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The application of mathematical modeling to biological fluids is of utmost importance, as it has diverse applications in medicine. The peristaltic mechanism plays a crucial role in understanding numerous biological flows. In this paper, we present a theoretical investigation of the double diffusion convection in the peristaltic transport of a Prandtl nanofluid through an asymmetric tapered channel under the combined action of thermal radiation and an induced magnetic field. The equations for the current flow scenario are developed, incorporating relevant assumptions, and considering the effect of viscous dissipation. The impact of thermal radiation and double diffusion on public health is of particular interest. For instance, infrared radiation techniques have been used to treat various skin-related diseases and can also be employed as a measure of thermotherapy for some bones to enhance blood circulation, with radiation increasing blood flow by approximately 80%. To solve the governing equations, we employ a numerical method with the aid of symbolic software such as Mathematica and MATLAB. The velocity, magnetic force function, pressure rise, temperature, solute (species) concentration, and nanoparticle volume fraction profiles are analytically derived and graphically displayed. The results outcomes are compared with the findings of limiting situations for verification. [ABSTRACT FROM AUTHOR]

Abstract The use of renewable energy sources is leading the charge to solve the world’s energy problems, and non-Newtonian nanofluid dynamics play a significant role in applications such as expanding solar sheets, which are examined in this paper, along with the impacts of activation energy and solar radiation. We solve physical flow issues using partial differential equations and models like Casson, Williamson, and Prandtl. To get numerical solutions, we first apply a transformation to make these equations ordinary differential equations, and then we use the MATLAB-integrated bvp4c methodology. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. In addition to numerical and tabular studies of the skin friction coefficient, Sherwood number, and local Nusselt number, important components of the flow field are graphically shown and analyzed. Consistent with previous research, this work adds important new information to the continuing conversation in this area. Through the examination of dimensionless velocity, concentration, and temperature functions under varied parameters, our work explores the physical properties of nanofluids. Comparing the Casson nanofluid to the Williamson and Prandtl nanofluids, it is found that the former has a lower velocity. Compared to Casson and Williamson nanofluid, Prandtl nanofluid advanced in heat flux more quickly. The transfer of heat rates are $$25.87\%$$ 25.87 % , $$33.61\%$$ 33.61 % and $$40.52\%$$ 40.52 % at $$Rd =0.5, Rd=1.0$$ R d = 0.5 , R d = 1.0 , and $$Rd=1.5$$ R d = 1.5 , respectively. The heat transfer rate is increased by $$6.91\%$$ 6.91 % as the value of Rd rises from 1.0 to 1.5. This study is further strengthened by a comparative analysis with previous research, which is complemented by an extensive table of comparisons for a full evaluation.

Darcy-Forchheimer model has been used to consider the mathematical and statistical aspects of Prandtl nanofluid flow on a stretched curvy geometry, with homogenic-heterogenic reactions, nonlinear radiation, exponential heat, Joule heating, velocity slip, and convective heat conditions. An account of entropy significance has been given to boost the applicability of the study. The 4-5th ordered numerical tool, Runge-Kutta-Fehlberg, has been employed to establish the plots for the considered flow. ANOVA and Taguchi optimisation technique is used to obtain the optimal condition in enhancing the heat transfer rate for modelled mathematical problem. Here, the study reveals that the increasing homo-heterogenic strength parameters foster the concentration profile. The study also found that the thermal curves are positively affected by the radiation parameter and the temperature differential parameter. In addition to this, graphical portraits of isotherms and streamlines have been given to characterise the flow and heat pattern. Taguchi method reveal that first level of Prandtl number, magnetic parameter, Weissenberg number, heat source parameter and third level of curvature parameter, produce maximum Nusselt number. Heat source parameter has large contribution of about 49.45% among the other parameters and Prandtl number has the least contribution of about 1.4% for optimisation. [ABSTRACT FROM AUTHOR]

