Hydrodynamic Boundary Layer Thickness Research Articles

Mathematical Modeling on Magnetohydrodynamics Upper Convected Maxwell Fluid Flow Past a Flat Plate Using Spectral Relaxation Approach

The focus of this treatise is on the scrutiny of MHD heat and mass transfer motion of upper convection Maxwell (UCM) fluid in a thermally radiated flat plate. The models of the flow equations were considered when there is a temperature difference: the Cattaneo-Christov model and the Soret-Dufour mechanisms. A magnetic field of firmness was inflicted in opposition to the flow. The flow inspection is controlled by partial differential equations (PDEs). Suitable similarity variables were utilised on the PDEs to obtain a set of nonlinear ordinary differential equations (ODEs). The simplified set of ODEs shall be answered by exploiting the spectral relaxation method (SRM). The SRM is a numerical technique that solves differential equations by utilising the repetition of a sequence of operations that is iterated iteratively by first decoupling the coupled systems of equations. The magnetism was found to decline the velocity and hydrodynamic boundary layer due to the Lorentz force. The Deborah number was found to enhance the velocity contour. The Eckert number was discovered to improve the temperature profile due to the production of heat energy within the boundary layer. An increase in Prandtl number was found to enhance the hydrodynamic and thermal boundary layer thickness. The local skin friction and Nusselt number were found to be elevated by the increase in the Dufour parameter.

A Model on Free Convective Casson Fluid Flow Past a Permeable Vertical Plate with Gyrotactic Microorganisms

This study focused on free convective MHD Casson fluid flow past a permeable vertical plate with microorganisms and Soret-Dufour mechanisms. The physical model is governed by system of partial differential equations (PDEs). The set of PDEs were overhaul into a nonlinear ordinary differential equation by considering acceptable similarity variable. The spectral relaxation method (SRM) was utilized to obtain solution to the overhaul nonlinear ordinary differential equations. This study examined critically the significant of the Gyrotactic microorganisms on Casson fluid locomotion Also, the significance of parameters such as thermal radiation, Soret-Dufour mechanisms, Brownian motion, Joule heating and so on are examined on velocity, concentration, motile microorganism and temperature profiles. The Casson parameter was found to intensify the velocity contour and the hydrodynamic boundary layer thickness. A higher magnetic parameter gives rise to Lorentz force which declines the motion of the fluid. There is higher temperature within the boundary layer because of the presence of thermal radiation.

Development of the hydrodynamic and thermal boundary layers of forced convective laminar flow through a horizontal tube with a constant heat flux

The development of the thermal and hydrodynamic boundary layers of laminar flow through a smooth horizontal tube was investigated for forced convective conditions. The tube was heated at a constant heat flux and had an inner diameter of 4 mm and a length of 6 m. Numerical simulations were conducted using ANSYS Fluent 19.3 with a highly structured mesh and accurate temperature-dependent thermophysical properties. Laminar flow of air, water, and a mixture of 75% ethylene glycol (by weight fraction) in water was simulated, which covered a Reynolds number range of 630 to 2000 and Prandtl number range of 0.7–70. Theoretical methods were also used to obtain the relevant boundary layer growth parameters, such as velocity profiles, velocity gradient, vorticity, temperature profiles, Nusselt numbers, as well as hydrodynamic and thermal boundary layer thicknesses. Subsequently, methods were proposed to determine the development of both boundary layers in the entrance region, the locations at which the boundary layers merged, and the location at which the flow became fully developed. By analysing the velocity and temperature profiles, it was found that our conventional understanding of the merging of the boundary layers in internal tube flows had to be modified. Three distinct regions were identified in the hydrodynamic entrance region of laminar forced convective tube flow: (1) hydrodynamic inlet region, (2) shear stress developing region, and (3) vorticity adjustment region. Furthermore, three distinct regions were identified in the thermal entrance region: (1) thermal inlet region, (2) convective diffusion region, and (3) conductive diffusion region.

