Falkner Skan Research Articles

Streamwise-Elongated Sinusoidal Roughness Elements with Enhanced Laminarizing Effect on Three-Dimensional Boundary Layer

As a laminar flow control device for delaying the crossflow-induced transition of a three-dimensional boundary layer, sinusoidal roughness elements (SREs) are placed in a Falkner–Skan–Cooke boundary layer, and the resultant laminarizing effect is numerically investigated in comparison with discrete roughness elements (DREs). Because SREs are elongated in the streamwise direction and designed to avoid flow tripping, the critical height of SREs is much higher than that of DREs. Moreover, the wake flow behind SREs efficiently generates and sustains crossflow vortices that are not dangerously unstable against secondary instabilities but able to strongly distort the mean crossflow profile into a less unstable one. By measuring this mean flow distortion by SREs and DREs, the laminarizing effect is compared among them. It is shown that the effect of SREs is higher than that of DREs and can be enhanced by choosing the appropriate height, angle, and wavelength depending on the local boundary-layer profile.

Entropy generation and heat transfer characteristics in MHD Casson fluid flow over a wedge with viscous dissipation and thermal radiation

AbstractThis investigation outlines the significance of MHD Falkner–Skan flow of non‐Newtonian fluid flow over a wedge. To study the non‐Newtonian flow, Casson fluid is taken. Additionally, this study explores heat and mass transport under the influence of viscous dissipation, Joule heating, and thermal radiation. This heat and mass transport investigation is carried out under the influence of thermal and concentration slip factors. Moreover, the entropy generation is also computed using the second law of thermodynamics. A set of nonlinear partial differential equations arises from the mathematical formulation of the problem. Similarity variables are then introduced to achieve a similarity solution. The leading differential equations are solved numerically using the Runge–Kutta‐4 method in conjunction with shooting techniques. Graphical representations are employed to demonstrate the physical significance of relevant parameters. The investigation presents and discusses the impact of various parameters on velocity, temperature, concentration and entropy profiles for three different positions of the wedge: stationary, forward‐moving, and backward‐moving. The main conclusions of the study are as follows: (1) Enhancing the magnetic parameter and wedge angle parameter leads to higher fluid velocity, (2) elevating both the Eckert and magnetic number results in a rapid escalation of fluid energy, (3) Skin friction coefficient increases gradually with an increase in the power law Falkner–Skan parameter.

Computational micropolar model of hybrid nanofluid flow across a wedge

Non-Newtonian flow from wedges is a vital problem in coatings, polymer processing etc. Hence the main aim is to investigate the convective boundary layer flow of micropolar hybrid nanofluid (MHN)from a wedge. It is assumed that the wedge surface is isothermal. The Eringen model is employed to define micropolar fluid. Nanoscale Tiwari–Das formulations are used to study the specific effects of nanoparticles and volume fractions. The dimensionless boundary value problem arises by suitable coordinate transformations as a system of nonlinearly coupled ordinary differential equations. The so-called Falkner Skan flow case is resolved. Dimensionless, transformed, coupled momentum, microrotation, boundary layer equations are resolved using the numerical scheme MATLAB bvp4c. The parameteric investigations are performed on hybrid nanofluids using water as the basis liquid and varying volume fractions from 0 to 8%. Affirmation with previous work is done. The consequence of Eringen micropolar parameter, Hartree pressure gradient parameter, nanoparticle volume fraction, Eckert number, heat absorption (sink) parameter, on the flow and physical characteristics are visualized graphically and in tables. With rising material factor and volume fraction of nanoparticle, temperature is markedly enhanced. Increases in volume fraction dampen angular velocity close to the wedge surface, while farther from the wall, the opposite effect is seen. Temperature and thermal boundary layer thickness are both greatly enhanced by increasing Eckert number. With an increase in the volume proportion of nanoparticles, velocity decreases. Heat generation raises temperatures while a heat sink lowers them. The pressure gradient parameter increases skin friction while decreasing Nusselt number. Nusselt number for hybrid nanofluid is higher than for nanofluid.

Falkner–Skan boundary layer flow of a fluid with pressure-dependent viscosity past a stretching wedge with suction or injection

In this paper we study the boundary layer flow of a particular class of non-Newtonian fluids, namely piezo-viscous fluids, past a stretching wedge (Falkner–Skan flow). These fluids are essentially Navier–Stokes fluids with pressure-dependent viscosity. We consider the two-dimensional steady flow and introduce some similarity variables, so that the problem is reduced to a third-order nonlinear BVP. We solve the problem numerically by means of a shooting procedure combined with Newton’s method, and we find local similarity solutions. We investigate the behaviour of the velocity field, of the drag and of the thickness of the boundary layer for different values of the involved parameters, making a comparison between the piezo-viscous and the Newtonian case. We find that, depending on some material parameters, the effect of pressure is that of reducing or increasing the boundary layer with respect to the Newtonian case.

