Non-dimensional Form Research Articles

Effects of Heat Transport Characteristics and Chemical Reaction in Unsteady Flow of Williamson Fluid and Entropy Generation: The Keller‐Box Numerical Scheme

ABSTRACTThe study of heat and mass transport in non‐Newtonian fluid flow over a stretching surface accompanying relevant characteristics is important in several engineering and industrial processes like annealing and thinning of copper wires, aerodynamic extrusion of plastic and rubber sheet, glass fiber, and so forth. Based on significant practical applications, the objective of this investigation is to assess the time‐dependent flow of Williamson fluid influenced by porous sheet stretching in exponential manner accompanied by thermal and mass transport and entropy generation. Various factors affecting fluid flow, thermal and mass transport (viscous dissipation, non‐linear radiation, porous media, chemical reaction, and heat source) are considered. The regulating PDEs are turned into ODEs in nondimensional form utilizing adequate similarity transformation relations. The problem is solved numerically on MATLAB adopting the Keller‐Box scheme. On fluid flow, temperature, and concentration distribution the effects of relevant parameters are depicted by drawing sketches and discussed. Besides, second law analysis is also evoked in the study in terms of entropy generation accompanying the Bejan number. Moreover, quantities of physical significance such as skin friction coefficient, Sherwood number, and Nusselt number are computed, compared with prior research and found in excellent agreement. It is concluded that temperature profile magnifies due to radiation and heat generation effects. The reaction coefficient and order of the reaction exhibited opposite effects on concentration profile. It is also concluded that entropy production reduces with increasing slips and temperature difference parameter, while opposite effect is observed due to Brinkman number. Furthermore, it is observed that skin‐friction coefficient at the surface decreases with velocity slip and non‐Newtonian parameter however, trend is reversed due to unsteadiness parameter. The results of the study may find applications of practical importance in engineering fields such as designing heat exchangers, cooling processes, improving energy storage systems, and so forth.

Velocity slip impact on the squeezing flow of water–CNTs and kerosene–CNTs nanofluids over a Riga surface

In various industrial vis-a-vis technological applications, the study of squeezing flow of carbon nanotube (CNT) nanofluid past along the Riga surface is vital. The present article aims in examining the role of velocity slip in two distinct cases considering the flow of CNT–water and CNT–kerosene nanofluid over a Riga-squeezing surface. The Riga surface generates Lorentz force and provides a platform for enhancing heat transfer characteristics. Additionally, the inclusion of dissipative heat along with radiant heat and irregular heat supplier energies the heat transport phenomenon. The mathematical model endorsed with dimensional features is transmuted into nondimensional form by the implementation of similarity rules, and the results of several flow profiles are manipulated by employing the traditional “Runge–Kutta fourth-order” along with shooting scheme. The parametric analysis is presented through graphs, and the validation with earlier investigation is depicted via tabular form representing the numerical results in the particular case.

Entropy Analysis in Magnetized Blood-Based Hybrid Nanofluid Flow via Parallel Disks

Magnetized hybrid nanofluid combined with ferrite and silver in a blood-based liquid presents their vital role in several aspects such as artificial heart pumping system, drug delivery process, the flow of blood in the artery, etc. This is because the high heat transportation rate of the nanofluid is caused by the inclusion of nanoparticles. The current investigation is based on the characteristic of particle concentration comprised of Fe3O4 and Ag in the base liquid blood that passed in between two infinite parallel disks filled with porous matrix. The electrically conducting fluid associated to maximum of 1.5% of volume concentration from each of the solid particles affects the flow phenomena. However, the impact of thermal radiation vis-à-vis the heat dissipation provides efficient heat transport properties with the inclusion of the effective thermal conductivity assumed from the Hamilton-Crosser model. The proposed conductivity model describes the role of particle shapes on the enhanced thermal properties. Further, numerical treatment is obtained for the transformed designed problem following similarity rules that are used for the conversion of the governing equations into their non-dimensional form. The computation of various flow profiles leads to get the entropy generation due to the irreversibility processes. Along with the fluid velocity and temperature distributions, the study is carried out for the entropy as well as the computation of Bejan number and afterwards the simulation of the shear and heat transportation rate are also depicted graphically. The main finding of the proposed study is that solid particle concentrations have a substantial impact to increasing fluid velocity in magnitude, resulting in a narrower wall thickness at both channel walls. Thermal radiation was shown to be more effective at increasing entropy generation and Bejan value.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Flow regimes in the evolution of a hot buoyant vortex dipole

