scholarly journals Falkner–Skan Equation with Heat Transfer: A New Stochastic Numerical Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Imran Khan ◽  
Hakeem Ullah ◽  
Hussain AlSalman ◽  
Mehreen Fiza ◽  
Saeed Islam ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Prandtl Number ◽  
Thermal Boundary Layer ◽  
Feedforward Neural Networks ◽  
Numerical Approach ◽  
Deborah Number ◽  
Squeezing Flow ◽  
And Performance

In this study, a new computing model is developed using the strength of feedforward neural networks with the Levenberg–Marquardt method- (NN-BLMM-) based backpropagation technique. It is used to find a solution for the nonlinear system obtained from the governing equations of Falkner–Skan with heat transfer (FSE-HT). Moreover, the partial differential equations (PDEs) for the unsteady squeezing flow of heat and mass transfer of the viscous fluid are converted into ordinary differential equations (ODEs) with the help of similarity transformation. A dataset for the proposed NN-BLMM-based model is generated in different scenarios by a variation of various embedding parameters, Deborah number ( β ) and Prandtl number (Pr). The training (TR), testing (TS), and validation (VD) of the NN-BLMM model are evaluated in the generated scenarios to compare the obtained results with the reference results. For the fluidic system convergence analysis, a number of metrics such as the mean square error (MSE), error histogram (EH), and regression (RG) plots are utilized for measuring the effectiveness and performance of the NN-BLMM infrastructure model. The experiments showed that comparisons between the results of the proposed model and the reference results match in terms of convergence up to E-05 to E-10. This proves the validity of the NN-BLMM model. Furthermore, the results demonstrated that there is an increase in the velocity profile and a decrease in the thickness of the thermal boundary layer by increasing the Deborah number. Also, the thickness of the thermal boundary layer is decreased by increasing the Prandtl number.

scholarly journals Thermal Boundary Layer in Flow due to an Exponentially Stretching Surface with an Exponentially Moving Free Stream

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Krishnendu Bhattacharyya ◽  
G. C. Layek
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Prandtl Number ◽  
Thermal Boundary Layer ◽  
Stretching Sheet ◽  
Free Stream ◽  
Velocity Ratio ◽  
Absorption Increase ◽  
Exponentially Stretching

A numerical investigation is made to study the thermal boundary layer for flow of incompressible Newtonian fluid over an exponentially stretching sheet with an exponentially moving free stream. The governing partial differential equations are transformed into self-similar ordinary differential equations using similarity transformations in exponential forms. Then those are solved numerically by shooting technique using Runge-Kutta method. The study reveals that the momentum boundary layer thickness for this flow is considerably smaller than the linear stagnation point flow past a linearly stretching sheet. The momentum and thermal boundary layer thicknesses reduce when the velocity ratio parameter increases. For the temperature distribution, in addition to the heat transfer from the sheet, the heat absorption at the sheet also occurs in certain situations and both heat transfer and absorption increase with the velocity ratio parameter and the Prandtl number. The temperature inside the boundary layer significantly decreases with higher values of velocity ratio parameter and the Prandtl number.

scholarly journals Velocity Slip and Thermal Jump on Maxwell Fluid with Non-Fourier Cattaneo-Christov Heat Flux Using SRM Solutions

2018 ◽  
Vol 7 (4.10) ◽  
pp. 233
Author(s):  
K. Gangadhar ◽  
K. V. Ramana ◽  
B. Rushi Kumar
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Flux ◽  
Differential Equations ◽  
Slip Velocity ◽  
Maxwell Fluid ◽  
Deborah Number ◽  
Stretching Cylinder ◽  
Zero Curvature ◽  
Curvature Parameter

The influence of the heat transfer within a boundary layer flow and magneto hydro dynamic slip flow of a Maxwell fluid over a stretching cylinder is analyzed and discussed in the present article. The effects of viscous dissipation and thermal jump are assumed. The procedure of heat transfer through hypothesis of Cattaneo-Christov heat flux is considered. We converted non-linear partial differential equations for mass, momentum and energy into a system of coupled highly non linear ordinary differential equations with proper boundary conditions by the help of suitable similarity transformations. The succeeding ordinary differential equations are solved by using Spectral relaxation technique. The solution is obtained in zero curvature parameter as well as non-zero curvature parameter.  i.e. for flow above a flat plate and flow above a cylinder. The flow and heat transfer attributes are witnessed to be encouraged in an elaborate mode by Prandtl number, thermal jump parameter, thermal relaxation parameter, Deborah number, slip velocity parameter, Eckert number and the magnetic parameter. Our findings reveal that one of the possible ways to decrease the Deborah number by boosting fluid velocity. It is also perceived that in the case of flow over a stretching cylinder, the momentum boundary layer thickness and the velocity of the fluid increases. Furthermore, an increase in slip velocity factor reduces the magnitude of skin friction.  

