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This paper signifies the consequence of the thermal diffusion (Soret) and diffusion-Thermo (Dufour) effect on MHD stagnation point flow and transportation of thermal energy near a stretching plate. However, the inclusion of thermal radiation, heat generation/absorption, and chemical reaction enriches the profiles significantly. By using similarity transformation, the set of leading equations is altered into a set of dimensionless ordinary differential equations. After that, the aim is to convert boundary layer equations into dimensionless ones so that the set of converting equations is computed by using bvp4c solver on MATLAB software. The governing nonlinear equations were resolved with Runge–Kutta fourth-order technique numerically. The significance of radiation, magnetic field, suction, Schmidt number, velocity ratio parameter, heat absorption coefficient, permeability, thermo-diffusion number, diffusion-thermo number on momentum gradient, energy gradient and concentration gradient distribution is analyzed. The parameters, diffusion-thermo number Du and thermo-diffusion number Sr have opposite behaviors on both temperature and concentration fields. Here, also, it is observed that the heat sink parameter has a noticeable influence on the fluid temperature distribution profile. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

This study emphasizes the dual exposition of the impact of convective condition and the significant effects of magnetic field, heat source and velocity slip on hybrid nanofluid flow over the exponentially shrinking sheet. The hybrid nanofluid is regarded as a contemporary variety of nanofluid; consequently, it is utilized to enhance the efficiency of heat transfer. By applying the Tiwari–Das model, the key objective of the current research is to investigate the impact of involving factors Biot number, Eckert number, volume fraction, magnetohydrodynamic (MHD), slip and suction on the temperature and velocity profiles. In addition, the Nusselt number and skin friction variations have been explored against the suction effect on the solid volume fraction of copper ϕ 2 ∈ [1–3%] and the Biot number Bi ∈ [1–3%]. The nonlinear partial differential equations are converted to a set of an ordinary differential equations by incorporating exponential similarity vectors. Eventually, ordinary differential equations are rectified utilizing MATLAB bvp4c solver. The results demonstrate the presence of dual exposition for suction with varying values of copper volume fraction and the Biot number. The heat transmission rate is observed to escalate in both cases as the strength of the Biot and Eckert numbers intensifies in the range of 1–3%. In summary, the results show that duality and non-unique solutions occur in the aiding flow situation when the suction S ≥ S ci , while there is no flow and unique solutions of hybrid nanofluid feasible when S < S ci . [ABSTRACT FROM AUTHOR]

Nanofluids may help enhance heat transfer if their thermal distribution is studied near lubricated surfaces. In view of the above scopes, this investigation examines the mass and thermal distribution of a magnified nanofluid flowing toward a lubricated surface. The mass equation describes the activation energy and chemical reaction, whereas the thermal equation is utilized to elucidate the heat source/sink impact. This investigation provides novel elements such as convective conditions, slip impact, and zero mass thermal movements near the wall. The boundary layer equations are used to construct the physical problem of the flow and are then converted into dimensionless forms by applying the necessary variables. Numerical solutions to transmuted equations are achieved using the RKF-45 technique and the shooting operation only for the power-law index m = 1 2 . The outcomes show that concentration and thermal profiles show deflection while the velocity profile has the opposite trend in the presence of slip factor. The study has significant practical value because of its widespread applicability in hydraulics, heat transfer, production, and polymer processing. [ABSTRACT FROM AUTHOR]

Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain S$$ S $$, and use the null field equation (NFE) of Green's representation formulation, where the source nodes Q$$ Q $$ are located on a pseudo‐boundary ΓR$$ {\Gamma}_R $$ outside S$$ S $$ but not close to its boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$. Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives uν$$ {u}_{\nu } $$ of the solutions on the boundary Γ(=∂S)$$ \Gamma \kern0.3em \left(=\partial S\right) $$ can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise q$$ q $$‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with O(10−4)$$ O\left(1{0}^{-4}\right) $$ can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing. [ABSTRACT FROM AUTHOR]

The influence of conjugate heat transfer based on radiation and magnetohydrodynamic utilization is an attractive research area, with progressive features; it has many applications in thermal engineering, heat exchangers, cooling phenomenon, magnetic cell separation, energy production, hyperthermia, etc. Following the motivating significances of the current research topic, this paper explores the influences of an electrically conducting and radiating fluid with internal friction, heat generation and thermal radiation embedded in a porous medium past a flat permeable plate. The boundary layer equations are dimensionally transformed and solved numerically using implicit finite difference technique called the Keller-box method. The effect of various fluid obeying parameters such as magnetic field, viscous dissipation and heat generation on the flow factors such as velocity and temperature are graphed and discussed in this paper. Growing heat source, energy inside the fluid boosts, thereby increasing the energy of the flow. Increase in generation of energy reduces the viscosity of the flow reduces thereby increasing the velocity of the flow particles. [ABSTRACT FROM AUTHOR]

In this study, magnetohydrodynamic cross fluid model is used to formulate the 2D boundary layer equations of fluid moving on the parabolic surface. The surface is assumed to be in vertical shape due to the convection. The underlying effects of viscous dissipation and chemical reaction are modeled to observe the transportation of heat and mass rate. To understand fluid behavior, the thermophysical properties are considered as variable because they have huge influence in food processing, viscometers, lubricants and various industrial works. The assumed geometry of fluid flows is similar in shape of bullet, submarine, aircraft and car's bonnet, namely paraboloid surface. The modeled equations of cross fluid with mentioned effects are obtained in form of PDEs and then converted these equations in form of ODEs by assuming set of scaling transformations. For the sake of numerical and graphical outcomes, the resulting equations are solved numerically on Matlab software via BVP4c method. Results achieved across velocity indicate that the decay happens by numerous values of Weissenberg number, power law index, viscosity parameter and Hartmann number. The temperature and concentration fields attained increasing influence by thermal conductivity coefficient, Eckert number and thermal diffusivity parameter. [ABSTRACT FROM AUTHOR]