Current study examined the magnetohydrodynamic (MHD) Prandtl nanofluid of a thermal double‐diffusive flow through an exponentially vertical surface in association with heat generation, and thermophoresis effect. The novelty of this study is due to the analysis of Prandtl nanofluid model with Soret mechanism and chemically responding fluids. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. The fluid flow phenomenon is characterized by nonlinear coupled differential equations involving two or more independent variables. A suitable numerical technique is used to handle the set of governing equations along with a stability and convergence analysis. According to recent study, the fluid velocity increases since all the parameters are set to higher levels. For the various parametric values, isotherms and streamlines have been explored. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. For instance, thermal radiation is crucial in designing sophisticated energy‐transformed systems that operate at high temperatures. On the other hand, the phenomenon of Soret is useful in separating isotopes in chemical engineering. An important findings of the current investigations can be treated as, radiative heat encourages fluid temperature distribution since it is the measure of the electromagnetic element radiates from the fluid particle that convert it into thermal radiation. These studies have several applications in the manufacturing and biomedical fields, petrochemical industries, automobiles, medical sciences, and various production processes in industries. [ABSTRACT FROM AUTHOR]

The current study examines the complex interaction between mass and thermal energy variations, considering factors including activation energy, chemical reactions, and thermal radiation across a nonlinear stretching sheet. The ternary nanofluid comprises titanium, single-walled carbon nanotubes, and multi-walled carbon nanotubes to improve the heat transfer characteristics of the fluidic system. Partial differential equations are utilized to represent the system’s dynamic behavior mathematically. Following this, similarity transformations are used to convert these governing partial differential equations into ordinary differential equations. The numerical solution for the transformed boundary value equations is derived using MATLAB. The graphical representations explain the results of these calculations, providing clear insights into the behavior of important variables in various conditions. It is concluded that an increase in the flexibility parameter improves the thermal behavior of a fluid, while an increase in the Prandtl parameter decreases it. Furthermore, applying a magnetic field to the dynamics of fluid heating enhances temperature and energy transfer.

Thermal radiations are extensively utilized in the medical sciences, and the investigation of thermal radiation combined with double diffusive phenomena is the burning research area because of its enormous applications in treatment of various diseases. Cilia play significant part in many physiological processes of animal and humans. A novel mathematical prodigy for Prandtl nanofluid with thermal radiations and double diffusion convection by implementing slip at boundaries is presented for cilia induce flow in an asymmetric microchannel. Mathematical scheme for the physiological flow is developed and then pertinent equations are designed exploiting low Reynolds number and long wavelength simplifications. Solution for the physiological nanofluid is gathered by emphasizing on the novel BVP4C technique in MATLAB and resulting outcomes are elaborated with aid of graphical illustrations. Pros and cons of various physical flow parameters like thermal slip parameters, Prandtl fluid parameters, Grashof parameter, Prandtl parameter , Brownian motion parameter, thermal radiation parameter, Brinkmann number, Soret parameter are examined on the velocity, magnetic force function, temperature profile, concentration and nanoparticles volume fraction. It is reported that slip parameter really effects the nanofluid transport within the ciliated microchannel, it reduces the fluid flow, diffusion phenomena of the nanoparticles in the ciliated microchannel surges when radiation parameter is strengthened. The reported investigation will be instrumental in heat radiation effect for regulation of blood circulation to cure the cancer tissues in multiple drug delivery systems and will pave the way for the designs of certain diagnostic and pharmacological devices.

Abstract Current study examined the magnetohydrodynamic (MHD) Prandtl nanofluid of a thermal double‐diffusive flow through an exponentially vertical surface in association with heat generation, and thermophoresis effect. The novelty of this study is due to the analysis of Prandtl nanofluid model with Soret mechanism and chemically responding fluids. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. The fluid flow phenomenon is characterized by nonlinear coupled differential equations involving two or more independent variables. A suitable numerical technique is used to handle the set of governing equations along with a stability and convergence analysis. According to recent study, the fluid velocity increases since all the parameters are set to higher levels. For the various parametric values, isotherms and streamlines have been explored. This suggested model is beneficial since it can significantly advance the domains of thermal and industrial engineering. For instance, thermal radiation is crucial in designing sophisticated energy‐transformed systems that operate at high temperatures. On the other hand, the phenomenon of Soret is useful in separating isotopes in chemical engineering. An important findings of the current investigations can be treated as, radiative heat encourages fluid temperature distribution since it is the measure of the electromagnetic element radiates from the fluid particle that convert it into thermal radiation. These studies have several applications in the manufacturing and biomedical fields, petrochemical industries, automobiles, medical sciences, and various production processes in industries.

