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In the current paper, we investigate the effect of spin slip conditions for the Couette-Poiseuille flow of a couple stress fluid between two parallel plates. In the study, the motion of the fluid is considered to be steady, incompressible and unidirectional. At the interface between the fluid and the plates (both the upper and lower), we consider the non-zero tangential and couple stress spin slip relationships as boundary conditions. We present analytical expressions for the velocity profile, volume flow rate, vorticity, and couple stresses. This paper discusses the numerical influence of the spin slip parameter, velocity slip parameter, couple stress parameter, and pressure gradient on the velocity, vorticity, couple stresses, and volume flow rate. Our results show that the presence of the spin slip parameter reduces the velocity, couple stress and volume flow rate of the fluid, while it enhances the vorticity. The limiting cases for each problem align well with previously published results regarding the vanishing of couple stresses at the boundaries. This research helps both researchers and engineers in understanding how to control the conditions to achieve an efficient fluid flow, particularly in applications involving couple stress effects, such as microfluidics systems, lubrication technology, and polymeric suspensions. [ABSTRACT FROM AUTHOR]

The aim of the current study is to investigate the stability analysis in case of the linear as well as nonlinear of a thermo-solutal chemically reactive couple stress fluid under uniform magnetic field convection. Investigations have been conducted on the impact of chemical reaction and external vertical magnetic field on the commencement of double diffusive convection in couple stress fluid between infinite horizontal parallel plates. Darcy's modified law governs the flow in porous media and the Oberbeck-Boussinesq approximation is accurate. For modelling the momentum equation, the modified Darcy equation with the time derivative and inertia terms is utilized. Expressions for the Rayleigh numbers with finite amplitude, oscillatory, and stationary states are found in accordance with the regulating factors. Graphics are used to illustrate how the couple-stress parameter, solute Rayleigh number, Vadasz number and diffusivity ratio affect stationary, oscillatory and finite-amplitude convection. Stationary, oscillatory and finite-amplitude convection are found to be stabilized by the couple-stress parameter and the solute Rayleigh number. The normal mode analysis method is utilized to look into the linear stability of flow dynamics after the nonlinear mathematical problem has been linearized. When stationary and finite-amplitude modes are present, the diffusivity ratio has a destabilizing effect; when oscillatory convection is present, it has a dual effect. Oscillatory convection develops earlier when the Vadasz number is higher. The couple-stress parameter and diffusivity ratio both increase with increasing solute Rayleigh number values, but the heat and mass transfer decreases as these values rise. Using double Fourier series, a generalized weakly nonlinear stability analysis is done. The research illustrates how various regulating parameters support and destabilize the flow dynamics. The influence of finite-amplitude convection on stability is also examined. Furthermore, the best conditions for stationary and oscillatory convection are based on altering the couple stress flow stability by controlling the applied magnetic field. [ABSTRACT FROM AUTHOR]

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 aim of the current study is to investigate the stability analysis in case of the linear as well as nonlinear of a thermo-solutal chemically reactive couple stress fluid under uniform magnetic field convection. Investigations have been conducted on the impact of chemical reaction and external vertical magnetic field on the commencement of double diffusive convection in couple stress fluid between infinite horizontal parallel plates. Darcy's modified law governs the flow in porous media and the Oberbeck-Boussinesq approximation is accurate. For modelling the momentum equation, the modified Darcy equation with the time derivative and inertia terms is utilized. Expressions for the Rayleigh numbers with finite amplitude, oscillatory, and stationary states are found in accordance with the regulating factors. Graphics are used to illustrate how the couple-stress parameter, solute Rayleigh number, Vadasz number and diffusivity ratio affect stationary, oscillatory and finite-amplitude convection. Stationary, oscillatory and finite-amplitude convection are found to be stabilized by the couple-stress parameter and the solute Rayleigh number. The normal mode analysis method is utilized to look into the linear stability of flow dynamics after the nonlinear mathematical problem has been linearized. When stationary and finite-amplitude modes are present, the diffusivity ratio has a destabilizing effect; when oscillatory convection is present, it has a dual effect. Oscillatory convection develops earlier when the Vadasz number is higher. The couple-stress parameter and diffusivity ratio both increase with increasing solute Rayleigh number values, but the heat and mass transfer decreases as these values rise. Using double Fourier series, a generalized weakly nonlinear stability analysis is done. The research illustrates how various regulating parameters support and destabilize the flow dynamics. The influence of finite-amplitude convection on stability is also examined. Furthermore, the best conditions for stationary and oscillatory convection are based on altering the couple stress flow stability by controlling the applied magnetic field.

