Width Of Boundary Layer Research Articles

Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion

Abstract Modern heat transport processes such as fuel cells, hybrid engines, microelectronics, refrigerators, heat exchangers, grinding, coolers, machining, and pharmaceutical operations may benefit from the unique properties of nanoliquids. By considering Al 2 O 3 {{\rm{Al}}}_{2}{{\rm{O}}}_{3} nanoparticles as a solo model and Al 2 O 3 – Cu {{\rm{Al}}}_{2}{{\rm{O}}}_{3}{\rm{\mbox{--}}}{\rm{Cu}} as hybrid nanocomposites in a hyperbolic tangent fluid, numerical simulations for heat and mass transfer have been established. To compare the thermal acts of the nanofluid and hybrid nanofluid, bvp4c computes the solution for the created mathematical equations with the help of MATLAB software. The impacts of thermal radiation, such as altering thermal conductivity and cross-diffusion, as well as flow and thermal facts, including a stretchy surface with hydromagnetic, and Joule heating, were also included. Furthermore, the hybrid nanofluid generates heat faster than a nanofluid. The temperature and concentration profiles increase with the Dufour and the Soret numbers, respectively. The upsurge permeability and Weissenberg parameter decline to the velocity. An upsurge variable of the thermal conductivity grows to the temperature profile. Compared to the nanofluids, the hybrid nanofluids have higher thermal efficiency, making them a more effective working fluid. The magnetic field strength significantly reduces the movement and has a striking effect on the width of the momentum boundary layer.

Modeling heterogenous crowd evacuation on stairs in high-rise buildings using a fine discrete floor field cellular automaton model: Accounting for speed and boundary layer variations

The article investigates the impact of crowd heterogeneities on stair evacuation dynamics in high-rise building. Using a fine discrete floor field cellular automaton (FFCA) model, the study explores the often-overlooked influence of crowd variations in mobility and behaviors. The simulation, set in a 21-story staircase, incorporates variations in speeds and boundary layer widths to replicate the heterogenous crowd dynamics. Key findings reveal that the heterogenous group, considering speed variations, exhibit prolonged stair evacuation time, with children and elderly individuals finishing last. Interactions among individuals with different mobilities lead to a 24.1% blocking effect on young adults' evacuation, while young adults enhance the evacuation of children and elderly by 19.0% and 23.8%, respectively. Children on the inner side of stairs lag behind due to the boundary layer, and the entire heterogeneous crowd's evacuation time decreases by 7.0%, highlighting the efficiency boost from boundary layers. A robust linear correlation is identified between specific outflow and the average desired speed for the heterogeneous group, with a consistent density of 0.46ped/m2 near the exit. The proposed model's validity is supported by trajectories consistent with prior observations and empirical data from homogenous crowd evacuation. The study advances our understanding of evacuation behaviors in heterogenous crowds within high-rise buildings during emergencies.

Practical robust fixed-time containment control for multi-agent systems under actuator faults

This article considers a containment control problem for nonlinear multi-agent systems with nonautonomous leaders under directed topology, where each follower may experience various actuator faults and unknown external disturbances. A practical robust fixed-time fault-tolerant controller is proposed that eliminates the adverse effects of disturbances and actuator faults while achieving the containment control objective within a fixed-time. Independent of initial states, a sufficient condition of practical robust fixed-time stabilization is derived, further relaxing the constraint of robust fixed-time stabilization. The combination of sliding-mode control and event-triggered mechanism in this approach provides better control performance. One advantage of this method is that the sliding-mode band size is no longer dependent on the sampling interval but rather on the triggering function. Another advantage is that the resource costs for handling faults are effectively reduced. Additionally, an adaptive boundary layer technique is designed to avoid chattering phenomenon of the discontinuous controller, which utilizes adaptive parameters to autonomously adjust the width of boundary layer. Zeno behavior analysis and a simulated example are also presented.