This paper presents exact, analytical, and numerical solutions to the two‐dimensional Casson fluid boundary layer flow over a moving wedge with varying wall temperature. The boundary layer flow of the Casson fluid with varying wall temperature is governed by a system of partial differential equations called Prandtl boundary layer equations modified by Casson fluid. By applying similarity transformations the governing system of partial differential equations is reduced to a system of nonlinear ordinary differential equations called as the Falkner–Skan equation modified by Casson fluid flow with heat transfer (C‐FSEHT). In the beginning, an exact solution of the C‐FSEHT is obtained for the particular values of physical parameters (i.e., β=−1$\beta = -1$, Pr=cc+1$\text{Pr} = \frac{c}{c+1}$, N=0$N = 0$, see nomenclature) in terms of two standard functions, namely error function and exponential function. Thus, obtained exact solution is then modified to obtain the analytical solution of C‐FSEHT for general values of physical parameters, in terms of power series. The analysis of the asymptotic behavior of the problem, when the wedge velocity is very large (λ→∞$\lambda \rightarrow \infty$), is performed using the Dirichlet series. A comparative analysis is performed using the Chebyshev collocation technique (CCT) to validate the obtained results in all the scenarios. The effect of governing parameters, which are the Casson parameter c$c$, Hartree pressure gradient parameter β$\beta$, moving wedge parameter λ$\lambda$, Prandtl number Pr$Pr$, and wedge temperature parameter N$N$ on the skin friction coefficient, temperature coefficient, velocity profiles, and temperature profiles is discussed in detail. Multiple solutions are found analytically for fixed values of governing parameters. The fact that an increase in the value of the Casson parameter (c$c$) reduces the thickness of both velocity and temperature boundary layers, is also validated during the study. [ABSTRACT FROM AUTHOR]

The study of microbial flow with nanofluids involves investigating the behavior and interactions of microorganisms in the presence of nanofluids. Nanofluids are colloidal suspensions containing nanoparticles dispersed in a base fluid, and they have unique properties that can significantly impact microbial flow dynamics. This research focuses on the simultaneous effects of melting heat, Joule heating, chemical reaction, and thermal radiation on an aligned magnetohydrodynamic Williamson nanofluid (WNF) with the Cattaneo–Christov heat flux model (CCHFM) comprising microorganisms near the leading edge having stagnation flow characteristics. A melting approach is employed to study the behavior of fluid flow and heat transfer in the presence of these combined phenomena. A mathematical model is developed based on the governing equations, including conservation equations for mass, momentum, energy, concentration and microbes. To analyze the system, a set of nonlinear partial differential equations is transformed into ordinary differential equations (ODEs) by using similarity measures. The solutions of these nonlinear ODEs are obtained via the RK-4th method in MATLAB, and the results are presented in graphs and tables. Boundary layer equations for microbe propagation, fluid temperature, nano-inclusion volume fraction, and fluid velocity are formulated and discussed for various influential parameters. A higher melting parameter M b results in thicker boundary layers for velocity and temperature profiles, accompanied by an increased concentration profile for α = π 4 . Notably, the trend is more pronounced in α = π 4 compared α = π 2 . The impact of Nb and Nt on the motile profile exhibits a positive influence, leading to an increased boundary layer thickness for the gyrotactic microorganisms. The numerical analysis in Table 1 reveals that the melting heat parameter has a more pronounced impact on the motile configuration, as evidenced by the respective rates of change (- 0.076783 , - 0.337954 , 0.597858 , 0.893040) for key physical quantities compared to other configurations. The convergence and validation of the numerical solutions ensure the accuracy and reliability of the findings, establishing a foundation for further research in this area. [ABSTRACT FROM AUTHOR]

Employing variable horizontal and vertical shifts for the unperturbed solution, a new perturbation method is proposed. The new method is named as the shift perturbation method. The existing perturbation methods are interpreted with respect to the horizontal and vertical shifts for the unperturbed solution. The new method is applied to some well-known vibrational problems. The method is capable of producing admissible approximate solutions for a wide range of problems such as the free and forced vibrations of the Duffing equation, equations with quadratic and cubic nonlinearities, variable amplitude nonlinear problems, and boundary layer problems. A variant of the method also produced valid solutions for large perturbation parameters. [ABSTRACT FROM AUTHOR]

The laminar conjugated natural convection heat transfer from a hollow cylinder is a common problem in engineering. The accurate calculation of the hot-spot temperature helps to monitor and evaluate the operating status of the equipment. In this paper, first, a mathematical model of the conjugated natural convection from a hollow cylinder is established. The boundary layer integral equations under arbitrary heat flux distribution on the surface are derived, and the optimal velocity and temperature profiles corresponding to different Prandtl numbers are determined. By combining the integral equations and the thermal network model, the hot-spot temperature and the average surface temperature of a hollow cylinder are solved iteratively. Subsequently, a computational fluid dynamic (CFD) simulation based on finite volume method is carried out to verify the accuracy of the integral solution method. The maximum relative error of the computed results of the two methods is 1.35%. The integral solution method performs much better in computation speed, increasing by 90 times. It is also found that as the thermal conductivity of solid materials decreases, although the hot-spot temperature increases, the average surface temperature remains basically unchanged. Finally, based on the lumped thermal equivalent circuit, the correlation of the Nusselt number corresponding to the hot-spot temperature is predicted. The parameters in the correlation are estimated by regression orthogonal design. The results show that under a wide range of solid thermal conductivity and fluid Prandtl numbers, the maximum error of the correlation is 6.02%. Therefore, this novel method is feasible and significant for calculating hot-spot temperature. [ABSTRACT FROM AUTHOR]