On the sensitivity of heat transfer in the stagnation-point boundary layer to free-stream vorticity

1963 ◽  
Vol 16 (4) ◽  
pp. 497-520 ◽  
Author(s):  
S. P. Sutera ◽  
P. F. Maeder ◽  
J. Kestin
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Stagnation Point ◽  
Thermal Boundary Layer ◽  
Free Stream ◽  
Stokes Equations ◽  
Point Boundary ◽  
Oncoming Flow ◽  
Flat Plates

Experiments have given evidence of strong sensitivity of the stagnation-point heat transfer on cylinders to small changes in the intensity of free-stream turbulence. A similar effect on local heat-transfer rates to flat plates has been measured, but only when a favourable pressure gradient is present. In this work it is theorized that vorticity amplification by stretching is a possible, and perhaps the dominant, underlying mechanism responsible for this sensitivity. A mathematical model is presented for a steady, basically plane stagnation flow into which is steadily transported disturbed unidirectional vorticity having the only orientation susceptible to stretching. The resulting velocity and temperature fields in the stagnation-point boundary layer are analysed assuming the fluid to be incompressible and to have constant properties. By means of iterative procedures and electronic analogue computation an approximate solution to the full Navier-Stokes equations is achieved which indicates that amplification by stretching of vorticity of sufficiently large scale can occur. Such vorticity, present in the oncoming flow with a small intensity, can appear near the boundary layer with an amplified intensity and induce substantial three-dimensional effects therein. It is found that the thermal boundary layer is much more sensitive to the induced effects than the velocity boundary layer. Computations indicate that a certain amount of distributed vorticity in the oncoming flow causes the shear stress at the wall to increase by 5%, while the heat transfer there is augmented by 26% in a fluid with a Prandtl number of 0.74. Preliminary computations reveal that the sensitivity of the thermal boundary layer increases with Prandtl number.

An integral approach to the calculation of skin friction and heat transfer in the laminar flow of a high-Prandtl-number power-law non-Newtonian fluid past a wedge

1991 ◽  
Vol 69 (2) ◽  
pp. 83-89 ◽  
Author(s):  
G. Ramamurty ◽  
K. Narasimha Rao ◽  
K. N. Seetharamu
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Prandtl Number ◽  
Skin Friction ◽  
Layer Thickness ◽  
Power Law ◽  
Boundary Layer Thickness ◽  
Thermal Boundary Layer ◽  
Integral Approach ◽  
High Prandtl Number

An integral approach to the theoretical analysis for the skin friction of a non-Newtonian, power-law-fluid flow over a wedge is presented, when the inertia terms in the boundary-layer equations are small but need consideration. The method adopted for the solution of the equations considers an integrated average value of the inertia terms in the momentum equation. The values of the velocities and the boundary-layer thickness obtained from the hydrodynamic analysis are used for the calculation of the thermal-boundary-layer thickness. A linear velocity profile is assumed for the flow field within the thermal boundary layer as the fluids chosen for the analysis are high-Prandtl-number fluids. The results of the skin friction and the rates of the heat transfer are tabulated for a number of values of the flow behaviour index, n, varying from 0.05 to 5.0. This analysis is applicable to viscous polymer solutions having high Prandtl numbers.

A Boundary-Layer Treatment of Laminar-Film Condensation

1959 ◽  
Vol 81 (1) ◽  
pp. 13-18 ◽  
Author(s):  
E. M. Sparrow ◽  
J. L. Gregg
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Prandtl Number ◽  
Vertical Plate ◽  
Film Condensation ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Prandtl Numbers ◽  
Mathematical Techniques

The problem of laminar-film condensation on a vertical plate is attacked using the mathematical techniques of boundary-layer theory. Starting with the boundary-layer (partial differential) equations, a similarity transformation is found which reduces them to ordinary differential equations. Energy-convection and fluid-acceleration terms are fully accounted for. Solutions are obtained for values of the parameter cpΔT/hfg between 0 and 2 for Prandtl numbers between 1 and 100. These solutions take their place in the boundary-layer family along with those of Blasius, Pohlhausen, Schmidt and Beckmann, and so on. Heat-transfer results are presented. It is found that the Prandtl-number effect, which arises from retention of the acceleration terms, is very small for Prandtl numbers greater than 1.0. Low Prandtl number (0.003–0.03) heat-transfer results are given in Appendix 2, and a greater effect of the acceleration terms is displayed.

Instantaneous Heat Flux Simulation of a Motored Reciprocating Engine: Unsteady Thermal Boundary Layer With Variable Turbulent Thermal Conductivity

2013 ◽  
Vol 136 (3) ◽  
Author(s):  
Abdalla Agrira ◽  
David R. Buttsworth ◽  
Mior A. Said
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Boundary Layer ◽  
Heat Flux ◽  
Prandtl Number ◽  
Thermal Boundary Layer ◽  
Combustion Engine ◽  
Turbulent Viscosity ◽  
Spark Ignition Engine ◽  
Turbulent Prandtl Number

Due to the inherently unsteady environment of reciprocating engines, unsteady thermal boundary layer modeling may improve the reliability of simulations of internal combustion engine heat transfer. Simulation of the unsteady thermal boundary layer was achieved in the present work based on an effective variable thermal conductivity from different turbulent Prandtl number and turbulent viscosity models. Experiments were also performed on a motored, single-cylinder spark-ignition engine. The unsteady energy equation approach furnishes a significant improvement in the simulation of the heat flux data relative to results from a representative instantaneous heat transfer correlation. The heat flux simulated using the unsteady model with one particular turbulent Prandtl number model agreed with measured heat flux in the wide open and fully closed throttle cases, with an error in peak values of about 6% and 35%, respectively.

