_{1}

This work deals with incompressible two-dimensional viscous flow over a semi-infinite plate ac-cording to the approximations resulting from Prandtl boundary layer theory. The governing non-linear coupled partial differential equations describing laminar flow are converted to a self-simi- lar type third order ordinary differential equation known as the Falkner-Skan equation. For the purposes of a numerical solution, the Falkner-Skan equation is converted to a system of first order ordinary differential equations. These are numerically addressed by the conventional shooting and bisection methods coupled with the Runge-Kutta technique. However the accompanying energy equation lends itself to a hybrid numerical finite element-boundary integral application. An appropriate complementary differential equation as well as the Green second identity paves the way for the integral representation of the energy equation. This is followed by a finite element-type discretization of the problem domain. Based on the quality of the results obtained herein, a strong case is made for a hybrid numerical scheme as a useful approach for the numerical resolution of boundary layer flows and species transport. Thanks to the sparsity of the resulting coefficient matrix, the solution profiles not only agree with those of similar problems in literature but also are in consonance with the physics they represent.

Ever since its first appearance in literature [

The resulting set of ordinary differential equations rich in numerical and physical challenges and supplemented by boundary conditions can then be numerically addressed by means of initial value Ordinary Differential Equation (ODE) techniques. Two-dimensional flow over a surface creates a boundary layer as fluid particles move more slowly near the surface than near the free stream. With a similarity transformation, the boundary layer equation is converted to a set of ordinary differential equations known as the Blasius equation [

The shooting algorithm is often regarded as a practical way to numerically handle the F-S boundary value problem. Many of the algorithms reported in literature require an initial guess of the shooting angle which should lead to acceptable results after an iterative numerical procedure. However an additional initial condition is required in order to replace the condition at infinity. To facilitate the numerical procedure, the initial boundary value problem is converted to a set of coupled nonlinear ODEs which are amenable to a straightforward iterative numerical solution by any of the ODE solvers like the Runge-Kutta method.

Boundary layer flow involving heat and energy transfer over a plate is very important in several engineering applications, for example, in polymer industry where plastic is produced or in general, for all cases involving the processing of sheet-like materials. In virtually all these cases, the sheet moves parallel to its own plane and induces the motion of the closest surrounding fluid as well as creates a boundary layer scenario. Some interesting physics involving the kinematics of the speed of the plate, stretching, contraction, heating and cooling have been observed to take place at this level (Shalini and Choudhary [

This paper obtains the integral discrete analog of the energy equation by the way of the Green’s second identity and an appropriate complementary differential equation based on the Green’s function of the Laplace differential operator. Hybridization is achieved by retaining the integral discretization of the conventional boundary element method coupled with a finite-element type element-by-element determination of the scalar profile. This mechanism unlike the classical boundary element formalism permits “interaction” between the problem domain and its boundary by the intermediaries of its element based implementation as well as the accompanying interacting grid points. The resulting hybrid numerical procedure has therefore an implicit “local support” as well as an enhanced accuracy. These features are associated with both the Finite Element Method (FEM) and Boundary Element Method (BEM) by virtue of a straightforward numerical formulation as well as the subsequent element- driven implementation. We intentionally adopt this numerical route partly to address the scarcity of information on the boundary integral solutions for flows over a flat plate and also to seek for simpler numerical techniques to handle the species transport component of boundary layer flows.

The goal of the present investigation is to present a robust algorithm capable of producing faithful results for the numerical solution of the Falkner Skan (F-S) equation and accompanying energy equation. The application of boundary integral numerical techniques for the computation of F-S equation accompanied by heat and/or mass transfer is rare in scientific literature. Its emergence however is buoyed by the desire to seamlessly link the problem boundaries with its domain in order to relax the “boundary-only” approach of the BEM technique.

Consider two dimensional flow of an incompressible viscous fluid with heat transfer over a flat plate. The x-axis is taken along the plate while the y-axis is considered normal to the plate. In order to simplify the numerical solution, the following assumptions are made namely:

1) the fluid flow is considered laminar, stable and at steady state;

2) all body forces are neglected and all thermal properties are not temperature dependent.