This article aims to achieve exact and analytical solutions for the classical Falkner–Skan equation with heat transfer (FSE‐HT). Specifically, when the pressure gradient parameter β=−1 $\beta =-1$, there already exists a closed‐form solution in the literature for the Falkner–Skan flow equation. The main purpose here is to extend this case to obtain a closed‐form solution to the heat transport equation with the solubility condition Pr=1 $Pr=1$. An algorithm is presented and is found to be new to the literature that enriches the physical properties of FSE‐HT. It is shown that for the moving wedge parameter λ>1 $\lambda \gt 1$, the momentum and temperature equations show multiple solutions analytically. The skin friction coefficient and the heat transfer rate are also obtained in analytical form. The thus‐obtained solution is then adapted to derive an analytical solution applicable to a wide range of pressure gradient parameters β $\beta $ and Prandtl numbers Pr $Pr$. Furthermore, an asymptotic analysis is conducted, focusing on scenarios where the moving wedge parameter becomes significantly large (λ→∞ $\lambda \to \infty $). Nevertheless, in all the above‐mentioned cases, the skin friction coefficient (f″(0) $f^{\prime\prime} (0)$) and the heat transfer rate (θ′(0) $\theta ^{\prime} (0)$) are compared with the direct numerical solutions of the boundary layer equations, and it is found that the results are in good agreement. These solutions provide a benchmark and shed light on further studies on the families of FSE‐HT. [ABSTRACT FROM AUTHOR]

The present project highlights the behavior of the unsteady heat transfer phenomenon developing along a horizontal surface in terms of both Strouhal and Prandtl numbers. Based on the changes that occur in the gouverning equation of the studied problem, an adequate analytical law of the velocity is proposed to solve unsteady momentum equation. This result presented a good agreement with Rayleigh's exact solution and numerical solutions of Blasius and Williams-Rhyne for diferents values of Strouhal numbers adopted in this study. The obtained velocity expression is included in the unsteady energy equation in order to establish the temperature profile for all considered Strouhal and Prandtl numbers. Taking into account the existence of the velocity-temperature coupling in the heat boundary layer equation, the proposed formula is used to solve unsteady energy equation for all Strouhal and Prandtl numbers. As the main results, a new analytic expression of the local heat transfer coefficient for all Strouhal and Prandtl numbers is established and interesting curves are plotted to explain the heat transfer evolutions from diffusion flow to the convective flow. [ABSTRACT FROM AUTHOR]

This paper provides a comparative numerical analysis of unsteady hybrid nanofluid (Ag + TiO 2 /H2O) and nanofluid (TiO2/H2O) flows through a permeable linear stretching cylinder placed in a porous medium has been presented in the presence of oblique Lorentz force. The flow regulating boundary layer equations has been governed due to the sway of the suction, viscous-Ohmic dissipation, heat source, thermal radiation and first-order velocity slip with imposing convective boundary conditions on the surface. The self-similar transformation has been employed to convert the governing coupled nonlinear PDEs into a set of ODEs and finally solved them numerically by fifth-order Runge–Kutta method with shooting technique. Finally, a comparative study has been made on the thickness of the momentum and thermal boundary layers due to the effect of governing parameters for simple and hybrid nanofluids. Further, a numerical observation on local skin friction and thermal-transport coefficients is presented with the help of tables. The result indicates that the progressiveness of the velocity profile reduces for the unsteadiness parameter, Hartree pressure gradient, velocity slip parameter, inclination parameter, porous medium parameter, magnetic parameter and suction parameter while improving for velocity ratio parameter and temperature profiles increases with Biot number, Eckert number, heat generation parameter and thermal radiation parameter, but decreases for Hartree pressure gradient and velocity ratio parameter. This work has found various applications in high-temperature and cooling processes, paints, conductive coatings, medicines, space technology, biosensors, conductive coatings and so on. [ABSTRACT FROM AUTHOR]

We consider in a smooth and bounded two dimensional domain the convergence in the L 2 norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, under general regularity on the initial conditions of the Euler equations, that assuming the dissipation of the energy of the solution of the Navier-Stokes equations in a Kato type boundary layer, then the inviscid limit holds. [ABSTRACT FROM AUTHOR]