Characteristics of MWCNT, SWCNT, Cu and water based on magnetized flow of nanofluid with Soret and Dufour effects induced by moving wedge: Consequence of Falkner–Skan power law

Soret and Dufour effects are utilized in designing efficient heat exchangers, optimizing separation processes, and understanding biological transport phenomena. A wedge-shaped structures aid in aerodynamics, optics, seismic resistance, and biological studies. Due to these application the consequences Soret and Dufour effects on flow of nanofluid due to wedge have been investigated. The characteristics of water ( H 2 O ) as a base fluid and single-walled carbon nanotubes ( SWCNTs ) , multi-walled carbon nanotubes ( MWCNTs ) , and copper ( Cu ) nanoparticles are taking into account. The influence of thermal radiation and heat source/sink are part of this investigation. Similarity variables are used to sort out the complexity of fundamental governing equations. Maple via Runge–Kutta–Fehlberg method (RKF45) is used to solve resulting system of ordinary differential equations (ODEs). The graphical and tabulated results of velocity, temperature, and concentration profiles are plotted against different parameters involved in this study. The findings indicate that modifying the moving wedge factor reduces the temperature distribution. By increasing value of magnetic constraint the velocity distribution increased, the temperature and concentration distribution both are increased for varying the value of Dufour, and Soret and source/sink parameters.

Entropy generation due to MHD Falkner–Skan flow of Casson fluid over a wedge: A numerical study

AbstractThis study highlights the significance of entropy generation in the Falkner–Skan flow of Casson fluid past a wedge. To investigate the energy analysis, the governing equations include the heat transport equation in the presence of internal heat source, and the energy transport accounts for heat dissipation using viscous dissipation and Joule heating effect. The mathematical formulation of the problem leads to a set of nonlinear coupled partial differential equations. To obtain a similarity solution, similarity variables are introduced. The resulting differential equations are solved numerically using the shooting technique in conjunction with the Runge–Kutta–Fehlberg 45 (RKF‐45) method. Graphical representations are utilized to demonstrate the physical significance of the relevant parameters. The study analyzes the impact of various parameters on the velocity, temperature, and entropy distributions for three wedge positions: stationary, forward‐moving, and backward‐moving. The results show that an increase in the wedge angle parameter and Casson parameter leads to an increase in fluid velocity, while fluid entropy increases rapidly with an increase in the Brinkmann number, power law Falkner–Skan parameter, and Reynolds number. Moreover, with an increment in the Prandtl and Eckert number, the Nusselt number coefficient decelerates for both static and moving wedge.

Control of linear instabilities in two-dimensional airfoil boundary layers

The present study deals with the control of linear, modal and non-modal growth of perturbations in a trapezoidal wing boundary layer. The asymmetric NACA 2515 airfoil is chosen for the analysis. The boundary layer velocity profiles, at different locations on the airfoil, are obtained using the Falkner–Skan equations. The non-modal linear stability theory is used to obtain the amplification of the initial perturbations, for which the state-space approach is adopted. The control is brought about using a Linear Quadratic Regulator (LQR) controller by the means of addition of fluid momentum in the wall-normal direction from the wing surface. Maximum transient energy growth is calculated across a range of wavenumber pairs, and control is applied to the pair with the highest energy growth. In this numerical analysis, we show that the transient growth is higher towards the trailing edge because of the adverse pressure gradient of the flow. Upon control actuation, the reduction in the transient energy growth of perturbations is achieved up to a maximum of 54.6%.