Cattaneo-Christov Heat and Mass Transfer Flux Across Electro-Hydrodynamics Blood-Based Hybrid NanoFluid Subject to Lorentz Force

This study is unique in that it considers both the magnetic field and the electromagnetic force in the flow direction. The imposed magnetic field causes the fluid flow to slow down, whereas the electric force factor increases the fluid velocity and temperature.The principle aim of the present study is addressing electro-hydrodynamic blood-based hybrid nanofluid behaviour when it passes through a vertical stretchable areas.The theories of the Cattaneo-Christov model were investigated with thermal radiation possessing heat generation in this study. Partial differential equations were used to solve the problem. To look into systems of coupled nonlinear differential equations, the set of equations was simplified using appropriate variables. The answer to simplified equations is obtained using a novel numerical method known as the Spectral relaxation method (SRM). Because it can decouple and linearize the set of equations, the SRM is preferred. The SRM uses the Gauss-Seidel approach as itsfoundation. The analysis of equations were done numerically utilizing spectral relaxation techniques. The outcomes are (i) Higher electric field factor is noticed to enhance both velocity and temperature profile; (ii) Increase in Eckert number is observed to increase both thermal and hydrodynamic boundary layer thickness; (iii) A higher value of K extends the hole and allows for higher fluid flow by increasing the fluid velocity; (iv) Increase in Pr depreciates both velocity and temperature profile; and (v) A higher value of Sc depreciates the skin friction and Sherwood number

Chemically reactive Maxwell nanoliquid flow by a stretching surface in the frames of Newtonian heating, nonlinear convection and radiative flux: Nanopolymer flow processing simulation

AbstractThe effects of a chemical reaction and radiative heat flux in a nonlinear mixed thermo-solutal convection flow of a viscoelastic nanoliquid from a stretchable surface are investigated theoretically. Newtonian heating is also considered. The upper-convected Maxwell (UCM) model is deployed to represent the non-Newtonian characteristics. The model also includes the influence of thermal radiation that is simulatedviaan algebraic flux model. Buongiorno’s two-component nanofluid model is implemented for thermophoretic and Brownian motion effects. Convective thermal and solutal boundary conditions are utilized to provide a more comprehensive evaluation of temperature and concentration distributions. Dimensionless equations are used to create the flow model by utilizing the appropriate parameters. The computed models are presented through a convergent homotopic analysis method (HAM) approach with the help of Mathematica-12 symbolic software. Authentication of HAM solutions with special cases from the literature is presented. The impact of various thermophysical, nanoscale and rheological parameters on transport characteristics is visualized graphically and interpreted in detail. Temperatures are strongly enhanced with Brownian motion and thermophoresis parameters. Velocity is boosted with the increment in the Deborah viscoelastic number and mixed convection parameter, and the hydrodynamic boundary layer thickness is reduced. A stronger generative chemical reaction enhances concentration magnitudes, whereas an increment in the destructive chemical reaction reduces them and also depletes the concentration boundary layer thickness. Temperature and concentration are also strongly modified by the conjugate thermal and solutal parameters. Greater radiative flux also enhances the thermal boundary layer thickness. Increasing the Schmidt number and the Brownian motion parameter diminish the concentration values, whereas they elevate the Sherwood number magnitudes,i.e.enhance the nanoparticle mass transfer rate to the wall.

Entropy analysis of nonlinear radiative Casson nanofluid transport over an electromagnetic actuator with temperature-dependent properties

This paper examines the transport of nonlinear radiative Casson nanoliquid past a vertical electromagnetic actuator with temperature-dependent transport properties and entropy analysis. The model incorporates the Grinberg-term containing the influence of Lorentz force impinged by the actuator and the impact of the thermo-migration and Brownian motion of nanoparticles. The main equations are translated into non-dimensional forms via appropriate dimensionless variables and then solved using Chebyshev collocation method and verified by Galerkin type of weighted residual method and existing studies found in literature. From the study, the reactions of the embedded physical quantities of interest on the fields of velocity, temperature, nanoparticles concentration and entropy generation number are communicated graphically and deliberated upon. The implications of the results are that the modified Hartmann number aids the fluid flow as the Richardson number boosts the hydrodynamic boundary layer thickness whereas the converse occurs with Casson fluid material parameter. Also, the entropy generation is enhanced in the existence of viscous dissipation and suction while it declines with thermophoretic influence. Conclusively, it is found that the rate of entropy generation can be reduced by augmenting the magnitudes of the thermophoresis and Brownian motion parameters.

