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Nanofluids have improved thermophysical properties compared to conventional fluids, which makes them promising successors in fluid technology. The use of nanofluids enables optimal thermal efficiency to be achieved by introducing a minimal concentration of nanoparticles that are stably suspended in conventional fluids. The use of nanofluids in technology and industry is steadily increasing due to their effective implementation. The improved thermophysical properties of nanofluids have a significant impact on their effectiveness in convection phenomena. The technology is not yet complete at this point; binary and ternary nanofluids are currently being used to improve the performance of conventional fluids. Therefore, this work aims to theoretically investigate the ternary nanofluid flow of a couple stress fluid in a vertical channel. A homogeneous suspension of alumina, cuprous oxide, and titania nanoparticles is formed by dispersing trihybridized nanoparticles in a base fluid (water). The effects of pressure gradient and viscous dissipation are also considered in the analysis. The classical ternary nanofluid model with couple stress was generalized using the fractal–fractional derivative (FFD) operator. The Crank–Nicolson technique helped to discretize the generalized model, which was then solved using computer tools. To investigate the properties of the fluid flow and the distribution of thermal energy in the fluid, numerical methods were used to calculate the solution, which was then plotted as a function of various physical factors. The graphical results show that at a volume fraction of 0.04 (corresponding to 4% of the base fluid), the heat transfer rate of the ternary nanofluid flow increases significantly compared to the binary and unary nanofluid flows. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the entropy production characteristics of two non-miscible fluids in an inclined porous channel with temperature-dependent thermal conductivity and viscosity. The porous region of the channel is divided in two regions. In region-1 and region-2, the Couple stress and Newtonian fluid take place due to constant pressure gradient, respectively, under the influence of a uniform magnetic field. Here, the Darcy–Brinkman model is used for the flow of immiscible fluid through the porous media. In this work, we used a semi-analytical method named as homotopy analysis method (HAM) to solve the coupled nonlinear ordinary differential equations. The goal of the considered problem is to examine the consequences of a variety of thermophysical parameters, including Hartmann number, varying viscosity parameter, varying thermal conductivity parameters, and Grashof number on the characteristics of entropy generation, Bejan number distribution, thermal behavior and flow characteristics of non-miscible couple stress and Newtonian fluid passing through a porous channel. The novel aspect of this study is the formation of entropy and Bejan number as a result of non-miscible Newtonian and couple stress fluids with varying thermal conductivity and viscosity in porous media. In terms of rheological investigation, a semi-analytical simulation for changeable thermal and flow properties in an immiscible Newtonian and couple stress fluid via an inclined porous channel is a brand-new concept, and the behaviors of such flows have not been examined yet. From this study, it is concluded that on raising the variable thermal conductivity, Hartmann number and the permeability of the porous medium, the flow velocity, thermal characteristics and entropy generation number decrease. The authors also come to the significant conclusion that non-miscible Newtonian and couple stress fluids have larger entropy production numbers, flow velocities, and temperature profiles for higher values of Grashof number, variable viscosity parameter, and couple stress parameter. The findings of this work have also been graphically validated through the previously established work. The results of the present analysis can be used in petroleum industry, lubrication theory, etc. [ABSTRACT FROM AUTHOR]

The purpose of this proposed research is to examine the two-dimensional couple stress fluid flow over an extensible cylinder. The flow rate was determined by convective boundary layer constraints and the presence of a magnetic field. A nonlinear convective expression is used to study the heat transfer process in the vicinity of the cylindrical surface, which has widespread applications in engineering and industrial sectors. Incorporating a thermal radiation source into the heat transfer process increases the effect of dissipative heat. The behavior of the flow is determined by its mathematical structure, which is then translated into ordinary differential equations by making suitable assumptions about similarity variables and stream function. The findings indicate that as the dimensionless couple stress parameter increases, fluid movement intensifies. Conversely, an increase in the Hartmann number (M) leads to a decrease in velocity. Additionally, varying the curvature parameter results in higher values for both temperature and fluid velocity profiles. The coefficient of skin friction rises with the curvature parameter but decreases with the Grashof number. Furthermore, the heat transfer rate increases with higher curvature and decreases with the Grashof number, respectively. The present study includes a comparison with existing research to reinforce the proposed model. [ABSTRACT FROM AUTHOR]

The present study investigates the dynamics of a biomagnetic couple stress fluid within an anisotropic porous channel where the channel walls are stretchable. This study examines the flow behavior under the influence of an external magnetic field generated by a magnetic dipole. Appropriate dimensionless parameters are introduced to simplify the equations of the problem. A suitable numerical approach based on the Spectral Quasi-Linearization Method is utilized to obtain a solution to the problem. In this work, influence of several important parameters like the anisotropic permeability ratio, couple stress parameter, anisotropic angle, Darcy number, ferromagnetic interaction parameter, Reynolds number, and Prandtl number are examined. The results indicate that ferromagnetic interaction parameter and couple stress parameter significantly impact heat transfer and fluid flow. Permeability ratio and angle also affect the flow dynamics. Furthermore, the coefficient of skin friction and rate of heat transfer were examined, varying the couple stress and ferromagnetic interaction parameters. The findings demonstrate that an existence of magnetic dipole and anisotropic permeability significantly influences the flow and thermal properties of ferrofluids, providing valuable insights for optimizing heat transfer and controlling fluid flow in diverse engineering and medical applications. [ABSTRACT FROM AUTHOR]

