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This work examines, accounting for viscous dissipation, an analytical simulation of mixed convection heat transfers in a Williamson blood base nanofluid flow on a stretched Riga plate (RP). The Lorentz force, which influences fluid dynamics, is produced by the alternating magnetic fields that comprise the RP. The flow characteristics are further complicated by the blood base Williamson nanofluid’s (WN) non-Newtonian behavior. The governing partial differential equations (PDEs) for momentum and energy are transformed into a set of nonlinear ordinary differential equations (NODEs) by similarity transformations. We solve these equations analytically using the homotopy analysis method (HAM). We investigate in detail how the temperature and velocity profiles (VPs) are affected by significant parameters as the Williamson parameter (WP), viscous dissipation, volume friction of nanoparticles, modified Hartmann number (MHN), heat generation parameter and Biot number (BN). The findings show that the thermal conductivity (TC) of nanoparticles is increased, improving the efficiency of heat transfer. Moreover, a Lorentz force generated by the RP considerably alters the temperature and flow fields. With regard to the design and optimization of thermal systems utilizing complex boundary conditions (BCs) and non-Newtonian nanofluids, this work offers significant new insights.

Here the Cattaneo–Christov double diffusion model explores the mixed convective flow of third-grade nanoliquid on a stretchable surface with Riga device diverging from the traditional Fourier and Fick’s law. The model incorporates entropy optimization and Soret–Dufour effects, offering a unique perspective on heat and mass transfer phenomena. By employing relevant transformations, the complex partial differential equations are converted into more manageable ordinary differential systems. An optimal analysis method is then applied to solve the resulting nonlinear differential system, shedding light on the intricate interplay of various physical variable. Through the utilization of plots, the study delves into the impact of these physical variable, providing insights into the behavior of the system under different conditions. This comprehensive approach not only enhances our understanding of the underlying mechanisms governing the convective flow of nano-liquids, but also highlights the significance of considering nonclassical models in thermal and mass transport studies. The key finding of this study is that fluid velocity enhances for material parameters due to low viscosity. Temperature and nanoparticle concentration enhance for higher values of Dufour and Soret numbers, respectively. For higher estimations of Reynold number, entropy of the system decreases.

Our paper is consecrated to show the influence of variable fluid properties in EMHD non-Newtonian power-law fluid along a moving Riga plate. Slip velocity phenomenon is considered at the surface which is convectively heated. Entropy analysis is elaborated employing thermodynamic second relation. The governing nonlinear PDEs are altered into ODEs through adequate propinquity transformations which have been solved numerically via the shooting method with the fourth-order Runge–Kutta algorithm through Mathematica software (bvp4c). Characteristics of different basic parameters on velocity, temperature, entropy generation and Bejan number are highlighted through graphs. The outcomes exhibit that the minimum entropy rate in the flow system can be obtained either with rising viscosity parameter and slip parameter or declining dimensionless parameter and thermal conductivity parameter. The entropy rate is minimal for dilatant fluid when compared to pseudo plastic fluid with the most governing parameters. Contrast behavior on the thermal field is noticed for larger values of viscosity parameter and thermal conductivity parameter.

The ultra-high significances of thermal radiation, magnetic field and activation energy in thermal enhancement processes allow significant applications in chemical and mechanical engineering, modern technology and various thermal engineering eras. The improvement in energy resources and production became one of the major challenges for researchers and scientists for sustained development in industrial growths. Beside this, the bioconvection assessment in nanomaterials conveys prestigious applications in biotechnology like bio-sensors, enzymes, petroleum industry, bio-fuels and many more. In view of such renewable applications, present exploration discloses unsteady two-dimensional flow of third-grade nanomaterial accommodating gyrotactic microorganisms induced by unsteady stretched Riga sheet in porous medium. The formulated flow problem is further scrutinized by utilizing the chemical reaction, activation energy, thermal radiation and magnetic aspects. The convective Nield constraints are further subjected in the current investigation. Apposite transformations are used to condense the nonlinear developed problem into dimensionless ordinary form. The numerical solution of such similar flow problem is presented via shooting technique. The detailed graphical illustrations of the dimensionless temperature, nanoparticles concentration, velocity and motile microorganisms for physical significance of diverse relevant parameters are deliberated. Furthermore, numerical data of local Sherwood, Nusselt and motile density numbers is designated in tabular form. Study accentuated that velocity increases for higher modified Hartmann and material constants, while the effects of buoyancy ratio and bioconvected Rayleigh numbers are rather opposite. The temperature, microorganism and concentration distributions were enhanced for unsteady parameter. It is also acknowledged that the concentration distribution is enhanced for activating the energy number. Moreover, the microorganism distribution enhances for concentration difference and magneto-porous constants, while bioconvected Lewis and Peclet numbers show conflicting trend.