Analysis of MHD boundary layer flow of a viscous fluid past a stretching sheet employing the Legendre wavelet method

The current work introduces a novel approach to analyse the non-linear Falkner–Skan equation from fluid dynamics by utilising the Legendre wavelet method. In this paper, we relied on the Legendre wavelet to generate a new functional integration matrix and presented a novel approach known as the Legendre wavelet approach (LWA). The primary goal of this study is to provide a unified method via LWA to compute an approximate solution to a non-linear differential equation for a viscous fluid over a stretching sheet. The Legendre wavelet method has been used for the first time to solve the stretching sheet problem. A viscous fluid across a stretching sheet has been explored for its stable laminar magnetohydrodynamic (MHD) flow, transportation of mass and heat characteristics in the context of an intermittent magnetic field. By specifying appropriate non-dimensional parameters, the governing boundary layer equation evolves into a dimension devoid of the Falkner–Skan equation. Using the Legendre wavelet approach, the resulting nonlinear equation modified to an algebraic equation is addressed. With LWA, the undershoots and overshoots for a set of factors are noted. The boundary layer thickness appears to decrease with increasing pressure gradient and magnetic field parameters. The graphical interpretation of the boundary layer flow’s features is done for a range of physical parameter values. To ensure the accuracy of the findings, local skin friction has been tracked and contrasted with other techniques that are already documented in the previous research.

An MHD boundary layer flow of Casson fluid due to a moving wedge analyzed by the Bernoulli wavelet method

AbstractThe 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.

Exact and analytical solutions for self‐similar thermal boundary layer flows over a moving wedge

AbstractThis 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 , 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 . 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 , 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 and Prandtl numbers . Furthermore, an asymptotic analysis is conducted, focusing on scenarios where the moving wedge parameter becomes significantly large (). Nevertheless, in all the above‐mentioned cases, the skin friction coefficient () and the heat transfer rate () 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.

Enhancing heat transfer in MHD Falkner's‐Skan flow with thermal radiation, free convection and dusty fluid between parallel plates

AbstractFalkner's‐Skan flows are one‐dimensional flows widely used in fluid dynamics and boundary layer theory. They optimize heat exchangers and cooling systems, enhance turbine and compressor blades in turbomachinery, and provide consistency and quality in semiconductor fabrication. Additionally, they assist in pollution management and environmental impact assessments. The variety of Falkner's‐Skan flows clarifies fluid flow phenomena and their practical applications. This article aims to explore the impact of surface temperature on the flow behavior of dusty fluid in Falkner's‐Skan flow. The study focuses on the flow of dust particles in a channel formed by two infinite parallel plates. The study assumes that the particles are round and uniformly dispersed in the fluid. Furthermore, the article takes into account the effect of radiation on the energy equation. With the findings of this study, we hope to gain a better understanding of the dynamics of Falkner's‐Skan flow and contribute to the development of effective strategies for managing the flow of dusty fluids. The right plate's movement at free stream velocity causes the fluid to flow. Partial differential equations are used to represent the behavior of the flow. The Poincare‐Light Hill Technique yields exact answers. Visual representations of the temperature and velocity curves show the effects of different factors. It is possible to create graphic pictures of the fluid and dust particles using Mathcad‐15. Furthermore, critical fluid characteristics for engineers, such as skin friction and heat transfer rate, are analyzed and tabulated. These evaluations include the heat transfer rate at the wedge surface and the influence of this enhancement on surface viscous drag forces.

Numerical and optimal thermal analysis of non-Newtonian fluids flow over a wedge by using response surface methodology (RSM): Sensitivity analysis

The numerical and optimal thermal analysis of Casson fluid flow over a wedge is investigated in this work. To perform numerical and sensitivity analysis we have convert third order non-linear PDEs into set of ODEs by using similarity transformations along with boundary conditions. These equations are simulated using numerical technique Keller Box method and then compared the values of skin friction coefficient and Nusselt number with those existing in literature and found excellent agreement. Then by using these numerical data we have performed sensitivity analysis by using Response Surface Methodology (RSM). ANOVA tables are generated and developed a correlations between input parameters and output responses. The coefficient of determination for output responses skinfriction coefficient and Nusselt number are 96.50% and 99.51% respectively which shows the best fitted correlations. Finally, we have performed sensitivity analysis which reflects that skinfriction coefficient is more sensitive to Falkner–Skan exponent (m) and there is no effect of Prandtl number (Pr) on skinfriction coefficient. It is also concluded that Nusselt number is sensitive to Prandtl number (Pr) and obtains its optimum value with variation in Prandtl number (Pr).

Prediction model based on artificial neural network and bivariate spectral quasi-linearization method for compressible turbulent boundary-layer flow over a smooth flat surface

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.