In this study, we explore the analysis of peristaltic flow with heat transfer occurring within the gap between coaxial inclined tubes. The inner tube's wall is rigid, while the outer tube's wall features a sinusoidal wave propagating through it. The cylindrical system is employed to formulate the problem. The flow is characterized using continuity, momentum, and energy equations. We apply the assumption of long wavelength and the low Reynolds number approximation to simplify the nonlinear governing equation, subsequently solving it through perturbation techniques. We investigate the impact of crucial parameters, such as the magnetic field, porous media, slipping conditions, and others, on the peristaltic flow of a couple stress fluid. Our focus lies on assessing their influence on axial velocity, pressure gradient, and flow streamlines. The outcomes are visually presented through graphical representations. Notably, an increase in the slipping parameter results in a reduction of fluid velocity, attributed to the reverse slipping of the flow. The introduction of a magnetic field leads to an augmentation of the pressure gradient. Moreover, elevating the peristaltic amplitude and heat source induces the formation of a vortex within the flow. The presence of porous media leads to an increase in the pressure difference of the fluid flow. The primary objective of this research is to enhance our understanding of the peristaltic motion of non-Newtonian fluid dynamics, specifically incorporating a couple stress fluid. This contributes to a deeper understanding of crucial fluids, such as blood, within the human circulatory system. The implications extend to biological and industrial applications like magnetic resonance imaging (MRI) and radiosurgery, advancing our scholarly understanding of fluid behavior, especially in non-Newtonian scenarios. [ABSTRACT FROM AUTHOR]

Ternary hybrid nanoliquid is employed in various engineering and science applications including electronic heaters, pharmaceutical, nuclear safety, and solar energy production. Hence, the current study focused on the thermal and entropy analysis of water-based ternary couple stress hybrid nanoliquid and couple stress hybrid nanoliquid flow in an oblique channel. The significance of heat flux, porous media, buoyant force, magnetic field, and convective condition on the couple stress ternary liquid has been scrutinized. The governing partial differential equations are transferred into ordinary differential equations with the help of dimensionless terms. In the next step, the ordinary differential equations were solved by utilizing the RKF-45 numerical method. The outturn elucidated that the inflated Brinkman number greatly promotes the thermal field. The flow profile declines with enhanced permeability parameters. The outcome of the scrutiny of the couple stress parameter concludes that highly reinforced by ternary fluid than hybrid couple stress nanofluid. [ABSTRACT FROM AUTHOR]

The present flow problem analyzes the impact of radiative heat and mass transfer with inclined magnetic field on thermal exchange and entropy production of two immiscible nature of electrically conducting couple stress fluid in a static porous saturated conduit. In this model, the lower and upper porous regions of the rectangular channel are occupied by two different types of couple stress fluids. The static horizontal parallel plates of the porous duct are completely isothermal and the flow of immiscible fluid through the porous duct develops because of a constant pressure gradient at the entry zone of the duct. The Brinkman model is utilized in the modeling of fluid flow through porous saturated domain and Rosseland's approximation is utilized to compute the radiative thermal exchange effect on nonmiscible couple stress fluid. In this work, authors have analyzed the effect of various thermo-physical parameters such as the Hartmann number (Ha), permeability parameter (Da), Schmidt number (Sc), Soret number (Sr) and couple stress parameter ( s i , i = 1 , 2) on the entropy generation characteristics, Bejan number distribution, thermal behavior, concentration distribution and flow characteristics of immiscible couple stress fluid which passes through the porous channel. The parameters Ha, Da, s i , Sc, Sr correspond to magnetic field effect, permeability of porous media, couple stress, mass diffusion and thermal diffusion, respectively. The most significant findings of this research work are as follows: • In a porous saturated channel, the immiscible couple stress fluid velocity, entropy production number and thermal profile get enhanced on increasing the couple stress parameter. • On increasing the Hartmann number and decreasing the permeability of porous region, the thermal properties and entropy production number both decrease. • The couple stress fluid's concentration field and Bejan number distribution get decreased on enhancing Soret number Sr and Schmidt number Sc. • The entropy generation near the wall of the channel rapidly increases on increasing the Schmidt number and Soret number. The emerging finding of this research work exhibits excellent agreement with previously published work. This research work can be utilized in food processing, petroleum products and chemical process. [ABSTRACT FROM AUTHOR]

The objective of current study is to analyze the magnetohydrodynamic (MHD) natural convection flow of couple-stress fluid and entropy generation (second law analysis) inside the cavity with heated fin by using the Galerkin finite element method. The heated fin has been placed in middle of the heated bottom wall, the temperature T c is fixed on the vertical walls and an insulated situation is considered at the top wall. The irreversibility ratio and heat flux inside the enclosure are examined through entropy generation and Bejan heatlines, respectively. The obtained results show that the intensity of circulations of streamlines decrease from 3.9744 to 0.03148 and heatlines from 7 to 4.5 due to increasing couple-stress parameter (K) from 0 to 1.5. It is noted that the heatlines bowls appear due to enhanced buoyancy-driven flow against high Rayleigh number. The reverse relation is noted between the entropy generation and the Hartmann number. The overall heat transfer rate along with the bottom and vertical side walls increase 111.65 % and 75.8297 % , respectively, with increasing the fin height from 0 to 0.25. Obtained results could help to optimize heat transfer in engineering systems like heat exchangers, cooling processes of devices, chemical reactors, and solar collectors. [ABSTRACT FROM AUTHOR]

The present study aims to perform a comparative analysis of the plane Couette flow of a couple stress fluid under the influence of magnetohydrodynamics (MHD) using two different methods: the Optimal Auxiliary Function Method (OAFM) and the Homotopy Perturbation Method (HPM). The couple stress fluid is known for its non-Newtonian behavior, where the fluid's response to shear is influenced by the presence of internal microstructure. The OAFM and HPM are utlized to solve the governing equations of the couple stress fluid flow under MHD. The OAFM is a numerical technique that involves introducing an auxiliary function to simplify the equations, leading to an easier solution procedure. On the other hand, HPM is an analytical method that employs a series solution. The comparative analysis focuses on examining the accuracy, efficiency, and convergence behavior of the two methods. Various flow parameters such as the couple stress parameter, the magnetic parameter, and the velocity ratio are considered to investigate their influence on the flow behavior. Furthermore the HPM solution was compared with the OAFM solution using different graphs and tables. It reveals that the solution obtained by HPM is batter than OAFM solution. [ABSTRACT FROM AUTHOR]