Multiscale analysis of nano-powder compaction process using the FEM–MD technique

In this paper, a continuum–atomistic (FEM–MD) multiscale technique is developed for modeling the compaction process of metallic nano-powders. Nano-powders are assumed as the continuum body that are modeled using the nonlinear finite element method (FEM). Representative volume element (RVE) consists of a number of nanoparticles to reflect the discrete nature of nano-powders. The atomistic RVEs are modeled using the molecular dynamics (MD) method to capture the plastic deformation of nanoparticles under the compaction process. The MD analysis of atomistic RVEs provides the feasibility of modeling the friction and interaction of nanoparticles. A parametric study is performed over the atomistic RVEs to evaluate the influence of various parameters, such as the type of boundary conditions, width of boundary layer, size and number of nanoparticles, and compaction velocity on the densification behavior of nano-powders. The effect of die-wall friction is investigated on the compaction behavior of nano-powders by evolution of die-wall friction coefficient for different values of compaction velocity and temperature. The accuracy of multiscale analysis is verified by comparing the numerical results with those of experimental data for the uniaxial compaction test. The performance of proposed computational method is finally presented in modeling the micro-cylindrical bush and micro-tablet components during the compaction process of micro-scale metallic nano-powders.

Boundary Layers of Circumplanetary Disks around Spinning Planets. I. Effects of Rossby Waves

Gas giant planets are believed to accrete from their circumplanetary disks (CPDs). The CPDs usually involve accretion through the boundary layer (BL) in the vicinity of planets. Prior studies have concentrated on the BL of nonspinning planets. We investigate the influence of planetary spin on the wave behaviors within the BL. The rotation profile in such BLs would show a sharp transition from the rigid rotation to the Keplerian rotation. We examine the angular momentum transport in these BLs in terms of linear perturbation analysis. We find that the global inertia-acoustic mode associated with spinning planets would give rise to the inflow of angular momentum and the accretion of gas. In this work, we identify a new kind of global mode, namely the Rossby mode. The Rossby mode can lead to the outflow of angular momentum and the decretion of gas from a spinning planet. The Rossby mode provides a negative feedback that regulates the planetary spin and mass. We compare the growth rate of the two modes as a function of the width of BL, the Mach number, and the spin rate of planets. Our results reveal the underlying hydrodynamic mechanism of terminal spins and asymptotic mass of the giant planets.

Multilayered membrane spacer: does it enhance solution mixing?

AbstractThe present review systematically investigates and illustrates the effect of multilayered membrane spacers on the features of fluid dynamics that influence all performance metrics. Multilayer spacers are frequently composed of three sets of filaments (i.e., top, middle, and bottom layers), which has the benefit of increasing mass transfer and decreasing membrane surface fouling when compared to ordinary monolayer (e.g., extruded spacer) and two‐layer spacers. The review found that the multilayer spacer's middle layer disperses primary flow to the thin side spacers placed near the membrane's surfaces. The thin side spacers will then form narrow passageways to keep the solution in situ for as long as mass transfer is achievable. The employment of thin spacers close to the membranes at satisfactory operational conditions (e.g., adequate flow velocity) results in swirling flows and incorporation of transverse and longitudinal eddies near to the membranes, reducing the boundary layer's width and making the associated ion concentration domain at the membranes much more consistent. The concept and implementation of multilayer geometry in feed channels appears to be promising, since a multilayered spacer can function at a lower maximum flow velocity than normal two‐layer spacers, saving operational energy while minimizing concentration gradients at the membrane surfaces. Furthermore, the multilayer structure's durability and mechanical strength may help to reduce membrane deformation and maintain long processes. Future studies might look at significantly reducing spacer thickness for industrial uses. © 2023 Society of Chemical Industry (SCI).

An anomalous method of using a camera in determining the velocity distribution within a boundary layer

The current study proposes a cost-effective technique wherein a camera can be used anomalously in tandem with the working of a pitot tube in determining the velocity distribution within a boundary layer. The technique involves the traversal of a pitot tube at a uniform velocity inside the width of boundary layer and simultaneously imaging the change in meniscus location of liquid column in the corresponding manometer, inside a wind tunnel set-up. The study aims in establishing a relation between the traversal velocity and the free stream velocity. The proposed approach holds the advantage of providing velocity values at equally spaced locations within the boundary layer, with the number of values depending on the velocity of retraction (traversal away from the surface) as well as the frame rate of recording. Experiments were performed at different combinations of retraction velocity (Vr) and Fan Speed for establishing the method. An optimum velocity at which the pitot tube should be retracted was determined based on convergence of trend of the variation in instantaneous meniscus locations to a standard reference trend. Results show that dimensionless optimum velocity (V¯o) defined as the optimum ratio between tunnel velocity (Ut) and Vr is related to the Fan Speed (FS) by V¯o∝ FS0.6. Additionally, with regard to the manual traversing performed, the probability density function plots show that traversing manually at V¯≫V¯o introduces higher non-uniformity in the results, thereby preferable to perform the traversal closer to values of V¯o. Using the results obtained from the developed imaging technique, the velocity variation within the boundary layer was studied. The technique was successful in precisely locating the prominent points inside a boundary layer for a particular flow condition. Finally as a case study, the imaging approach was also utilized in determining the effect of roughness change across the seam of a cricket ball on its aerodynamics. This case study also shows that the technique is successful in obtaining the flow characteristics of a general boundary layer as well as a flow separation case.

