .

On solving Eq. (11), an analytical solution for concentration is derived.

The skin friction coefficient (C f – shear stress) is represented graphically, while the Nusselt number (Nu – heat transport) and Sherwood number (Sh – mass transport) are represented in tabulated form and are obtained by the following expressions:

(50) C f = h ′ ( x ) d u d y y = h ,

(51) Nu = h ′ ( x ) d θ d y y = h ,

(52) Sh = h ′ ( x ) d ϕ d y y = h .

4 Results and discussion

The primary purpose of this section is to analyse the impact of pertinent parameters, including material parameters of the Eyring–Powell fluid (A and B), angle of inclination (γ), variable viscosity (α 1), velocity slip parameter (β 1), magnetic parameter (M), non-uniform parameter (m), variable thermal conductivity (α 2), thermal slip parameter (β 2), wall rigidity parameter (E 1), wall stiffness parameter (E 2), wall-damping parameter (E 3), concentration slip parameter (β 3) on the velocity (u), temperature (θ), concertation (ϕ), and stream function (ψ). Graphs are drawn to analyse the effects of the relevant parameter mentioned above using the MATLAB 2023a programming language. Furthermore, the skin friction coefficient, Nusselt number, and Sherwood number were analysed.

4.1 Velocity profiles

Figure 2(a)–(h) depicts the modulation of velocity profiles for various fitting parameters. As shown in Figure 2(a), an elevation in the Eyring–Powell fluid parameter A results in a corresponding increase in the axial velocity. On the other hand, it can be observed from Figure 2(b) that the fluid parameter B displays an inverse pattern, leading to a reduction in velocity. Figures 2(c) and (d) shows the impact of parameters α 1 and β 1 on the velocity profiles. Observations indicate that both the coefficient of variable viscosity and velocity slip parameters positively influence by enhancing the velocity profiles. The magnetic parameter has a divergent impact on the velocity profiles, as shown in Figure 2(e). The velocity decreases as the magnetic parameters are increased. The effect of the parameter m on the velocity profiles is shown in Figure 2(f). It was found that increasing the non-uniform parameters causes the velocity profile to improve. The parameter shown in Figure 2(g) contributes significantly to creating the velocity profiles. When γ increases, the resulting velocity profile is also enhanced. Figure 2(h) compares the velocity profiles with rigidity, stiffness, and wall-damping parameters. The observations indicate that increased rigidity and stiffness parameters increase the velocity profiles. This indicates that the fluid flow becomes faster and more robust. In contrast, an increase in the wall-damping parameter reduces the velocity profiles. This indicates that the damping effect exerted by the channel walls causes the fluid transition to become slower and more restricted.

Figure 2 Variation of the velocity profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, t = 0.1, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, M = 2, and m = 0.1. (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) velocity slip parameter, (e) magnetic parameter, (f) non-uniformity parameter, (g) angle of inclination, and (h) wall parameters.
Figure 2

Variation of the velocity profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , α 1 = 0.02, t = 0.1, ϵ = 0.3 , β 1 = 0.2, M = 2, and m = 0.1. (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) velocity slip parameter, (e) magnetic parameter, (f) non-uniformity parameter, (g) angle of inclination, and (h) wall parameters.

4.2 Temperature profiles

The impact of various parameters on temperature is graphically represented in Figures 3(a)–(k). The increasing value of the Eyring–Powell fluid parameter A improves the temperature profiles, as shown in Figure 3(a); however, in Figure 3(b), the temperature profiles decrease as the material parameter B increases. The influence of temperature on the viscosity is depicted in Figure 3(c), which demonstrates that when the parameter α 1 is increased, the temperature profiles diminish, but the temperature profiles exhibit a dual behaviour: the parameter α 2 increases, as shown in Figure 3(d). The varying temperature profiles for β 1 and β 2 are shown in Figures 3(e) and (f), respectively. An increase in the β 1 parameter causes a reduction in temperature profiles, whereas an increase in the β 2 parameter leads to an elevation in temperature profiles. In addition, Figure 3(g) demonstrates that the temperature change depends on the magnetic parameters. The magnetic parameter brings a magnetic field into the system and affects heat transfer. The observation indicates that the magnetic field’s intensity and presence directly influence the temperature distribution. As the parameter M increases, the temperature profiles decline. Figure 3(h) shows that as the non-uniform parameter is increased, the temperature profiles also improve. Figure 3(i) shows the effect of the Brinkman number on temperature; increasing Br significantly influences the temperature. As Br increases, the temperature of the system increases significantly. Furthermore, insights into the correlation between the angle of inclination and fluid temperature are shown in Figure 3(j). An increase in the angle of inclination results in a corresponding increase in the temperature of the fluid. Thus, system temperature profiles are directly affected by the angle of inclination. Figure 3(k) shows increased rigidity and stiffness parameters increase the temperature. As the wall-damping value increases, the figure demonstrates that temperature profiles move in opposite directions.