In this paper, the Poiseuille flow of non-isothermal couple stress fluid of Reynolds model between two heated parallel inclined plates has been studied. The strongly nonlinear coupled ordinary differential equations have been investigated using the Asymptotic Homotopy Perturbation method (AHPM) and the Optimal Homotopy Asymptotic Method with DJ Polynomials (OHAM-DJ). Employing both these techniques, the approximate solutions for velocity profile, temperature distributions, shear stress, average velocity and volumetric flow rate have been achieved. The results obtained using both these methods are in the form of infinite series, hence the results can be obtained easily. Comparison of various results, obtained through these methods, is carried out. Here, it is worth to mention that the results gained by both these approaches are in excellent resemblance. The effect of different parameters on the flow problem has been studied numerically as well as presented graphically. [ABSTRACT FROM AUTHOR]

Peristaltic transport of couple stress fluid with heat transfer is investigated through flexible walls of the channel furnished with hair‐like structures. Locomotion of biological fluid is the result of the simultaneous propagation of metachronal waves (MCWs) and peristaltic waves. MCW emerges due to multi‐movement of cilia, while the elastic‐walls of the channel are responsible for the peristaltic wave. The two‐dimensional rheological flow experiences the resistive behavior of applied magnetic fields, besides, the role of viscous dissipation and entropy generation. A closed‐form solution is achieved for the nonlinear coupled differential equation, which is vetted through graphs. Finally, the theoretical analysis introduces the manufacture of a ciliate bronchial tube that can efficiently; remove mucus out of the lungs and bloodstream. [ABSTRACT FROM AUTHOR]

AbstractThe study reports an application of homotopy perturbation method to investigate entropy generation in the steady micropolar couple stress fluidics inside a vertical porous medium channel. The channel is composed of two isothermal parallel walls maintained at different temperatures. The governing equations are framed in the Cartesian system. The boundary value problem depicting the thermofluidic configuration is treated by the homotopy perturbation method (HPM) and for its validation, the 4th-order Runge-Kutta scheme is also invoked. The quantities of interest, viz. skin friction, wall couple stress, Nusselt number computed by HPM, and 4th order R-K method are found to be in good agreement. After getting the velocity, temperature, and microrotation regime, the entropy involved with the thermodynamic process was modeled to facilitate future thermodynamic optimizations. [ABSTRACT FROM AUTHOR]

One significance of the fluid flow in a curved channel is the generation of centrifugal forces. However, nothing is known on the hydrodynamic study of a couple stress fluid flowing in a porous curved channel. The proposed work, concerns with the flow of couple stress fluid in a porous curved channel and analysing the generating centrifugal force. Here, the flow of fluid in the porous curved channel takes place due to the constant pressure gradient. In this work, the effect of the radius of curvature and permeability parameters are examined on the flow velocity and temperature profiles of the considered fluid. The aim of this study is to analyse the flow pattern of the couple stress fluid, measure the rate of heat transfer, and volumetric flow rate, when fluid flow through porous curved channel. The proposed model is useful in the biofluid mechanics, nuclear reactors, and in oil and chemical industries. The present model, i.e., the flow of couple stress fluid in a porous curved channel, governs by the highly non-linear coupled differential equations which are solved by using Homotopy Analysis Method (HAM). Through this work, it is concluded that couple stress fluid's velocity and temperature is smaller in magnitude than that of the Newtonian fluid. Further, it is noticed that the fluid's flow velocity and temperature in porous curved channel is reduced significantly on increasing the curvature parameter, permeability parameter, and on reducing the couple stress parameter. [ABSTRACT FROM AUTHOR]

We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions. [ABSTRACT FROM AUTHOR]

This study assesses the dynamics of electromagnetohydrodynamic (EMHD) flow of couple stress fluid through a circular cylinder. The flow is assumed to be driven by electromagnetic force and applied pressure gradient. A comprehensive theoretical framework is developed to solve the Poisson–Boltzmann equation (under the Debye–Hückel approximation) for the electric potential within the electric double layer and the momentum equation for the fluid flow under suitable boundary conditions. The analytical expressions are obtained for velocity and volume flow rate. It is observed that the present results of the couple stress fluid (CSF) model strongly match with those reported in the literature for a Newtonian fluid. The outcome of our analysis reveals that the velocity accelerates with an increase in couple stress and electric field parameters, while it decreases with Hartmann number in the absence of lateral electric field. The study finds major applications in chemical processing and mixing, development of biochips for drug delivery and biomedical engineering. [ABSTRACT FROM AUTHOR]

Linear and weakly nonlinear stability analyses of thermosolutal convection in a couple‐stress fluid with effects of helical force and rotation are performed. The governing nondimensional equations are solved using the normal modes. We have shown the effect of the helical force parameter, solutal Rayleigh number, Couple stress parameter, Lewis number, Taylor number, and Prandtl number on stationary and oscillatory convection regions and presented graphically. Solutal Rayleigh number, Couple stress parameter, Lewis number, and Taylor number have a stabilizing effect on the system whereas the helical force parameter has a destabilizing effect on the system. To study heat transport by convection we have derived the Ginzburg–Landau equation. [ABSTRACT FROM AUTHOR]

To examine the performance of the rough-porous Rayleigh step slider bearing lubricated with couple stress fluid considering the effect of MHD forms the crux of this article. The bearing surface roughness is assumed to be longitudinal as well as transversal. The expression for longitudinal and transversal roughness are derived by using the stochastic random variable. The modified Darcy's law is used to derive the pressure in the porous region, The space between the slider filled with couple stress fluid. The stochastic Reynolds equation is derived applying the Christensen's stochastic approach. Following this the expression for load carrying capacity, fictional force and coefficient of friction are derived. It is observed that work load and frictional force increases when compared with plate without roughness and porous and decreases for coefficient of friction. [ABSTRACT FROM AUTHOR]