Vortices and vortex dipoles play a critical role in turbulence, facilitating scalar mixing, diffusion, and energy dissipation. When a temperature gradient is present, buoyancy effects become significant, and buoyant vortices and dipoles emerge as characteristic features of such flows. In this study, we examine the evolution of a buoyant vortex dipole (BVD) arising from a temperature difference between the vortex dipole and the surrounding fluid. Using the Oberbeck–Boussinesq approximation, we derive the governing equations in non-dimensional form to capture the essential physics. The computational domain is periodic, and simulations are performed using the open-source spectral solver Dedalus. The interplay of thermal diffusion, viscous diffusion, and buoyancy drives the evolution of various coherent structures. Based on these interactions, we identify four distinct topological features and classify the evolution of the BVD into six regimes: (a) thermal diffusion-dominated regime, (b) viscous diffusion-dominated regime, (c) balanced diffusion regime, (d) weak wake street regime, (e) buoyancy-driven transition regime, and (f) multiple tertiary wakes regime. This classification provides a comprehensive framework to understand the dynamics of BVDs under varying physical influences.

Heat and Mass Transfer in Unsteady MHD Casson Fluid Flow Over a Semi-Infinite Vertical Plate Through Porous Medium with Dissipative and Radiative Effects

Nowadays, Casson fluid and its diverse applications are highly effective across a vast array of industrial, biological and environmental sectors, due to its unique flow properties. This study focuses on the numerical investigation of the heat and mass transfer characteristics of an unsteady incompressible Casson fluid flow over a semi-infinite vertical flat plate by considering free convection effects. The fluid is conducting the electrical current as it moves through the porous medium. The formulation of the governing equation is based on the use of this phenomenon. In the model, the formulated governing equations are converted into non-dimensional form first, and the MHD mathematical model is analyzed using the boundary conditions. The scheme of finite difference method is implemented to solve the model, where concentration profiles are also discussed. Stability and convergence criteria are applied for the accuracy of numerical techniques, and the effects of various parameters on skin friction, rate of heat and mass transfer have been examined significantly. The results indicate that the temperature of the plate increases with the increase in the value of the magnetic parameter for the Casson fluid but the reverse is observed in its limiting case. The present results are compared and checked with previously published results and a remarkable conclusion is given at the end of the study.

Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors

In this paper, we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behaviors of trajectories in the phase space of the classical double pendulum system. To successfully quantify the degree of chaos with this tool, we first derive Hamilton’s equations of motion for the problem in nondimensional form and demonstrate how they can be expressed compactly using matrix algebra. Once the dynamical equations are obtained, we conduct a parametric study based on the system’s total energy and other key parameters such as the lengths and masses of the pendulums, as well as gravity, to determine the extent of the chaotic and regular regions in the phase space. Our numerical results reveal that, for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Furthermore, we characterize how chaos grows and decays within the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.

Magnetohydrodynamic orientation effects on Soret and Dufour phenomena in inclined corrugated triangular cavities with non-Newtonian fluids

This research delves into the intricate influence of magnetohydrodynamic orientation on Soret and Dufour effects within inclined, corrugated triangular cavities containing non-Newtonian fluids, underscoring the impacts of magnetic alignment, cavity inclination, and fluid rheology on convective transport dynamics. By systematically transforming the governing partial differential equations into non-dimensional forms using selected similarity variables, the study applies the finite element method (FEM) for computational analysis. The research intricately dissects the influence of multiple interdependent physical parameters on flow morphology, concentration, isotherms, and local Nusselt numbers, which serve as a barometer for the system's heat transfer efficacy. Critical variables under scrutiny for include magnetohydrodynamics (0≤Ha≤103), buoyancy-driven convective forces (0≤Nχ≤20), the non-Newtonian nature of Casson fluid (0.1≤β≤1), as well as the cross-diffusion effects epitomized by the Soret (−15≤Sχ≤15) phenomena. Additional dimensionless parameters, such as the Lewis (0.1≤Lε≤50) and Rayleigh numbers (102≤Ra≤105), further characterize the thermal and concentration fields within the cavity, alongside the role of internal heat generation/absorption (−10≤Δ≤10) mechanisms for fixed value of Darcy number (λd=10−3). The results show that the Casson parameter subtly affects the distribution of thermal energy and particles, which in turn influences flow patterns and convection. In contrast, the Soret parameter has a direct effect on concentration gradients, regulating the layering of solutes within the fluid. It has been found that inclined MHD orientation effects create variations in magnetic fields, which disrupt fluid velocity and affect heat transfer rates. These effects reshape temperature contours, altering isotherm patterns and local thermal gradients. The study underscores the complex interplay of non-linear factors that collectively govern the efficiency of heat and mass transfer processes in non-Newtonian fluids subjected to magnetothermal and buoyancy forces, with broad implications for optimizing industrial and natural convection systems.