Figure 3 Variation of temperature profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, t = 0.1, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, and Br = 2. Temperature profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) magnetic parameter, (h) non-uniformity parameter, (i) Brinkmann number, (j) angle of inclination, and (k) wall parameters.
Figure 3 Variation of temperature profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, t = 0.1, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, and Br = 2. Temperature profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) magnetic parameter, (h) non-uniformity parameter, (i) Brinkmann number, (j) angle of inclination, and (k) wall parameters.
Figure 3

Variation of temperature profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , α 1 = 0.02, t = 0.1, ϵ = 0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, and Br = 2. Temperature profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) magnetic parameter, (h) non-uniformity parameter, (i) Brinkmann number, (j) angle of inclination, and (k) wall parameters.

4.3 Concentration profiles

Figure 4(a)–(n) shows the influence of significant parameters of peristalsis on the concentration profiles. These figures aid in comprehending the relationship between parameter variations and concentration distribution within the system, providing valuable insights into various concentration-dependent applications. These figures represent the relationship between the considered parameter and the resulting concentration profiles. The concentration distribution throughout the system is affected by changes in several factors, which researchers can better understand by looking at these graphical representations. The figures provide valuable insights for designing and optimising the processes that involve concentration-dependent phenomena, such as chemical reactions, mass transfer operations, or fluid mixing. Researchers can use the figures to identify the optimal parameter values that result in desired concentration profiles or to analyse the system’s sensitivity to various parameter variations. The relationship between Eyring–Powell fluid parameter A and concentration profiles is illustrated in Figure 4(a). An inverse relationship is noted between the concentration profiles and the parameter A, whereas the opposite trend has been observed for the material parameter B (Figure 4(b)). According to Figure 4(c) and (d), an increasing parameter α 1 enhances the concentration profiles, whereas the increasing value of α 2 has the opposite effect. The higher concentration in Figure 4(e) and (f) results from increased velocity slip and thermal slip parameters. Contrary behaviour has been observed for the concentration slip parameter. As the concentration slip parameter increases, the concentration profiles decrease (Figure 4(g)). Figure 4(h) illustrates the variation in concentration as a function of varying magnetic parameters. An increase in the magnetic parameter enhances the concentration profiles. Figure 4(i) demonstrates that concentration profiles decrease as the non-uniform parameter increases. In Figure 4(j), a significant variation in concentration profiles for the Br can be observed. As the Brinkman number increases, the concentration profiles decrease. The variations in concentration profile for various Schmidt and Soret numbers are depicted in Figure 4(k) and (l), respectively. Both parameters behave similarly, i.e. increasing both parameters results in a decrease in concentration profiles. Further, as the inclination angle increases, the concentration profiles also decrease (Figure 4(m)). Figure 4(n) shows the effect of the rigidity, stiffness, and wall-damping factors on the concentration profiles. Changes in these parameters affect the concentration distribution throughout the system, and this information is crucial. Figure 4(n) demonstrates that increased rigidity and stiffness parameters decrease fluid concentration. Conversely, an increase is observed in the concentration profiles as the value of the wall-damping parameter increases – also, the increased wall-damping parameter results in a greater fluid concentration within the system.