The boundary layer equations can now be reduced to (Chow [

In the above system of equations, u and v represent velocities in the x and y directions respectively, T is the fluid temperature

nal flow velocity and the pressure gradient are given by:

nal velocity and b is a function of the flow geometry. The pressure gradient for a flat plate can be made favorable or adverse by assigning a positive or negative value to the variable m. Equations (1a)-(1c) are transformed by applying the non-dimensionalizations adopted by Falkner and Skan [

A nondimensional stream function

The velocity components become

And

A governing equation for f can be found by substituting these non-dimensionalized variables into the momentum equation to yield

where

Equation (6) is the well known Falkner-Skan equation. It comes with the following boundary conditions:

and reduces to the Blasius equation for

Over the years there have been several solutions of the F-S equation, it is not possible to go through all of them in depth. Among them are a host of numerical techniques involving finite differences, finite elements, spectral methods, adomian’s polynomials and perturbation methods. A comprehensive list of these can be found in Parand et al. [

We start by assuming a shooting angle say

1) Guess

2) If at a particular value of

3) Once this range is identified, we continue with the iteration until

4) We can then further refine our computation by applying the secant method.

Having then obtained the value of f and the derivatives as functions of

Equation (1c) describes the energy aspect of the boundary layer flow. The assumption of incompressibility decouples the energy equation from the continuity and momentum equations.

The boundary conditions for velocity and temperature fields are given as

In order to solve the energy equation, the velocity components u and v are first obtained from the momentum and continuity equations and then substituted in terms of their similarity variables

where

where

Equation (10) paves the way for the introduction of a hybrid boundary-integral domain-discretized numerical procedure for a typical Blasius problem and its variations. Our major goal here is to achieve its integral representation by the way of the Green’s second identity.

The first step in this route is to propose a complementary differential equation of the type:

Equation (12) is interpreted physically to represent a one-dimensional diffusion subjected to a dirac-dellta type unit input. This characterizes a response known as the free-space Green function:

where K is an arbitrary constant, representing the longest element in the problem domain, and

where

The resulting integral equation yields:

where

As can be observed, Equation (16) is a typical Boundary Element Method (BEM) equation. However, its element-driven numerical implementation is the essence of the hybrid procedure adopted in this study. The problem domain

Equation (18) is the interpretation of Equation (16) in elemental context where the variable “e” stands for an element and

Equation (19) can be expressed in a compact matrix indicial form to yield a system of element equations:

Equation (20) provides both the primary variable and its flux at each node of the problem domain and unlike FEM, it does not determine the spatial derivative of the primary variable by an approximation procedure.

1) Example 1

We test the formulation developed herein by computing the velocity profiles

flows across obstacles or blunt bodies, the fluid kinetic energy is dissipated by viscous drag and the fluid is deprived of enough energy to continue along its path. As a result, it halts at some “separation” point after which it reverses and departs from the surface. This is intuitive picture is confirmed by observing the point of inflection in the profile for

2) Example 2

The code developed herein is again validated by solving an example given by Chow [

In order to be consistent with Chow [

3) Example 3

We solve the Falkner-Skan equation (Equation (6)) together with the energy equation (Equation (10)) to quantify the effect of different pressure gradients

surface.