In this work, authors have studied peristaltic flow of immiscible type of couple stress and Newtonian fluid via an asymmetric horizontal channel filled with porous material by taking slippage into account at the channel wall which was ignored in previous published literature under the influence of thermal radiation and magnetic field. The study of peristaltic flow of considered immiscible fluids will play an important role in maximizing system efficiency, reducing energy losses, and improving performance of various fluid transport. In the present problem, the linearized form of governing equations has been solved by direct method with the use of Mathematica 14 software to obtained the closed form solutions of various flow variables. In this work, the impacts of various influencing factors on the model's flow, thermal, and concentration properties have been analyzed graphically. The work presented here leads to the significant result that large Schmidt number values in a porous filled duct reduces the mass diffusivity. Another key finding of this work is that the formation of entropy is reduced when low permeability porous material is used. Our research concludes that entropy creation in the asymmetric porous channel is decreased when the slip length increases on the channel's static borders. • We examine entropy generation, and concentration in peristaltic flow of immiscible fluids in a porous channel. • We obtain the closed form solutions for flow velocity, temeperature and concentration. • The concentration of the immiscible fluid diminishes as the Soret numbers increase. • The entropy generation number in the porous material filled channel can be reduced by raising the slip parameter. • This model is useful in thermal fusion reactors, cooling systems, and petroleum products. [ABSTRACT FROM AUTHOR]

In the article, we aim to investigate the well-posedness of solution and the regularity of the global attractor for the couple stress fluid in saturated porous media with the local thermal non-equilibrium effect. To be more specific, we firstly show the existence and uniqueness of global weak solution to the model by making use of the standard Galerkin method. Second, relying on verifying the uniformly compact condition required, we prove the existence of the global attractor of the model in the space where the weak solution resides. Finally, we improve the regularity of global attractor by uniformly compact condition and obtain the C ∞ attractor for the model. [ABSTRACT FROM AUTHOR]

In recent years, a great deal of interest has been generated in modern micro‐ and nanotechnologies for micro/nano‐electronic devices. These technologies are increasingly utilizing sophisticated fluid media to enhance performance. Among the new trends is the simultaneous adoption of nanofluids and biological micro‐organisms. Motivated by bio‐nanofluid vertical channel oxygenators in medical engineering, in the current work, a mathematical model is developed to examine the flow of mixed convective couple‐stress nanofluids in a vertical channel with a transverse magnetic field, fluid viscosity that changes with temperature, and thermal conductivity. The non‐Newtonian model follows Brownian motion and heat spread by nanoparticles in a fluid under coupled stress. Highly linked, nonlinear regulating equations are translated into nondimensional equations using relevant variables. The governing equations are then turned into a form with no dimensions. The Keller‐box technique, a second‐order finite difference method for solving second‐order equations, is used to solve them numerically. On the other hand, the effects of different non‐Newtonian flow parameters, such as the couple stress fluid parameter, the magnetic parameter, the variable fluid viscosity, the variable thermal conductivity parameters, the Brinkman number, the nanofluid and buoyancy parameters, and the rate of chemical reaction parameter, are carefully studied. The velocity, temperature, and concentration fields are calculated over a wide range of possible values for the relevant parameters. [ABSTRACT FROM AUTHOR]

In this article, the exact solution for a couple stress fluid flow past a fluid sphere filled with a couple stress fluid is considered using interfacial slip on the boundary. The velocity is expressed with regard to the stream function. The external velocity and internal velocity are alongside the drag coefficient. It was noticed that with the increases in slip parameter, the drag coefficient is decreasing and the inflow pattern has more circulations with more area appearing near the poles and internal flow disappears gradually. The special cases of viscous fluid, no-slip condition, and solid sphere are evaluated. A good understanding of the current and literature outcomes have been obtained, including the special case of a viscous fluid past a couple stress fluid sphere. [ABSTRACT FROM AUTHOR]

This study investigates the thermal characteristics and irreversibility analysis of the combined electro-osmotic and electromagnetic flow of a couple stress fluid through a microfluidic channel with constant wall heat flux boundary condition. The analytical solutions for electromagnetohydrodynamic (EMHD) flow velocity, temperature field and entropy production distribution are obtained using the physically relevant boundary conditions. Results show the effects of several factors, including couple stress parameter, Hartmann number, lateral/transverse electric field parameter and Brinkman number on the temperature transport and entropy generation, particularly, the details of how the temperature, Nusselt number and entropy generation vary with these parameters. The results demonstrate that the entropy generation in the channel is strongly dependent on the flow velocity and thermal energy distribution. In this analysis, the primary focus is placed on the impact of the couple stress parameter on EMHD flow velocity, heat transfer and entropy generation. The results reveal that when the couple stress physical parameter increases, the magnitude of temperature and Bejan number decreases. However, the Nusselt number and entropy production rates have opposite effects towards the channel walls. The present research has enormous practical significance and can design efficient thermal micro/nanoequipment. [ABSTRACT FROM AUTHOR]

This study provides the exact solutions of couple stress fluid (CSF) in a channel. The CSF is bounded by two plates in which the lower plate is moving with constant velocity Uo and the upper plate is fixed. The influence of the external pressure gradient is considered on the CSF fluid. To transform the classical CSF model, we have introduced three approaches to fractional derivatives, (a) Caputo (b) Caputo–Fabrizio (CF), and (c) Atangana–Baleanu (AB) fractional definitions. Exact solutions have been obtained using the Laplace and finite Fourier sine transforms jointly. Furthermore, the effect of different fractional derivatives is compared, and the results are shown in graphs. Moreover, parameters that affect the CSF motion are discussed in the graphs using the computational software MATHCAD. The most important outcome of the given study is the comparison of Caputo, CF, and AB fractional models with the classical CSF model. From the comparison, it can be noticed that AB fractional model discusses the dynamics of the CSF with a good memory effect as compared to Caputo and CF fractional operators. The CSF model can be reduced to Newtonian fluid as a limiting case and also investigated solutions in the absence of external pressure. Finally, Skin friction is evaluated for lower and upper plates, and the obtained results presented in tabular form. [ABSTRACT FROM AUTHOR]