Explaining Browns paradox in NdFeB magnets from micromagnetic simulations

We present the results of a systematic investigation into how the coercivity and maximum energy product of NdFeB permanent magnets are affected by magnetic and geometric microscopic properties. The results are based on numerical micromagnetic simulations carried out with the open source numerical framework MagTense. We considered artificially generated realistic microstructures for which we can control the number of crystal grains as well as the thickness of the intergrain region. Based on variations of the exchange constant, easy axis orientation, grain boundary width and intergrain material properties, the results indicate that while all of these can contribute to a reduction of coercivity, the easy axis orientation has the largest influence. For this, if the easy axis orientation is distributed within a cone with an opening angle of 15°, that is enough to reduce the coercivity by 1 T. Regarding the maximum energy product, the width of the grain boundary layer as well as the easy axis orientation were seen to have the largest influence, with the exchange constant only very weakly influencing the maximum energy product. Our analysis thus methodically clarifies the different factors contributing to the reduction of the value of the coercivity and maximum energy product when compared to the theoretical limit, a discrepancy known as Brown’s paradox.

Scaling in Rayleigh–Bénard convection

We consider the Nusselt–Rayleigh number problem of Rayleigh–Bénard convection and make the hypothesis that the velocity and thermal boundary layer widths, $\delta _u$ and $\delta _T$ , in the absence of a strong mean flow are controlled by the dissipation scales of the turbulence outside the boundary layers and, therefore, are of the order of the Kolmogorov and Batchelor scales, respectively. Under this assumption, we derive $Nu \sim Ra^{1/3}$ in the high $Ra$ limit, independent of the Prandtl number, $\delta _T/L \sim Ra^{-1/3}$ and $\delta _u/L \sim Ra^{-1/3} Pr^{1/2}$ , where $L$ is the height of the convection cell. The scaling relations are valid as long as the Prandtl number is not too far from unity. For $Pr \sim 1$ , we make a more general ansatz, $\delta _u \sim \nu ^{\alpha }$ , where $\nu$ is the kinematic viscosity and assume that the dissipation scales as $\sim u^3/L$ , where $u$ is a characteristic turbulent velocity. Under these assumptions we show that $Nu \sim Ra^{\alpha /(3-\alpha )}$ , implying that $Nu \sim Ra^{1/5}$ if $\delta _u$ were scaling as in a Blasius boundary layer and $Nu \sim Ra^{1/2}$ (with some logarithmic correction) if it were scaling as in a standard turbulent shear boundary layer. It is argued that the boundary layers will retain the intermediate scaling $\alpha = 3/4$ in the limit of high $Ra$ .

Unfolding flow features of MHD hybrid nanofluid (Ag–Al2O3–H2O) and mono nanofluid (Al2O3–H2O) flow over exponentially expanded sheet soaked in a Darcy–Forchheimer absorbent medium co-existing non-uniform heat generation/absorption