In this study, we examine the movement of two hard spheres aligned in a straight line within an incompressible couple stress fluid under the impact of the magnetic field. Both objects have distinct shapes and move along an axis connecting their centers with varying velocities. As a first step, an incompressible analytical analysis is performed on a fluid with couple stress properties around an axially symmetric particle. Using the superposition principle, a general solution is developed for couple stress fluid flows over two moving objects. In order to achieve the boundary conditions, the boundary collocation strategy is applied to the surfaces of the two spheres. A set of tables and graphs illustrates numerical estimates of the dimensionless drag forces acting on two spherical objects. In addition, a drop in Hartmann number or an increase in couple stress viscosity will increase the dimensionless drag force on each spherical particle.

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]

In this work, we have analysed the impact of a transverse magnetic field on the steady flow of an incompressible conducting couple stress fluid within a rectangular channel of uniform cross-section, incorporating suction or injection at the lateral walls. In order to get the velocity ‘w’ along the axis of the rectangular tube, we ignore the induced electric and magnetic fields. To find w, we apply the standard hyper stick and no slip boundary conditions. The velocity w and temperature θ were calculated using Fourier series. The velocity distribution compared for various magnetic parameter values by taking suction velocity zero and the results are in good agreement (99.99 %) with the existing results. The volumetric flow rate and skin friction are obtained and the effects of physical parameters like magnetic parameter, Reynolds number and couple stress parameter on this are studied through graphs.

This study focuses on exploring the influence of homogeneous and heterogeneous chemical reactions on a couple of stress fluids surrounding a permeable inclined stretching cylinder. The impact of a uniform magnetic field and porous media is also considered in the fluid model. It is assumed that the diffusion coefficients for chemical species A and B are of similar magnitudes and that the heat released during the chemical reaction is negligible. The governing partial differential equations (PDEs) are evolved and transformed into ordinary differential equations (ODEs) using adequate similarity alterations. These ODEs are subsequently solved using the shooting technique in conjunction with the fourth-order Runge-Kutta method, implemented through MATLAB software. Results are presented through graphs and tables depicting the velocity, temperature, and concentration fields. Furthermore, numerical findings for the skin friction coefficient and Nusselt number are discussed. The concentration field experiences a decline as homogeneous-heterogeneous reactions intensify, attributable to the heightened dispersion of concentrations across the system, resulting in a more intricate distribution pattern.

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.

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]

This study focuses on exploring the influence of homogeneous and heterogeneous chemical reactions on a couple of stress fluids surrounding a permeable inclined stretching cylinder. The impact of a uniform magnetic field and porous media is also considered in the fluid model. It is assumed that the diffusion coefficients for chemical species A and B are of similar magnitudes and that the heat released during the chemical reaction is negligible. The governing partial differential equations (PDEs) are evolved and transformed into ordinary differential equations (ODEs) using adequate similarity alterations. These ODEs are subsequently solved using the shooting technique in conjunction with the fourth-order Runge-Kutta method, implemented through MATLAB software. Results are presented through graphs and tables depicting the velocity, temperature, and concentration fields. Furthermore, numerical findings for the skin friction coefficient and Nusselt number are discussed. The concentration field experiences a decline as homogeneous-heterogeneous reactions intensify, attributable to the heightened dispersion of concentrations across the system, resulting in a more intricate distribution pattern. [ABSTRACT FROM AUTHOR]