This paper analyzes the influence of mixed convective fourth grade nanofluid flow by a stretchable Riga device in the presence variable thermal conductivity and mass diffusivity. Heat and mass transportation are considered with Cattaneo–Christov (CC) model. Thermal radiation and dissipation are also taken in the energy expression. Suitable transformation is employed to reduce partial differential system into nonlinear ordinary system. The governing nonlinear expression is solved via optimal homotopy analysis method. Impact of different physical variables is discussed via graphs. Velocity profile is enhanced for higher values of cross viscous parameter and fourth grade fluid variable. Fluid temperature enhances for higher estimation of thermal relaxation parameter but reverse behavior is seen for solutal concentration variable on nanoparticle concentration.

This paper numerically simulates the nanofluid flow over a thermally expanding Riga plate. Buongiorno model for nanofluid is employed to investigate the contribution of Brownian motion and thermophoretic force on the nanoflow. Magnetohydrodynamics (MHD) of viscous nanofluid through a porous medium is characterized with the help of Darcy–Forchheimer’s model. In addition, the simultaneous effects of activation energy and chemical reaction have been incorporated. Moreover, highly nonlinear coupled differential equations are formulated which highlight the influence of viscous dissipation and heat generation. A numerical solution is achieved with the help of the Range–Kutta fourth-order (RK4) method combined with the shooting technique. Finally, the role of emerging parameters is studied via performing the numerical simulation which reveals that the momentum boundary layer of nanofluid shrinks due to the porous medium. Whereas, thermal boundary layer expands for all variables, except for the Prandtl number. Finally, mass transfer rated suffers due to Schmidt number.

The main purpose of this study is to scrutinize the axisymmetric flow of second-grade nanofluid with variable viscosity near a stagnation point under the influence of the Cattaneo–Christov double diffusion model. A Riga plate is assumed to be the source of the three-dimensional flow. The consequences of the anisotropic slip conditions and the Buongiorno model are included in the formulation of the constituting equations of the fluid flow. The highly nonlinear ordinary differential equations are developed with the executions of the relevant similarity transformations on the problem’s governing equations. For the asymptotic analysis, the appropriate expansions are implemented on the nonlinear system of ordinary differential equations. The bvp4c technique of the MATLAB package is implemented on the nonlinear ordinary equations for the numerical solutions. In order to explore the impact of the parameter slip factor on the pattern of velocities, temperature, and concentration profiles, several graphs are plotted. From this study, we conclude that the normalized velocities boost up with the amplification in the slip factor parameter but the temperature profile exhibits a declining behavior. The concentration field exhibits the accelerating behavior relative to the slip parameter.

Riga plate is known to create a crossing electric and magnetic field to generate a wall-parallel Lorentz force. The significance of Casson nanofluid flow past a Riga plate is observed in the sphere of engineering, such as polymer extrusion, food manufacturing, plastic films, oil reserves and geothermal manufacturing. Researchers are interested in this model because of its potential use in biological rheological models. As Casson nanofluid flows are of great interest, this study aims to investigate the three-dimensional magnetohydrodynamics (MHD) flow with heat and mass transport of Casson nanofluid over a flat Riga plate. As a novelty, this study also includes the effectiveness of wall velocity slip, activation energy, nonlinear radiation, and temperature and space-dependent heat source/sink. Suitable similarity transformations have been employed to generate the dimensionless ordinary differential equations (ODEs) from the partial differential equations (PDEs) regulating the fluid flow problem. The transformed nonlinear boundary value problem is then solved numerically using the in-built routine “bvp4c” in MATLAB. The visual demonstrations are provided for the effects of various significant physical factors on the flow, heat and mass distributions. On the other hand, wall shear stress and rates of heat and mass transport at the surface are measured and displayed numerically in tabular form. The findings indicate that the fluid velocity in both directions slows as the velocity slip parameter increases. However, the velocity profile is escalated with the boost of modified Hartmann number. An increase in heat source parameters leads to decrease the heat transmission rate at the wall. The higher values of the radiation parameter result in a better wall heat transmission rate. Further, the rate of mass transport drops when the activation energy parameter is hiked.