Near-space hypersonic vehicles could encounter significant rarefied nonequilibrium effects during the flight through atmosphere, which largely influence the gas-surface momentum and heat transfer. In this paper, hypersonic boundary layer over a flat plate with velocity slip, temperature jump, and shear nonequilibrium effects is theoretically considered. The slip boundary conditions and nonlinear transport relations are embedded into the boundary-layer equations to describe the flow. Local similar solutions are derived, and key parameters for characterizing slip and shear nonequilibrium effects are determined. The velocity-slip and temperature-jump effects are determined by [ (2 − σ u) / σ u ] M a e / R e x and [ (2 − σ T) / σ T ] M a e / R e x respectively, and the shear nonequilibrium effect is characterized by M a e 2 / R e x . The obtained boundary-layer solutions are compared with the Navier–Stokes solutions for a Mach 4.5 slip flow, and the results of Direct Simulation Monte Carlo for a Mach 10 rarefied flow, good agreements are achieved. The separate and combined effects of velocity slip, temperature jump, and shear nonequilibrium on boundary-layer solutions and momentum/heat transfer are clarified. The results show that both the slip and shear nonequilibrium effects cause the boundary layer to become thinner and decrease the skin friction and Fourier heat conduction. However, with including sliding friction, the total heat flux might even increase as the slip degree increases. These results provide valuable insight into the boundary-layer characteristics of hypersonic near-continuum flows. [ABSTRACT FROM AUTHOR]

As a typical nonlinear wave, forward-leaning waves can be frequently encountered in the near-shore areas, which can impact coastal sediment transport significantly. Hence, it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately, especially for the turbulent boundary layer. In this work, the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied, and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer. Then, a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function. Consequently, an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation. The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature. [ABSTRACT FROM AUTHOR]

Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved. [ABSTRACT FROM AUTHOR]

The current study adopts a Bernoulli wavelet technique to explore the magnetohydrodynamic (MHD) boundary layer flow of a Casson fluid through a wedge. A nanofluid across a wedge has been investigated for its stable laminar MHD flow, heat, and mass transport characteristics. By choosing worthwhile non‐dimensional parameters, the governing boundary layer equation is reformed into a dimensionless Falkner–Skan equation. The consequent nonlinear equation is addressed via the Bernoulli wavelet method. For a range of different values of the physical parameters, the nature of the boundary layer flow is visually observed. Fluid velocity rises with rise in the values of Casson parameter, wedge parameter, and magnetic parameter. Local wall skin friction increases with increase in wedge parameter and magnetic parameter values. To ensure the accuracy of the findings, local wall skin friction is estimated and tested with other techniques that are already reported in the existing research. [ABSTRACT FROM AUTHOR]

This work leaves to explore the stability of a 2‐D, time free, thermal boundary layer of graphene oxide ethylene glycol (GO‐EG,) where GO‐blood is taken as base nanofluids over a stretchable surface by executing a magnetic field (MHD), coupled stress effects, heat source/sink, and sun‐oriented radiations. This work is done due to the reason that nanofluids have a higher heat conductivity level rather than regular liquids. Nonlinear partial differential equations (N‐PDEs) are modeled utilizing the notable boundary layer equations. Dimensionless ordinary differential conditions (ODEs) are made from the created NPDEs by utilizing dimensional analysis and group theoretic approach. We examine the obtained ODEs through the notable homotopy analysis technique (HAM). The impacts of different parameters on temperature distribution and velocity profiles are shown graphically. However, some tables are constructed in support of the presented study to confirm the stability and convergence of the solutions. The main intentions in this work are to underline the enhancement of traditional heat transfer's thermal conductivity. [ABSTRACT FROM AUTHOR]

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O (ν 1 / 4) order terms in L ∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν = 0 , are nonlinearly unstable when ν > 0 , provided ν is small enough, up to O (ν 1 / 4) terms in L ∞ norm. [ABSTRACT FROM AUTHOR]

It has recently been established [Naghshineh et al. IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the series itself. Here, we provide exact and explicit analytical expressions for these parameters, including that associated with wall shear, thus completing the exact analytical solution. The resulting solution to the Sakiadis problem is summarized here for direct use. [ABSTRACT FROM AUTHOR]

Purpose: This work aims to concentrate on the mixed convection of the stagnation point flow of ternary hybrid nanofluids towards vertical Riga plate. Aluminum trioxide (Al2O3), silicon dioxide (SiO2) and titanium dioxide (TiO2) are regarded as nanoparticles, with water serving as the base fluid. The mathematical model incorporates momentum boundary layer and energy equations. The Grinberg term for the viscous dissipation and the wall parallel Lorentz force coming from the Riga plate are taken into consideration in the context of the energy equation. Design/methodology/approach: Through the use of appropriate nonsimilar transformations, the governing system is transformed into nonlinear nondimensional partial differential equations (PDEs). The numerical method bvp4c (built-in package for MATLAB) is used in this study to simulate governing equations using the local non-similarity (LNS) approach up to the second truncation level. Findings: Numerous graphs and numerical tables expound on the physical properties of the nanofluid temperature and velocity profiles. The local Nusselt correlations and the drag coefficient for pertinent parameters have been computed in tabular form. Additionally, the temperature profile drops while the velocity profile increases when the mixed convection parameter is included to oppose the flow. Originality/value: The fundamental goal of this work is to comprehend how ternary nanofluids move towards a vertical Riga plate in a mixed convective domain with stagnation point flow. [ABSTRACT FROM AUTHOR]

We introduce a scalar elliptic equation defined on a boundary layer given by Π2 × [0, ztop], where Π2 is a two dimensional torus, with an eddy vertical viscosity of order zα, α∈ [0, 1], an homogeneous boundary condition at z = 0, and a Robin condition at z = ztop. We show the existence of weak solutions to this boundary problem, distinguishing the cases 0 ≤ α < 1 and α = 1. Then we carry out several numerical simulations, showing the ability of our model to accurately reproduce profiles close to those predicted by the Monin–Obukhov theory, by calculating stabilizing functions. [ABSTRACT FROM AUTHOR]