This study unfolds a comparative flow analysis of a hybrid nanofluid (Ag–Al2O3–H2O) and a single nanoparticle-based mono-nanofluid (Al2O3–H2O) through an exponentially permeable expanded sheet. The sheet is drenched in a non-Darcy absorbent medium under the prevalence of variable magnetic flux and non-uniform heat absorption/generation . Similarity transformations are engaged to the leading system of PDEs to transform the equations into ordinary BVP. To gain the analytical solution, DTM is engaged and numerical solution is produced by fifth-order RKF with shooting process. One of the interesting observations depicts that the fluid velocity for Ag–Al2O3–H2O decreases rapidly in comparison to Al2O3–H2O with the escalation of magnetic field parameter and porosity parameter . Whereas the thermal boundary layer width grows quickly for hybrid nanofluid with the augmentation of the porosity parameter and inertial parameter . Moreover, under the control of space and temperature reliant heat generation parameters the temperature distribution for hybrid nanofluid is better in contrast to mono-nanofluid. Also, it is noteworthy that the rate of heat transport decays by 9.56% for Al2O3–H2O while 11.44% for Ag–Al2O3–H2O as and the rate of shear stress is lessened by 45.71% and 37.93% for Al2O3–H2O and Ag–Al2O3–H2O respectively while varies from 0.0 to 1.6.

Double-diffusive transport in multicomponent vertical convection

Motivated by the ablation of vertical ice faces in salt water, we use three-dimensional direct numerical simulations to investigate the heat and salt fluxes in two-scalar vertical convection. For parameters relevant to ice-ocean interfaces in the convection-dominated regime, we observe that the salinity field drives the convection and that heat is essentially transported as a passive scalar. By varying the diffusivity ratio of heat and salt (i.e., the Lewis number $Le$), we identify how the different molecular diffusivities affect the scalar fluxes through the system. Away from the walls, we find that the heat transport is determined by a turbulent Prandtl number of $Pr_t\approx 1$ and that double-diffusive effects are practically negligible. However, the difference in molecular diffusivities plays an important role close to the boundaries. In the (unrealistic) case where salt diffused faster than heat, the ratio of salt-to-heat fluxes would scale as $Le^{1/3}$, consistent with classical nested scalar boundary layers. However, in the realistic case of faster heat diffusion (relative to salt), we observe a transition towards a $Le^{1/2}$ scaling of the ratio of the fluxes. This coincides with the thermal boundary layer width growing beyond the thickness of the viscous boundary layer. We find that this transition is not determined by a critical Lewis number, but rather by a critical Prandtl number $Pr\approx 10$, slightly below that for cold seawater where $Pr=14$. We compare our results to similar studies of sheared and double-diffusive flow under ice shelves, and discuss the implications for fluxes in large-scale ice-ocean models. By coupling our results to ice-ocean interface thermodynamics, we describe how the flux ratio impacts the interfacial salinity, and hence the strength of solutal convection and the ablation rate.

Influence of melting heat transfer and chemical reaction on the flow of non-Newtonian nanofluid with Brownian motion: Advancement in mechanical engineering

The aim of the present study is to investigate the melting heat and mass transport characteristics on the stagnation point flow of Powell–Eyring nanofluid over a stretchable surface because melting is so important in many processes, such as Permafrost melting, magma solidification, and thawing of frozen grounds, are all examples of soil melting and freezing around the heat exchanger coils of a ground-based pump. The developing mathematical model under the boundary layer flow in terms of differential equations is solved through a numerical algorithm using a boundary value problem solver bvp4c/shooting technique with the help of MATLAB software. The impact of emerging parameters on the velocity profile, temperature profile, and concentration profile is elaborated graphically. The profile and boundary-layer width rate for the value stretching parameter less than one rises when A enhances while the thickness of boundary layer velocity profile for the value stretching parameter greater than one decreases as A. The velocity function shows a decrement response for M, while the opposite behavior is seen against the concentration field. Furthermore, the numeric data for the friction factor and Nusselt number are demonstrated in tabular form, and the result shows a remarkable agreement with the previously published data.

The performance evolution of hybrid nanofluid flow over a rotating disk using Cattaneo–Christov double diffusion and Yamada–Ota model

The article addresses the problem related to Darcy–Forchheimer flow past a rotating disk with radial stretching and shrinking. The effects of viscous dissipation, convective heating and heating due to porous medium are considered; the heat and mass transfer expressions are modeled according to Cattaneo–Christov double diffusion theory. Additionally, a carbon nanotubes-based hybrid nanofluid is taken as working fluid. The mathematical model for existing problem is quite complex which is reduced to a simpler form (non-dimensional) by utilizing Von-Kármán similarity transformation. Then, these non-dimensional expressions are solved using Runge–Kutta–Fehlberg technique via shooting method. The impact of the sundry variables is illustrated via both graphs and tabular data, and the same is explained in detail. The result elucidates that the thermal boundary layer width of stretching rotating and shrinking rotating disks increase by rising the porosity parameter, Eckert number and thermal relaxation parameter. Moreover, a hike in the value of thermal relaxation parameter from 0.1 to 0.4 causes a reduction of 22.36% in heat transfer rate in stretching rotation, whereas a boost in the value of solutal relaxation parameter from 0.2 to 0.8 is responsible for 4.24% and 0.54% increment in mass transfer rate in stretching rotation and shrinking rotation, respectively.