Figure 4 Variation in the concentration profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, t = 0.1, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, Br = 2, Sc = 1, Sr = 1, and β 3 = 0.2. Concentration profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) concentration slip parameter, (h) magnetic parameter, (i) non-uniformity parameter, (j) Brinkmann number, (k) Schmidt number, (l) Soret number, (m) angle of inclination, and (n) wall parameters.
Figure 4 Variation in the concentration profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, t = 0.1, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, Br = 2, Sc = 1, Sr = 1, and β 3 = 0.2. Concentration profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) concentration slip parameter, (h) magnetic parameter, (i) non-uniformity parameter, (j) Brinkmann number, (k) Schmidt number, (l) Soret number, (m) angle of inclination, and (n) wall parameters.
Figure 4

Variation in the concentration profiles when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , α 1 = 0.02, t = 0.1, ϵ = 0.3 , β 1 = 0.2, M = 2, m = 0.1, α 2 = 0.02, β 2 = 0.2, Br = 2, Sc = 1, Sr = 1, and β 3 = 0.2. Concentration profiles for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) coefficient of variable thermal conductivity, (e) velocity slip parameter, (f) thermal slip parameter, (g) concentration slip parameter, (h) magnetic parameter, (i) non-uniformity parameter, (j) Brinkmann number, (k) Schmidt number, (l) Soret number, (m) angle of inclination, and (n) wall parameters.

4.4 Trapping phenomenon

Trapping is essential in studying the peristaltic transport of biological fluids. It serves to illustrate the creation of boluses through closed flow patterns. Figures 5–10 show the physiological parameters within these flow patterns. Flow patterns for various values of the fluid parameter A are depicted in Figure 5. The figure illustrates that an increase in A leads to the generation of larger boluses. On the other hand, it can be observed from Figure 6 that the fluid parameter B displays an inverse relationship, whereby an increase in B results in a decrease in the creation of boluses. The impact of the parameters α 1 and β 1 on the trapped bolus is analysed in Figures 7 and 8. The presented graphs indicate that an increase in viscosity and velocity slip parameters results in a higher occurrence of bolus formations. Figure 9 demonstrates that increasing the magnetic parameter reduces the quantity of trapped boluses. Figure 10 presents an analysis of the impact of non-uniform parameters. The figure indicates that an increase in non-uniform parameters results in an increase in the bolus size.

Figure 5 Variation of streamlines for (a) A = 0.01 and (b) A = 0.02.
Figure 5

Variation of streamlines for (a) A = 0.01 and (b) A = 0.02.

Figure 6 Variation of streamlines for (a) B = 1.5 and (b) B = 2.5.
Figure 6

Variation of streamlines for (a) B = 1.5 and (b) B = 2.5.

Figure 7 Variation of streamlines for (a) α 1 = 0.01 and (b) α 1 = 0.04.
Figure 7

Variation of streamlines for (a) α 1 = 0.01 and (b) α 1 = 0.04.

Figure 8 Variation of streamlines for (a) β 1 = 0.1 and (b) β 1 = 0.2.
Figure 8

Variation of streamlines for (a) β 1 = 0.1 and (b) β 1 = 0.2.

Figure 9 Variation of streamlines for (a) M = 2 and (b) M = 2.5.
Figure 9

Variation of streamlines for (a) M = 2 and (b) M = 2.5.

Figure 10 Variation of streamlines for (a) m = 0.1 and (b) m = 0.2.
Figure 10

Variation of streamlines for (a) m = 0.1 and (b) m = 0.2.

4.5 Skin friction coefficient

Figure 11 depicts an analysis of the graphical representation of the skin-friction coefficient and its variation according to various physiological constraints. The following are the primary discoveries: Figure 11(a) illustrates that an increase in the fluid parameter A reduces skin friction. The data suggest a negative correlation exists between parameter A and the skin friction coefficient. Figure 11(b) shows that an increase in the fluid parameter B leads to an increase in skin friction, distinct from the trend observed for parameter A. The data indicate a direct relationship between parameter B and the skin friction coefficient. Figure 11(c) demonstrates a consistent skin-friction coefficient regardless of changes in viscosity. The results suggest that variations in viscosity exert a negligible influence on skin friction under the given conditions. According to Figure 11(d), it can be inferred that there is a positive correlation between the velocity slip parameter and skin friction coefficient; an increase in the slip velocity is correlated with a decrease in skin friction. According to Figure 11(e), the magnetic field parameter’s resistive function causes a reduction in velocity, which consequently impacts the skin friction coefficient. It can be inferred that the existence of a magnetic field exerts an impact on the kin friction, resulting in a decrease in its intensity. Figure 11 is a comprehensive graphical representation that sheds light on the fluctuations of the skin friction coefficient under various physiological constraints. The results of this study emphasise the influence of fluid parameters, velocity slip, and magnetic field parameters on skin friction.