The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher--Kolmogorov--Petrovsky--Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event. [ABSTRACT FROM AUTHOR]

The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances. [ABSTRACT FROM AUTHOR]

This paper presents a numerical integration method for the solution of 'Singularly Perturbed Differential-Difference Equations' having dual layers. It is well known that when we use already existing numerical methods to solve such problems, we get oscillatory or unsatisfactory results unless we take a very small step size, which is time-consuming and costly. To get a numerical solution for such a problem, first, the delay and advanced parameters present in the SPDDE are approximated by Taylor's series to get an equivalent 'Singularly Perturbed Differential Equation' of second order. Second, an asymptotically equivalent first-order differential equation is obtained from SPDE using Taylor's transformation. Composite Simpson's 1/3 rule is implemented to get a three-term recurrence relation. The Thomas algorithm is applied to get the solution of the tri-diagonal system of equations. Several model examples are tested and it was found that the numerical solution approximates the available/exact solution very well. [ABSTRACT FROM AUTHOR]

Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended formof the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from t to t1/2, in agreement with experiments. [ABSTRACT FROM AUTHOR]

The symmetries of the governing equations of atmospheric flows constrain the solutions. The present study applies those symmetries identified from the governing equations to the atmospheric boundary layers under relatively weak stratifications (stable and unstable). More specifically, the invariant solutions are analyzed, which conserve their forms under possible symmetry transformations of a governing equation system. The key question is whether those invariant solutions can rederive the known vertical profiles of both vertical fluxes and the means for the horizontal wind and the potential temperature. The mean profiles for the wind and the potential temperature in the surface layer predicted from the Monin–Obukhov theory can be recovered as invariant solutions. However, the consistent vertical fluxes both for the momentum and heat no longer remain constant with height, as assumed in the Monin–Obukhov theory, but linearly and parabolically change with height over the dynamic sublayer and the above, respectively, in stable conditions. The present study suggests that a deviation from the constancy, though observationally known to be weak, is a crucial part of the surface-layer dynamics to maintain its symmetry consistency. Significance Statement: The atmospheric flows are governed by a differential equation system, which is often difficult to solve in any satisfactory manner, either analytically or numerically. However, without solving them explicitly, many insights can be obtained by examining the "symmetries" of the governing equations. The study suggests that basic vertical profiles of the mean state of the atmospheric boundary layer is more strongly constrained by the symmetry consistency than suggested by standard similarity theories. [ABSTRACT FROM AUTHOR]

In this work, we present an investigation about the sources of dissipation in adiabatic boundary layers of non-ideal compressible fluid flows. Direct numerical simulations (DNS) of transitional, zero-pressure gradient boundary layer flows are performed for two fluids characterized by different complexity of the fluid molecules, namely, "air" and siloxane MM. Different sets of thermodynamic free-stream boundary conditions are selected to evaluate the influence of the fluid state on both the frictional loss and the dissipation mechanisms. The thermophysical properties of siloxane MM are calculated with a state-of-the-art equation of state. Results show that the dissipation due to both time-mean strain field, irreversible heat transfer, and turbulent dissipation differs significantly depending on both the molecular complexity of the fluid and its thermodynamic state. The dissipation coefficient calculated from the DNS results is then compared against the one obtained using a reduced-order model (ROM), which solves the two-dimensional boundary layer flow equations for an arbitrary fluid [M. Pini and C. De Servi, "Entropy generation in laminar boundary layers of non-ideal fluid flows," in 2nd International Seminar on Non-Ideal Compressible Fluid Dynamics for Propulsion and Power (Springer, 2020), pp. 104–117]. Results from both the DNS and the ROM show that low values of the overall dissipation are observed in the case of fluids made of simple molecules, e.g., air, and if the fluid is at a thermodynamic state in the proximity of that of the vapor–liquid critical point. [ABSTRACT FROM AUTHOR]

We present a new formulation and novel scaling relations for natural convection boundary layers on horizontal surfaces with weak transpiration velocities Vi, for intermediate Schmidt numbers (or Prandtl numbers Pr) 1 < S c < 20. Using characteristic scales derived from integral boundary layer equations, we define a new similarity variable η = (y / x) (R e x / G r x) 1 / 4 / S c 1 / 5 that has Vi within it, where Rex and Grx are the local Reynolds and Grashof numbers based on Vi and the concentration difference Δ C , respectively. Using η, and the dimensionless transpiration velocity ξ = (R e x / G r x 1 / 5) 5 / 4 as a non-similarity variable, we obtain novel, reduced, non-similar boundary layer equations. Numerical solution of these reduced boundary layer equations using a local non-similarity approach gives us the profiles of horizontal velocity and concentration, boundary layer thicknesses, wall shear stress, and mass flux as functions of Sc, for various ξ in the range ξ < 1. Using normalizing functions, we then obtain scaling relations for these parameters; we observe extended similarity where the non-similarity parameter ξ itself appears in the obtained similarity scaling relations. [ABSTRACT FROM AUTHOR]

This paper demonstrates a computational method to solve equations for a thermoacoustic boundary layer in a gas-filled tube subjected to a temperature gradient axially. With applications to numerical simulations of thermoacoustic systems, the purpose is to establish a method to evaluate numerically a velocity vb at the edge of a boundary layer directed into a core region in the outside of the layer. The computational method exploits a series expansion in terms of Chebyshev polynomials. Solved is a benchmark example for which analytical solutions are available. Assuming that a quiescent gas is started by an impulse and an acoustic field in the core region is given, an axial velocity and a temperature disturbance in the boundary layer are sought, from which vb is derived. Comparing the numerical solutions obtained against the analytical ones, it is found that the acoustic field and ultimately vb are well obtained with good accuracy. It is thus concluded that the present computational method will apply to numerical simulations based on the boundary-layer theory. [ABSTRACT FROM AUTHOR]