ETA | THETA (Water) | THETA (Air) | THETA (Mercury) |
---|---|---|---|

0.0 | 2.492125794007539e+00 | 8.447204393523394e−01 | 1.590312919160360e−01 |

0.2 | 2.462355058308143e+00 | 8.415713570872477e−01 | 1.588372308240696e−01 |

0.4 | 2.373335460484374e+00 | 8.321298830686987e−01 | 1.582552306776364e−01 |

0.6 | 2.227093784233705e+00 | 8.164363434640637e−01 | 1.572865710889394e−01 |

0.8 | 2.028960739477595e+00 | 7.945994919944929e−01 | 1.559347048787274e−01 |

1.0 | 1.788649937053652e+00 | 7.668282291715085e−01 | 1.542062712279741e−01 |

1.2 | 1.520416801040675e+00 | 7.334588328766877e−01 | 1.521119759779887e−01 |

1.4 | 1.241963447127322e+00 | 6.949747553076039e−01 | 1.496672519226826e−01 |

1.6 | 9.720985910703718e−01 | 6.520158586471457e−01 | 1.468926071688203e−01 |

1.8 | 7.276662217804996e−01 | 6.053743732612509e−01 | 1.438135829564430e−01 |

2.0 | 5.206809086516063e−01 | 5.559759550031819e−01 | 1.404602761336933e−01 |

2.2 | 3.566630988506530e−01 | 5.048459351649483e−01 | 1.368664335344950e−01 |

2.4 | 2.347306897891762e−01 | 4.530629579196684e−01 | 1.330681878381211e−01 |

2.6 | 1.492625237176419e−01 | 4.017042805185697e−01 | 1.291025641353412e−01 |

2.8 | 9.232452391111272e−02 | 3.517885864924077e−01 | 1.250059284332115e−01 |

3.0 | 5.589354335869892e−02 | 3.042228066667908e−01 | 1.208125613629297e−01 |

3.2 | 3.325104509879356e−02 | 2.597589366877494e−01 | 1.165535172771315e−01 |

3.4 | 1.945347585929216e−02 | 2.189652581795439e−01 | 1.122558753144024e−01 |

3.6 | 1.117096998582093e−02 | 1.822140751083792e−01 | 1.079424181979113e−01 |

3.8 | 6.275075530117414e−03 | 1.496856026534212e−01 | 1.036317042666051e−01 |

4.0 | 3.436137186162650e−03 | 1.213855303912711e−01 | 9.933844472321433e−02 |

4.2 | 1.828983434297532e−03 | 9.717240505548914e−02 | 9.507407097087726e−02 |

4.4 | 9.443509429017163e−04 | 7.679047697864817e−02 | 9.084737716526684e−02 |

4.6 | 4.722762839707529e−04 | 5.990393038300477e−02 | 8.666514429542689e−02 |

4.8 | 2.285173542443050e−04 | 4.612922091095485e−02 | 8.253268397088569e−02 |

5.0 | 1.068876947235345e−04 | 3.506328099202136e−02 | 7.845427258073902e−02 |

5.2 | 4.829660560941896e−05 | 2.630637744010499e−02 | 7.443347258086852e−02 |

5.4 | 2.106843520979321e−05 | 1.947925940474980e−02 | 7.047335433844905e−02 |

5.6 | 8.868730517078944e−06 | 1.423485434172388e−02 | 6.657663957651691e−02 |

5.8 | 3.601002579859497e−06 | 1.026515874772753e−02 | 6.274578832953298e−02 |

6.0 | 1.409822047348595e−06 | 7.304168818693924e−03 | 5.898304828547878e−02 |

6.2 | 5.320501508627313e−07 | 5.127754192377550e−03 | 5.529048085813975e−02 |

6.4 | 1.934976921784940e−07 | 3.551341493118904e−03 | 5.166997387916475e−02 |

6.6 | 6.780127529555688e−08 | 2.426179863340110e−03 | 4.812324718963083e−02 |

6.8 | 2.288531971029030e−08 | 1.634834926563655e−03 | 4.465185483504732e−02 |

7.0 | 7.439846878228148e−09 | 1.086420167936023e−03 | 4.125718590230307e−02 |

7.2 | 2.329169611818925e−09 | 7.119395908217890e−04 | 3.794046504615431e−02 |

8.0 | 1.531944294889122e−11 | 1.137597139026569e−04 | 2.547185242930064e−02 |

8.2 | 3.967376461870550e−12 | 6.929173122026577e−05 | 2.255756134505500e−02 |

8.4 | 9.891069356139590e−13 | 4.153084860454167e−05 | 1.972518478059964e−02 |

9.0 | 1.219831347302892e−14 | 7.953196131768619e−06 | 1.172001240857885e−02 |

9.1 | 2.609719526824824e−15 | 4.329965222000307e−06 | 9.215053457123028e−03 |

9.4 | 5.352314906956821e−16 | 2.233845522374445e−06 | 6.791194566114655e−03 |

9.6 | 1.032785996399164e−16 | 1.039735141047852e−06 | 4.447873658772147e−03 |

9.7 | 1.670444532507259e−17 | 3.699233178487305e−07 | 2.184407994935732e−03 |

9.9 | 0 | 0 |

ETA | THETA (Water) | THETA (Air) | THETA (Mercury) |
---|---|---|---|

0.0e−00 | 2.069252719243075e+00 | 7.856234525655620e−01 | 1.499825426547470e−01 |

1.0e−01 | 2.069191533269493e+00 | 7.856169804321249e−01 | 1.499821438116730e−01 |

1.5e−01 | 2.069073708364376e+00 | 7.856045169473743e−01 | 1.499813757522288e−01 |