In this paper, the study investigates the problem of the creeping flow of a non-Newtonian couple stress fluid through a linearly porous walled slit within a Darcy porous medium. The well-established approach, the Inverse Method approach, was employed to attain the precise solution of the governing equations. The inverse technique is utilized to transform the momentum equations into stream functions. The physical parameters are all computed: longitudinal velocity, transverse velocity, fractional reabsorption, leakage flux, axial pressure, volume flow rate, and mean pressure. The data was also graphed, indicating that the initial flow rate affects longitudinal velocity but not transverse velocity. The streamlines are greatly affected by the initial flow rate, resulting in straighter and more uniform patterns as the flow rate increases. Parameters like permeability and the couple stress parameter also have an impact on physical quantities. Because of the porosity parameter, backward flow occurs at the slit's end. The novelty of this research lies in the investigation of couple stress fluid flow within both linear channel walls and a porous medium. The present research focuses on how kidney disease affects fluid flow in renal tubules. It's critical because fibers, lipids, and waste particles can clog or disrupt these channels, lowering kidney performance. Understanding this helps in the management of kidney disease and provides insights for better biomedical engineering in the development of improved renal medicines.

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]

In this paper, the study investigates the problem of the creeping flow of a non-Newtonian couple stress fluid through a linearly porous walled slit within a Darcy porous medium. The well-established approach, the Inverse Method approach, was employed to attain the precise solution of the governing equations. The inverse technique is utilized to transform the momentum equations into stream functions. The physical parameters are all computed: longitudinal velocity, transverse velocity, fractional reabsorption, leakage flux, axial pressure, volume flow rate, and mean pressure. The data was also graphed, indicating that the initial flow rate affects longitudinal velocity but not transverse velocity. The streamlines are greatly affected by the initial flow rate, resulting in straighter and more uniform patterns as the flow rate increases. Parameters like permeability and the couple stress parameter also have an impact on physical quantities. Because of the porosity parameter, backward flow occurs at the slit's end. The novelty of this research lies in the investigation of couple stress fluid flow within both linear channel walls and a porous medium. The present research focuses on how kidney disease affects fluid flow in renal tubules. It's critical because fibers, lipids, and waste particles can clog or disrupt these channels, lowering kidney performance. Understanding this helps in the management of kidney disease and provides insights for better biomedical engineering in the development of improved renal medicines. [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]

This paper investigates the effects of quadratic convection on the electro-magneto-hydrodynamically driven flow of couple stress fluid in a microchannel. The integration of quadratic convection effects and electro-magnetic hydrodynamics in microchannels offers high potential for advancing microfluidic technologies and opening new avenues of innovation. In microfluidic systems, the couple stress fluids describe more accurately the flow mechanisms. The current mathematical modeling offers more details on the impacts of viscous dissipation and joule heating. Analytical series solutions for the temperature and velocity profiles are used with a homotopy perturbation technique. The impacts of all the parameters are explored using tables and graphs. The dynamics of velocity and temperature distribution are analyzed in detail. Furthermore, the Nusselt number values are computed and shown using funnel charts. From the graphical results it is concluded that the simultaneous impact of quadratic convection and thermal Grashof number enhances the entire velocity profile over the whole channel. The velocity and temperature profile are also observed to increase with increasing Brinkman number. The thermal profile is considerably increased by the Hartmann number, the Grashof number, and the quadratic convection parameter. The thermal profile always has a peak near the center of the channel. The thermal profile is reduced by the couple stress fluid variable parameter. The present study can be of assistance to novel advancements in various fields such as lab-on-a-chip devices, chemical synthesis, and thermal management. [ABSTRACT FROM AUTHOR]

