Spreadsheets Solutions of the Boundary Layer Equations

2004 ◽  
Author(s):  
Ahmad Fakheri
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Fluid Mechanics ◽  
Undergraduate Students ◽  
Classical Problem ◽  
Boundary Layer Equations ◽  
Specialized Software ◽  
Layer Flow ◽  
Basic Course ◽  
The Impact

A classical problem in fluid mechanics and heat transfer is boundary layer flow over a flat plate. This problem is used to demonstrate a number of important concepts in fluid mechanics and heat transfer. Typically, in a basic course, the equations are derived and the solutions are presented in tabular or chart from. Obtaining the actual solutions is mathematically and numerically too involved to be covered in basic courses. In this paper, it is shown that the similarity solution and the solution to boundary layer equations in the primitive variables can easily be obtained using spreadsheets. Without needing much programming skills, or needing to learn specialized software, undergraduate students can use this approach and obtain the solution and study the impact of different parameters.

Mixed Convection Boundary Layer Flow of Viscoelastic Fluids Past a Sphere

2013 ◽  
Vol 336 ◽  
pp. 57-63 ◽  
Author(s):  
Anisah Dasman ◽  
Abdul Rahman Mohd Kasim ◽  
Nurul Farahain Mohammad ◽  
Aurangzaib Mangi ◽  
Sharidan Shafie
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Viscoelastic Fluid ◽  
Classical Problem ◽  
Convection Boundary Layer ◽  
Boundary Layer Equations ◽  
Keller Box Method ◽  
Convection Boundary ◽  
Layer Flow ◽  
Newtonian Viscous Fluid

The mixed convection boundary layer of a viscoelastic fluid past a sphere with constant temperature is discussed. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. The governing non-similar partial differential equations are first transformed into dimensionless forms and then solved numerically using the Keller-box method by augmenting an extra boundary condition at infinity. Numerical results are presented for different values of the viscoelastic and mixed convection parameters K and , respectively. It is found that for cases of cooling sphere and heating sphere, the boundary layer separates from the sphere. To the best of our knowledge, this important classical problem has not been studied before for the case of a viscoelastic fluid. Thus, the results are original and new for this type of fluids.

scholarly journals Combined Effects of Inclined Lorentzian forces and Melting heat transfer on Radiating Flow of Carbon Nanofluid past a Stretching Cylinder

2018 ◽  
Vol 7 (4.10) ◽  
pp. 1070
Author(s):  
Sreenivasulu P. ◽  
Poornima T.
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Magnetic Field Effect ◽  
Stretching Cylinder ◽  
Melting Heat ◽  
Boundary Layer Equations ◽  
Melting Heat Transfer ◽  
Highly Nonlinear ◽  
Layer Flow ◽  
Prandtl Boundary Layer

An analysis was made to study the effects of non-uniform heat source or sink and aligned magnetic field effect on boundary layer flow of carbon nanofluid past a stretching cylinder with melting heat transfer and radiation. The Prandtl boundary layer equations are transformed into highly nonlinear ordinary differential equations utilizing similarity variables. The final resolved system is explained with shooting method. The upshots of governing factors on the velocity, temperature, surface skin friction and rate of heat transfer are discoursed with the help of graphs. MWCNT performance is better than SWCNT in case of fluid flow. SWCNTs  transfers more heat from the fluid rather than multi wall tubes.Comparison of our numerical results with available literature works and shown a good agreement. 

scholarly journals Melting Heat Transfer and MHD Boundary Layer Flow of Eyring-Powell Nanofluid Over a Nonlinear Stretching Sheet with Slip

2019 ◽  
Vol 24 (1) ◽  
pp. 161-178 ◽  
Author(s):  
N. Vijaya Bhaskar Reddy ◽  
N. Kishan ◽  
C. Srinivas Reddy
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Volume Fraction ◽  
Skin Friction Coefficient ◽  
Physical Parameters ◽  
Local Skin ◽  
Layer Flow ◽  
Nonlinear Stretching ◽  
The Impact

Abstract The steady laminar incompressible viscous magneto hydrodynamic boundary layer flow of an Eyring- Powell fluid over a nonlinear stretching flat surface in a nanofluid with slip condition and heat transfer through melting effect has been investigated numerically. The resulting nonlinear governing partial differential equations with associated boundary conditions of the problem have been formulated and transformed into a non-similar form. The resultant equations are then solved numerically using the Runge-Kutta fourth order method along with the shooting technique. The physical significance of different parameters on the velocity, temperature and nanoparticle volume fraction profiles is discussed through graphical illustrations. The impact of physical parameters on the local skin friction coefficient and rate of heat transfer is shown in tabulated form.

