where the subscripts and tlou" denote the inner (boundary layer) and outer (wave-like) solutions, respectively. By applying the Laplace transform to Eq.. Writing this[r]

89 Read more

Abstract
In this paper the laminar **flow** of Newtonian conducting fluid produced by a moving **plate** in presence of transverse magnetic field is investigated. The basic equation governing the motion of such **flow** is expressed in non-dimensional form. Analytic solution of the governing equation is obtained by Laplace transformation. Numerical solution of the dimensionless equation is also ob- tained with the help of Crank-Nicholson implicit scheme. Velocity profiles of the corresponding problem are shown in the graphs.

(Received 8 January 2012; revised 7 July 2012)
Abstract
A new scaling analysis has been performed for the unsteady natural convection boundary layer under a downward facing inclined **plate** with uniform heat flux. The development of the thermal or viscous boundary layers is classified into three distinct stages including an early stage, a transitional stage and a steady stage, which is clearly identified in the analytical as well as in numerical results. Earlier scaling shows that the existing scaling laws of the boundary layer thickness, velocity and steady state time scales for the natural convection **flow** on a **heated** **plate** of uniform heat flux provide a very poor prediction of the Prandtl number dependency. However, those scalings performed very well with Rayleigh number and aspect ratio dependency. In this study, a modified Prandtl number scaling is developed using a triple-layer integral approach for Prandtl number larger than unity.

Show more
17 Read more

The study of heat transfer analysis is an interesting research area due to its potential applications in engineering, such as nuclear plants, combustion modeling, heat exchangers, cooling systems design, various propulsion devices for air craft’s, chemical engineering and electronics etc. Moreover, heat transfer by thermal radiation plays a significant role on the heat transfer characteristics where high temperature is occurred. Furthermore, the **flow** field is influenced noticeably with the effect of magnetic field. Considering its wide applications in science and engineering, a large number of theoretical, numerical and experimental works have been conducted by many investigators. Pop and Na [1] developed a mathematical model in free convection **flow** for arbitrary inclined **flat** **plate** embedded in a porous medium to analyze the behavior of the **flow** and heat transfer characteristics. The effects of conduction-radiation on natural convection boundary layer **flow** of viscous incompressible fluid over an isothermal horizontal **plate** studied by Hossain and Takhar [2]. Hossain et al. [3] analyzed the effect of radiation on natural convection **flow** in incompressible fluid along a uniformly **heated** vertical **plate**. Abdel-Naby et al. [4] investigated radiation effects on MHD unsteady free convection **flow** over a vertical **plate** with variable surface temperature. Ali et al. [5] employed the implicit finite difference method to analyze the effect of radiation and viscous dissipation on conjugate free

Show more
Ukai, T., Kontis, K. and Yang, L. (2018) **Flow** structure **generated** by laser-induced blast wave propagation through the boundary layer of a **flat** **plate**. Aerospace Science and Technology, 78, pp. 569-573.
There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.

15 Read more

The Boltzmann kinetic equation is valid for all ranges of Knudsen number [14], while the Navier-Stokes (N.S.) system is suitable to give us acceptable results for the continuum **flow** only, (where 𝐾𝑛 = 𝜆
𝐿 is the Knudsen number that measures the rarefaction of any gas molecules and represents the ratio between the mean free path λ to a characteristic length L). For this purpose, we utilize coupled systems of non-stationary BGK kinetic equations, one for each component of the neutral gas. The radiation force effect is inserted into the force term of the Boltzmann kinetic equation. These procedures are done by applying the Liu-Lees model for two-side Maxwell non-equilibrium distribution functions using the moment method. Moreover, the manner of the macroscopic characteristics of the non-homogenous gas is estimated for different radiation force strength according to different fixed, rigid **plate** temperatures. The temperature and concentration are, in turn, substituted into the related non- equilibrium distribution function. This approach permits us to investigate the manner of the equilibrium, non- equilibrium, and non-stationary distribution functions for different magnitudes of the molar fraction parameters.

Show more
17 Read more

The following years saw large strides in the studying and understanding of boundary-layer flows. Since boundary-layer solutions provide information about drag and heat transfer at the surface of an object in a fluid stream, the solutions to these boundary layers allow for the calculation of drag and lift forces, as well as heat transfer from the solid surface. Thermal boundary layers are typically classified according to the velocity field into either forced or natural flows. Forced-**flow** boundary layers are due to an imposed velocity field, while natural-**flow** boundary layers are due to velocity gradients that arise from buoyant forces. During the 20th century boundary layers were analyzed with great mathematic rigor, resulting in multiple additions to the Blasius **flat**-**plate** solution as well as in the description of the boundary layers of other geometries.