Dual similarity solutions because of mixed convective flow of a double-nanoparticles hybrid nanofluid: critical points and stability analysis

Purpose The purpose of this article is to study the steady laminar magnetohydrodynamics mixed convection stagnation-point flow of an alumina-graphene/water hybrid nanofluid with spherical nanoparticles over a vertical permeable plate with focus on dual similarity solutions. Design/methodology/approach The single-phase hybrid nanofluid modeling is based on nanoparticles and base fluid masses instead of volume fraction of first and second nanoparticles as inputs. After substituting pertinent similarity variables into the basic partial differential equations governing on the problem, the authors obtain a complicated system of nondimensional ordinary differential equations, which has non-unique solution in a certain range of the buoyancy parameter. It is worth mentioning that, the stability analysis of the solutions is also presented and it is shown that always the first solutions are stable and physically realizable. Findings It is proved that the magnetic parameter and the wall permeability parameter widen the range of the buoyancy parameter for which the solution exists; however, the opposite trend is valid for second nanoparticle mass. Besides, mass suction at the surface of the plate as well as magnetic parameter leads to reduce both hydrodynamic and thermal boundary layer thicknesses. Moreover, the assisting flow regime always has higher values of similarity skin friction and Nusselt number relative to opposing flow regime. Originality/value A novel mass-based model of the hybridity in nanofluids has been used to study the foregoing problem with focus on dual similarity solutions. The results of this paper are completely original and, to the best of the authors’ knowledge, the numerical results of the present paper were never published by any researcher.

Spectral relaxation computation of electroconductive nanofluid convection flow from a moving surface with radiative flux and magnetic induction

Abstract A theoretical model is developed for steady magnetohydrodynamic viscous flow resulting from a moving semi-infinite flat plate in an electrically conducting nanofluid. Thermal radiation and magnetic induction effects are included in addition to thermal convective boundary conditions. Buongiorno’s two-component nanoscale model is deployed, which features Brownian motion and thermophoresis effects. The governing nonlinear boundary layer equations are converted to nonlinear ordinary differential equations by using suitable similarity transformations. The transformed system of differential equations is solved numerically, employing the spectral relaxation method (SRM) via the MATLAB R2018a software. SRM is a simple iteration scheme that does not require any evaluation of derivatives, perturbation, and linearization for solving a nonlinear system of equations. Effects of embedded parameters such as sheet velocity parameter$\lambda$, magnetic field parameter$\beta$, Prandtl number$Pr$, magnetic Prandtl number$Prm$, thermal radiation parameter$Rd$, Lewis number$Le$, Brownian motion parameter$Nb$, and thermophoresis parameter$Nt$ on velocity, induced magnetic field, temperature, and nanoparticle concentration profiles are investigated. The skin-friction results, local Nusselt number, and Sherwood number are also discussed for various values of governing physical parameters. To show the convergence rate against iteration, residual error analysis has also been performed. The flow is strongly decelerated, and magnetic induction is suppressed with greater magnetic body force parameter, whereas temperature is elevated due to extra work expended as heat in dragging the magnetic nanofluid. Temperatures are also boosted with increment in nanoscale thermophoresis parameter and radiative parameter, whereas they are reduced with higher wall velocity, Brownian motion, and Prandtl numbers. Both hydrodynamic and magnetic boundary layer thicknesses are reduced with greater reciprocal values of the magnetic Prandtl number Prm. Nanoparticle (concentration) boundary layer thickness is boosted with higher values of thermophoresis and Prandtl number, whereas it is diminished with increasing wall velocity, nanoscale Brownian motion parameter, radiative parameter, and Lewis number. The simulations are relevant to electroconductive nanomaterial processing.

Cooling capacity of magnetic nanofluid in presence of magnetic field based on first and second laws of thermodynamics analysis

ABSTRACT In this numerical study, the cooling capacity of water/Cu magnetic nanofluids in a horizontal duct exposed to a magnetic field has been investigated. Both the first and second laws of thermodynamics analysis are conducted. All simulations have been performed for the volume fraction of 5%, Reynolds number (Re) of 100, and the Hartmann numbers (Ha) of 0, 20, 50, and 80. Temperature, velocity profiles, Nusselt number (Nu), Lorentz force (F), and thermal, frictional, and magnetic irreversibility have been studied by changing Ha. The received results indicate that as the Ha increases, the hydrodynamic boundary layer thickness decreases, while the length of the hydrodynamic entrance length increases. In addition, as the Ha increases in the range of 0 to 80, the local Nu number increases by about 7.5%, and the local wall temperature drops to 2.7%. The second law of thermodynamic analysis showed that as the Ha increases, the frictional irreversibility augments, while the thermal irreversibility decreases. Finally, the order of thermal irreversibility is larger than that of frictional and magnetic irreversibility.