The aim of this study is to explore the effects of radiative-induced magneto-Prandtl nanofluid on peristaltic waves, in conjunction with viscous dissipation and double-diffusivity convection caused by slip boundaries over an asymmetric channel, using the long wavelength and low but finite Reynold number approximation. The application of thermal radiation and double diffusion is significant in medical research, particularly for treating skin-related ailments through the use of infrared radiation technique. Additionally, infrared radiation can be utilized for medical treatment to restore thermal regulation homeostasis. This study integrates peristaltic motion theory, heat flux using linear approximation, and thermal radiation to analyze the flow problem which incorporates the situation with small temperature difference. The mathematical framework is based on partial differential equations that are further calculated through Numerical Solutions. The numerical solution is calculated using built-in command in the software Mathematica 11 and MATLAB. The impact of various physical parameters on the temperature profile, pressure rise, velocity profile, solute (species) concentration of thermal radiation is illustrated graphically. The significant finding of the study is that the heat radiation effect on blood circulation enhances the temperature which may help to destroy the cancer tissues in drug distribution. [ABSTRACT FROM AUTHOR]

Entropy generation in nanofluid flows is a complex phenomenon influenced by various factors, including the presence of nanoparticles, temperature gradients, and other effects such as magnetic fields and heat generation. Minimizing entropy generation is crucial for improving the efficiency of nanofluid-based heat transfer systems. In this study, we analyze the irreversibility in cubic autocatalytic Prandtl nanofluid flow by a porous stretchable sheet. To model the nanofluid flow, we consider the effects of Darcy-Forchheimer, thermophoresis, and Brownian motion. Additionally, we account for the impacts of magnetic fields, heat generation, and radiation. Under the implementation of second thermodynamics approach, the entropy generation mechanism is evaluated. Under the certain flow assumptions, the mathematical model is constructed for which numerical treatment is followed via the Runge-Kutta-Fehlberg method (RKF-45) implemented in the Mathematica package. The novel-interpreted outcomes are physically observed in view of flow parameters. Numerical studies are conducted to examine engineering quantities. The results show that fluid velocity decreases with higher magnetic and Forchheimer variables. Temperature increases with greater radiation and heat generation parameters. Concentration and temperature fields are enhanced with higher estimations of the thermophoresis variable. Entropy increases with advanced magnetic variables and the Brinkman number, while the opposite effect is observed for the Bejan number. [ABSTRACT FROM AUTHOR]

Abstract The present analysis aims to compute the non-similar solution of a chemically reactive Prandtl nanofluid across an elongated surface in a permeable medium, with a uniform magnetic field applied perpendicular to the surface. The Buongiorno model is used to study the random motion and thermophoresis of the nanoliquid by considering factors such as viscous dissipation, Joule heating, and multiple slips. By using appropriate transformations and a non-similarity approach, the governing flow equations are scaled down to the second level and numerically analyzed using the MATLAB bvp4c algorithm. The profound effects of the dimensionless parameters are illustrated graphically for the velocity, temperature, and solutal profiles. The impact of drag force, thermal, and solutal transmission at the interface of the deformable sheet are inspected in tabulated form. The results of this study revealed that higher Prandtl and elastic parameters lead to an increase in fluid velocity but a rise in thermal slip parameter results in the opposite behavior. An increase in Hartmann number and porosity parameters upsurges drag force, whereas an increase in the slip parameter reduces it. The research is validated by comparing it to previous studies, and a strong correlation is observed. It is examined that the percentage (%) error with both comparative papers is between −0.0017181 and 0.0006.377 which is almost negligible. Thus, confirming the reliability and precision of the formulated problem. [ABSTRACT FROM AUTHOR]