A singularity in thermal boundary-layer flow on a horizontal surface

1992 ◽  
Vol 242 ◽  
pp. 419-440 ◽  
Author(s):  
P. G. Daniels
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Numerical Solutions ◽  
Rational Numbers ◽  
Velocity Fields ◽  
Boundary Layer Equations ◽  
Layer Flow ◽  
Buoyancy Flows ◽  
Temperature And Velocity Fields

A thermal boundary layer, in which the temperature and velocity fields are coupled by buoyancy, flows along a horizontal, insulated wall. For sufficiently low local Froude number the solution terminates in a singularity with rising skin friction and falling pressure. The structure of the singularity is obtained and the results are compared with numerical solutions of the horizontal boundary-layer equations. A novel feature of the analysis is that the powers of the streamwise coordinate involved in the structure of the singularity do not appear to be simple rational numbers and are determined from the solution of a pair of ordinary differential equations which govern the flow in an inner viscous region close to the wall. Modifications of the theory are noted for cases where either the temperature or a non-zero heat transfer are specified at the wall.

Non-Iterative Solution of Ordinary and Partial Differential Equations Using Spreadsheets

2006 ◽  
Author(s):  
Ahmad Fakheri ◽  
Mohammad H. Naraghi
Keyword(s):  
Undergraduate Students ◽  
Iterative Solution ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Thomas Algorithm ◽  
Specialized Software ◽  
Attractive Option ◽  
Basic Course ◽  
The Impact

Typically, in a basic course, the equations are derived and the solutions are presented in tabular or chart from. Spreadsheets provide an attractive option, as most students have access to and are familiar with their use. In this paper, a classic algorithm, called Thomas algorithm, used for the solution of set of algebraic equations whose coefficient matrix is tri-diagonal, is adopted to spreadsheets. It is used to obtain solutions to a number of classical problems in fluid mechanics and heat transfer non-iteratively, and in some cases where the governing equations are non-linear via some iterations. Without needing much programming skills, or needing to learn specialized software, undergraduate students can use this approach and easily obtain the solution to many otherwise difficult problems and study the impact of different parameters.

Boundary layer flow and melting heat transfer of Prandtl fluid over a stretching surface by considering Joule heating effect

2019 ◽  
Vol 15 (2) ◽  
pp. 337-352 ◽  
Author(s):  
K. Ganesh Kumar ◽  
M.R. Krishnamurthy ◽  
Rudraswamy N.G.
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Stretching Surface ◽  
Content Type ◽  
Melting Heat ◽  
Boundary Layer Equations ◽  
Melting Heat Transfer ◽  
Layer Flow ◽  
Linear Radiation

PurposeThe purpose of this paper is to study the impact of Joule heating on boundary layer flow and melting heat transfer of Prandtl fluid over a stretching sheet in the presence of fluid particles suspension. The transformed boundary layer equations are solved numerically by RKF-45 method. The influence of the non-dimensional parameters on velocity and temperature growths in the boundary layer region is analyzed in detail and the results are shown graphically. The results indicate that the larger estimation ofαandβreduces for both velocity and temperature profile. Further, the rate of heat transfer decreases by increasing melting parameter.Design/methodology/approachThe converted set of boundary layer equations is solved numerically by RKF-45 method. Obtained numerical results for flow and heat transfer characteristics are deliberated for various physical parameters. Furthermore, the skin friction coefficient and Nusselt number are also presented.FindingsIt is found that the heat transfer rates are advanced in the occurrence of non-linear radiation camper to linear radiation. Also, it is noticed that velocity profile increases by increasing Prandtl parameter but establishes opposite results for temperature profile.Originality/valueThe authors intend to analyze the boundary layer flow and melting heat transfer of a Prandtl fluid over a stretching surface in the presence of fluid particles suspension. The governing systems of partial differential equations have been transformed to a set of coupled ordinary differential equations by applying appropriate similarity transformations. The reduced equations are solved numerically. The pertinent parameters are discussed through graphs and plotted graphs. The present results are compared with the existing limiting solutions, showing good agreement with each other.