Figure 11 Variation in the skin friction coefficient when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , \gamma =\frac{\text{π}}{4}, α 1 = 0.02, M = 2, ϵ = 0.3 {\epsilon }=0.3 , β 1 = 0.2, and m = 0.1. Skin friction coefficient for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) velocity slip parameter, and (e) magnetic parameter.
Figure 11

Variation in the skin friction coefficient when E 1 = 0.4, E 2 = 0.1, E 3 = 0.01, A = 0.01, B = 2, x = 0.2, F = 2, γ = π 4 , α 1 = 0.02, M = 2, ϵ = 0.3 , β 1 = 0.2, and m = 0.1. Skin friction coefficient for (a) material parameter A, (b) material parameter B, (c) coefficient of variable viscosity, (d) velocity slip parameter, and (e) magnetic parameter.

4.6 Nusselt and Sherwood numbers

According to the data, an increase in the fluid parameter A reduces the Nusselt number. This implies that the parameter A and the Nusselt number have an inverse relationship. Unlike the behaviour noted for parameter A, the Nusselt number is enhanced with an increase in the fluid parameter B. A direct relationship between parameter B and the Nusselt number is observed. Table 1 presents numerical data indicating that an elevation in the variable viscosity results in a favourable effect on the Nusselt number. The data suggest a positive correlation between higher variable viscosity values and increased Nusselt numbers. According to the data, an increase in the variable thermal conductivity results in a detrimental impact on the Nusselt number. It can be inferred that an increase in the variable thermal conductivity values results in a reduction in the Nusselt number. Based on the data, it can be concluded that the Nusselt number is increased by thermal slip. The observation indicates that thermal slip enhances the convective heat transfer, resulting in increased Nusselt numbers. The Nusselt number is observed to increase as the magnetic parameter increases in magnitude. This suggests that increasing the magnetic parameter improves convective heat transfer, increasing the Nusselt numbers. The Brinkman number exhibits an inverse relationship with magnetic parameters.

Table 1

Nusselt number for various values of (a) A, (b) B, (c) α 1, (d) α 2, (e) β 2, (f) M, and (g) Br

A B α 1 α 2 β 2 M Br Nu
0.1 −77.2078
0.3 −82.2242
0.5 2.0 0.02 0.02 0.2 2.0 2.0 −87.2387
0.7 −92.2531
2.0 −74.9533
2.2 −73.1938
2.4 −71.4978
2.6 −69.8698
0.01 −75.3596
0.02 −74.9533
0.03 −74.5479
0.04 −74.1435
0.01 −73.1977
0.02 −74.9533
0.03 −76.7089
0.04 −78.4645
0.00 −75.0924
0.02 −75.0785
0.04 −75.0646
0.06 −75.0507
2.0 −74.9533
2.1 −64.7306
0.01 0.2 2.2 −56.1418
2.0 0.02 2.3 −48.8684
2.0 −74.9533
2.5 −94.7913
3.0 −115.069
3.5 −135.787

Table 2 offers valuable insight into the numerical variations in Sherwood numbers under various constraints. The Sherwood number measures the efficiency of mass transmission in a system. Several significant trends can be observed by analysing the data depicted in Table 2. Increasing the fluid parameter A results in an increase in the Sherwood number. This indicates that increasing the parameter A increases the system’s mass transfer rate. On the other hand, an increase in the fluid parameter B results in a decrease in the Sherwood number. This indicates that the parameter B negatively affects mass transfer efficacy. Second, as viscosity varies, the Sherwood number decreases. This suggests that the mass transfer rate decreases as the fluid viscosity changes. In the case of variable thermal conductivity, an enhancement positively impacts the Sherwood number, contrary to the trend mentioned above. This means that a higher thermal conductivity improves the system’s mass transfer efficiency. The Sherwood number is not significantly affected by the concentration slip parameter. This is the third point to consider. The mass transfer rate remains unaffected by variations in the concentration slip parameter. In addition, an increase in magnetic parameters leads to a decrease in the Sherwood number. The observed phenomenon suggests that a magnetic field impacts the efficacy of mass transfer, decreasing the Sherwood number. The numerical analysis indicates that the Sherwood number increases as the Schmidt number and Soret numbers increase. Schmidt number is a dimensionless parameter that quantifies the ratio of momentum diffusivity to mass diffusivity. In contrast, the Soret number is another dimensionless parameter that represents the ratio of thermal diffusivity to mass diffusivity. The correlation between higher values of the Sherwood number and improved mass transfer efficiency indicates the significant contribution of mass diffusion in enhancing the process. Overall, the findings presented in Table 2 cast light on the impact of various parameters on Sherwood number, providing valuable insights into the mass transfer behaviour and system efficiency under varying constraints.

