Stationary Flat Plate Research Articles

Three-dimensional upper-convected Maxwell nanofluid transport due to cross-diffusion and stratification: Numerical solutions

Cross-diffusion and stratification (thermal/solutal) effects on the transport of non-Newtonian nanofluid are analyzed in this investigation. Upper-convected Maxwell model is chosen to depict the non-Newtonian behavior. The flow is considered to be bounded by a stationary flat plate from one side. This phenomenon is formulated mathematically under boundary layer approximation which is further solved by the numerical procedure of Runge-Kutta-Fehlberg along with Newton-Raphson shooting scheme. Effects of appearing parameters are elaborated with the help of tables and figures. It is expected that the stratification and diffusion processes will help in the storage and transport of energy through the moving non-Newtonian nanofluid. It is found that Soret number and solutal stratification have similar impacts on temperature. Moreover, Dufour number and thermal stratification have adverse impacts on temperature/concentration.

Potential impacts of Cattaneo–Christov model of heat flux on the flow of Carreau–Yasuda fluid with mixed convection over a vertical stationary flat plate

Heat transfer in Carreau–Yasuda fluid with mixed convection, effect of Cattaneo–Christov model of heat flux past a vertical flat plate has been studied in this paper. Using appropriate transformations, governing PDEs are reduced to higher order non-linear non-dimensional ODEs and subsequently these are solved using “bvp4c” package of MATLAB. The Carreau–Yasuda fluid is used to explore the behaviour of fluids having shear-thinning and shear-thickening characters for several values of the parameter called power law exponent involved in the fluid model. The effects of different physical parameters, such as Weissenberg number(We), thermal relaxation parameter(α), Carreau–Yasuda fluid parameter(d), mixed convection parameter(λ) and Prandtl number(Pr) on velocity and temperature has been investigated and depicted through graphs. Results reveals that for lower value of Pr velocity of fluid enhances with higher value of λ and reverse effect is witnessed for larger Pr. Also, velocity rises for shear-thinning fluid(STNF) and decreases for shear-thickening fluid(STKF) with We and d. Temperature exhibits growing behaviour for Cattaneo–Christov model of heat flux near the plate in comparison with Fourier's model, whereas velocity exhibits growing behaviour with α. For larger We, temperature in Carreau–Yasuda model upsurges for STKF and falls for STNF. The surface drag-force displays higher values for both STNF and STKF with growth in λ. The cooling rate rises/declines with We for STNFs/STKFs.

Numerical investigation on the thermal performance of a flat plate solar collector using ZnO & CuO water nanofluids under Egyptian weathering conditions

Stationary flat plate solar collectors (FPSC) absorber plate and outlet water-nanofluid temperatures are investigated under steady state and Egyptian weather conditions. A 3D computational fluid dynamics (CFD) model, is developed to predict the outlet and absorber plate temperatures at 1-h interval sampling rate through the day and its effect on thermal performance of solar collectors. Simulation model is verified and validated by comparing the results to other previous research. In the research, the working fluids investigated are: water (H2O), zinc oxide (ZnO) nanofluid, and copper oxide (CuO) nanofluid water based with three volume concentration fraction (VF) of 0.05, 0.10 and 0.15% nanoparticles varied for various mass flow rates of 0.0125 and 0.025 kg s−1. The investigation revealed that the best achievement is obtained by using H2O–CuO nanofluid with average efficiency about 81.64% at mass flow rate of 0.0125 kg s−1 and VF of 0.15%. Using H2O–ZnO nanofluid resulted in an average efficiency of about 77.64% at mass flow rate of 0.0125 kg s−1 and VF of 0.15%, which is compared to the best achievement for water fluid of 60.21%. The results present the significant enhancement of the average thermal performance efficiency due to the usage of nanofluids.

Hydromagnetic Blasius-Sakiadis flows with variable features and nonlinear chemical reaction

Time-dependent, two-dimensional Sakiadis flow of quiescent fluid is considered. The flow is induced by stationary flat plate via uniform free-stream (Blasius flow). The variable conductivity and viscosity ratio parameters and non-linear chemical reaction are employed in the mathematical equations. Similarity variables are employed in the governing transport expressions to convert into the ordinary differential system. The transformed system is computed numerically by employing Runge-Kutta scheme via shooting criteria. Results of concentration, velocity and temperature distributions are studied through plots. Moreover, mass and heat transfer rates and friction factor have been discussed in detail. The constraint of chemical reaction slow down the friction-factors and heat transportation rates for the Sakiadis-Blasius flow situations and enhances the mass transportation rate in both cases. Rate of mass transportation is smaller in Sakiadis flow as comparative to Blasius flow. The present results of the heat transfer rate are matched with the literature and excellent agreement is noticed.