Show more
38 Read more

(e)
Fig.4.6Flow visualisation of fow over backfacing step at Re2
As the uniformly flowing fluid **suddenly** experiences a sudden change of geometry in its **flow** direction, then the **flow** tries to accommodate for the change in geometry. When the **flow** is trying to adjust to the change of geometry, it results in the complex **flow** phenomena. In this case, when the **flow** approaches the wall of the step the **flow** deviates from its path and get separated along the wall. The separated **flow** over the wall touches the **flat** surface of the step at a distance of 3cms from the wall. After touching the surface it gets re circulated and touches the wall and gets mixed with the free stream **flow**. In the process of recirculation it forms a vortex near the wall which is of the length 3cms.The initial separated layers down the step mixed with the separated layers over the wall at a distance of 4cms from the rear end of the complete step thus resulting in secondary recirculation zone and vortex. The primary and secondary vortices formed are responsible for the pressure drag.

Show more
12 Read more

Dyer J.R. [1] in this paper author used the concept of a theoretical and experimental study of laminar natural- convective **flow** in **heated** vertical duct. The ducts are open ended and circular in cross section and their internal surfaces dissipate heat uniformly. Temperature and velocity fields and the relationship between Nusselt and Rayleigh numbers were obtained by solving the governing equations by step-by-step numerical technique. Two Rayleigh numbers are introduced expressing in terms of the uniform heat flux and the other in terms of the mean wall temperature. The effect of the prandlt number on the relationship between the Nusselt and Rayleigh numbers is discussed. Three inlet conditions were examined they all gave the same Nusselt relationship at small Rayleigh numbers. It is also observed that the difference between the Nusselt numbers obtained at large Rayleigh number were only small. Experimentally determined Nusselt numbers with air as the convicted fluid, agreed satisfactorily with the theoretical relationship.

Show more
Keywords — Boundary layer, Critical height, Free convection, Heat flux, Isothermal **plate**, Turbulent **flow**, Vertical **plate**.
I . INTRODUCTION
Boundary layer of natural convection **flow** has extensively been studied for long years. Many experimental and theoretical investigations have been carried out to study the behaviour of the development of boundary layer **flow** on various geometries, for instance, cylinder, sphere and **flat** **plate** with different boundary conditions. In fact, the natural convection on a vertical **plate** has received more attention because the phenomenon of free convection is employed in many engineering applications, for example, cooling industrial equipments or circuit boards in electric models.

Show more
oscillating flows of a rotating second grade fluid in porous medium. Riaz et al. (2015) have investigated the flows of generalized second grade fluid **generated** by an oscillating **flat** **plate**.
The aim of this paper is to study the motion of a viscoelastic incompressible and electrically conducting fluid near an infinite oscillating **flat** **plate** in presence of a transverse magnetic field fixed relative to the fluid. Fourier Sine transforms and Laplace transform techniques have been used to solve the basic equations. The effects of magnetic field, elasticity of the fluid, frequency of oscillation of the **plate** and the time of commencement of motion on the velocity field have been studied.

Show more
4. C ONCLUSIONS
This article conducted an experimental and numerical simulation based on the RNG k-ε turbulence model for determining the cooling process of the **heated** Aluminium **plate** surface, with the help of a TJIM and involving 9 models. They investigated the effect of the nozzle-nozzle distance, nozzle-**plate** distance, and the Re number on the convective heat transfer rate for deriving the heat transfer coefficient, Nu number and the thermal enhancement factor. The major conclusions derived from the simulation study helped in comparing and validating all results obtained from the experiments, and determining the major factors which affected the heat transfer rate, Nu number and the distribution of static pressure. The arrangement of the jets (i.e., jet position) of this TJIM indicated that the first model was the best model for determining the heat transfer coefficients and the Nu number when S/D=

Show more
20 Read more

Oluwole Daniel MAKINDE 1 and Winifred Nduku MUTUKU 2 This paper examines the effect of the complex interaction between the electrical conductivity of the conventional base fluid and that of the nanoparticles under the influence of magnetic field in a boundary layer **flow** with heat transfer over a convectively **heated** **flat** surface. Three types of water based nanofluids containing metallic or non-metallic nanoparticles such as copper (Cu), Alumina (Al 2 O 3 ) and Titania (TiO 2 ) are investigated. Using a similarity analysis of the model transport equations followed by their numerical computations, the results for the nanofluids velocity, temperature, skin friction and Nusselt number are obtained. The effects of various thermophysical parameters on the boundary layer **flow** characteristics are displayed graphically and discussed quantitatively. It is observed that the presence of nanoparticles greatly enhance the magnetic susceptibility of nanofluids as compared to the convectional base fluid.