Table 2

Sherwood numbers for various values of (a) A, (b) B, (c) α 1, (d) α 2, (e) β 2, (f) Sc, (g) M, and (h) Sr

A B α 1 α 2 β 3 Sc M Sr Sh
0.1 77.2078
0.3 82.2242
0.5 2.0 0.02 0.02 0.2 1 2.0 1 87.2387
0.7 92.2531
2.0 74.9533
2.2 73.1938
2.4 71.4978
2.6 69.8698
0.01 75.3596
0.02 74.9533
0.03 74.5479
0.04 74.1435
0.01 73.1977
0.02 74.9533
0.03 76.7089
0.04 78.4645
0.00 74.9533
0.02 74.9533
0.04 74.9533
0.06 74.9533
0.2 14.9907
0.4 29.9813
0.6 44.972
0.01 0.8 59.9626
2.0 0.02 2.0 74.9533
0.01 0.2 2.1 64.7306
0.2 2.2 56.1418
2.3 48.8684
0.1 7.49533
0.3 22.486
0.5 37.4767
0.7 52.4673

5 Conclusion

This study emphasises the Eyring–Powell fluid model in a non-uniform channel to analyse the peristaltic transport of blood in the human vascular system. Understanding blood flow regulation and dysfunction requires incorporating diverse effects mirroring natural phenomena, especially in small arteries. The discoveries provide crucial insights into peristaltic blood flow properties in the circulatory system. This knowledge could aid diagnosis and treatment of vascular diseases and promote scientific research.

From the model, some important key findings are summarised as follows:

  • Higher Schmidt and Soret numbers correlate with a better Sherwood number, indicating a more effective mass transfer process.

  • Magnetic fields significantly reduce frictional forces at the interface of solids and fluids, as evidenced by the decrease in the skin friction coefficient in their presence.

  • Higher variable viscosity increases the Nusselt number, indicating improved convective heat transfer efficiency, while increased variable thermal conductivity is related to a reduced Nusselt number, indicating decreased efficiency.

  • The increase in rigidity and stiffness parameters increases temperatures, which is essential in material processing. In contrast, the wall-damping parameter reduces temperature, which is crucial for thermal regulation in electronics.

  • High-velocity and thermal slip parameters increase concentration, which is essential in chemical reactions and fluid transport systems requiring efficient mixing. The concentration slip parameter decreases concentration profiles, which is helpful in pharmaceutical manufacturing and environmental monitoring where accurate concentration gradients are needed.

  • Increasing fluid parameter A increases velocity, which is critical for efficient fluid transport, while fluid parameter B decreases the velocity. Understanding these relationships is essential for maintaining controlled flow rates in precision manufacturing and chemical processing.

  • Increasing the non-uniform parameters leads to an expansion of the bolus size, indicating its important influence on fluid dynamics and transport mechanisms in the system.

Overall, the model provides vital insights into peristaltic blood flow properties and the factors influencing hemodynamics in the human circulatory system.

Acknowledgements

This study was supported by funding from the Prince Sattam bin Abdulaziz University (project number PSAU/2024/R/1445).

  1. Funding information: This study was supported by fundingfrom the Prince Sattam bin Abdulaziz University (project number PSAU/2024/R/1445).

  2. Author contributions: Manjunatha Gudekote: supervision, writing – original draft, and visualisation; Rajashekhar Choudhari: modelling, software, and visualisation; Prathiksha Sanil: conceptualisation, modelling, methodology, writing – original draft, software, and visualisation. Dharmendra Tripathi: conceptualisation and software; Hanumesh Vaidya: methodology and visualisation. K S Nisar: supervision, writing – original draft, software, and resources. Kerehalli Vinayaka Prasad: conceptualisation and resources.

  3. Conflict of interest: The authors state that they have no financial conflicts of interest or personal relationships that might have influenced the findings presented in this paper.