The effect of chemical reaction and external vertical magnetic field on the onset of the double diffusive convection in couple stress fluid between infinite horizontal parallel plates has been studied. The effectiveness of vertical magnetic field and chemical reaction were gauged by determining the values of Chandrashekhar number (Q) and Damk hler number in terms of other controlling parameters and shown their effect on stability of system through graphs. The entire investigation is performed in two parts: linear and weakly non-linear stability analysis. A comparative study is presented in stationary case of linear stability analysis for four types of bounding surfaces: (a) Realistic bounding surfaces i.e. Rigid-Rigid, Rigid-Free and Free-Rigid (R/R, R/F and F/R) (b) Non Realistic bounding surface i.e. Free-Free (F/F). However, oscillatory case and weakly non-linear stability analysis are restricted for Free-Free (F/F) boundary surfaces. Graphical representations are used to illustrate how different parameters affect stationary, oscillatory, finite-amplitude states and the amount of heat and mass transfer. By analysing the linear stability analysis, it is observed that the onset of convection is more dominant in oscillatory case than stationary. The stability criteria for Q came out as (in decreasing order) F/F>F/R>R/R>R/F which is different from the criteria came out for rest of the controlling parameters i.e F/R>R/R>F/F>R/F in stationary case. It is also reported that the Q, Couple stress parameter (C) and ratio of heat capacities on heat transfer is responsible for the delay of the onset of convection while (impact of chemical reaction) enhances the onset of convection. Non-linear stability analysis using the truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of Q, Lewis number Le, and solute Rayleigh number RaS, increased the amount of heat and mass transfer while C decreased. We also draw streamlines, isotherms, isohalines and magnetic streamlines for different time intervals (unsteady) i.e. for (0.01, 0.03, 0.009, 0.006) and showed the pattern of the onset of convection.

The problem of the axisymmetric steady flow of an incompressible couple stress fluid within two concentric spheres is analyzed analytically in this study. In the interior of the solid sphere, translation takes place with a uniform velocity, and in the exterior, translation takes place with a fixed velocity. A study of slip conditions was carried out on the surfaces of the two spheres. In terms of stream functions, modified Bessel functions provide an analytical solution to the problem. As the outer sphere remains stationary, the inner sphere experiences the normalized drag force. A graphic representation of drag is provided based on the slip and spin slip parameters, the couple stress parameter, the separation distance, and the MHD parameter. In addition, the results are compared with earlier results for special cases of translation of one rigid sphere through an unbounded magnetic medium, viscous fluid flow, and Couple stress fluid without MHD between two concentric spheres.

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]

The effect of chemical reaction and external vertical magnetic field on the onset of the double diffusive convection in couple stress fluid between infinite horizontal parallel plates has been studied. The effectiveness of vertical magnetic field and chemical reaction were gauged by determining the values of Chandrashekhar number (Q) and Damköhler number (X) in terms of other controlling parameters and shown their effect on stability of system through graphs. The entire investigation is performed in two parts: linear and weakly non-linear stability analysis. A comparative study is presented in stationary case of linear stability analysis for four types of bounding surfaces: (a) Realistic bounding surfaces i.e. Rigid-Rigid, Rigid-Free and Free-Rigid (R/R, R/F and F/R) (b) Non Realistic bounding surface i.e. Free-Free (F/F). However, oscillatory case and weakly non-linear stability analysis are restricted for Free-Free (F/F) boundary surfaces. Graphical representations are used to illustrate how different parameters affect stationary, oscillatory, finite-amplitude states and the amount of heat and mass transfer. By analysing the linear stability analysis, it is observed that the onset of convection is more dominant in oscillatory case than stationary. The stability criteria for Q came out as (in decreasing order) F/F>F/R>R/R>R/F which is different from the criteria came out for rest of the controlling parameters i.e F/R>R/R>F/F>R/F in stationary case. It is also reported that the Q, Couple stress parameter (C) and ratio of heat capacities on heat transfer γ is responsible for the delay of the onset of convection while x (impact of chemical reaction) enhances the onset of convection. Non-linear stability analysis using the truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of Q, Lewis number Le, and solute Rayleigh number Ras, increased the amount of heat and mass transfer while C decreased. We also draw streamlines, isotherms, isohalines and magnetic streamlines for different time intervals (unsteady) i.e. for (0.01, 0.03, 0.009, 0.006) and showed the pattern of the onset of convection. [ABSTRACT FROM AUTHOR]