This study examined the relationship between stresses and couple stresses within the context of the couple stress fluid theory. The application of the effective medium approach, based on the Brinkman equation, is also discussed under the effect of slip and spin slip on the surface of the solid sphere immersed in porous couple stress fluid. The motion is generated by moving a solid sphere at a constant speed. Using Stokesian assumptions, the equation of motion ignores nonlinear terms. The surface conditions for slip and spin slip of couple stress fluids have been applied. Tables and graphs are used to represent the normalized drag force exerted by the fluid flow on the solid sphere. According to the numerical results, the drag force increases monotonically with permeability and couple stress. As a result, the results obtained are consistent with those found in the literature for viscous fluids, and the special cases are determined. [ABSTRACT FROM AUTHOR]

The present level of literature on the subject matter indicates that nothing is known on the heat transfer across the couple stress rheological fluid flowing over a vertical avenue with Robin (mixed) wall conditions. The obtained conservation equations of the model are solved through DTM (differential transform method) and RPM (regular perturbation method). The nondimensional parameters obtained are a couple stress parameter, Brinkman number, mixed convection parameter, and Biot number. The computations reveal that flow acceleration and thermal enhancement is induced with increasing mixed convection parameter and Brinkman number for symmetric and asymmetric conditions. Increasing couple stress parameters dwindle the velocity and temperature for symmetric and asymmetric cases. The large values of the mixed convection parameter and Brinkman number increase the Nusselt values at the left wall and reduces at the right wall. The mass flow rate is augmented with the mixed convection parameter and Brinkman number but it is reduced with the couple stress parameter. The DTM, RKSM, and RPM solutions are in good agreement. [ABSTRACT FROM AUTHOR]

• Creeping sinusoidal flowing and diffusion of solute material in incompressible couple-stress fluid are examined. • Both homogeneity and heterogeneity chemical interactions are taken into considered. • Surface properties, long wavelength hypothesis and dynamic periphery situations are elaborated. • Mean active diffusion factor is improved with wall features and amplitude ratio. • Suggesting peristalsis is promoted dispersion. A mathematical model has been constructed to examine the creeping sinusoidal flowing and diffusion of a solute material of an incompressible couple-stress fluid across a porous medium with wall features in the existence of heterogeneous-homogeneous chemical reaction. The mean effective diffusion factor is calculated using the long wavelength hypothesis, Taylor's limit criteria, and dynamic periphery restrictions. The graphs have been used to study the effects of important constraints on the mean effective diffusion coefficient. The mean effective diffusion factor is found to rise with wall characteristics and amplitude ratio, indicating that peristalsis causes an increase in scattering. Additionally, scattering increased with the permeability constraint but decreased with the couple-stress constraint, homogeneous reaction rates, and heterogeneous reaction rates. [ABSTRACT FROM AUTHOR]

The impact of non-uniform temperature gradients on Rayleigh-Darcy convection in a composite system of couple stress fluid is discussed. The composite system is bounded by stress-free surfaces and adiabatically insulated, and the fluid-porous layers are coupled by employing appropriate interfacial boundary conditions. To determine the eigen value, the regular perturbation method is used. The effect of dimensionless parameters on Rayleigh-Darcy convection is analysed graphically, and it is demonstrated that the couple stress parameter and couple stress viscosity ratio stabilise the system, while the opposite effect is observed for the Darcy number and thermal diffusivity ratio. [ABSTRACT FROM AUTHOR]

The influence of linear, parabolic and inverted parabolic temperature gradients on the onset of double-diffusive Rayleigh-Darcy convection is theoretically investigated. The composite system is constrained horizontally by adiabatic and free-free thermal boundaries, and appropriate interfacial boundary conditions are used to connect fluid-porous layers. The regular perturbation approach is used to determine the critical Rayleigh number expression for different temperature gradients. Graphs are used to investigate the significance of a variety of dimensionless characteristics. The couple stress parameter, couple stress viscosity ratio, solute Rayleigh number, and solute diffusivity ratio clearly have a stabilizing effect on the system, whereas the Darcy number and thermal diffusivity ratio destabilize it. [ABSTRACT FROM AUTHOR]

This paper deals with the effect of three‐dimensional magnetohydrodynamic flow for a couple of stress fluids on an exponentially stretching sheet. The magnetic field is implemented normally to the surface. To observe the transfer of heat phenomenon, the Cattaneo–Christov flux model of heat is employed. Using similarity transformation, the substantial differential equations are reformed into ordinary differential equations. Eventually, the effects of different physical parameters are studied graphically. The drawback in Fourier heat flux model is removed by adding a new paramter known as thermal relaxation time by Cattaneo. This perimeter allows heat transportation by way of propagation of waves thermally at a defined speed. After this, the Cattaneo law is further modified by Christov–Christov to replace the ordinary derivative along Oldroyd's upper‐convective derivative. [ABSTRACT FROM AUTHOR]

Peristalsis is the most dynamic phenomenon that is significant in the biomedical domain and offers considerable promise in microscale fluids. For the past few years, this biomimetic (peristaltic) phenomenon attracted the attention of the research community because of its massive applications in numerous medical and industrial domains. In the present study, the steady, laminar rheology of biological fluid from a biomimetic (peristaltic) channel is considered. The rheological equations are expressed in the Cartesian system, and the porosity effects are modelled (added) on a body force term of the momentum equation. The current analysis depends on the creeping phenomena and long wavelength. The closed-form solutions are acquired by using integration on the rheological equations subject to suitable boundary conditions. The rheology is controlled by two embedded parameters, the porosity and non-Newtonian parameters. It is shown using graphs that the magnitude of axial velocity is strongly affected by the larger strength of porosity (Darcy's number) and non-Newtonian (couple stress) parameters. We noticed the impacts of involved rheological constraints on pumping and trapping phenomena in the light of porous effects. Additionally, the wavy pattern of sinusoidal waves is utilised in the present analysis to increase the efficiency of the peristaltic pump. By increasing the porosity impacts, the magnitudes of velocity profile and pressure gradient of a biofluid are increased. The porosity has a dynamic role in the augmentation of peristaltic pumping. The impacts of couple stress parameter reduces the viscous effects. These outcomes may be useful in bioengineering (drug delivery schemes) and chemical processes. [ABSTRACT FROM AUTHOR]