2.0e−01 | 2.068855844676721e+00 | 7.855814705382397e−01 | 1.499799555166314e−01 |

2.5e−01 | 2.068498285885078e+00 | 7.855436440060658e−01 | 1.499776244372829e−01 |

3.0e−01 | 2.067954810393538e+00 | 7.854861423495924e−01 | 1.499740808218717e−01 |

3.5e−01 | 2.067172674258566e+00 | 7.854033744935965e−01 | 1.499689800406409e−01 |

4.0e−01 | 2.066092676273593e+00 | 7.852890559234335e−01 | 1.499619346596096e−01 |

8.5e−01 | 2.030199602313743e+00 | 7.814682042919943e−01 | 1.497263098068126e−01 |

1.0e+00 | 2.000323063325884e+00 | 7.782545349433063e−01 | 1.495278988570549e−01 |

1.45e+00 | 1.816385400454253e+00 | 7.577425559579142e−01 | 1.482563286269803e−01 |

1.5e+00 | 1.785222467100916e+00 | 7.541336152591621e−01 | 1.480316501721723e−01 |

2.0e+00 | 1.348751990329582e+00 | 6.984991481214543e−01 | 1.445286355094697e−01 |

2.5e+00 | 7.793759434718216e−01 | 6.034302823219873e−01 | 1.383384701503052e−01 |

3.0e+00 | 3.118782910768406e−01 | 4.771199363942041e−01 | 1.295527997056951e−01 |

3.5e+00 | 8.042907190037835e−02 | 3.431033731642481e−01 | 1.190605116572901e−01 |

4.0e+00 | 1.297935097026406e−02 | 2.255838164632527e−01 | 1.079062957053351e−01 |

4.5e+00 | 1.313544737464662e−03 | 1.367022471479897e−01 | 9.673225694802133e−02 |

5.0e+00 | 8.391513562757727e−05 | 7.665932514849577e−02 | 8.579017445437877e−02 |

9.0e+00 | 1.014674261019077e−12 | 3.865780010045579e−05 | 1.314903449253200e−02 |

1.0e+010 | 0 | 0 |

4) Example 4

Next we consider the dimensionless energy difference distributions by comparing the total energy content in the free stream and the boundary layer. When a fluid particle originally in the free stream of temperature

Equation (21) gives the impression that the kinetic energy and by default the free stream velocity is fully recovered. But this is not true for real fluids with non-vanishing kinematic viscosity

The second part on the right hand side of Equation (22) “recovers” the kinetic energy component of the total energy and

It can be shown that the dimensionless energy difference between freestream and the wall can be represented as:

Figures 5(a)-(d) confirms that for a fluid with a smaller Prandtl number, where the thermal conductivity is

relatively large, more heat is conducted out of the fluid element than is produced within it by friction, hence the overall energy within its boundary layer becomes lower. This is also combined with the fact, that for a case where the fluid is approaching separation there is a further decrease in the overall energy resulting from a decrease in kinetic energy which further hinders the further advance of the fluid particle. This will in some cases deplete the available boundary layer energy to the extent that it becomes much smaller than that of the free stream and yields a negative value for the dimensionless energy difference.

A robust numerical algorithm involving an element-driven hybridization of the boundary integral theory has been applied to the solution of the energy equation for physical relevant flows of the Falkner-Skan equation. To this author’s knowledge, this is probably one of the few times such an approach is taken for this type of problem. By relying on this hybrid numerical technique, we report highly accurate numerical results without resorting to complex numerical integrations. There are a few issues to emphasize here. First the hybrid procedure permitted Equation (10) to be solved in a 1-D domain. There is no need to seek or device an equivalent analog of the same problem in a 2D domain as would be the case for a classical application of the Boundary Element Method (BEM). Such a transformation will give a false sense of accuracy, because the rigor incurred in order to achieve a BEM domain-reduction is more often than not compensated for by accuracy. The second point is that by dealing directly with such problems without an undue resort to approximations and by discretizing the problem domain we open the door for handling problems involving heterogeneity and nonlinearity (Onyejekwe and Onyejekwe [

Okey Oseloka Onyejekwe, (2016) Incompressible Flow and Heat Transfer over a Plate: A Hybrid Integral Domain-Discretized Numerical Procedure. Applied Mathematics,07,1426-1439. doi: 10.4236/am.2016.713123