This study explores the flow properties of a couple stress fluid with the consideration of variable viscosity and a uniform transverse magnetic field. Under the effect of irreversible heat transfer, a steady fluid flow has taken place between two parallel inclined plates. The fluid flows due to gravity and the constant pressure gradient force. The plates are fixed and isothermal. The governing equations have been solved analytically for velocity and temperature fields. The total rate of heat flow and volume flow across the channel, skin friction, and Nusselt number at both plates are calculated and represent the impacts of relevant parameters through tables and graphs. The findings show that velocity, temperature, and the total rate of heat flow across the channel are enhanced by increasing the couple stress parameter and the viscosity variation parameter, while increasing the values of the Hartmann number reduces them.

Partially ionised fluids subject to an external magnetic field results in a completely different dynamical behaviour. Very few studies are conducted to explore the fascinating properties of Hall current and Ion slip in fluid flows. Flow over a Riga plate with MHD Ion and Hall currents, nonlinear thermal radiation is still rare. We aim to analyse the inclined stagnation point slip flow of Couple Stress fluid over a stretched horizontal Riga plate to close this gap. The flow problem is modelled using Navier stokes theory along with energy conservation and then principal partial differential equations are converted into ordinary differential equations. The governed system is solved mathematically by numerical technique RK Fehlberg method. The behaviour of fluid is described through graphs including fluid’s motion and energy. Local surface skin friction coefficients and local heat and mass flux at surface are the concerned physical magnitudes which are described through graphs and tables and are highly recommended in practical solicitations in numerous production plants and manufacturing workshops. Thermally radiative MHD flow with Ion Hall currents are broadly used in thermal controlling systems, spectro scopy, microchip technology, Magnetohydrodynamic energy conversion campaigns, exploration and expansion for high & low efficiency release, power-driven & power generation structures, MHD accelerations & thrusters, and drift regulator in hypersonic and automobiles.

The creeping rotational motion of a magnetohydrodynamic couple stress fluid between two concentric spheres under the impact of slippages. The slippage conditions are applied on the surface of the inside sphere. In addition, the couple stresses on the boundary are assumed to vanish. The analytical solution to the problem is used to obtain the field functions of velocity, tangential stress, and couple stresses. The wall correction factor experienced by the fluid on the inner solid sphere is evaluated and plotted. However, in the presence of a magnetic field, the eddy current caused by rotating particles produces a torque that tends to rise because the assumption of the torque direction is the opposite direction of magnetic induction. Also, the first couple stress parameter, the angular velocity ratio, and slip condition did an improvement in the torque. On the other side, the wall correction factor slows down with slippage on the inner sphere and the size parameter. All results give limiting cases with no slippage and viscous fluid.

Gold nanoparticles have a variety of applications in medical sciences, including photothermal (PT), gene transfection agents, radiosensitizing, drug delivery, therapeutic etc. Especially in a lab, gold metallic nanoparticles are commonly utilized as a tracer to indicate the existence of certain proteins or DNA in a sample and identify the different antibiotics. They are biocompatible and can deliver heat energy to tumor cells via various therapeutic techniques. Because cancer cells are microscopic, appropriately, nanosized gold nanoparticles were infused into the bloodstream for infiltration and found very effective. Keeping in mind the applications of gold nanoparticles, this paper aims to consider the Magnetohydrodynamics thermofluids flow of blood containing gold nanoparticles. More precisely, the couple stress Casson fluid flow is taken as blood flow through a porous medium. This flow regime is modeled in terms of classical partial differential equations. These equations are nondimensionalized using suitable non-dimensional variables. The dimensionless partial differential equations are fractionalized using Caputo’s fractional derivatives definition with the application of Fick’s and Fourier’s laws. Laplace and Fourier’s transforms are used to accomplish exact solutions for temperature, concentration, and velocity profiles. The skin friction, Nusselt and Sherwood numbers are calculated and displayed in tabular form. The impact of several physical parameters on temperature, concentration and velocity profiles of thermofluids are shown graphically. The obtained results show that the blood velocity is a decreasing function of the couple stress parameter (λ) and the Casson parameter (β), which is very useful for controlling blood flow during magnetic therapy for treating pain in the back and foot and bleeding during surgeries. It is observed that the heat transfer rate of blood using gold nanoparticles increases up to 36%, which is very useful in hyperthermic tumor treatment.