scholarly journals Boundary layer flow over a moving plate in MHD Jeffrey nanofluid: A revised model

2018 ◽  
Vol 189 ◽  
pp. 02005
Author(s):  
S M Zokri ◽  
N S Arifin ◽  
A R M Kasim ◽  
N F Mohammad ◽  
M Z Salleh
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Temperature Profile ◽  
Graphical Representation ◽  
Numerical Approach ◽  
Deborah Number ◽  
Physical Parameters ◽  
Moving Plate ◽  
Perfect Agreement ◽  
Jeffrey Nanofluid

The flow and heat transfer of magnetohydrodynamic (MHD) Jeffrey nanofluid induced by a moving plate is examined numerically. The formulation is established by using the revised model of passively controlled boundary layer instead of actively, which is more realistic physically. The similarity transformation variables are used to transform the partial differential equations into a set of ordinary differential equations before solving it via numerical approach called as the Runge-Kutta Fehlberg method. Graphical representation of the physical parameters over the temperature profile is deliberated. Temperature profile is slowed down due to the parameters of Deborah number and plate velocity while the reverse trend is observed for thermophoresis diffusion parameter. The Brownian motion has shown an insignificant outcome on the temperature profile. A comparison with the earlier publication has been conducted and a perfect agreement between the data is detected.

scholarly journals Numerical Solutions for Convective Boundary Layer Flow of Micropolar Jeffrey Fluid with Prescribe Wall Temperature

2020 ◽  
Vol 26 (3) ◽  
pp. 286-298
Author(s):  
Noraihan Afiqah Rawi ◽  
Nor Athirah Mohd Zin ◽  
Asma Khalid ◽  
Abdul Rahman Mohd Kasim ◽  
Zaiton Mat Isa ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Skin Friction ◽  
Convective Boundary Layer ◽  
Boundary Layer Flow ◽  
Deborah Number ◽  
Jeffrey Fluid ◽  
Convective Boundary ◽  
Layer Flow

The steady two dimensional convective boundary layer flow of micropolar Jeffrey fluid past a permeable stretching sheet is studied in this paper. The governing boundary layer equation in the form of partial differential equations are transformed into nonlinear coupled ordinary differential equations and solved numerically using an implicit finite-difference scheme known as Keller-box method. The effects of Prandtl number, Deborah number, and material parameter with the boundary condition for microrotation, n = 0 (strong concentration of microelements) on the velocity, microrotation, temperature profiles as well as the skin friction and heat transfer coefficients are presented and discussed. An excellent agreement is observed between the present and earlier published results for some special cases. The results revealed that, the effect of Deborah number and stretching parameter are increased the heat transfer coefficient while the opposite trend is observed for the effects of material and velocity slip parameters. It was also observed that, the values of skin friction increased with the increment on the values of all studied parameters.

Comparative study of SiO2 and TiO2 nanoparticles on flow and heat transfer of dusty fluid over a stretching sheet

2019 ◽  
Vol 15 (5) ◽  
pp. 990-1005 ◽  
Author(s):  
K. Ganesh Kumar ◽  
M. Archana
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Comparative Study ◽  
Thermal Boundary Layer ◽  
Nonlinear Ordinary Differential Equations ◽  
Content Type ◽  
Dusty Fluid ◽  
Similarity Transformations ◽  
Flow And Heat Transfer

Purpose The purpose of this paper is to model the boundary layer flow and heat transfer of dusty fluid with suspended nanoparticles over a stretching surface. The effect of multiple slip and nonlinear thermal radiation is taken into the account. Adequate similarity transformations are used to obtain a set of nonlinear ordinary differential equations to govern formulated problem. The resultant non-dimensionalized boundary value problem is solved numerically using the RKF-45 method. The profiles for velocity and temperature, which are controlled by thermophysical parameters, are presented graphically. Based on these plots, the conclusion is given and the obtained numerical results are tabulated. Observed interesting fact is that the SiO2-water nanoparticles show a thicker thermal boundary layer than TiO2-water nanoparticles. Design/methodology/approach The governing partial differential equations are approximated to a system of nonlinear ordinary differential equations by using suitable similarity transformations. An effective fourth–fifth-order Runge–Kutta–Fehlberg integration scheme numerically solves these equations along with a shooting technique. The effects of various pertinent parameters on the flow and heat transfer are examined. Findings Present results have an excellent agreement with previous published results in the limiting cases. The values of skin friction and wall temperature for different governing parameters are also tabulated. It is demonstrated that the SiO2-water nanoparticles show a thicker thermal boundary layer than TiO2-water nanoparticles. It is interesting to note that the dusty nanofluids are found to have higher thermal conductivity. Originality/value This paper is a new work related to comparative study of TiO2 and SiO2 nanoparticles in heat transfer of dusty fluid flow.

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