Irreversibility analysis on the radiative buoyancy flow toward stagnation point through water conveying three kinds of nanoparticles past a heated vertical flat plate with the ramification of Hall effects

Recent advancements in nanotechnology have created a tremendous platform for the development of the improved performance of ultrahigh coolants known as nanofluids for several industrial and engineering technologies. The present research peruses an inspection of irreversibility analysis of mixed convective flow near a stagnation point provoked by ternary hybrid nanoparticles through a vertical heated flat plate with the Hall effects. Water conveying alumina (Al2O3), silver (Ag) and titanium oxide (TiO2) nanoparticles experiencing convectively heated as appropriate in the engineering or industry are investigated. The leading equations are non-dimensionalized using relevant similarity variables and then numerically cracked via utilizing the bvp4c solver. The impressions of different pertinent parameters on the axial velocity, transverse velocity and temperature profile along with heat transfer and drag force are discussed carefully. Double solutions are observed in the opposing flow; however, a single solution is obtained for the assisting flow. Also, the results indicate that due to nanofluid, the velocity boundary layer thicknesses decrease and the thermal boundary layer width upsurges. Further, the flow and the characteristics of heat transfer can be controlled using a magnetic field.

Robust self-organising fuzzy sliding mode-based path-following control for autonomous underwater vehicles

A robust self-organising fuzzy sliding mode control law steers autonomous underwater vehicles (AUVs) to track a predefined planar path at a constant speed without temporal specifications. An intelligent methodology has been adopted for path-following control to handle varying parametric uncertainties in vehicle dynamics and also conquers stringent preliminary condition constraints in several path-following control strategies illustrated in the literature. Robust controller design builds on a fusion of sliding mode control theory and fuzzy logic technique with an adaptation mechanism to tune boundary layer width and hitting gain. This novel strategy proposes two distinct tuning procedures: the first method commonly uses absolute error and their derivative as fuzzy input variables in a two-dimensional fuzzy logic rule structure. Herein, skew symmetry property is utilised in rule base structure to derive a single input fuzzy variable based on the signed distance technique, drastically reducing two-dimensional fuzzy logic rules. Since the second method provides substantial reductions in rule inferences through the use of the fuzzy rule's mirror image and the Lyapunov approach for tuning purposes, the resulting guidance control law yields fast convergence of the path-following error trajectory towards zero along with the elimination of chattering problem. Simulation results illustrate the effectiveness and robustness of the proposed control law to achieve favourable tracking performance with a high accuracy.

Darcy–Forchheimer flow of CNTs-H2O nanofluid over a porous stretchable surface with Xue model

This investigation includes a three-dimensional Darcy–Forchheimer flow model and the heat transfer phenomenon of H2O-CNTs nanofluid for a two-way stretchable surface. Xue’s proposed thermal conductivity model is employed. The numerical analysis scheme is applied to solve the transformed PDEs. The outline of velocities, temperature, surface drag forces and Nusselt number against relevant variables are portrayed. From this study, it has been noted that with an increase in Eckert numbers along both directions, two patterns were obtained for temperature curves, the initial temperature outlines increased and after that they decreased. Moreover, the width of the thermal boundary layer for H2O-MWCNT nanofluid was more compared to H2O-SWCNT nanofluid. To validate the existing code, numerical outcomes were compared to the earlier published data.