Squeezing or squeeze flows have tremendous applications in applied fields, like engineering, biomedical sciences and rheological studies. This paper demonstrates the squeezing flow of kerosene-based nanoliquids between parallelly aligned plates, with a Riga-type fixed lower boundary. The effects of dispersing two types of copper-functionalized carbon nanotubes (CNTs), single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) are examined. Using appropriate transformations, a self-similar ordinary differential system is derived from the governing model of partial differential equations and substantial boundary conditions. Using the homotopy analysis method (HAM) and the $bvp4c$ package, analytical and numerical estimates are obtained, respectively. For higher values of the squeezing parameter, dimensionless temperature increases, while velocity patterns are upside-down. Moreover, increments in dimensionless parameters representing Riga constituent width, modified Hartmann number, nanoparticle concentration and radiation parameter improve heat transfer rates and thermal boundary layer thickness, however, adversely affect velocity. Despite enhanced friction effects, numerical results show that the Nusselt number increases as nanoparticle loads and radiation parameters increase. This suggests that convection rates are improved over conduction rates. Excellent agreement is found between analytical and numerical evaluations. Apparently, it is noticed that MWCNTs perform better than SWCNTs.

Owing to the practical importance of nanofluids and their adjustable thermal capabilities, this study intends to develop a robust generalized differential quadrature local linearization (GDQLL) algorithm for examining realistically the heat and mass aspects of electrically conducting nanofluids during their non-Darcian laminar motion nearby a convectively heating vertical surface of an active electromagnetic actuator (i.e., Riga plate). By invoking Wakif–Buongiorno model and Oberbeck–Boussinesq approximations along with other generalized transport laws (i.e., Cattaneo–Christov and non-Fick’s laws) and Grinberg’s concept, a set of gigantic partial differential equations is stated appropriately in the sense of the boundary layer approximations for describing exhaustively the present EMHD mixed convective nonhomogeneous flow under the passive control strategy of nanoparticles within the nanofluidic medium. Operationally, the dimensionless differential forms of the governing boundary equations are derived properly by introducing reasonable mathematical adjustments into the preliminary formulation. In this case, the differential complexity of the leading differential structure is reduced to a nonlinear coupled system of ordinary differential equations, whose discrete numerical solutions are computed perfectly via a well-structured GDQLL algorithm. As foremost outcomes, it is demonstrated that the nanofluid motion and its surface thermal enhancement rate can be reinforced significantly through the thermal strengthening in the convective heating and mixed convective process as well as via the electromagnetic improvement in the driven aspect of Lorentz’s forces.

This study of electro-magneto-hydrodynamics has great significance due to its numerous applications like chromatography, fluid pumping, micro coolers and fluid stirring thermal reactors and flow regulation in fluidics systems. Subject to upper functions on electrical magnetic field, the consequences of electromagnetic initiation into ternary nanofluids flow through the Riga plate were noticed. Furthermore, this ternary hybrid nanofluid flow was based on the effect on slip condition, uniform heat source, convective energy and thermal radiation. The ternary hybrid nanofluid was built from the scattering of silver, copper and copper oxide nanoparticles by this base fluid blood. This phenomenon had been formed by the model of the system in partial differential equations; it was made easy in the dimensionless nonlinear structure of ordinary differential equations by employing comparison substitutions. This result on the acquired set of the differential equations was simulated over the bvp4c method. It has been discovered that the ternary hybrid nanofluid velocity is essentially lower along the differing numbers of permeable media, although it amplifies along the upshot on the Hartmann number. Moreover, an enhanced heat transport rate of up to 14% was marked for the triple nanoparticle nanofluid by relating it to another nanofluid and establishing an excellent behavior on triple nanoparticle nanofluids.

This paper reports the mass and energy transmission characteristics of an electrically conducting mixed convective nanofluid flow past a stretching Riga plate. An additional effect of viscous dissipation, Arrhenius activation energy and heat source is also studied. The energy and mass transmissions are evaluated by a zero-mass flux of nanoparticle and convective boundary conditions. Buongiorno’s relations are proposed for the Brownian motion and thermophoretic diffusion. The similarity substitutions are employed to derive the non-dimensional set of modeled equations. The obtained set of equations is numerically processed via parametric continuation method (PCM). Several flow factors affecting the velocity, energy, and mass distributions are graphically discussed. It has been perceived that the fluid velocity field declines with the influence of velocity power index (m), while improves with the upshot of modified Hartmann number (Q). The effect of Schmidt number and chemical reaction diminishes the concentration profile $\phi (\eta )$. Furthermore, the energy curve enhances with the effect of thermophoresis factor, Biot and Eckert number.