Investigation of Sutterby fluid flow through elliptic multi-stenosed artery: analytical solutions of blood flow problem

PurposeThe presence and progression of stenosis disturb the normal circulation of blood through an artery and cause serious consequences. The proposed investigation is aimed to assess non-Newtonian characteristics of blood in an elliptical artery having stenosis. The blood is taken as Sutterby fluid flowing via a multi-stenosed elliptical cross-section artery.Design/methodology/approachThe analytical solution of a mathematical model representing the considered problem is extracted in a non-dimensional form by utilizing the perturbation technique under the mild stenosis assumptions.FindingsThe graphical nature of these results is examined and discussed comprehensively for different physical parameters. The height and shape of stenosis are noted to have prominent effects on flow velocity. The wall shear stress and flow velocity attained high values in the stenotic portion of the artery. The non-uniform stenosis is observed to create higher resistance to the flow than the uniform stenosis. Further, a high disorder is noticed in the constricted region of the artery by streamlines analysis.Research limitations/implicationsThe manuscript completely comprehends the blood’s non-Newtonian flow in the arteries of elliptical shape having multiple stenoses. The present study is about the properties of non-Newtonian blood flow through an elliptical artery with many stenoses. The Sutterby fluid model is used to describe the blood’s non-Newtonian nature. By utilizing presumptions of mild stenosis, the mathematical model’s non-linearity is decreased, and the perturbation method is applied to generate the resulting equations.Practical implicationsThe presence of stenosis can significantly impact the circulation of blood flow. When an artery becomes narrowed, it can create a constriction or obstruction in the flow path of blood, which can lead to several important fluid dynamics phenomena, i.e. increased velocity, shear stress, pressure drop, etc. The presence of stenosis can cause various damages and complications in the affected blood arteries and surrounding tissues, resulting in heart attacks or diseases like atherosclerosis.Originality/valueThe work presented in the manuscript was not published earlier in any form.

A scientific report on heat transfer analysis due to generated and absorbed heat over an oscillatory stretching sheet: a finite difference-based study

A non-Newtonian liquid motion across a stretchable surface is relevant to various industrial applications, including extruding plastic sheets and stretching plastic films. In connection with this, the impact of endothermic and exothermic chemical reactions on the flow of rate-type liquid via an oscillatory stretching sheet in the presence of permeable media with the Maxwell liquid model is elucidated in the present study. Scientists and engineers may enhance the efficiency of chemical reactions or heat transmission by optimising system flow and investigating the effect of reactions on flow dynamics. The present study’s governing partial differential equations (PDEs) are transformed into their non-dimensional form using similarity variables. The resulting equations are numerically solved using the finite difference method (FDM). Outcomes disclose that the temperature profile declines as the activation energy and unsteadiness parameters increase. The influence of the Maxwell and unsteadiness parameters on the fluid’s velocity with respect to time is represented. The rise in the values of chemical reaction parameter upsurges the thermal profile. As the activation energy and unsteadiness parameters upsurge, the thermal profile declines. As the chemical reaction parameter and the ratio of oscillation frequency to stretching rate values rise, the concentration profile falls.