Ramification of Hall and Mixed Convective Radiative Flow towards a Stagnation Point into the Motion of Water Conveying Alumina Nanoparticles Past a Flat Vertical Plate with a Convective Boundary Condition: The Case of Non-Newtonian Williamson Fluid

Heat transfer technologies are experiencing rapid expansion as a result of the demand for efficient heating and cooling systems in the automotive, chemical, and aerospace industries. Therefore, the current study peruses an inspection of mixed convective radiative Williamson flow close to a stagnation point aggravated by a single nanoparticle (alumina) from a vertical flat plate with the impact of Hall. The convective heating of water conveying alumina (Al2O3) nanoparticles, as appropriate in engineering or industry, is investigated. Using pertinent similarity variables, the dominating equations are non-dimensionalized, and after that, via the bvp4c solver, they are numerically solved. We extensively explore the effects of many relevant parameters on axial velocity, transverse velocity, temperature profile, heat transfer, and drag force. In the opposing flow, there are two solutions seen; in the aiding flow, just one solution is found. In addition, the results designate that, due to nanofluid, the thickness of the velocity boundary layer decreases, and the thermal boundary layer width upsurges. The gradients for the branch of stable outcome escalate due to a higher Weissenberg parameter, while they decline for the branch of lower outcomes. Moreover, a magnetic field can be used to influence the flow and the properties of heat transfer.

Two-Dimensional Finite Element Method for Modeling Edge Effects in Cross-Ply Laminates

A 2-dimensional finite element (FE) method, based on an approximate analytical solution, is developed to define accurately stress fields and boundary layer width in symmetrically laminated composites. A better insight into the edge effect and delamination failure of cross-ply laminates is attained through a physical explanation of the mathematical FE solution. While some other solutions failed, the present FE solution obtains accurate results using a limited number of elements, and therefore reduces computer storage and computation time. A perturbation technique is employed to derive shape functions for the free edge boundary region, which properly describes the physics of sudden exponential growth of stress field in the vicinity of the edge. An original and comprehensive FE computer program is developed, using these specific shape functions, special numerical integration, and mesh generation algorithms. Results for bidirectional laminates under uniaxial extension are presented and compared with finite difference, FE, and approximate analytical solution.
 HIGHLIGHTS
 
 Defining accurately stress field in free edge region is the main challenge
 Shape functions based on perturbation solution yield accurate results
 Boundary layer width is proportional to geometric ratio of laminates
 Finer mesh in boundary region is required for small geometric ratio laminates
 A huge number of finite elements are needed to determine narrow boundary width
 
 GRAPHICAL ABSTRACT

Switching function parameter variation analysis of a quasi-sliding mode controlled induction motor drive

Sliding mode control is a nonlinear, robust control that is having better load disturbance rejection capability, less parameter sensitivity and fast dynamic response. Conventional sliding mode control introduces high chattering that can degrade the induction motor (IM) drive system responses. Hence, a quasi-sliding mode controller (Q-SMC) using a hyperbolic tangent function coupled with equivalent control is designed for robust speed control of vector-controlled IM drive in this work. This work focuses on the effect of variation of the switching function parameters of the Q-SMC on the performance of the drive. Extensive simulations are performed using MATLAB/Simulink software, and the switching function parameters are adjusted across a wide range and its impact on motor performance is studied qualitatively and quantitatively, with accompanying graphical results and various transient parameters. It is observed that a Q-SMC controller with a larger boundary layer width has less overshoot, less steady-state error, and a lower current THD. It is also observed that even though a high gain Q-SMC controller responds quickly, the percentage overshoot for high gain systems is likewise large. Hence, if the boundary layer width and switching gain parameters are optimized, a Q-SMC speed controller is a promising choice for a high-performance IM drive.

Simulation Research on the Effect of Reynolds Number on the Aerodynamic Characteristics of High Lift Device

Based on the consideration of grid factors and turbulence model factors, the confluence boundary layer and aerodynamic characteristics of the NHLP-2D airfoil are studied. Compared with the experimental results, the grid distribution near the confluence boundary layer is important for calculating the width of the wake area. The calculation results of the structured and unstructured grids are basically the same when the surface grid distribution is the same. On the basis of the mesh refinement in the wake area, the aerodynamic characteristics calculated by the SA turbulence model and the distribution of the total pressure coefficient in the confluence boundary layer are in good agreement with the experimental results. The width of the confluence boundary layer and the wake strength decrease with the increase of Reynolds number, the lift coefficient increases with the increase of Reynolds number, but the drag and pitching moment have opposite pattern. The maximum lift of the airfoil increases as the Reynolds number increases. As the Reynolds number increases, the stall angle of attack of the airfoil gradually increases and then remains stable with the increase of the Reynolds number.