Earlier research work related to the unsteady flow of thin liquid film on a heated stretching cylinder has considered the boundary‐layer approximation under the assumption that the film thickness is thick. In this article, we have considered the full set of Navier–Stokes equations to study the unsteady film development over a porous heated stretching cylinder with the assumption that film height either coincides or lies within the boundary‐layer thickness. The effects of magnetic field, suction/injection, and cooling or heating of the cylinder have been considered for investigation. The governing set of coupled nonlinear partial differential equations is solved numerically by the implicit finite difference scheme (Crank–Nicolson). It was found that the film thinning rate diminishes with rising values of the porosity parameter and Hartmann number. It is also noted that the film thickness is enhanced for injection whereas opposite results are observed for suction through the porous cylinder. The thermocapillary effect showed that the thinning rate of the liquid film decreases or increases according to the surface of the cylinder being heated or cooled. It is observed that continuously increasing the stretching speed of the cylinder produces faster thinning of the liquid film. The temperature of the liquid film is the maximum at the surface of the cylinder and it decreases toward the free surface when the cylinder is heated. But the opposite phenomenon occurs when the cylinder is cooled. [ABSTRACT FROM AUTHOR]

The study of the boundary layer is considered one of the most important theories in the field of heat and mass transfer because of its important explanation that shows us the behavior of different surfaces while they are under the influence of the flow accompanied by different thermal forces. The study of corrugated surfaces is one of the engineering applications, such as flow in heat exchangers or solar cells or cooling processes during surface heat treatments. This model is also used in medical applications such as flow in arteries or movement in the intestines. So, the work deals with investigating the boundary layer surrounding a three‐dimensional sinusoidal pipe; the boundary layer was assumed to be filled with a hybrid nanofluid consisting of water +Cu nanoparticles as the main fluid, supported by a small concentration of Al2O3 or Ag nanoparticles. The boundary layer is described by a set of nonlinear partial differential equations due to the continuity, momentum, and energy equations, which are transformed into a set of dependently coupled nonlinear ordinary differential equations. The obtained system of equations was solved using numerical techniques. The behavior of the boundary layer under the varying types and concentrations of nanoparticles and the influence of the magnetic field has been depicted by a set of graphs and tables. With reference to some results, it is found that using 5% of nanoparticles of aluminum oxide raises the rate of cooling by 8% and using 5% of silver nanoparticles increases it by 5%. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the effect of non-uniform heat generation and viscous dissipation on the boundary layer flow of a power-law nanofluid over a nonlinear stretching sheet. Within the thermal domain, the analysis considers both thermal radiation and variable thermal conductivity. Through the use of similarity transformations, the governing boundary layer equations are transformed into a system of ODEs. The spectral collocation method (SCM) with shifted Vieta-Lucas polynomials (VLPs) is implemented to give an approximate expression for the derivatives and then use it to numerically solve the proposed system of equations. By employing this technique, the system of ODEs is converted into a system of nonlinear algebraic equations. The dimensionless temperature, concentration, and velocity are graphically presented and analyzed for various values of the relevant governing parameters. Through the presented graphical solutions, we can see that the main outcomes indicate that an increase in the power law index, thermal conductivity parameter, and radiation parameter leads to a noticeable decrease in the local Nusselt number, with reductions of around 0.05 percent, 0.23 percent, and 0.11 percent, respectively. In contrast, the Prandtl parameter demonstrates an opposing effect, elevating the local Nusselt number by about 0.1 percent. We validated the accuracy of the numerical solutions by comparing them in some special cases with existing literature. [ABSTRACT FROM AUTHOR]

Within the framework of a three-dimensional boundary layer, the region of developed turbulent flow of an incompressible fluid under the action of a longitudinal unfavorable (positive) pressure gradient (hereinafter, referred to as UPG) in the diffuser is simulated. For a turbulent flow regime, the Reynolds-averaged Navier–Stokes equations in the boundary layer approximation are closed using a differential turbulence model based on the introduction of turbulent viscosity and the Kolmogorov–Prandtl hypotheses. Using the nonlinear least squares method (NLSM), correlation dependencies are found for the initial experimental data, which are further used in the numerical simulation. The calculated and experimental velocity profiles are compared in the region close to separation. Based on the numerical solutions of the turbulent boundary layer equations, the mechanisms of the interaction of the flows in the presence of transverse pressure gradients in an incompressible fluid are studied. [ABSTRACT FROM AUTHOR]

The compressible two-dimensional turbulent flow solutions at an arbitrary point in time and space by incorporating the mass, momentum, and energy conservation equations over a smooth flat surface and parallel free stream with unfavorable pressure gradient are studied. The Falkner–Skan transformation is applied to the turbulent boundary-layer equations and related boundary conditions, and the resulting nonlinear coupled system of partial differential equations is solved by the bivariate spectral quasi-linearization method. Moreover, to predict the thermal distribution of the flow, an artificial neural network model has been developed with the Nusselt number as target values. Several plots have been depicted, it is evaluated that the mean squared error value is 6.41 × 10−7, the overall coefficient of determination (R) is 0.997 52, and the average error rate is 0.68% for the said model, indicating the attainment of high accuracy for estimation. [ABSTRACT FROM AUTHOR]