The role of thermal radiation on the boundary layer past a stationary flat plate with constant surface boundary condition

This study is devoted to investigating the importance of thermal radiation on the boundary layer over a horizontal surface considering classical constant surface boundary condition. The mathematical model consists of coupled two-dimensional partial differential equations which are transformed to the set of ordinary differential equations via the similarity transformation. The final set of dimensionless equations is solved numerically using Runge Kutta Fehlberg (RKF45) method in Maple software. The significant effect of the thermal radiation is examined using four fluids namely; water, Sulphur oxide, air and mercury whose respective Prandtl numbers are 7, 2, 0.72 and 0.044. The influence of other prominent parameters affecting the flow formation and temperature profile is demonstrated using tables and graphs. The results indicated that the thermal boundary layer thickness could be increase by reducing the Prandtl number. The results also showed that increasing the thermal radiation parameter has a positive impact on the boundary layer thickness. The heat transfer rate could be improved by increasing thermal radiation or decreased by increasing the values of the Prandtl number. Regarding the temperature gradient, an observable increasing is seen far from the flat plate with the growing of thermal radiation whereas the opposite trend is true near the plate surface.

Unsteady boundary layer flow and heat transfer of a viscous fluid over a semi-infinite stationary flat plate: Blasius flow problem

In this paper, a class of similarity solution for the unsteady Navier–Stokes equations which describe the two-dimensional unsteady laminar boundary layer flow of a viscous incompressible fluid over a semi-infinite flat stationary plate surface has been derived. Here, the boundary layer is formed owing to an uniform free stream flow over this plate surface and the unsteadiness in this flow dynamics comes only from the similarity solution which is not yet available in the open literature. For this similarity solution the governing boundary layer equations (momentum and thermal) transform into a set of non-linear ordinary differential equations which conceive two physical parameters, namely, unsteadiness parameter γ and Prandtl number Pr. The impact of these two parameters on the unsteady boundary layer flows as well as on the heat transfer analysis have been discussed in detailed. The present analysis confirms the non-existence of the boundary layer solutions after a certain negative value of γ, while it continues for a large positive value of γ and specifically for the value of γ = 0, we get the steady boundary layer solution for the Blasius flow problem. Here, we have considered the whole range of the Pr values, which includes both the limiting cases Pr → 0 and Pr → ∞, in the discussion of the heat transfer analysis. For these two limiting cases, two approximated separate mathematical formulae for the heat transfer solutions have been derived in terms of the well-known bessel, error and hypergeometric functions, whose results for any given value of γ within the solution domain are well fitted with the corresponding numerical results.

Mixed Convective Conditions on Natural Convection of MHD Blasius and Sakiadis Flows with Variable Properties and Nonlinear Chemical Reaction

An unsteady two-dimensional boundary layer slip flow of a viscous incompressible fluid moving plate in a quiescent fluid (Sakiadis flow) and the flow-induced over a stationary flat plate by a uniform free stream (Blasius flow) are investigated simultaneously, from a numerical point of view. The variable thermal conductivity, viscosity ratio parameter, and nonlinear chemical reaction are used in the governing equations. Similarity equations of the governing transport equations are converted into an ordinary differential equation. The transformed equations are solved numerically using the Runge-Kutta method via the shooting technique. Sample results for the dimensionless velocity, temperature, and concentration distributions are studied through graphically. Moreover, friction factor, heat, and mass transfer rates have been discussed in detail. The chemical reaction parameter decelerates the friction factor and heat transfer rates for the Sakiadis and Blasius flow cases and enhances in mass transfer rate in both cases. The rate of mass transfer is higher in Blasius flow compared with Sakiadis flow. The present results of the heat transfer rate are compared with the published results are found to be in good agreement.