Effect of suction or blowing on velocity and temperature distribution of flow over a flat plate

In this paper, laminar, incompressible flow with constant pressure gradient and constant wall temperature over a flat plate for several values of fluid suction and blowing parameters at β = −0.1978, m = −0.09 and β = 0.66, m = 0.5 respectively were studied numerically. The governing nonlinear partial differential equations are transformed to ordinary differential equations by using a similarity variable and then solved numerically adopting Runge–Kutta integration method. Effects of continuous suction and blowing with very low ratio of constant velocity to free stream neglecting the mass transfer on the velocity, temperature profiles and skin friction are presented. The results indicated that the skin friction factor is increased with increase the value of the suction for given Reynolds number. In order to reduce the skin friction, the laminar flow and avoid transition the blowing have been applied. The effect of fluid blowing parameter on velocity profile in hydrodynamic boundary layer are the certain incidence angles. Results show that the hydrodynamic boundary layer thickness is increased due to the blowing. It can be noted that the delaying separation is led to increase of the maximum lift. It was observed that increasing the maximum lift by maintain laminar flow to avoid transition and delaying separation to reduce skin friction is existing by suitable magnitude of suction or blowing. It is concluded that the skin friction is decreasing due to apply the blowing over a flat plate.

Influence of perforated triple twisted tape on thermal performance characteristics of a tube heat exchanger

One of the challenging tasks in the development of heat exchangers is to select an appropriate geometry that will consume minimum energy. This could be achieved by designing a suitable enhancing device which will boost the thermal performance characteristics of heat exchangers. Twisted tape swirl generator, a heat transfer enhancement device equipped in a tube heat exchanger would induce turbulence and superimposed vortex motion causing a reduction in the hydrodynamic or thermal boundary layer thickness which results in the improvement of convective heat transfer. In this study, a new design of perforated triple twisted tape (PTTT) insert was applied as a swirl flow device with four different porosities (Rp) of 1.2, 4.6, 10.4 and 18.6% to enhance the convective heat transfer of a tube heat exchanger. The effects of porosity (Rp) of PTTT swirl generators on heat transfer, fluid friction and thermal performance characteristics of a tube heat exchanger were experimentally investigated. The experiments were conducted under constant heat flux condition using air as the working fluid in turbulent flow regime for variation in Reynolds number (Re) from 7250 to 49,800. The Nusselt number of the tube installed with PTTT inserts for porosity ranging from 1.2 to 18.6% was found to be 88–320% higher, whereas, the friction factor was achieved to be 112–355% higher in comparison to the plain tube. The highest heat transfer result of 320% was obtained at a porosity of 4.6% with PTTT insert compared to the smooth tube. Despite the significant enhancement in heat transfer, the friction factor was obtained to be 355% higher in comparison to the plain tube at a porosity of 4.6% with PTTT insert. The experimental results demonstrated that Nusselt number, friction factor, and thermal enhancement efficiency (TEE) were increased with decreasing the porosity of the tape inserts except for 1.2%. The heat transfer performance was evaluated based on constant blower power and the values were found to be 1.13–1.5 relative to the plain tube. The maximum TEE of 1.5 was achieved using PTTT insert with a porosity of 4.6%. Based on the experimental data, the empirical correlations of heat transfer, friction factor and TEE were developed in terms of Rp, Re and Prandtl number (Pr) to predict the heat transfer, friction factor and TEE. The optimal design of PTTTs based on the evaluation of thermal performance with suitable porosity could be an excellent heat transfer enhancement device for many industrial applications.

MHD Flow and Heat Transfer of SiC-TiO2/DO Hybrid Nanofluid due to a Permeable Spinning Disk by a Novel Algorithm

This study intends to semi-analytically investigate the steady 3D boundary layer flow of a SiC-TiO2/DO hybrid nanofluid over a porous spinning disk subject to a constant vertical magnetic field. Here, the novel attitude to single-phase hybrid nanofluid model corresponds to considering nanoparticles and base fluid masses to compute solid equivalent volume fraction, solid equivalent density, and also solid equivalent specific heat at constant pressure. The basic PDEs are transformed into dimensionless ODEs using Von Karman similarity transformations, which are then solved numerically using bvp4c function. Results indicate that mass suction and magnetic field effects diminish all hydrodynamic and thermal boundary layer thicknesses. Finally, a significant report is presented to investigate quantities of engineering interest due to governing parameters’ effects.