When a static fluid encounters a moving boundary, it experiences a large shear and forms a boundary layer. A self-similar solution of the boundary-layer equations for such flow was first revealed by Sakiadis in 1961. Despite the ubiquity of this type of flow, there are so far no published numerical simulations. In this article, we use OpenFOAM, a widely used open source software, to conduct a numerical simulation of the isothermal Sakiadis flow. The results are in good accord with the theoretical solution except near the leading edge, where the boundary-layer approximations are not fulfilled. We present that the boundary layer thickness is not zero at the beginning of the boundary-layer flow, although this condition has been extensively used. Currently, in boundary-layer research different definitions of boundary layer thickness are being employed. We also show that depending on the definition used, self-similarity appears at different stream-wise positions. The widest range of self-similarity can be obtained by using the definition of momentum thickness. Finally, we also present a new self-similar solution in wall normal direction near the leading edge. These results obtained from the simulation might well be applicable to many other boundary-layer flows, such as the Blasius flow. [ABSTRACT FROM AUTHOR]

A mathematical analysis for a steady, incompressible, laminar Carreau nanofluid flow over a porous cone and wedge is considered. The novelty of this research work is to scrutinize an exponential space-dependent Cattaneo–Christov heat flux and quadratic Rosseland approximation effects. Two-phase nanofluid model, i.e., Brownian motion, and thermophoretic effects are included. This study also employs non-linear thermal radiation and convective surface boundary conditions to investigate heat transport phenomena. The Carreau nanofluid boundary layer equations are derived using the standard boundary layer approximations. By applying the similarity transformations, the managing set of partial differential equations (PDEs) becomes a system of connected non-linear ordinary differential equations (ODEs). The resultant non-linear ODEs are solved numerically utilizing the numerical method, particularly the bvp4c function in MATLAB. The significances of the fluid motion are visualized by sketching several Carreau nanofluid flow's relevant parameters using graphic outputs. The effects of a wide range of thermophysical variables on liquid properties including velocity, temperature, concentration, skin-friction, Nusselt number, and Sherwood number are investigated and discussed. This study's findings are compared to previously announced results within a limited range, where strong validation was seen. The results demonstrate that the opposite mechanism is observed with repercussion of Weissenberg number (We) on velocity profile and concentration, the enhancement of We increases the momentum boundary layer while it reduces the concentration boundary layer. The outcomes could be applied to the cooling of equipment, electronics, and various industrial units. The two-dimensional Carreau nanofluid over porous cone and wedge is considered. Heat sources that are exponentially space-dependent and quadratic are exposed to the flow. The addition of the Cattaneo-Christov heat flow enhances the novelty and the chemical reaction term broadens the scope of existing work. A MATLAB built-in function is used to integrate coupled equations with boundary conditions numerically using a bvp4c approach, and their results are in good agreement with the prevailing results available in the literature. [ABSTRACT FROM AUTHOR]

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast "far away" from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions. [ABSTRACT FROM AUTHOR]

The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities. [ABSTRACT FROM AUTHOR]

This study explores the consequences of free convective laminar two-dimensional flow with the effects of thermal radiation of power-law non-Newtonian water–Cu nanofluids over the frustum of a cone whose sides are wavy shaped. The flow is described by some well-known equations of fluid dynamics, such as the equation of continuity, momentum and energy equation having the Rosseland diffusion model for radiation effect. These governing equations are transformed into non-dimensional boundary-layer equations, which are solved by using the marching order implicit finite difference method. The numerical results are obtained by varying pertinent parameters, such as the radiation–conduction parameter R d , the surface heating parameter θ w , power-law index n and nanoparticle volume fraction ϵ . The numerically calculated findings are reported with respect to the velocity and temperature distributions, streamlines, isotherms, local skin friction coefficient and average Nusselt number using graphs. According to the obtained results, R d , ϵ and θ w accelerate the convection process. Hence, velocity, skin friction, as well as the rate of heat transfer enhance. On the other hand, an increasing power-law index decreases the convection process. As a result, velocity, skin friction as well as the rate of heat transfer reduce. The present study of non-Newtonian nanofluid with radiation effect over a truncated wavy cone is a novel study. The motto of the existing numerical effort is to observe the situations considering non-Newtonian nanofluid fluids and how the considered parameters reflected their behaviour with non-Newtonian viscosity and radiation–conduction effects. The obtained result of this present study is validated with the work of Pop and Na (Int. J. Nonlinear Mech.34, 925, 1999). The findings of their investigation of free convective flow in a vertical cone are compared with the current work and the comparison showed an excellent agreement. [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness theory in lower regular Sobolev space for the Prandtl-type equation arising from the inhomogeneous incompressible MHD equation in dimension two. Due to the degeneracy of horizontal dissipation, the well-posedness result of MHD boundary layer equation is established in higher regular Sobolev space rather than lower one generally. By using the anisotropic Sobolev inequality and characteristic line method, the well-posedness result of inhomogenous MHD Prandtl-type equation can be established under the H 2 − framework. [ABSTRACT FROM AUTHOR]