The purpose of this article is to examine the effect of surface roughness on porous curved annular plates lubricated with couple stress fluid in the presence of magnetic field. The MHD-Stochastic Reynolds equation is derived using Christensen’s stochastic model and applied to predict the squeeze film characteristics of porous curved annular plates. The expressions for squeeze film pressure, load-carrying capacity, and squeeze film time are obtained analytically, the results are discussed for various values of operating parameters, and are plotted graphically. It is found that the squeeze film characteristics of porous curved annular plates are improved using a non-Newtonian fluid in the presence of an external magnetic field. The effect of roughness parameter is to increase (decrease) the squeeze film attributes for azimuthal (radial) roughness configuration as compared to the smooth case. Furthermore, the effect of permeability parameter is to decrease the pressure, load-carrying capacity, and squeeze-film time as compared to the non-porous case.

The current study concentrated on the analysis of entropy generation in three dimensional mixed convection flow of couple stress nanofluid under the effect of high order velocity slip flow, passive and active control conditions. The problem advances more significance with the effect of non-Fourier’s heat, and non-Fick’s mass flux model to scrutinize the heat and mass transport rate. The high-order partial differential equations obtained from the designed problem are transmuted into higher order ordinary differential equations using similarity conversion and lastly solved by using bvp4c. Through this investigation how governing parameters have affected the flow problem were examined and offered in graphical and tabular form. It is discovered that the occurrence of couple stress inflow problem upsurges the velocity of the fluid and on the other hand it causes a reduction in the entropy production. Likewise, it is detected that temperature improves by snowballing the Hartman parameter. Moreover, the temperature and concentration disseminations are greater all for the situation of an active control while correlated with a case of passive control. Also, the same is true for numerical results of Nusselt number and Sherwood number that is due to an active control is greater than all for the case of passive control. Furthermore, this theoretical and numerical deliberation bears the potential to beneficial in the field of bio-fluid dynamics.

This study explores the flow properties of a couple stress fluid with the consideration of variable viscosity and a uniform transverse magnetic field. Under the effect of irreversible heat transfer, a steady fluid flow has taken place between two parallel inclined plates. The fluid flows due to gravity and the constant pressure gradient force. The plates are fixed and isothermal. The governing equations have been solved analytically for velocity and temperature fields. The total rate of heat flow and volume flow across the channel, skin friction, and Nusselt number at both plates are calculated and represent the impacts of relevant parameters through tables and graphs. The findings show that velocity, temperature, and the total rate of heat flow across the channel are enhanced by increasing the couple stress parameter and the viscosity variation parameter, while increasing the values of the Hartmann number reduces them. [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]

The rising industrial demand for technological advancement in improving their working fluids and thermal augmentation has prompted studies on various non-Newtonian fluids and nanofluids. These play momentous role in many engineering devices and electronics development. Therefore, this study seeks to examine the thermal performance of ternary hybridized magneto-couple stress nanofluids in permeable media with nonlinear thermofluidic properties. The dispersion of suspended cobalt ferrite CoFe2O4, titanium dioxide TiO2 and magnesium oxide MgO nanoparticles occurs in H2O conventional solvent. With isothermal energy, the tri-hybrid nanofluid is controlled by stretchy velocity, magnetic field, gravity and Ohmic heating. An invariant transmutation of the partial derivative model is offered using similarity variables. The complete solutions to the model are obtained via Chebyshev technique coupled with integrated collocation method. From the graphically presented results, it is revealed that strong heat transfer is provided by ternary CoFe2O4+Ti02+MgO nanofluids compared to CoFe2O4+Ti02 and CoFe2O4 nanofluids. Also, the nanofluid concentration volume fraction enhanced thermal distribution and performance in the system.