Synergistic influence of gyrotactic microorganisms and bimolecular reaction on bidirectional tangent hyperbolic fluid with Nield boundary conditions: A biomathematical model

In biomedical engineering, the behavior of gyrotactic microorganisms with non-Newtonian fluids such as tangent hyperbolic fluids improve the design of targeted drug delivery systems. In this system control over microorganism movement is essential. The present study deals with the synergistic influence of gyrotactic microorganisms and bimolecular reactions on the bidirectional flow of tangent hyperbolic fluids under Nield boundary conditions. Further, the flow characteristic of the non-Newtonian fluid is enhanced by incorporating the impact of thermal radiation, heat sources, Brownian motion, and thermophoresis. The presentation of these phenomena is vital for an extensive range of applications, including industrial processes, biomedical engineering, and environmental management. The analysis employs advanced mathematical modeling which needs suitable transformation rules to get the non-dimensional form and further numerical simulation is presented with the assistance of the “shooting-based fourth-order Runge–Kutta technique”. The results are depicted for the several contributing factors via the built- in-house function bvp4c in “MATLAB”. The authentication of the study with the prior research is a benchmark to precede further research in this direction. However, the outstanding results are; the fluid velocity is controlled by increasing non-Newtonian Weissenberg number whereas the velocity slip shows dual characteristics on the axial velocity distribution. Further, the motile microorganism profile is controlled by the enhanced bioconvection Lewis number.

Characteristics of induced magnetic field on the time-dependent MHD nanofluid flow through parallel plates

Abstract The heat transfer characteristics of an unsteady magnetohydrodynamic flow through non-conducting infinite vertical parallel plates are presented in this investigation. The flow is subjected to an induced magnetic field, and the base fluid water contains carbon nanotubes (CNTs), in particular multi-wall carbon nanotubes, to present the behaviour of the nanofluid. The aim is to examine the effect of the applied magnetization and CNT concentration on the heat transport performance of the system. However, suitable transformation rules are adopted for the re-designing of the proposed design model into its non-dimensional form. This transformed system is then solved analytically following the standard transformations. The influence of key parameters, including the Hartmann number (Ha), the angle of inclination of the magnetic field, thermal buoyancy, heat source, and the concentration of CNTs in the nanofluid, on the flow phenomena is analysed. The consequences reveal that the occurrence of the inclined magnetic field affects the flow and heat transfer characteristics significantly. Additionally, the introduction of CNTs to the nanofluid enhances the heat transfer performance due to their unique thermal properties. The study demonstrates that enhanced Ha and CNT concentration augments the heat transfer rate.

Comparative analysis on the radiative time-dependent water/kerosene-based Cu nanofluid through squeezing Riga plates with heat dissipation: Spectral quasilinearization technique

The time-dependent thermal management along with convective flows are vital in various heat transfer applications in engineering systems such as in automotive cooling systems, and industrial heat exchangers. Because of enhanced thermal conductivity, nanofluids are widely considered for advanced cooling systems such as electronics, aerospace, geothermal energy extraction, etc. The current analysis presents comparative results of the radiative, time-dependent flow of water/kerosene-based Copper nanofluids between squeezing Riga plates focusing on heat dissipation. Both the plates are embedded within a porous matrix and the influence of non-uniform heat source/sink and thermal convective boundary conditions is examined. Riga plates, generally utilized for their ability to generate electromagnetic fields provide greater control over the fluid flow. The problem designed with the inclusion of aforesaid factors is transformed into a non-dimensional form for the utilization of appropriate similarity functions. Further, the impacts of several effective terms on the flow profiles are presented followed by the numerical solution of the profile obtained using the spectral quasilinearization method. Moreover, some of the outstanding findings are; an increase in the fluid velocity is marked for the separation of the plates but the rate of enhancement in the case of kerosene is more pronounced than that of water. Further, irrespective to the type of fluids, the heat transfer rate enhances for the increasing heat fluid which provides the variation of thermal radiation.