AERODYNAMICS OF JETS INTERACTING WITH A FLAT SURFACE

In this paper, the development features of a single free jet of hightemperature nitrogen interacting with a flat surface were studied. Calculation of the heat exchange process during heating by the attacking jets is very difficult to implement analytically due to complexity of the gas-dynamic processes occurring both in a single jet and in a system of jets interacting with the metal. The computational difficulties are aggravated by the fact that when interacting with the surface the jet as such disappears. The flat (fan) flow interacts with the surface: form, aerodynamic properties and thermal state of the flow strongly differ from those of the original jet. The studies were conducted on the basis of numerical simulation in the FloEFD software and computing complex for multiphysical simulation based on solution of the equations of gas dynamics and heat transfer. The solved system of equations consisted of Navier-Stokes equations, equations of energy and continuity and was supplemented by k – ε turbulence model. A three-dimensional model was developed for simulation, the necessary properties, initial and boundary conditions were specified. In the study of aerodynamics of a single high-temperature jet interacting with the surface, the main defining values were: nitrogen flow rate from the nozzle U0 , nitrogen temperature T, internal diameter of the nozzle d0 , distance from the nozzle section to the surface h, distance from the critical point (point of intersection of the jet axis with the surface) along the flow radius r. Data on the gas velocity decrease as the jet develops due to the loss of initial energy to engage the motionless surrounding gas in motion, is presented. The studies have shown that increase in the initial velocity of gas outflow brings the area of higher velocities closer to the surface both in the jet itself and in the fan jet. This factor contributes to heat transfer intensification. In addition, high speeds increase the total thickness of the fan flow and reduce the thickness of hydrodynamic boundary layer, which increases with distance from the critical point.

Mathematical Modelling of Hydromagnetic Casson non-Newtonian Nanofluid Convection Slip Flow from an Isothermal Sphere

Abstract In this article, the combined magnetohydrodynamic heat, momentum and mass (species) transfer in external boundary layer flow of Casson nanofluid from an isothermal sphere surface with convective condition under an applied magnetic field is studied theoretically. The effects of Brownian motion and thermophoresis are incorporated in the model in the presence of both heat and nanoparticle mass transfer convective conditions. The governing partial differential equations (PDEs) are transformed into highly nonlinear, coupled, multi-degree non-similar partial differential equations consisting of the momentum, energy and concentration equations via appropriate non-similarity transformations. These transformed conservation equations are solved subject to appropriate boundary conditions with a second order accurate finite difference method of the implicit type. The influences of the emerging parameters i.e. magnetic parameter (M), Buoyancy ratio parameter (N), Casson fluid parameter (β), Brownian motion parameter (Nb) and thermophoresis parameter (Nt), Lewis number (Le), Prandtl number (Pr) and thermal slip (ST) on velocity, temperature and nano-particle concentration distributions is illustrated graphically and interpreted at length. Increasing viscoplastic (Casson) parameter decelerates the flow and also decreases thermal and nano-particle concentration. Increasing Brownian motion accelerates the flow and enhances temperatures whereas it reduces nanoparticle concentration boundary layer thickness. Increasing thermophoretic parameter increasing momentum (hydrodynamic) boundary layer thickness and nanoparticle boundary layer thickness whereas it reduces thermal boundary layer thickness. Increasing magnetohydrodynamic body force parameter decelerates the flow whereas it enhances temperature and nano-particle (species) concentrations. The study is relevant to enrobing processes for electric-conductive nano-materials, of potential use in aerospace and other industries.