Abstract The addition of nanoparticles in the base fluid increases the thermal conductivity of the fluid but the stability of the fluid gets affected due to the presence of nanoparticles in the base fluid. The addition of motile microorganisms in the nanofluid increases the stability of the nanofluid by enhancing thermal conductivity as well as mass transport. In this perspective, the current investigation deals with the study of the behavior of microorganisms in bioconvection over the fluid variables in the non-Darcy background. A two-dimensional Prandtl-nanofluid moving along an inclined stretching flat surface subject to inlet and outlet rate of flow to the surface under the action magnetic drag force normal to the surface. The non-linear partial differential equations are transmuted to ODEs through appropriate similarity transformations. The MATLAB inbuilt bvp4c solver has been employed to generate the solution of the transformed ODEs. The stability of the finite difference scheme is demonstrated by a stability test, and the influence of various parameters on the nanofluid velocity, thermal as well as concentration, and density of motile microorganisms is depicted using graphs for each of them. Additionally, interested physical quantities, shearing stresses, rate of thermal diffusion, rate of mass diffusion, and rate of motile microorganism at the surface, have been plotted by varying the controlling parameters. Growth in the shear stress as well as the rate of thermal diffusion has been detected for the increasing values of δ and β . However, the Sherwood number decays for augmented values of δ . The findings could be utilized to increase the thermal efficiency of heat exchangers in order to maintain thermal balance management in small heat-density equipment and gadgets, as well as the thermal efficiency of microbial fuel cells, enzyme biosensors, and microfluidics devices. [ABSTRACT FROM AUTHOR]

The focus of this study is to examine cross-diffusion effects on radiative and reactive Prandtl nanofluid due to a convectively heated stretching surface. The present flow problem is considered under the assumption that the external magnetic field is applied to study the nanofluid behavior and thermal radiation considered, for a complete understanding of magnetohydrodynamics of the flow. The governing system of the nonlinear partial differential equations of the flow problem is remodeled with suitable similarity invariants and obtained the system of nonlinear ordinary differential equations. The transformed ODEs with their modified boundary conditions deal with the employment of the Runge-Kutta technique using computation shooting schema with MATLAB software. The consequence of numerous leading parameters on the fluid velocity, temperature, concentration, and engineering quantities profiles are discoursed and presented through graphical illustration. Moreover, the current work is also equated with the available work for limiting cases. It is found that the external magnetic field diminishes the motion of fluid and improvement in heat transfer. Concentration performance improvises on cross diffusion, thermophoresis, and Brownian motion impacts. [ABSTRACT FROM AUTHOR]

The two-dimensional boundary layer flow of a Prandtl nanofluid was explored in the presence of an aligned magnetic field over an inclined stretching/shrinking sheet in a non-Darcy permeable medium. To transform the PDEs of the leading equations into ODEs, a coupled boundary value problem was formed and suitable similarity functions were used. To obtain numerical answers, an efficient code for the Runge–Kutta technique with a shooting tool was constructed with a MATLAB script. This procedure is widely used for the solution of such problems as it is efficient and cost-effective with a fifth-order accuracy. The significance of immersed parameters on the velocity, temperature, concentration, and bioconvection is shown through figures. Furthermore, the physical parameters of the skin friction coefficient and the Nusselt numbers are demonstrated in tables. The declining behavior of the flow velocity against the porosity parameter K p and the local inertia co-efficient F r is shown, and the both parameters of the Darcy resistance and Darcy–Forchheimer resistance are responsible for slowing the fluid speed. The increasing values of the Schmidt number S c decrease the concentration of the nano entities. [ABSTRACT FROM AUTHOR]

The two-dimensional boundary layer flow of a Prandtl nanofluid was explored in the presence of an aligned magnetic field over an inclined stretching/shrinking sheet in a non-Darcy permeable medium. To transform the PDEs of the leading equations into ODEs, a coupled boundary value problem was formed and suitable similarity functions were used. To obtain numerical answers, an efficient code for the Runge–Kutta technique with a shooting tool was constructed with a MATLAB script. This procedure is widely used for the solution of such problems as it is efficient and cost-effective with a fifth-order accuracy. The significance of immersed parameters on the velocity, temperature, concentration, and bioconvection is shown through figures. Furthermore, the physical parameters of the skin friction coefficient and the Nusselt numbers are demonstrated in tables. The declining behavior of the flow velocity against the porosity parameter Kp and the local inertia co-efficient Fr is shown, and the both parameters of the Darcy resistance and Darcy–Forchheimer resistance are responsible for slowing the fluid speed. The increasing values of the Schmidt number Sc decrease the concentration of the nano entities.