An analytical exploration of low Reynolds number cilia-driven flow of Johnson-Segalman fluid with heat transfer effects

This article discusses the properties of heat transfer on Johnson-Segalman fluids in a complex wavy channel made of metachronal wavy cilia. Such complex wall structures can be used in the design of biomimetic systems. The movement of fluid through a two-dimensional complex wave channel produced by the metachronal wave of cilia is characterized as laminar and incompressible. Firstly, the system of equations is transformed from a fixed frame of reference to a wavy frame of reference. Secondly, the system of equations of motion (EOM) is transformed into a non-dimensional form by using the scaling factors. Mathematical analysis has been conducted using long wavelength as well as low Reynold numbers assumptions. The perturbation approach is used to solve the simplified governing equations for the axial velocity, temperature, stream function, pressure rise, and heat transfer coefficients. The equations representing axial velocity, pressure rise, and the streaming function are illustrated, and the reason for observed changes in different physical aspects are explained using basic theoretical principles. A solution for small values of the Weissenberg numbers is generated for the resulting nonlinear system. The current study investigates the effects of different boundaries and rheology on fluid flow.The velocity profile increases near the channel center with higher Q (time-mean flow rate) and e¯ (slip parameter), but decreases with greater Weissenberg number (We). The Brinkman number (Br), Q, and e¯ directly influence the temperature distribution, while we have the opposite effect. The size and number of trapped boluses increase with higher Q but decrease with rising We and e¯. It is noteworthy that the overall number of trapped zones rises throughout the complex wavy path.

New effectual configuration of bistable nonlinear energy sink

The study presents a new configuration of nonlinear energy sinks (NESs) which is adaptable to function as either stable or bistable NES. The proposed NES is based on the spring-loaded inverted pendulum (SLIP) in which a torsional stiffness element couples the SLIP to the linear oscillator (LO). The bistable configuration provides a critically stable position when the SLIP is vertically aligned with respect to the LO motion. At this critical stability position, the SLIP NES incorporates pre-stored potential energy which generates the bistability characteristics resembling that of a stiffness-based bistable NES. The equations of motion of the coupled LO with the SLIP NES are derived based on the Euler–Lagrange method in non-dimensional form. The parameters of the considered SLIP NESs are optimized to achieve an optimum energy absorption from the LO. The proposed B-SLIP NES is also applied to suppress seismic ground motion and forced torsional vibrations. The obtained numerical simulation and analytical response results verify the robustness of the B-SLIP NES in vibration suppression performance compared with the tuned mass damper and the cubic stiffness NES.

Numerical Assessment of the Thermal Performance of Microchannels with Slip and Viscous Dissipation Effects.

Microchannels are widely used across various industries, including pharmaceuticals and biochemistry, automotive and aerospace, energy production, and many others, although they were originally developed for the computing and electronics sectors. The performance of microchannels is strongly affected by factors such as rarefaction and viscous dissipation. In the present paper, a numerical analysis of the performance of microchannels featuring rectangular, trapezoidal and double-trapezoidal cross-sections in the slip flow regime is presented. The fully developed laminar forced convection of a Newtonian fluid with constant properties is considered. The non-dimensional forms of governing equations are solved by setting slip velocity and uniform heat flux as boundary conditions. Model accuracy was established using the available scientific literature. The numerical results indicated that viscous dissipation effects led to a decrease in the average Nusselt number across all the microchannels examined in this study. The degree of reduction is influenced by the cross-section, aspect ratio and Knudsen number. The highest reductions in the average Nusselt number values were observed under continuum flow conditions for all the microchannels investigated.

The effect of aspect ratio and mass ratio on the flow-induced flutter of a thin flexible sheet

This study experimentally investigates the flow-induced flutter of a thin flexible sheet, focusing on how the sheet's aspect ratio and mass ratio affect its stability and flutter characteristics in the post-critical regime. The flutter frequency of the sheet was obtained using hotwire measurements, while flutter amplitude and mode shape were acquired through high-speed imaging. The flowfield around the flapping sheet was analyzed using particle image velocimetry (PIV). Based on experimental observations, we report the onset of flutter as a subcritical bifurcation with hysteresis. The dynamic characteristics of the sheet play a significant role in its flutter instability, with the onset and cessation of flutter occurring at a frequency close to the sheet's second-mode natural frequency. The results show that both aspect ratio and mass ratio significantly affect the critical wind speed and flutter characteristics in the post-critical regime. Both flutter frequency and amplitude decrease as the aspect ratio decreases. PIV measurements in various planes reveal the highly three-dimensional nature of the flow. Results from off-axis PIV show a pair of counter-rotating spiral vortices in the wake that oscillate and change orientation with the sheet's movement. Additionally, a theoretical analysis was conducted to derive an approximate analytical relationship between the aspect ratio and critical wind speed. Experimental results aligned well with theoretical predictions for sheets with low aspect ratios (aspect ratio ≤1) but deviated for sheets with higher aspect ratios (aspect ratio >1). The relevant scaling parameters have also been explored to represent the experimental data in a non-dimensional form.