Mechanism of heat transfer in Falkner–Skan flow of buoyancy-driven dissipative hybrid nanofluid over a vertical permeable wedge with varying wall temperature

Heat transport issues of wedge-shaped flow on hybrid nanofluids are of special significance by reason of their importance in manufacturing uses including heat exchangers solar panels, electronic equipment cooling, drying processes, and air heaters. Consequently, the current investigation investigates the behavior of dissipative Single-walled Carbone nanotubes (SWCNT) — Multi-walled Carbon nanotubes (MWCNT) hybrid nanofluid in the buoyant flow towards a radiative vertical permeable wedge subjected to variable wall temperature considering heat source/sink. By selecting appropriate similarity conversions, the resultant flow regulating nonlinear boundary layer PDEs are subsequently transformed into coupled nonlinear ODEs. After that, the boundary layer Boussinesq approximations are numerically resolved. The significant outcome of the current investigation is the heat transport which is higher for linear radiation, volume fraction, and uniform heat source parameters. The momentum is controlled with wedge angle and volume fraction of nanotube particles. The momentum boundary layer is upsurged with mixed convection parameters for an assisting flow. Changes to the Nusselt number and fluid flow rate are measured for flow-controlling parameters. Furthermore, the results are compared and effectively validated with previously reported literature results.

Impact of magnetic dipole on unsteady ferromagnetic Casson nanofluid flow over a static/moving wedge in the presence of viscous dissipation

This article mainly focused on magnetic dipole effects on unsteady two-dimensional ferromagnetic Casson nanofluid Falkner–Skan flow along a static/moving wedge under the effects of viscous dissipation. A nanofluid features i.e. Brownian motion and thermophoresis are also observed for static (λ = 0) and moving wedge ( and ). A class of appropriate similarity functions is utilized to convert time-dependent nonlinear partial differential equations (PDEs) into a set of ordinary differential equations (ODEs) and then computed numerically by applying two distinct techniques Runge–Kutta–Fehlberg (RKF) in Maple as well as bvp4c in Matlab. For limited cases, we also provided a comparison with already published work and found in good agreement. Impact of various controlling parameters active in the velocity, temperature, and concentration are observed and presented extensively. In addition, for engineering aspects the numerical computations and graphical presentations for Nusselt and Sherwood numbers (the wall heat and mass fluxes) are also performed for various wedge movements. All the three profiles ( and ) vary for a static (λ = 0) and moving wedge ( and ). Velocity and thermal profiles decelerates for rising ferromagnetic interaction and unsteady parameter. Furthermore, it was noticed that fluid velocity gets smaller for a static wedge.

Effects of freestream turbulence on the secondary instability of the roughness-induced crossflow vortex in swept flat plate boundary layers

In a three-dimensional boundary layer with spanwise constant flow and chordwise acceleration, which is unique to the vicinity of the leading edge of a swept wing, an inviscid instability induces the crossflow vortex (CFV), around which a secondary instability further occurs to yield a turbulent transition. It is well known that surface roughness induces stationary CFVs and that CFVs are receptive to freestream turbulence (FST), but the interference of both effects remains unclear. To determine the transition mechanisms in mixed roughness and FST environments, we performed direct numerical simulations of a Falkner–Skan–Cooke boundary layer under various conditions considering the presence of cylindrical roughness and the peak FST wavelength. Without roughness, the FST with a long wavelength comparable to the most unstable mode promotes CFV induction, resulting in an earlier transition. Conversely, the short-wavelength FST interacts with a wake vortex induced by the presence of roughness, yielding a hairpin vortex as a secondary instability. We discuss the main production terms of the disturbance energy associated with each of the type-1 and -2 secondary instabilities, which are accompanied by finger and hairpin vortices, respectively. The rapid turbulent transition via the type-1 secondary instability is due to the spanwise velocity gradient of the developed CFV, whereas that via the type-2 is due to a wall-normal gradient and is similar to the fully developed wall turbulence. The wavelength of FST played a key role in inducing a high-frequency secondary instability on the CFV or the CFV itself, and the receptivity to FST differed between the CFV and wake vortex.

Water thermal enhancement in a porous medium via a suspension of hybrid nanoparticles: MHD mixed convective Falkner's-Skan flow case study

Keeping in mind the whispered applications of Falkner's-Skan Flows, the present hydrothermal scrutinization intended to explore the two-dimensional flow pattern and thermal characteristics featuring the steady mixed convective motion of a radiating homogeneous hybrid nanofluidic mixture (Ag+TiO2)−H2O (i.e., a homogeneous aquatic mixture containing spherical silver and titanium dioxide nanoparticles) over a non-movable wedge surface. By combining Darcy's-Brinkman and single-phase models, the conservation equations are stated properly in the case where the hydrothermal impacts of specific variable heat and magnetic sources are taken into account effectively in the present flow problem. After many simplifications and rearrangements, the derived boundary layer equations are handled numerically with the help of a robust iterative GDQM-NRT algorithm, whose accurate results are presented graphically and tabularly. As the main findings, it is evidenced that the thermal buoyancy forces, as well as Lorentz's and Darcy's forces, act as assisting factors, which exert a cooling impact throughout the hybrid nanofluidic mixture (Ag+TiO2)−H2O due to the imposed axial pressure gradient. Energetically, it is demonstrated that the silver nanoparticles play a noteworthy role in the enhancement of the heat transfer rate at the wedge surface with a strengthening in the resulting surface viscous drag forces.