The geometry under consideration for this flow problem, namely paraboloid surface is similar to aircraft and car bonnet, bullet shape, front shape of submarine, rockets and these types of are useful in manufacturing of formula F1 cars and airplanes. The flow of fluid over paraboloid surface has several applications in space science, industries, mechanical works and architecture. The aim of this study is to explore the Cattaneo-Christov heat and mass flux model in steady 2D couple stress fluid flow with variable thermo-physical characteristics near the parabolic surface. The effects of enthalpy and activation energy are also taken into consideration. The stretching in the layers of fluid is assumed at the surface of body. The problem is amended by using the law of conservation of momentum, energy and mass transfer. The non-linear governing partial differential equations are converted into ordinary differential equations by using self-similar form with similarity transformation. The resulting ODE's are numerically solved by bvp4c method. The conclusion indicates that the fluid moves quickly due to higher inputs of couples stress parameter and buoyancy parameter while the viscosity coefficient creates restriction in the fluid motion. The transference analysis of temperature and concentration enhances for higher impacts of thermal conduction and mass diffusivity coefficient. The thermal concentration relaxation parameters are the source of decline in the concentration and temperature of fluid. The heat transmission rate is maximum and mass transfer is minimum due to raising values of Dramkholer number. The impression of all parameters on different fields is listed in discussion section.

It is generally considered that fractal-fractional order derivative operators are highly sophisticated mathematical tools that can be applied in a variety of physics and engineering situations to obtain real solutions. By using fractal-fractional derivatives, we can simultaneously investigate fractional order and fractal dimension. Due to extensive applications of fractal-fractional derivatives, in the present article fractal-fractional order model of non-linear Couple stress nanofluid has been analyzed. The homogenous mixture of base fluid and nanoparticles has been formed by the uniform dispersion of cadmium telluride nanoparticles in mineral transformer oil. Primarily, the classical mathematical model has been formulated via relative constitutive equations and then generalized by using fractal-fractional derivative operator. This model has been numerically solved using Crank-Nicolson technique. Using numerical solutions, various graphs are plotted to analyze how physical parameters alter Couple stress nanofluid rheology. As can be seen from the graphical study, couple stress slows down fluid velocity. Adding cadmium telluride nanoparticles to transformer oil increased its efficacy by 15.27%.

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]

This study explores the influence of an inclined magnetic field and variable viscosity on the entropy generation in steady flow of a couple stress fluid in an inclined channel. The walls of the channel are stationary and non-isothermal. The fluid flow is driven due to pressure gradient and gravitational force. Reynold's model for temperature-dependent viscosity was used. The dimensionless, non-linear coupled equations of momentum and energy was solved, and we obtained an analytical solution for the velocity and temperature fields. The entropy generation and Bejan number were evaluated. The variation of pertinent parameters on flow quantities was discussed graphically. The rate of volume flow, skin friction coefficient, and Nusselt number at the surfaces of the channel were calculated and their variations were discussed through surface graphs. From the results, it is noticed that the entropy generation rate can be minimized by increasing the magnetic field and the temperature difference parameters. The findings of the current study in some special cases are in precise agreement with the previous investigation. [ABSTRACT FROM AUTHOR]

This work solves the problem of thin-film withdrawal and drainage of a steady incompressible couple stress fluid on the outer surface of a vertical cylinder. The governing equations for velocity and temperature distributions are subjected to the boundary conditions and solved with the help of homotopy analysis method. The obtained expressions for flow profile, temperature profile, average velocity, volume flow rate, and shear stress confirmed that the thin-film flow of couple stress fluid highly depends on involved parameters say Stokes number St , vorticity parameter λ, couple stress parameter η, and Brinkman number Br presented in the graphical description as well.

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]