Unsteady MHD Rotating Jeffreys Fluid Flow Embedded in a Porous Medium Over an Infinite Perpendicular Plate

The analysis of the rotation outcomes for the viscous, incompressible, electrically conducting Jeffreys fluid in an infinite vertical plate through the revolution impacts has been explored. This is owing to the exponential exciting perpendicular absorbent plate embedded in the permeable medium by allowing for ramped surface temperature into the endurances of thermal radiation. The essential governing sets of equations for the flow are translated into non-dimensional form with insert appropriate parameters as well as variables; therefore the resultant equations are computationally resolved by the well-organizing Laplace transform methodology. The influences of numerous significant considerable parameters for the modelling on the velocity, temperature and concentration of the liquid, and the skin friction coefficient, Nusselt number and Sherwood number for together thermal conditions have been deliberated and exploring strongly by producing of graphical profiles and tabular format. This is determined that, with increasing quantities of the rotation, thermal radiation parameters, the fluid temperature as well as velocity enhances. Similarly, this is notified that, a mounting in porous parameter reasons to escalate fluid velocity in addition to concentration reversal results are noticed by the chemical reaction parameter.

Heat Transfer Characteristics of a Quiescent Fluid over a Moving Flat Surface

Optimizing thermal control methods and enhancing the efficiency of various engineering processes, such as the cooling of hot moving surfaces, is imperative. Therefore, it is crucial to investigate heat transfer and temperature distributions across moving surfaces in stationary fluids. In this study, the combined effects of viscous dissipation, temperature-dependent viscosity, and a moving surface in a quiescent fluid on the temperature and velocity profiles are investigated. A two-dimensional steady laminar boundary layer flow of an incompressible Newtonian fluid, driven by the movement of the surface where viscosity is an inverse function of temperature, is analyzed. The effects of flow parameters, including Reynolds number, Eckert number, Prandtl number, variable viscosity, and surface velocity on temperature and velocity profiles, are determined. The governing boundary layer partial differential equations are transformed into non-dimensional form and solved using the central finite difference numerical method, implemented in MATLAB software. The numerical results demonstrate that an increase in the Reynolds number and the variable viscosity parameter leads to an increase in the velocity profiles. Additionally, an increase in Reynolds number, Prandtl number, variable viscosity, and surface velocity leads to a decrease in temperature profiles. Finally, an increase in the Eckert number results in higher temperature profiles. Therefore, with suitable flow parameters, the temperature of the fluid can be regulated. These findings are useful in cooling hot sheets or metallic plates drawn over a quiescent fluid to obtain high-quality final products.

FLOWERS AEP: An Analytical Model for Wind Farm Layout Optimization

ABSTRACTAnnual energy production (AEP) is commonly used in objective functions for wind farm layout optimization. AEP is proportional to wind farm power production integrated over an annual distribution of free‐stream wind conditions. Physics‐based estimates of wind farm power production typically rely on low‐fidelity engineering wake models that approximate the steady‐state wind farm flow field. AEP estimates are then obtained by performing independent simulations for discrete wind conditions and using rectangular quadrature to account for each condition's expected frequency of occurrence. Depending on the number of simulated discrete wind conditions, this numerical integral could be hampered by poor accuracy or high computational costs. The FLOWERS AEP model instead poses an analytical integral of the engineering wake model over the variable wind conditions, yielding a closed‐form, analytical function for wind farm AEP. This paper derives the analytical functions for FLOWERS AEP and its derivatives with respect to turbine position, which are useful for gradient‐based wind farm layout optimization, in nondimensional form. We then analyze the benefits of the FLOWERS AEP model over conventional reference models, focusing on its low cost, adequate wake loss predictions, and smooth design space. Although the FLOWERS approach is found to predict the exact value of AEP with some error relative to the reference model (within 14% on average), it dramatically reduces computation time by an order of magnitude, produces a qualitatively similar design space at relatively low resolution, and yields comparable optimal layouts. This significant speed improvement is critical in layout optimization applications, where determining an optimal layout in an efficient manner is more important than precise AEP prediction.