This paper investigates the electric properties of gold nanoparticles mixed with a convection dielectric couple stress fluid inside a vertical cylindrical tube with moving endoscope in the presence of Hall currents and thermal radiation. Under the long wavelength approximation and the use of appropriate conversion relationships between fixed and moving frame coordinates, the exact solutions have been evaluated for temperature distribution, gold nanoparticles concentration, electrical potential function and nanofluid pressure, while analytical solution is found for the axial velocity using the homotopy analysis method. The results show that the presence of the electric field enhances the effects of Brownian motion parameter, thermophoresis parameter, radiation parameter, Hall currents and wave amplitude ratio on the axial nanofluid velocity, while it was found that its presence reduces the effects of couple stress parameter, thermophoresis diffusion constant and Brownian diffusion constant. [ABSTRACT FROM AUTHOR]

This paper concerns the analytical investigation of the axisymmetric and steady flow of incompressible couple stress fluid through a rigid sphere embedded in a porous medium. In the porous region, the flow field is governed by Brinkman's equation. Here we consider uniform flow at a distance from the sphere. The boundary conditions applied on the surface of the sphere are the slip condition and zero couple stress. Analytical solution of the problem in the terms of stream function is presented by modified Bessel functions. The drag experienced by an incompressible couple stress fluid on the sphere within the porous medium is calculated. The effects of the slip parameter, the couple stress parameter, and permeability on the drag are represented graphically. Special cases of viscous flow through a sphere are obtained and the results are compared with earlier published results. [ABSTRACT FROM AUTHOR]

This research aims to utilize the local thermal non-equilibrium (LTNE) model to discuss the heat transfer within a thermally developing region. This study delves into the effects of magnetohydrodynamics couple-stress fluid flow in a channel filled with porous medium within the framework of the LTNE model with consideration for axial conduction. The flow within this system is unidirectional, and the walls of the channel sustain a constant heat flux. Numerical solutions are obtained using the finite difference method. The computational and numerical findings are visually depicted through graphical representations. Our analysis concludes that as the couple stress parameter increases, the local Nusselt number decreases. Conversely, the Nusselt number rises with the Hartmann and Peclet numbers increase. Thermally fully developed condition is satisfied under the LTNE mode. Applications like heat exchangers or pipelines carrying heated fluids require careful consideration of thermal expansion and material stress due to their significant impact. Comprehending the impacts of non-equilibrium heat transfer can enhance the efficiency and effectiveness of heat exchanger designs. The present findings are validated by the existing literature's computational and experimental investigations. • Examine the effect of MHD couple-stress fluid in the thermal field forming under local thermal non-equilibrium (LTNE) with consideration for axial conduction. • A finite difference approach has been implemented to solve the coupled differential equation. • Hydrodynamics and convective heat transfer analysis have been discussed in this study. • Dimensionless temperature based on bulk mean temperature is discussed. • Fully developed conditions for the thermal field are satisfied under the LTNE model. • 3D visualization of temperature profiles is shown for the axial conduction effects. • As the couple stress parameter rises, the local Nusselt number decreases, while the Nusselt number increases with the Hartmann and Peclet numbers. • The present findings are validated by the existing literature's computational and experimental investigations. [ABSTRACT FROM AUTHOR]

Numerous disciplines, including fluid dynamics, heat transfer, geophysics, materials science, and environmental engineering, use Rayleigh numbers, both thermal and solute, extensively. They are crucial resources for foreseeing and comprehending fluid convective processes and their effects on many physical systems. In this analysis, we first derive the governing equations that describe the evolution of density perturbations in a stratified fluid system with a salinity gradient. Linear stability analysis is then applied to these equations to identify the conditions under which salt finger instability can occur. For the Rayleigh-Bénard convection of a chemically reactive pair stress fluid with rotation acting in the vertical direction, the non-autonomous Ginzburg-Landau equations are derived. For determining the amplitude for Ginzburg-Landau equations, the Lorenz model is introduced. Many physical parameters are taken into consideration when drawing the stationary and oscillatory curves. Heat transport is measured using the Nusselt number and Sherwood number, which are determined as a function of the lengthy time scale. The study provokes in understanding and predicting Rayleigh-Bénard convection contribute not only to our knowledge of ocean dynamics but also to our ability to address climate-related challenges and sustain marine ecosystems. [ABSTRACT FROM AUTHOR]

The heat transfer of the combined magnetohydrodynamic (MHD) and electroosmotic flow (EOF) of non-Newtonian fluid in a rotating microchannel is analyzed. A couple stress fluid model is scrutinized to simulate the rheological characteristics of the fluid. The exact solution for the energy transport equation is achieved. Subsequently, this solution is utilized to obtain the flow velocity and volume flow rates within the flow domain under appropriate boundary conditions. The obtained analytical solution results are compared with the previous data in the literature, and good agreement is obtained. A detailed parametric study of the effects of several factors, e.g., the rotational Reynolds number, the Joule heating parameter, the couple stress parameter, the Hartmann number, and the buoyancy parameter, on the flow velocities and temperature is explored. It is unveiled that the elevation in a couple stress parameter enhances the EOF velocity in the axial direction. [ABSTRACT FROM AUTHOR]