Show more
12 Read more

conduction modeling and numerical simulations are identical to the method outlined above for the infinite lateral domain.
The problem domain essentially models an infinite, repeating lateral array of **heated** elements. In addition to the nondimensional lateral parameter ζ ∗ , two new nondimensional parameters are needed to describe the problem. The spacing between the **heated** sections and width of the **heated** sections, both normalized by the conduc- tion thickness, describe the geometry of the problem domain. The solution recaptures the semi-infinite lateral extent solution when the spacing and width tend towards in- finity. As might be expected, the Nu increases as the width of the **heated** section de- creases. However, if the width of the **heated** section is larger than the spacing between the **heated** sections, the Nu decreases when compared to the semi-infinite problem domain. This happens because the temperature field above the plates approaches the two dimensional temperature field described by a **heated** surface of infinite lateral extent. In other words, the heat transfer everywhere approaches the canonical two dimensional heat transfer everyone knows and loves, shown in eqs. (2.44) and (2.44).

Show more
369 Read more

Figure 2 shows representative grids in the vicinity of the trailing edge in both zones. These grids were **generated** with an algebraic grid generator. Both the grids have the same spacing in the wall normal direction at the **plate** surface, Δn/D = 0.002. The grid in the wake zone transitions from curvilinear near the trailing edge to rectangular downstream. Beyond 13.87D, the wake grid coarsens gradually in the streamwise direction. In addition to reducing the computational costs incurred, this coarsening dissipates the wake to a degree that inviscid exit boundary conditions can be employed at the exit boundary of the wake zone. The wake grid was constructed with 741 grid points in the streamwise direction, 411 in the cross-stream direction and 256 in the spanwise direction (about 78 x 10 6 grid points). The wake grid used in Ref. 1 consisted of about 43 x 10 6 grid points. The additional grid points in the present simulation were used primarily to provide more resolution for the detached shear layers and are thus, for the most part, concentrated in the first 5.0D downstream of the trailing edge. The resolution achieved along the centerline in the three spatial directions at x/D = 10.0 is approximately Δx/η = 3.7, Δy/η = 2.2 and Δz/η = 2.1 where η is the computed Kolmogorov length scale at the same location. The grid spacing in the y direction increases gradually from the centerline to the upper/lower wake zone boundaries to Δy/η = 4.0.

Show more
20 Read more

In addition, the study of thermal radiation also included in this research.
According to Anbuchezhian et al. (2012), radiation comes from solar energy, and the resultant solar energized resources, such as wave power, wind, biomass, and hydroelectricity, all give an explanation for most of the accessible renewable energy that is present on the Earth. Meanwhile, thermal radiation refers to electromagnetic radiation **generated** by the thermal motion of charged particles in matter. It consists of ultraviolet rays, infra-red and light rays follows a nuclear explosion. The examples of major radiation exposure in real live events are Hiroshima and Nagasaki, Three

Show more
27 Read more

For all the patterns, the top view of the visualization re- sults show that there is an oval region after each valley as
shown in Figure 4. These regions are formed due to the fact that the **plate** at the valley is thicker than at the lead- ing edge, which causes an upwash **flow** so that the boundary layer is considerably developed in a compara- tively short distance. Within this region, the low mo- mentum fluid lifted up by upwash movement cannot pe- netrate the high momentum **flow** in the freestream. It causes the **flow** to be deflected down ward resulting in the entrainment of the high momentum fluid in a down- wash region and the appearance of a pair of counter ro- tating vortices. The vortices after the oval region break down to turbulence with no distinguishable vortex struc- ture. In other words, the occurrence of streamwise vor- tices increases mixing as it causes the entrainment of the high momentum fluid into the boundary layer in the downwash region. The increase of mixing is responsible in the early occurrence of turbulent **flow**.

Show more
Simple shear flow past a flat plate in an incompressible fluid of small viscosity. & Carrier, ,F0 The magnetohydrodynamic flow past a lat plate[r]

14 Read more

(Received 31 August 2006; revised 19 December 2007)
Abstract
We perform a two dimensional numerical simulation of transient natural convection in a **suddenly** differentially **heated** cavity in order to observe the initial transient flows, particularly the leading edge effect. The numerical results show that the pressure plays a key role in the origination and propagation of the leading edge effect and the deviation of the numerical solution from the theoretical solution is due to the neglect of the convection terms in the theoretical solution of the energy equation. Accordingly, the one dimensional conduction solution does not estimate the transport coefficients.

Show more
16 Read more

Turbulence Boundary Layer
In turbulence boundary layer the **flow** is unsteady and not smooth, but eddying. When specifying velocities, we must consider mean values over a small time interval and not instantaneous values as before. The distribution of mean velocity in any one time interval is the same as in another. Thus we can still draw velocity profiles, which have meaning. Due to the eddying nature of the **flow** there is a lot of movement of fluids between inner and outer layers of the regions. Thus the velocity near the wall will be higher than in a laminar boundary layer where the movement and energy transfer do not occur. The velocity gradient at the wall is consequently much higher so the skin friction and drag are also higher.

Show more
14 Read more