We develop an asymptotic theory of the compressible turbulent boundary layer on a flat plate, in which the mean velocity and temperature profiles can be obtained as exact asymptotic solutions of the boundary-layer equations, which are closed using functional relations of a general form connecting the turbulent shear stress and turbulent enthalpy flux to mean velocity and enthalpy gradients. In this part of the study, we consider the near-wall region that consists of viscous and logarithmic sublayers. The solution is constructed in the form of expansions in a small parameter ε that is proportional to the Mach number formed with the friction velocity and the speed of sound on the wall. Three characteristic flow regimes are possible in the viscous sublayer, which occur at small (including zero), moderate, and large negative wall heat flux. For the first two regimes, the flow is incompressible to the first approximation, while the compressibility is significant in the viscous sublayer on a strongly cooled plate. The Crocco integral in the logarithmic region is obtained, which in the zeroth-order approximation in ε gives the Waltz equation, but in contrast to it, the new relation describes well the dependence of temperature on velocity for any heat flux on the wall. Along with the constants known for incompressible flow, the theory contains two new universal constants, which are determined from a comparison with direct numerical simulation data for velocity and temperature. [ABSTRACT FROM AUTHOR]

The turbulent boundary layer on a flat plate in a supersonic flow is considered. The study consists of several parts. In this first one, we prove that the turbulent shear stress and turbulent enthalpy flux can be represented as functions of dimensionless parameters that are expressed in terms of the distance from the wall and mean velocity and enthalpy gradients. The existence of such functional relationships is a consequence of the fact that the flow on the plate is completely determined by a finite set of constant quantities: the free-stream parameters, the wall enthalpy, and the parameters specifying the dependence of viscosity on temperature and the equation of state for a perfect gas. The asymptotic structure of the relationships between the mean quantities is derived from the properties possessed by the solutions of the Navier–Stokes and energy equations. The established functional relations are closure relations of a general form, which together with the boundary-layer and energy equations give a well-posed problem for the velocity and enthalpy fields. For various characteristic flow regions in the boundary layer, this problem will be solved in subsequent parts of the study. This will make it possible to develop a theory of compressible boundary layers based on first principles, without involving particular hypotheses and approximate turbulence models. [ABSTRACT FROM AUTHOR]

In this paper, some rational high-accuracy compact finite difference schemes on nonuniform grids (NRHOC) are introduced for solving convection–diffusion equations. The derived NRHOC schemes not only can suppress the oscillatory property of numerical solutions but can also obtain a high-accuracy approximate solution, and they can effectively solve the convection–diffusion problem with boundary layers by flexibly adjusting the discrete grid, which can be obtained with the singularity in the computational region. Three numerical experiments with boundary layers are conducted to verify the accuracy of the proposed NRHOC schemes. We compare the computed results with the analytical solutions, the results of the rational high-accuracy compact finite difference schemes on uniform grids (RHOC), and the other schemes in the literature. For all test problems, good computed results are obtained with the presented NRHOC schemes. It is shown that the presented NRHOC schemes have a better resolution for the solution of convection-dominated problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of an unsteady 2-D MHD dynamic and thermal boundary-layer over a horizontal cylinder in mixed convection, in the presence of suction/injection, heat source/sink, and heat radiation fluid electrical conductivity is constant. The system of MHD equations of dynamic and temperature boundary-layer, which describe complex non-auto model problems, has been solved by a new approach. New variables and sets of parameters were introduced and transformed equations were obtained, in which the influence of the magnitude Z was explicitly retained. In order to close the system of equations, to the equations of the boundary-layers momentum equation was added. The solution of the obtained system of non-linear differential equations was performed numerically using the finite difference method, with the simultaneous application of the iteration method. By replacing the derivatives in the system of equations with the corresponding relations of finite differences, a system of linear derivative algebraic equations is obtained, which is solved by the three-diagonal method. As a concrete example of the introduced method, the effects of heat transfer in the MHD boundary-layers were considered, in the case of mixed convention, over a horizontal circular cylinder. The boundary conditions for temperature are defined by linear functions of longitudinal co-ordinates and time. Numerical results for different Eckart, Schmidt, and expanded Prandtl numbers and values: magnetic, dynamic, thermal parameters, temperature, and buoyancy parameters were obtained and presented. The obtained results were analyzed through the diagrams of changes in velocity and temperature and the diagrams of integral and differential characteristics of boundary-layers and the corresponding conclusions were given. [ABSTRACT FROM AUTHOR]

This paper aims to analyze the problem with the study of thermal and momentum transport with entropy generation in view of the second law of thermodynamics in Magneto hydrodynamics (MHD) micropolar fluid through porous medium under the consideration of the non-Darcy model, temperature-dependent viscosity and thermal conductivity. In practical situations at higher temperatures and high speed fluid flow, it becomes reasonable to consider variable fluid flow parameters. The governing boundary layer flow equations are first converted into a coupled system of the ordinary differential equations (ODE) under the assumption of differing plate temperatures by applying appropriate similarity transformations. A shooting method has been applied to solve ordinary differential equations numerically. The last effect of microrotation, magnetic field, variable viscosity coefficient, variable thermal conductivity, etc. on momentum and thermal transport has been depicted through various graphs. The table for skin friction coefficient and Nusselt number for ideal cases has been shown to validate the model by previous findings. It is seen that K and m enhance the velocity profile on their increment opposite to this M, δ , F and Da have been found to reduce the velocity profile. Table 3 is constructed for numerical values of skin friction coefficient and Nusselt number for different values of parameters where it can be concluded that magnetic parameter M has a tendency to enhance the skin friction and heat transfer, while variable viscosity parameters have a tendency to decline the skin friction and heat transfer. [ABSTRACT FROM AUTHOR]