This article addresses the influence of double-diffusivity convection in nanofluids in relation to peristaltic flow of magneto couple stress fluid in a nonuniform channel. Firstly, mathematical formulations of magneto couple stress fluid in the presence of double-diffusivity convection in nanofluids for 2-dimensional and 2-directional flows are described in detail. The highly nonlinear equations are simplified using approximation of long wavelength and low but finite Reynolds number. Exact solutions of stream function, axial induced magnetic field, axial pressure gradient, nanoparticle volume fraction temperature and concentration are computed. Graphical representations of numerous physical parameters of interest are presented to explain the conduct of flow quantities. It is observed from graphical plots that temperature profile and solutal concentration increases by increasing values of thermophoresis, Brownian motion, Soret and Dufour parameters. [ABSTRACT FROM AUTHOR]

The present paper addresses magnetohydrodynamic pulsating ow and heat transfer of two immiscible, incompressible, and conducting couple stress uids between two permeable beds. The ow between the permeable beds is assumed to be governed by Stokes' [28] couple stress uid ow equations, whereas the dynamics of permeable beds is determined by Darcy's law. In this study, matching conditions were used at the uid{ uid interface, whereas the B-J slip boundary condition was employed at the uid{porous interface. The governing equations were solved analytically, and the expressions for velocity, temperature, mass ux, skin friction, and rate of heat transfer were obtained. The analytical expressions were numerically evaluated, and the results are presented through graphs and tables. [ABSTRACT FROM AUTHOR]

Being motivated from the recent developments in biomicrofluidics, a mathematical model is presented to analyze the two-layered electrothermal flow via peristaltic propulsion of couple stress fluids caused by velocity and thermal slip conditions. An asymmetric microchannel is considered for the flow regime with different zeta potential moving with wave velocity. Couple stress fluid has been taken for aqueous solution to represent the non-Newtonian characteristics of physiological fluids. A lubrication approach with Debye H u ¨ ckel linearization is taken to obtain the analytical solution. Furthermore, heat transfer analysis is performed to analyze the thermal characteristics and variations in Nusselt number in the presence of thermal radiation. The study shows that the temperature reduces with increasing the electrical double layer thickness, thermal radiation, couple stress parameter, and magnitude of Brinkman number, however, it enhances with increasing the Joule heating and thermal slip effects. The rate of heat transfer enhances with Joule heating effects, while the trend is reversed due to the presence of two-layered electroosmotic flow and couple stress effect. [ABSTRACT FROM AUTHOR]

The impact of lateral walls and partial slip with different waveforms on peristaltic pumping of couple stress fluid in a rectangular duct with different waveforms has been discussed in the current article. By means of a wave frame of reference the flow is explored travelling away from a fixed frame with velocity c. Peristaltic waves generated on horizontal surface walls of rectangular duct are considered using lubrication technique. Mathematical modelling of couple fluid for three-dimensional flow are first discussed in detail. Lubrication approaches are used to simplify the proposed problem. Exact solutions of pressure gradient, pressure rise, velocity and stream function have been calculated. Numerical and graphical descriptions are displayed to look at the behaviour of diverse emerging parameters. [ABSTRACT FROM AUTHOR]

The presented article addresses the electro-osmotic peristaltic flow of a couple stress fluid bounded in an inclined asymmetric microchannel. The viscous dissipation, Joule heating and chemical reaction effects are employed simultaneously in the flow analysis. Heat and mass transfer have been studied under large wavelength and small Reynolds number. The resulting nonlinear systems are solved numerically. The influence of various dominant physical parameters is discussed for velocity, temperature distribution, concentration distribution and the pumping characteristics. Electro kinetic flow of fluids by micro-pumping through micro channels and micro peristaltic transport has accelerated considerable concern in accelerated medical technology and several areas of biomedical engineering. Deeper clarification of the fluid dynamics of such flow requires the continuous need for more delicate mathematical models and numerical simulations, in parallel with laboratory investigations. [ABSTRACT FROM AUTHOR]

The present article deals with theoretical analysis on the peristaltic transport of a couple stress fluid with lubrication on the walls of the channel. The walls are coated with thin layer of power-law lubricant and the condition arised by the coating is modelled by taking the continuity of velocity and shear stresses of both the fluids at the fluid–fluid interface. The flow problem is discussed in two-dimensional asymmetric medium assuming long wavelength and low Reynolds number. Closed form solutions for velocity, stream function and flow rate are discussed. The numerical solutions for pressure rise, trapping and reflux are discussed. The effects of lubricant parameter, couple stress parameter, phase difference, wave amplitudes and flow rate are discussed for symmetric and asymmetric channels. It is observed that by increasing the lubrication effects, pressure rise increases in the pumping region and for trapping, boluses disappear and streamlines become parallel to the walls of the channel. From the given flow problem, the limiting cases of no-slip and viscous flow are presumed and found to be in excellent agreement with the existing literature. [ABSTRACT FROM AUTHOR]

Swimming of spermatozoa through couple stress fluid in an asymmetric human cervical canal is investigated in the present theoretical analysis. A couple of fourth-order partial differential equations arising from the mathematical modelling of the proposed model is solved analytically. Flow variables like pressure gradient, propulsive velocity, mucus velocity and time mean flow rate are analysed for the pertinent parameters. Conspicuous features of the pumping characteristics are explored. It is found that pressure rise facilitates the motion of spermatozoa to fertilize an ovum in the female reproductive tract, whereas pressure drop by inverting the direction of spermatozoa controls the probability of pregnancy. Maximal propulsive velocity of the spermatozoa is reported in the absence of travelling waves along the cervical walls. Minute impact of phase difference on propulsive velocity is evident. An analogy of the current analysis with the existing literature is also made. [ABSTRACT FROM AUTHOR]