Simultaneous measurement of velocity and temperature fields during natural convection in a water-filled cubical cavity

Many advanced schlieren systems have been reported in the literatures and a great number of it are associated in gas flow study, commonly air. Present research proposes an efficient use of Z-type schlieren technique for simultaneous measurement of ray-averaged velocity as well as temperature fields beside qualitative flow visualization. Experiments are conducted to study natural convective flow in de-ionized water in a differentially heated cubical enclosure. The same set of transient schlieren images has been used for simultaneous measurement of velocity and temperature field. The velocity field is also determined by using laser based particle image velocimetry (PIV) to compare with schlieren based technique. The developed schlieren system has been used to obtain ray-averaged quantities, such as, thermal and hydrodynamic boundary layer thicknesses, temperature gradient and temperature fields, distribution of local Nusselt number along the thermally active walls, average Nusselt number and velocity field. In this work, the above quantities have been simultaneously determined for the first time using halogen projector lamp based Z-type calibration schlieren system. The results inspire the use of calibration schlieren system for the study of natural convective flow in water-filled cubical enclosures. The above results also provide experimental data as new benchmark for researchers to compare results for high Rayleigh number free convective flow with water as the working medium.

Effect of Wall Boundary Layer Thickness on Power Performance of a Recirculation Microbial Fuel Cell

Hydrodynamic boundary layer is a significant phenomenon occurring in a flow through a bluff body, and this includes the flow motion and mass transfer. Thus, it could affect the biofilm formation and the mass transfer of substrates in microbial fuel cells (MFCs). Therefore, understanding the role of hydrodynamic boundary layer thicknesses in MFCs is truly important. In this study, three hydrodynamic boundary layers of thickness 1.6, 4.1, and 5 cm were applied to the recirculation mode membrane-less MFC to investigate the electricity production performance. The results showed that the thin hydrodynamic boundary could enhance the voltage output of MFC due to the strong shear rate effect. Thus, a maximum voltage of 22 mV was obtained in the MFC with a hydrodynamic boundary layer thickness of 1.6 cm, and this voltage output obtained was 11 times higher than that of MFC with 5 cm hydrodynamic boundary layer thickness. Moreover, the charge transfer resistance of anode decreased with decreasing hydrodynamic boundary layer thickness. The charge transfer resistance of MFC with hydrodynamic boundary layer of thickness 1.6 cm was 39 Ω, which was 0.79 times lesser than that of MFC with 5 cm thickness. These observations would be useful for enhancing the performance of recirculation mode MFCs.

Multiphysics Modeling of Surface Finishing Performance in Pulsed-Waveform Electrochemical Machining

Electrochemical machining (ECM) is a manufacturing technology that allows metal to be precisely removed by electrochemical oxidation and dissolution into an electrolyte solution. ECM is suited for machining parts fabricated from “difficult to cut” materials and/or parts with complicated and intricate geometries. In ECM, the workpiece is the anode and the tool is the cathode in an electrochemical cell; by relative movement of the shaped tool into the workpiece, the mirror image of the tool is “copied” or machined into the workpiece. One notable difficulty with ECM is an inability to predict a priori the tool and process parameters required in order to satisfy the final specifications of the fabricated part [[1]]. Accordingly, there is potential value in development of a physical phenomenon-based design platform to predict optimal ECM tool shape. Such a capability is anticipated to dramatically shorten the process/tooling development cycle. A further goal of ECM is to simultaneously achieve a target surface finish on the machined part. It is thus of interest to develop the capability also to predict the distribution of local surface finish resulting from ECM processing. Modeling of the changes in local surface finish intrinsically operates on a different length scale from that of bulk material removal (μm, versus mm or cm), and thus is most easily treated separately. The physicochemical phenomena involved in the evolution of surface finish during ECM processing are strongly coupled, and include the electric field itself (primary current distribution), surface polarization and electrochemical kinetics (secondary current distribution), and fluid flow and mass transfer (tertiary current distribution). Of particular interest is modeling of pulsed-waveform ECM, for which significant practical advantages have been demonstrated [[2],[3],[4]]. While an extensive body of literature exists analyzing pulsed electrodeposition [[5],[6]], comparatively little work has been published to date on pulsed ECM [[7],[8]]. This talk will discuss recent modeling work seeking to develop a solid foundation for a predictive understanding of the surface finishing aspects of ECM processes. The work described herein encompasses time-dependent modeling of concentration profiles and other relevant physical quantities under the application of pulse(-reverse) current ECM waveforms, starting from simulations assuming a locally flat surface and working toward quantitative descriptions of the transient and steady-periodic behavior on structured substrates. Prior work (see, e.g., Ref. 3) has demonstrated the value in differential pulsed-ECM processing of surfaces with features of size comparable to or larger than the hydrodynamic boundary layer thickness (“macroprofiles”) versus surfaces with features much smaller than the boundary layer thickness (“microprofiles”). Figure 1 (left) plots a schematic of the pulsating concentration profiles at the end of the pulse on-time in forward-only pulsed ECM, assuming 100% current efficiency. Here, the pulse period (ton + toff) and the peak applied current density (jpeak) are assumed constant, with the curves corresponding to different values of ton. The saturation concentration of the dissolved metal is Cs, and the bulk metal concentration Cb equals zero. The quantity τ is the “transition time,” which is the value for ton for which the metal concentration at the surface rises exactly to Cs at the end of the forward pulse. As can be seen, for ton < τ, metal dissolution is not mass transfer limited, but for any ton > τ the rate of dissolution is constrained by the diffusion of the dissolved material, approaching the DC mass transfer rate for sufficiently high ton. Figure 1 (right) illustrates that preliminary simulations performed with COMSOL Multiphysics® exhibit behavior consistent with this qualitative concept.