  4. Data availability statement: All the data sets generated for the current study are available in the manuscript.

Appendix

C 1 = M 1 + B ,

C 2 = P − f M 2 ( 2 ( 1 + B ) β 1 C 1 2 sinh ( C 1 h ) + 2 C 1 cosh ( C 1 h ) ) ,

C 3 = C 2 2 ( 1 + B ) ,

C 4 = − C 1 C 3 h cosh ( C 1 h ) − C 1 2 C 3 h 2 sinh ( C 1 h ) + C 3 sinh ( C 1 h ) − 1 2 7 β 1 C 1 2 C 2 h sinh ( C 1 h ) − 1 2 β 1 C 1 3 C 2 h 2 cosh ( C 1 h ) 2 ( 1 + B ) β 1 C 1 2 sinh ( C 1 h ) + 2 C 1 cosh ( C 1 h ) ,

C 5 = C 2 3 ( 1 + B ) ,

C 6 = C 1 5 C 5 cosh ( C 1 h ) + C 1 6 C 5 h sinh ( C 1 h ) − 1 4 C 1 5 C 5 cosh ( 3 C 1 h ) − 1 12 β 1 C 1 6 C 2 3 sinh ( 3 C 1 h ) + β 1 C 1 7 C 2 3 h cosh ( C 1 h ) 2 ( 1 + B ) β 1 C 1 2 sinh ( C 1 h ) + 2 C 1 cosh ( C 1 h ) ,

C 7 = − 3 C 1 5 C 2 2 C 4 cosh ( C 1 h ) − 3 C 1 6 C 2 2 C 4 h sinh ( C 1 h ) + 1 4 3 C 1 5 C 2 2 C 4 cosh ( 3 C 1 h ) + 1 2 C 6 C 1 h cosh ( C 1 h ) + 1 2 C 6 C 1 2 h 2 sinh ( C 1 h ) − 1 2 C 6 sinh ( C 1 h ) + 1 16 3 C 5 C 1 6 h 2 sinh ( 3 C 1 h ) + 11 16 C 5 C 1 5 h cosh ( 3 C 1 h ) − 9 12 C 5 C 1 4 sinh ( 3 C 1 h ) − 2 C 1 6 C 5 h 2 sinh ( C 1 h ) − 5 4 C 1 5 C 5 h cosh ( C 1 h ) + 5 4 C 1 4 C 5 sinh ( 3 C 1 h ) + 5 32 C 1 4 C 5 e ( − C 1 h ) ,

C 8 = 8 C 1 6 C 2 2 C 4 ( sinh ( C 1 h ) ) 3 + 2 C 6 C 1 2 h sinh ( C 1 h ) + 1 2 C 5 C 1 7 h 2 cosh ( 3 C 1 h ) − 13 2 C 5 C 1 6 h sinh ( C 1 h ) + 9 4 C 5 C 1 6 h sinh ( 3 C 1 h ) − 3 2 C 5 C 1 7 h 2 cosh ( C 1 h ) ,

C 9 = − 3 C 1 7 C 2 2 C 4 h cosh ( C 1 h ) − 6 C 1 7 C 2 2 C 4 sinh ( C 1 h ) + 9 4 C 1 6 C 2 2 C 4 sinh ( 3 C 1 h ) + 3 2 C 6 C 1 2 h sinh ( C 1 h ) − 1 2 C 6 C 1 3 h 2 cosh ( C 1 h ) + 39 16 C 5 C 1 6 h sinh ( 3 C 1 h ) − 11 8 C 5 C 1 7 h 2 cosh ( C 1 h ) − 5 32 C 5 C 1 5 cosh ( 3 C 1 h ) − 1 8 C 1 8 C 5 h 3 sinh ( C 1 h ) − 21 8 C 1 6 C 5 h sinh ( C 1 h ) + 9 16 C 1 6 C 5 h 2 cosh ( 3 C 1 h ) − 5 32 C 1 4 C 5 e ( − C 1 h ) ,

C 10 = − C 7 + β 1 C 8 − β 1 ( 1 + B ) C 9 2 ( 1 + B ) β 1 C 1 2 sinh ( C 1 h ) + 2 C 1 cosh ( C 1 h ) ,

P = 8 π 3 ϵ ( − ( E 1 + E 2 ) cos ( 2 π ( x − t ) ) + 1 2 π E 3 sin ( 2 π ( x − t ) ) .

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Received: 2023-11-29
Revised: 2024-05-30
Accepted: 2024-06-30
Published Online: 2024-10-25

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2024-0026
Keywords for this article
magnetohydrodynamic; non-Newtonian fluid; long-wavelength; variable viscosity; Eyring Powell fluid
Creative Commons
BY 4.0

Articles in the same Issue

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
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  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
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