The present work aims to examine the effects of viscous dissipation and unsteadiness parameters on nonlinear convective laminar boundary layer flow of micropolar‐couple stress nanofluid past a permeable stretching sheet with non‐Fourier heat flux model in the presence of suction/injection variable. The unsteadiness in the flow, temperature, and concentration profile is caused by the time‐dependence of the stretching velocity, surface temperature, and surface concentration of the boundary layer flow. Similarity transformation is applied to transform the time‐ dependent boundary layer flow equations into the corresponding highly nonlinear coupled ordinary differential equations with appropriate boundary conditions. The robust numerical technique called Galerkin finite element method is used to solve the obtained dimensionless governing equations of the flow. The effects of Eckert number, unsteadiness parameter, suction/injection parameter, mixed convection parameter, material parameter, Schmidt number, and couple stress parameter on linear velocity, angular velocity, temperature, concentration, local skin friction coefficient, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed with the help of graphical and tabular form. Under special conditions, the present result is compared with the existing literature and revealed good agreement. Our result shows that as unsteadiness parameter boost, both heat and mass transfer rate rises. The present study has a significant application in material processing technology. [ABSTRACT FROM AUTHOR]

This paper looks at the MHD natural convection flow through a vertical porous channel of couple stress fluid nanoparticles under convective heat conditions at the wall of the channel. Incorporated Soret and Dufour effects lead to strong coupled and highly nonlinear differential systems. To describe accurately all the physical phenomenon, the nanoparticle’s volume fraction is taken into account. The useful technical of the similarity transformations is used to produce an ordinary system which is numerically solved. Graphical illustrations containing non-dimensional velocity, temperature, concentration and concentration of nanoparticles are extensively presented for different values of various thermophysical parameters. Main quantities of interest as Nusselt and Sherwood numbers at both the walls are tabulated. [ABSTRACT FROM AUTHOR]

In this paper, two-dimensional non-Newtonian couple stress fluid flow over the upper horizontal surface of a paraboloid (uhsp) (shaped like a submarine or any aerodynamical automobile) is investigated. At the freestream, a stretching of the fluid layer is assumed along with catalytic surface reaction which tends to induce the flow in the fluid-saturated domain. The problem is modeled by engaging laws of conservation for mass, momentum, heat and concentration. Velocity components are converted to stream functions and similarity transformations to reduce the dependent and independent variables in the partial differential equation describing the flow. Stream functions ideally satisfy continuity equation and transformation to reduce the PDEs to the system of coupled nonlinear ODEs. The numerical solution of these equations is obtained using the shooting-RKF method. The graphical results show that both the lateral and horizontal velocities decrease by increasing the couple stress material parameter and cause the temperature to rise. The thermal boundary layer decreases subject to the thickness parameter and has appositive effects on concentration boundary layer. Finally, numerical results have also been tabulated. [ABSTRACT FROM AUTHOR]

The present article deals with the effects of nanoparticles and magnetic field on the peristaltic flow of a couple stress fluid in an asymmetric channel with different wave forms. Mathematical modelling for 2-D and two directional flows of a couple stress fluid along with nanofluid are first given and then simplified under the assumptions of long wavelength and low Reynolds number approximation. After invoking these approximations we get coupled non-linear differential equations. The exact solutions of temperature distribution, the nanoparticle concentration, velocity, stream function and pressure gradient are calculated. Finally graphical results of various physical parameters of interest are discussed to examine the behavior of flow quantities. [ABSTRACT FROM AUTHOR]

The fundamentals of electroosmosis are the most fascinating mechanism to induce flow of ionic solution under the effect of external electric field. The electroosmotic flow of the couple stress fluid in a microchannel with an active membrane pumping is analysed in the present model. The upper wall experiences membrane kinematics, generates the pressure to drive the fluid in the microchannel. Governing equations are formulated based on the mass and momentum conservation, and electric potential distribution of ions in diffuse layer. The suitable boundary conditions for the model have been considered. An analytical approach is employed to derive the solutions under the lubrication approximation, and the Debye–Hückel approximation. The effects of electric double layer thickness, couple stress parameter, and Helmholtz-Smoluchowski velocity on the velocity profile, pressure difference, net flow rate, induced streamline velocity, and local wall shear stress are computed for discussion of the physical significance of the model and its applications. Results reveal that electroosmosis mechanisms (axial electric field and electric double layer thickness) has remarkable role in modulating the fluids (Newtonian and non-Newtonian) flow driven by the rhythmic propulsion of membranes in microchannel. It is also worth mentioning that non-Newtonian fluids (couple stress fluids) need more external forces (electric force) to be pumped as compared to Newtonian fluids. Furthermore, it can be analysed that thinner electric double layers (EDLs) elevate wall shear stress, whereas stronger electric fields result in decreased wall shear stress. The results of the model help in designing smart membrane based electric pump for the use in various microscale transport phenomena of bio microfluidics systems. [Display omitted] • A mathematical analysis for electroosmotic flow of couple stress fluid is presented. • The flow geometry is considered single membrane fitted microchannel. • The fluid is driven by the rhythmic propagation of the membrane. • The effects of external electric field and Debye length are computed for pumping mechanisms. • The results of the present study will be applicable in manufacturing of various microfluidics systems for micro scale transport phenomena. [ABSTRACT FROM AUTHOR]

In this paper, the effect of Coriolis force is explored on convective instability of a doubly diffusive incompressible couple stress fluid layer with gravity acting downward. A linear stability analysis is used to obtain the conditions for the onset of stationary and oscillatory convection in the closed form. Being a multiparameter instability problem, results for some isolated cases have been presented to illustrate interesting corners of parameter space. It is found that the neutral curve for oscillatory onset forms a closed-loop which is separate from the neutral curve for stationary onset indicating the requirement of three critical thermal Rayleigh numbers to specify the linear instability criteria instead of the usual single value. Besides, the simultaneous presence of rotation and the addition of heavy solute to the bottom of the layer exhibit an intriguing possibility of destabilizing the system under certain conditions, in contrast to their stabilizing effect when they are present in isolation. The implication of couple stresses on each of the aforementioned anomalies is clearly brought out. The spatial wavelength of convective cells at the onset is also discussed. [ABSTRACT FROM AUTHOR]