The current research investigates the effects of Brownian motion, thermophoresis, and multi buoyancy forces on MHD Prandtl fluid as it flows through a stretched cylinder with convective boundary conditions. Linear thermal conductivity is included in the energy equation's development. The nanofluid flows are important in cooling systems, modern nano-technology, electronic parts and heat exchangers. The primary objective of this research is to enhance heat transportation. The problem consists of nonlinear PDEs and these equations are transformed into ODEs by incorporating suitable similarity transformations. The shooting strategy is used to solve the problem numerically after converting the non-dimensional ordinary differential equations to a system of first-order ODEs. A comparison is made between the MATLAB-generated findings and those that have already been published. The numerical results of various parameters have been systematically organized into tabular representations. Graphs show how significant parameters affect the given velocity, temperature, concentration, and motile microorganism measurements. When the buoyancy ratio and bio-convective Rayleigh number increase, the velocity profile drops; however, the temperature profile shows increasing behavior for the thermal radiation parameter. The concentration profiles are enhanced for the higher inputs in curvature parameter, magnetic parameter, and activation parameter while the density of motile microbes' decay for enhanced inputs of bio-convective Lewis number. [ABSTRACT FROM AUTHOR]

The communication aims to analyze the conduct of rotating Prandtl nanofluid with the inclusion of effects like MHD, variable species diffusivity, nonlinear warm radiation, convective limit conditions, velocity slip and chemical reaction on fluid moving along a stretchable surface. Buongiorno model is implemented in orde to catch the Brownian movement and thermophoresis impacts by the presence of nanofluids. The effect of viscous dissemination and Joule heating is used in the current review. Using suitable similarity variables, the governing differential conditions with respect to momentum, energy, and concentration are improved into customary, ordinary differential equations and these equations numerically with the shooting technique. The ranges of different dimensionless parameters used in our study are Prandtl fluid parameter 0 ≤ α ≤ 3.5, elastic parameter 1 ≤ β ≤ 7, thermal radiation 1 ≤ Rd ≤ 2, temperature ratio parameter 1 ≤ θ w ≤ 1.6, Eckart number 0.3 ≤ Ec ≤ 0.7, velocity slip parameter 0 ≤ γ 1 ≤ 1, thermophoresis parameter 0.1 ≤ Nt ≤ 4, Brownian motion parameter 0.1 ≤ Nb ≤ 1.5, Prandtl number 1.7 ≤ Pr ≤ −2.6, magnetic parameter 0.1 ≤ M ≤ 0.8, chemical reaction parameter 0.2 ≤ δ ≤ 1, convection parameter 0.3 ≤ γ 2 ≤ 1.1, rotational parameter 0.1 ≤ λ ≤ 0.7, Schmidt number 0.1 ≤ Sc ≤ 0.7, variable species diffusivity 0.1 ≤ ϵ ≤ 0.5 which shows up during mathematical arrangement are shown as tables and charts. These results are enhanced with the data for the heat transfer rate and skin friction coefficients. It is seen that the temperature profile enhanced on account of an increase in temperature convection boundary and mass fraction field raises due to an improvement in species diffusivity parameter. [ABSTRACT FROM AUTHOR]

Magnetohydrodynamic flow of Prandtl nanofluid due to stretching/shrinking wedge with activation energy is investigated. The stagnation point flow is attributed with suction/injection in the presences of heat sources. Enhancement in thermal transportation of the base fluid attracted our attentions to increases thermal conductivity. Prandtl fluid formulation pertains to novelty of this work. The boundary layer approximations are considered for the governing equations to be contortioned in the form of partial differential equations. Appropriate similarity transforms are employed to attain the boundary value problem in ordinary differential form. Then Keller-box method is utilized to gain numerical suction for to provide results for flow field, thermal distributions and concentration field. The controlling parameters are varied in their suitable rang to exhibit their role as graphically and numerical. Velocity profile f ′(ζ) is increased directly with larger values of material parameter α while for elastic parameter β , it shows decreasing trend. The skin friction− f ′(ζ)is boosted with the increasing values of magnetic parameter. [ABSTRACT FROM AUTHOR]