This investigation uses the Tiwari and Das nanofluid model to enhance the heat transfer rate in Sutterby nanofluid over a Riga plate. The effects of heat source/sink, viscosity dispersion, and mass flow for water-based fluids are also considered in this work. Sutterby fluid has been utilized to investigate the rheological features of nanofluids. The transverse Lorentz force produced by the Riga plate assists in the flow down the plate by producing an electromagnetic field. The main aim of this investigation is to evaluate the presence of two different types of nanoparticles in water, specifically silicon carbide $(\mathrm{\text{SiC}})$ and copper $(\mathrm{\text{Cu}})$. Dimensionless variables are first used to convert the mathematical model into a non-dimensional form. The similarity approach is then used to further rewrite the non-dimensional partial differential equations into a set of similarity equations. The bvp4c function in MATLAB software provides a numerical solution to these equations. The effects on temperature and velocity profiles of many physical factors, including the Reynold number, heat source/sink, and Deborah number, have been analyzed and presented. Furthermore, using tables, a detailed analysis of the skin friction coefficient and local Nusselt numbers is conducted. The results show that convective flow is suppressed when solid nanoparticles are added to the base fluid. The velocity distribution improves as Deborah and Reynold’s numbers get a higher value. Also, the temperature field improves by incrementing exponential and thermal heat source/sink parameters.

Ferro hybrid nanofluids can be used in electronics and microelectronics cooling applications to reduce heat accumulation and efficiently remove surplus heat. These nanofluids aid to maintain optimum operating temperatures and reduce device overheating by enhancing the heat transfer rate. With this motivation, the aim of the present numerical analysis is to study the three-dimensional incompressible hybrid nanofluid flow over a slippery Riga surface by combining the Casson fluid model. Mathematical modeling is constructed with nanoparticles as Fe₃O₄ and CoFe₂O₄ with base fluid as water. The non-uniform heat source/sink and thermal linear radiation effects are taken into account with the Hamilton–Crosser thermal conductivity model. A system of nonlinear PDEs is produced by the proposed problem and then the relevant similarity variables are implemented to transform the set of partial differential equations and their accompanying boundary conditions into the coupled nonlinear differential equations with one independent variable. These modified ordinary differential equations (ODEs) were then successfully solved with the Runge–Kutta fourth-order method by combining via the shooting technique. With the aid of graphical representations, the effects of various influencing parameters were presented and analyzed comprehensively. Furthermore, the impacts of relevant parameters on heat transfer rates and shear stress are concisely discussed and illustrated in tabular forms. The significant findings include the enhancement of the radiation parameter increases the thermal boundary layer thickness and the thickness increases whenever surface experiences slippery conditions. The axial and transverse momentum of the fluid are controlled with the Casson parameter. An effective connection is noticed once the current numerical solutions are verified under particular conditions that were previously described.

The importance of non-Newtonian fluid (Casson fluid) in industry is increasingly appreciated. However, little is known about the flow rheology of Casson fluid flowing over a Riga plate. Thus, the purpose of this investigation is to examine the nature of entropy generation (EG) and heat transfer (HT) on Casson hybrid nanofluid flow past a Riga plate by considering the influences of magnetic field and thermal radiation. The Hamilton–Crosser (Model 1) and Xue model (Model 2) of thermal conductivity are incorporated for Casson hybrid nanofluid. The governing equations are solved by numerical methods i.e., bvp4c and shooting techniques. In the current framework, the comparative patterns for both models of temperature, velocities, EG and Bejan number are depicted due to the existing parameters. The domain of the pertinent parameters is taken as thermal radiation, $4\le Ra\le 7$; stretching parameter, $0.6\le \lambda \le 1$; Casson factor, $0.5\le \delta \le 2$; rotation parameter, $1\le {\Re }_t\le 4$and Hartmann number, $2\le Ha\le 11$. The outcomes show that the rise in volume fraction and thermal conductivity profile of Xue model (Model 2) is better than Hamilton–Crosser model (Model 1). Moreover, EG profiles are escalated with augmentation in values of Hartmann number and stretching parameter for both models. The results of the study are useful for predicting the rheology of right fluid, while it also assists in safeguarding the boundary layer (BL) separation, along with establishing a parallel force to the surface in assisting the domain of science and technology.