Evolutionary Padé Approximation for Heat and Mass Transfer Analysis of Falkner–Skan Flow of a Bio-Convective Casson Fluid

This study presents numerical work to investigate the Falkner–Skan flow of a bio-convective Casson fluid over a wedge using an Evolutionary Padé Approximation (EPA) scheme. The governing partial differential equations and boundary conditions of a Falkner–Skan flow model are transformed to a system of ordinary differential equations involving ten dimensionless parameters by using similarity transformations. In the proposed EPA framework, an equivalent constrained optimization problem is formed. The solution of the resulting optimization problem is analogous to the solution of the dimensionless system of ordinary differential equations. The solutions produced in this work, with respect to various combinations of the physical parameters, are found to be in good agreement with those reported in the previously published literature. The effects of a non-dimensional physical-parameter wedge, Casson fluid, fluid phase effective heat capacity, Brownian motion, thermophoresis, radiation, and magnetic field on velocity profile, temperature profile, fluid concentration profile, and the density of motile microorganisms are discussed and presented graphically. It is observed that the fluid velocity rises with a rise in the Casson fluid viscosity force parameter, and an increase in the Prandtl number causes a decrease in the heat transfer rate. Another significant observation is that the temperature and fluid concentration fields are greatly increased by an increase in the thermophoresis parameter. An increase in the Péclet number suppresses the microorganism density. Moreover, the increased values of the Prandtl number increase the local Nusslet number, whereas the skin friction is increased when an increase in the Prandtl number occurs.

Falkner–Skan slip flow of non-Newtonian fluid over a moving and nonlinearly radiated wedge with variable heat source/sink and viscous dissipation

Analyzed in this paper is an investigation of the impacts of the varying source of heat or thermal sink and viscous dissipation on Casson fluid flow from a moving wedge by considering thermal and momentum slips in a mathematical model. Regarding the regulation of concentration fields, we incorporated the Soret and Dufour phenomena. The outcomes are analyzed numerically by using the shooting technique. The governing parametric effects of the flow characteristics are revealed graphically. We also presented the local Sherwood and Nusselt number values for non-Newtonian fluid (Casson fluid) and Newtonian. It is discovered that the thermal and concentration boundary layer is more extensive in Casson fluid while equated with the Newtonian fluid. The thermal and mass transmission in Newtonian fluid is significantly larger than in non-Newtonian fluid. The heat source/sink, Eckert number and Dufour number boost the rate of thermal transport and depreciate the rate of mass transport for both the Newtonian and non-Newtonian fluid cases.

Heat transfer rate in Falkner–Skan fluid flow of ZnO-EG over a moving wedge: Intelligent backpropagated neural networks

In this study, the Falkner–Skan stream (FSS) of ZnO-EG across a moving wedge is investigated using artificial neural networks backpropagated using a Bayesian regularization strategy (ANN-BRS). The PDEs of the Falkner–Skan are converted into a set of ordinary differential equations (ODEs). The reference dataset is created using the mathematical solver in Mathematica and includes moving wedge boundaries, radiation boundaries, nanoparticle volume division boundaries and Falkner–Skan power-regulation boundaries for all proposed scenarios (ANN-BRS). Examined is the effect of practical cut off points on the stream field and temperature profiles, including the radiation limit, moving wedge limit, nanoparticle volume division and Falkner–Skan power. When nanoparticles are present, viscosity and heat conductivity rise, which can be physically explained. Increases in Falkner–Skan power have both positive and negative effects. When the settings are adjusted to their highest values, the maximum rate of heat transmission is realized. The effects of radiation boundary and so on are identical due to the linear augment. As a result, as the parameters are increased, the rate of heat transmission increases. Based on the intended data points that were obtained, the estimated answer is calculated for every scenario utilizing the testing, training and validation procedures. The mean square error data (MSE), error histogram and regression analysis are used to validate the performance of (ANN-BRS).