Thermal transport due to liquid jet impingement on superhydrophobic surfaces with isotropic slip

This paper presents an analytical investigation of thermal transport due to a steady, laminar, axisymmetric liquid jet impinging normally on a superhydrophobic (SHPo) surface maintained at constant surface temperature. At the liquid-surface boundary of the spreading thin film, an isotropic hydrodynamic slip and temperature jump are imposed to approximate the SHPo surface boundary condition. Applying an integral analysis within the thin film results in a system of differential equations which are solved numerically to obtain local hydrodynamic and thermal boundary layer thicknesses, thin film height, and local and radially averaged heat flux. The classical smooth hydrophobic scenario with no-slip and no-temperature jump showed excellent agreement with previous differential analysis of the same problem. The influence of varying temperature jump length on the local Nusselt number was obtained over a range of Reynolds and Prandtl numbers. Increasing temperature jump length results in a dramatic decrease in the local thermal transport near the impingement point. The greatest decrease occurs at small temperature jump lengths. Further, local and average Nusselt numbers are less influenced by the Reynolds and Prandtl numbers as temperature jump length increases. Overall, variations in the temperature jump length exert much more influence than variations in the hydrodynamic slip length.

MHD nanofluid flow over a rotating disk with partial slip effects: Buongiorno model

In this article, Von Kármán problem of infinite rotating disk is extended for the case where the space above the rough disk is equipped by an electrically conducting nanofluid. Buongiorno model is employed to incorporate Brownian motion and thermophoresis effects due to nanoparticles. Mass transfer process is accompanied with zero normal flux condition. Additionally, heat transfer mechanism is inspected through more general temperature jump conditions. Accurate similarity solutions are developed for full range of slip coefficients by means of standard shooting procedure. Also, a collocation based MATLAB routine is used for findings the solutions. The impact of parameters on the flow fields is depicted by plotting graphs. It is predicted that velocity distributions are inversely proportional to the magnetic field parameter. Velocity slip effect also presents opposition to the momentum transport and reduces hydrodynamic boundary layer thickness. It is also found that Brownian motion has negligible influence on the heat flux at rotating disk. Further, thermal and concentration boundary layer thicknesses are increased due to the existence of thermophoretic force. Axial velocity is negative due to the downward fluid motion caused by the disk rotation. The analysis for no-slip boundary can also be retrieved as a special case of the present model.

An analytical solution of turbulent boundary layer fluid flow over a flat plate at high Reynolds number

Abstract This is an analytical investigation of turbulent boundary layer flow over a flat plate in the range of Reynolds number between 10 6 and 10 9 . With the use of novel similarity variables, the governing partial differential equations are transformed to an ordinary differential equation with inconsistent coefficients and they are solved analytically via the optimal homotopy analysis method (OHAM). To achieve the convergence of analytical series solutions, residual errors have been calculated and minimized. In addition, the distribution of the velocity profile, friction coefficient and thickness of the boundary layer are calculated and compared with the pertained experimental results in the literature, where good agreements between them are observed. Based on the series solution, analytical expressions for the friction coefficient and the hydrodynamic boundary layer thicknesses were proposed and compared with previously existed expressions. It was shown that the proposed expressions are highly accurate for the high Reynolds number.