MHD Flow and Heat Transfer of Ferrofluid on Stagnation Point along Flat Plate with Convective Boundary Condition and Thermal Radiation Effect

This theoretical study deals with numerical solution of MHD convection laminar boundary layer flow and heat transfer on stagnation point along a stationary horizontal flat plate. For this purpose, ferrofluid which contains magnetite, Fe3O4 as a ferroparticles and water as a base fluid is considered. Ferrofluid has shown a particular achievement when the effect of external magnetic field applied, such as helping to control the properties of physical and flow of ferrofluid. The study starts with the formulation of the mathematical equations that governed the ferrofluid flow and heat transfer. The governing equation which is in the form of dimensional nonlinear partial differential equations are reduced to nonlinear ordinary differential equations by using appropriate similarity transformation and then solved numerically by using the Keller-box method. Numerical result is discussed in terms of pertinent effects that influence the ferrofluid flow and heat transfer like magnetic parameter, ferroparticles volume fraction parameter, Biot number and radiation parameter on velocity and temperature profiles. It is found that the temperature profile increase with an increase volume fraction of ferroparticles parameter, radiation parameter and Biot number and decrease with increasing magnetic parameter.

Hydrodynamics of a rigid stationary flat plate in cross-flow near the free surface

In this work, the flow normal to a rectangular flat plate with sharp angles in the presence of the free surface and gravity has been studied numerically, using a three dimensional (3D) large Eddy simulation methodology. Most of the previous studies on this geometry consider either periodic assumptions or laminar regime Re ∼ O(103). The numerical study described here, with a fully 3D simulation of the flat plate in turbulent regime Re ∼ O(105), had not been reported earlier. Important differences have been found with respect to the laminar case or when periodic boundary conditions are assumed. The influence of the gravity force and the free surface interface on the hydrodynamics of the problem is extremely relevant for industrial applications. The simulations are initially validated using a reference case with a plate with sharp corners and a single phase laminar 3D-periodic configuration. The time averaged drag force, the velocity, and pressure fields are compared as well against the case of the plate with smooth corners. Finally, in the case of the two-phase 3D numerical simulations, the drag force and the vorticity fields in the near wake of the plate are compared to recent experimental work [S. Satheesh and F. J. Huera-Huarte, “Effect of the free surface on a flat plate translating normal to the flow,” Ocean Eng. 171, 458–468 (2019)]. These experiments were used as a guideline for the computational setup; consequently, the values of most of the dimensionless parameters are the same. A second scenario where the free surface is replaced by a solid wall is considered. Two important observations show up after this research: first, the characteristic unsteady frequencies that are found associated with two periodic regimes disappear when 3D cases are evaluated. In such a scenario, no dominant frequencies are found. Second, a critical submergence depth (distance) between the upper part of the plate and the free surface (or solid wall) has been found, where the drag force shows a maximum value. These observations are discussed in relation to the pressure distribution, the vortex structures formed at the wake, and the gap flow formed between the plate and the free surface.

Direct numerical simulation and statistical analysis of stress-driven turbulent Couette flow with a free-slip boundary

The effects of free-slip boundary on shear turbulence are studied numerically using the direct numerical simulation (DNS) approach. The flow considered in this study is a stress-driven turbulent Couette flow between two flat boundaries. The top boundary has an imposed shear stress in the streamwise direction and a free-slip condition for the streamwise velocity fluctuation and the spanwise velocity, and the bottom boundary satisfies the no-slip condition. This type of flow has a mean flow pattern similar to the turbulent plane Couette flow between a stationary flat plate and a moving flat plate but exhibits considerable differences in turbulence statistics due to the effects of the free-slip boundary. Statistical analysis based on the DNS data and theoretical derivation based on Taylor series expansion show that near the free-slip surface the turbulence variances of the three velocity components vary as quadratic functions of the vertical distance from the boundary while the Reynolds shear stress exhibits a linear behavior, which are very different from the counterparts near the no-slip boundary. The free-slip surface condition also leads to zero horizontal vorticities at the surface but allows nonzero vertical vorticity in the meantime, leading to considerable differences in the near-boundary statistics of vorticities and coherent vortex structures. Comparison of three DNS runs with different grid resolutions shows that smoothly resolving the more energetic turbulent flow structures near the free-slip boundary requires a higher horizontal grid resolution than that used for resolving the structures near the no-slip boundary.

Experimental and Numerical Analyses of Multiple Jets Impingement Cooling for High-Power Electronics

Effective cooling of electronic equipment has emerged as a challenging and constraining problem of the 21st century. Jet impingement is a high-performance cooling technology for enhancing the heat transfer in thermal devices. This paper reports the experimental and numerical investigations carried out to study the heat transfer characteristics of multiple air jets impinging on a stationary flat copper plate. The software used for the numerical studies is a finite-volume-based software, ANSYS Fluent. An experimental setup was designed and fabricated. The results obtained from numerical studies were validated with that obtained from experimental studies. A copper plate, which is heated from the bottom, was used as the heat transfer surface of a simulated chip. The jets were produced by three sharp-edged circular orifices arranged in an in-line configuration. The nozzle-to-plate distance was varied as 1, 2, 4, 6, and 7 times the nozzle diameter ( $d$ ). The effects of Reynolds number (4000–8000) and jet-to-jet spacing on heat transfer characteristics were examined numerically. From the studies, it was found that for a Reynolds number of 5000, optimum cooling is obtained when nozzle-to-plate spacing ( $H$ ) is 6 times the orifice diameter. A correlation was proposed for Nusselt number in terms of Reynolds number, and it is valid for air as the cooling medium.