The HAM Solutions for Unsteady Boundary Layer Flow and Heat Transfer with Heat Source/Sink

2014 ◽  
Vol 887-888 ◽  
pp. 919-923 ◽  
Author(s):  
Jing Zhu ◽  
Zheng Liu
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Nonlinear Differential Equations ◽  
Physical Factors ◽  
Boundary Layer Equations ◽  
Flow And Heat Transfer ◽  
Analytical Approximations ◽  
Homotopy Analysis ◽  
Layer Flow ◽  
Source Sink

Considering the combined effects of the magnetic field and viscous dissipation, this paper investigates the problem of two-dimensional incompressible unsteady flow over a horizontal continuous stretching sheet. Due to the strongly nonlinear and various parameters of this problem, the governing boundary layer equations are transformed into a system of nonlinear differential equations through the similarity transformation, and then analytical approximations of solutions are derived by homotopy analysis method. In addition, the effects of physical factors on the flow and heat transfer characteristics are examed and discussed graphically.

Solution of Similarity Transformation Equations for Boundary Layers Using Spreadsheets

2004 ◽  
Author(s):  
Mohammad H. N. Naraghi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Fluid Mechanics ◽  
Flat Plate ◽  
Thermal Boundary Layer ◽  
Similarity Transformation ◽  
Present Approach ◽  
Published Data ◽  
Boundary Layer Equations ◽  
Transfer Courses

A spreadsheet based solution of the similarity transformation equations of laminar boundary layer equations is presented. In this approach the nonlinear third order differential equations, for both the hydrodynamic and the thermal boundary layer equations, are discretesized using a simple finite difference approach which is suitable for programming spreadsheet cells. This approach was implemented to solve the similarity transform equations for a flat plate (Blasius equations). The thermal boundary layer result was used to obtain the heat transfer correlation for laminar flow over a flat plate in the form of Nu = Nu(Pr,Re). The relative difference between results of the present approach and those of published data are less than 1%. This approach can be easily covered in the undergraduate. Fluid Mechanics and Heat Transfer courses. Also, it can be incorporated in graduate Viscous Fluid Mechanics and Convection Heat Transfer courses. Application of the present approach is not limited to the flat plat boundary layer analysis. It can be used for the solution of a number of similarity transformation equations, including wedge flow problem and natural convection problems that are covered in graduate level courses.

scholarly journals Magnetohydrodynamic slip flow and heat transfer over a nonlinear shrinking surface in a heat generating fluid

2020 ◽  
Vol 9 (2) ◽  
pp. 534
Author(s):  
Leli Deswita ◽  
Mohamad Mustaqim Junoh ◽  
Fadzilah Md Ali ◽  
Roslinda Nazar ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Slip Flow ◽  
Transfer Coefficients ◽  
Suction Parameter ◽  
Boundary Layer Equations ◽  
Heat Transfer Coefficients ◽  
Flow And Heat Transfer ◽  
Layer Flow ◽  
Shrinking Surface

In this paper, the problem of steady slip magnetohydrodynamic (MHD) boundary layer flow and heat transfer over a nonlinear permeable shrinking surface in a heat generating fluid is studied. The transformed boundary layer equations are then solved numerically using the bvp4c function in MATLAB solver. Numerical results are obtained for various values of the magnetic parameter, the slip parameter and the suction parameter. The skin friction coefficients, the heat transfer coefficients, as well as the velocity and temperature profiles for various values of parameters are also obtained and discussed. 

Convective Heat Transfer Analysis of Hydromagnetic Micropolar Fluid Flow Past an Inclined Nonlinear Stretching Sheet with Variable Thermo-Physical Properties

2020 ◽  
Vol 26 ◽  
pp. 63-77 ◽  
Author(s):  
Ephesus Olusoji Fatunmbi ◽  
Samuel Segun Okoya ◽  
Oluwole Daniel Makinde
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Micropolar Fluid ◽  
Strong Relationship ◽  
Skin Friction Coefficient ◽  
Heat Transfer Analysis ◽  
Convection Boundary Layer ◽  
Similarity Transformations ◽  
Layer Flow ◽  
The Impact

The current work examines mixed convection boundary layer flow and heat transfer attributes in hydromagnetic micropolar fluid past a heated inclined sheet which stretches nonlinearly along the direction of flow. The impact of variable thermo-physical characteristics of the fluid together with the influence of magnetic field, thermal radiation and viscous dissipation are also checked on the flow field. The modelled governing equations are translated from partial to ordinary differential equations via relevant similarity transformations and the resulting equations are subsequently solved numerically by means of shooting techniques in company with Runge-Kutta integration algorithms. The reactions of the skin friction coefficient, Nusselt number, dimensionless velocity as well as temperature to variations in the emerging controlling parameters are illustrated through different graphs. In the limiting situations, the results obtained exhibit a strong relationship with the existing related works in literature. The facts emanated from this study also reveal that the thickness of the thermal boundary layer grows widely with a rise in the Eckert number and Biot number parameters whereas increasing the material (micropolar) and thermal conductivity parameters have opposite effects on the rate of heat transfer.

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