Numerical Analysis of Heat Transfer and Flow Field Due to Slot Air Jet Impingement for the Cases of Moving Plate and Moving Nozzle

Heat transfer to/from long flat surfaces is often accomplished by employing arrays of impinging air jets. Both cases of the moving target plate and the jet issuing device could be other options, which have been investigated numerically in the present study. Heat transfer distribution on a hot stationary flat plate impinged by a slot air jet was first specified. Air jet Reynolds number at the nozzle exit plane and the nozzle-to-plate separation were Re = 5000 and H/W = 5, respectively. The effect of movement of the target plate and the nozzle exit plane was then separately examined on both the flow and thermal fields. The distribution of the local Nusselt number was presented for a range of the moving nozzle and the moving plate velocities. There was a slight change in the flow and thermal fields predicted for the stationary case when the relative velocity was small. However, the flow field and the corresponding Nusselt number distribution diverged from the symmetrical distribution, being different for the two cases of the moving nozzle and the moving plate. The results indicated that the average Nusselt number decreased more dramatically in the case of moving nozzle.

Homotopy Analysis Solution to Thermal Radiation Effects on MHD Boundary Layer Flow and Heat Transfer Towards an Inclined Plate with Convective Boundary Conditions

This paper investigates the Magnetohydrodynamic flow and heat transfer towards a stationary/moving plate with convective boundary conditions in presence of thermal radiation. The governing equations are simplified by similarity transformations which are then solved by homotopy analysis method (HAM). Effects of emerging parameters such as Prandtl number, local Grashof number, Magnetic parameter, Radiation parameter and the Biot number on the velocity field and temperature field are plotted and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate. The obtained results are compared with other results obtain in previous works so that the high accuracy of the present result is apparent.

Slot jet impingement heat transfer for the cases of moving plate and moving nozzle

Jet impingement heat transfer for the case of extended surfaces is usually accomplished by employing arrays of impinging jets. Both the cases of moving plate and moving nozzle could be other options to transfer heat to/from the extended surfaces, which has been investigated numerically at the present study. Heat transfer distribution on a hot stationary flat plate was first specified for an unconfined slot laminar air jet impinging on the plate. Air jet Reynolds number and nozzle-to-plate separation were Re = 500 and H/W = 5, respectively. The effects of the moving plate and moving nozzle were then separately examined on both the flow and thermal fields. The distribution of the local Nusselt numbers was presented for a range of moving nozzle and moving plate. The analysis revealed that the Nusselt number decreased as the velocity of the plate or the nozzle was increased, where the effect of the moving nozzle was relatively substantial.

Boundary layer flow past an inclined stationary/moving flat plate with convective boundary condition

In this study, the mathematical modeling for boundary layer flow and heat transfer past an inclined stationary/moving flat plate with a convective boundary condition is considered. Using a similarity transformation, the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations and are solved numerically using the shooting method. The obtained numerical solutions are compared with the available results in the literature and are found to be in excellent agreement. The features of the flow and heat transfer characteristics for various values of the angle of inclination, Prandtl number, local Grashof number and the Biot number are analyzed and discussed. It is found that the temperature of the stationary flat plate is higher than the temperature of the moving flat plate.

Laminar streak growth above a spanwise oscillating wall

The use of spanwise wall oscillations to attenuate the growth of laminar streaks within the incompressible Blasius boundary layer is investigated. As in the case of the flow above a stationary flat plate, studied by Leib et al. (J. Fluid Mech., vol. 380, 1999, pp. 169–203), free-stream convected gusts interact with the boundary layer to drive the streak growth. Spanwise wall oscillations can either reduce or increase the total energy of the laminar streaks, depending upon the wall oscillation amplitude and frequency, as well as the free-stream gust properties. Reductions in streak energies of up to 90 % are obtained, indicating that spanwise wall oscillations are an effective technique for attenuating the laminar streak growth. Therefore they may suppress secondary boundary-layer instabilities and delay transition. The laminar boundary-layer base flow matches the Blasius profile in the streamwise and wall-normal directions, while in the spanwise direction a generalized version of the classical Stokes layer profile (generated by a wall oscillating beneath a quiescent fluid) occurs, which evolves downstream due to non-parallel flow effects. Via a Wentzel–Kramers–Brillouin–Jeffreys analysis, this generalized Stokes layer is shown to approach the classical Stokes layer in the limit of large downstream distances or high-frequency plate oscillations. The laminar streaks forced by the generalized and the classical Stokes flows differ significantly, which implies that the choice of the spanwise base flow may affect the secondary instability and transition in this flow. The analysis also proves that the use of the classical Stokes layer as spanwise base flow, as employed by Hack & Zaki (Phys. Fluids. vol. 24 (3), 2012, 034101), is inappropriate.

An Improved Direct-Forcing Immersed Boundary Method for Fluid-Structure Interaction Simulations1

In the present work, we present an improved version of the direct-forcing immersed boundary (IB) method proposed in Wang and Zhang (2011, “An Immersed Boundary Method Based on Discrete Stream Function Formulation for Two- and Three-Dimensional Incompressible Flows,” J. Comput. Phys., 230(9), pp. 3479–3499). In order to obtain an accurate prediction of local surface force, measures have been taken to suppress the unphysical spatial oscillations in the Lagrangian forcing. A fluid-structure interaction (FSI) solver has been developed by using the improved IB method for the fluid and the finite difference method for the structure. Several flow problems are simulated to validate our method. The testing cases include flows over a stationary cylinder and a stationary flat plate, two-dimensional flow past an inextensible flexible filament, and three-dimensional flow past a flapping flag. The results obtained in the present work agree well with those from the literature.

Effects of radiation on Blasius slip flow of oxide nanofluids with Merkin boundary condition

The two-dimensional steady laminar forced convective boundary layer slip flow of oxide nanofluids past a radiating stationary semi-infinite flat plate is studied numerically. We assume that the plate is subjected to Newtonian heating (Merkin) boundary condition. The governing partial differential equations are converted into similarity equations using suitable similarity transformations. We chose three types of oxide nanofluids, namely, copper oxide, alumina, and titania and water as base fluid. The dimensionless friction factor, heat transfer rates, and the dimensionless velocity and temperature profiles are presented graphically and discussed. It is found that the controlling parameters strongly affect the fluid flow and heat transfer characteristics.

Stretching liquid bridges with moving contact lines: The role of inertia

Liquid bridges with moving contact lines are found in a variety of settings such as capillary feeders and high-speed printing. Although it is often assumed that the length scale for these flows is small enough that inertial effects can be neglected, this is not the case in certain applications. To address this issue, we solve the Navier-Stokes equations with the finite element method for the stretching of a liquid drop between two surfaces for non-zero Reynolds numbers. We consider an axisymmetric liquid bridge between a moving flat plate and either a stationary flat plate or a cavity. The contact lines are allowed to slip, and we evaluate the effect of the Reynolds number and contact angles on the transfer of liquid to the moving plate. In the case of two flat plates, we find that inertia forces the interface to map onto a similarity solution in a manner that shifts the breakup point toward the more wettable surface. Inertia and wettability are thus competing effects, with inertia driving fluid toward the surface with the higher contact angle and wettability driving fluid toward the surface with the lower contact angle. When a cavity is present, contact line pinning on the cavity wall biases breakup toward the cavity as the Reynolds number is increased, leading to improved cavity emptying. As the flat plate is made more wettable, a second pinch-off point can form near that plate, leading to a satellite drop and a reduction in liquid transfer to the plate. Therefore, higher liquid transfer is not always obtained with a more wettable substrate when inertia is present, in contrast to Stokes flow. We also compare our results to those obtained using a model based on the long-wave approximation and find good qualitative agreement, with the long-wave model overpredicting the amount of liquid transferred after breakup.

Vortex Simulations of the Flow-Field of a Flat Plate with a Non-Zero Angle of Attack

AbstractA Lagrangian style vortex simulation technique is used to study the flow fields past a stationary flat plate at various angles of attack in the range 1° to 90°. Time mean values of oscillating lift and drag coefficients, and the Strouhal number versus angle of attack are computed and compared with experimental results. Time-mean and root-mean-square values of stream-wise and transverse velocities in the wake region are also calculated. Self-similar defect velocity distribution is obtained far downstream. Owing to the interaction of free shear layers, highly root-mean-square values of velocities appear at the downstream vertex of the triangular low velocity region, which exits behind an inclined flat plate.