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Viscoelastic Walters' B fluid flows for three problems, stagnation-point flow, Blasius flow, and Sakiadis flow, have been investigated. In each problem, Cauchy equations are changed to a nondimensional differential equations using stream functions and with assumption of boundary layer flow. The fourth-order predictor-corrector finite-difference method for solving these nonlinear differential equations has been employed. The results that have been obtained using this method are compared with the results of the last studies, and it is clarified that this method is more accurate. It is also shown that the results of last study about Sakiadis flow of Walter's B fluid are not true. In addition, the effects of order of discretization in the boundaries are investigated. Moreover, it has been discussed about the valid region of Weissenberg numbers for the second-order approximation of viscoelastic fluids in each case of study.

Boundary layer flows of non-Newtonian fluids can be a beneficial approach for modeling several manufacturing processes of industry such as the aerodynamic extrusion of plastic sheets, the cooling of metallic plates, fabrication of adhesive tapes, application of coating and layers onto rigid substrates. Some of articles that have investigated these kinds of fluid flows are given in [

Constitutive equations of viscoelastic fluids usually generate higher-order derivative terms in the momentum equations that make them more difficult to solve in comparison with Newtonian fluids. There are some analytical and numerical methods for investigating these kinds of fluids such as homotopy analysis method [

In this paper, similarity solutions of three viscoelastic Walters’ B boundary layer flow problems, stagnation-point flow, Blasius flow and Sakiadis flow, are obtained and all of the equations are changed to nondimensional forms in the second section. The mentioned fourth-order predictor-corrector method for solving these nonlinear differential equations is presented in the third section, and the results are compared with last studies in the fourth section. It is obtained that this method is more accurate and straightforward. It is also shown that the results of last article about Sakiadis flow by Sadeghy and Sharifi [

In this section, nondimensional equations of motions for three different boundary layer flows, stagnation-point flow, Blasius flow, and Sakiadis flow, are obtained. These equations are solved in Section

Cauchy equations are employed for obtaining boundary layer equation of two-dimensional stagnation-point flow. Steady Cauchy equations are

These equations are the relation of components of stress tensor and velocity components. Because of requirement for derivatives of these stress components, these derivatives are presented.

By substituting these relations in (

On the basis of viscous flow theory, the change in pressure across boundary layer is

The

Order of dimensionless equation of motion, (

In addition to velocities and stream functions that are common parameters for investigating boundary layer problems, shear stress as another important parameter should obtained in the wall. Therefore, the following equation for shear is presented:

Equation of motion for Blasius flow can be obtained using (

The

Sakiadis flow for viscoelastic Walters’ B fluid has been investigated by Sadeghy and Sharifi [

Sakiadis flow is similar to Blasius flow and the only difference of these fluid flows is their boundary conditions that are presented in following relations:

In the next section, the predictor-corrector method for analysis of nondimensional equations of these flows is described.

In addition, shear stress on the solid boundary can be obtained similar to (

In this section, fourth-order predictor-corrector method that was proposed by Ariel [

It should be noted that because of the singularity of (

The auxiliary parameters

The key of the algorithm is to retrain the second-order derivative in (

The following equations should be used for approximating the first- and second-order derivatives:

These operators are employed for discretizing (

Therefore, we have a fourth-order predictor-corrector method for obtaining the values of

Assume that the amount of the initial guess of

The amounts of in

Nondimensional equation of Blasius flow (

Predictor:

Quantities of extra nodes that are necessary for solving above equations are obtained similar to those obtained for stagnation-point flow using Taylor series. So, we have

It is mentioned that the only difference between Sakiadis flow and Blasius flow is just in their boundary conditions. So, their predictor and corrector relations are same and we can use those of Blasius flow for solving (

The results of the solution of stagnation-point flow with the proposed fourth-order method are presented in Table

Variation of

Serth [ | |||||
---|---|---|---|---|---|

0 | 1.201388826 | 1.21349757 | 1.213566436 | — | 1.232587 |

0.05 | 1.294618079 | 1.29464226 | 1.294646606 | 1.294646 | 1.294646 |

0.1 | 1.369500992 | 1.36953449 | 1.369541015 | 1.369539 | 1.369538 |

0.2 | 1.560314529 | 1.58732493 | 1.587328125 | 1.587328 | 1.587332 |

0.3 | 1.088899532 | 1.13850445 | 2.110821533 | 2.110818 | — |

On the basis of Table

Moreover, the comparison of values of shear stress at solid boundary can be beneficial in this problem. On the basis of (

The boundary layers in stagnation-point flow for different Weissenberg numbers are compared in Figure

Nondimensional horizontal velocity of stagnation-point flow with respect to nondimensional parameter

As we see in the literature [

Effect of Weissenberg number of the stream line near stagnation point (solid line:

Quantities of

Variation of

0 | 0.3320573425 |

0.1 | 0.2970753479 |

0.2 | 0.2683211517 |

0.3 | 0.2444062805 |

0.4 | 0.2244654083 |

0.5 | 0.2068865966 |

0.6 | 0.1923710250 |

0.7 | 0.1804304504 |

0.8 | 0.1698255157 |

0.9 | 0.1598894882 |

1 | 0.1505795288 |

Figure

Nondimensional horizontal velocity of Blasius flow with respect to nondimensional parameter

The differences between the values of nondimensional horizontal velocity at any particular nondimensional parameter

Variation of

Variation of

0 | −0.44349197 |

0.2 | −0.45658203 |

0.4 | −0.47529945 |

0.6 | −0.50525421 |

0.7 | −0.530166503 |

The most important result of this study is shown in Figure

Nondimensional horizontal velocity of Sakiadis flow with respect to nondimensional parameter

One of the fundamental principles of numerical methods to obtain an arbitrary order of approximation is the sameness of order of discretization both in the entire domain and in the boundaries. The predictor-corrector method that is used in this study is a the fourth order method. So, boundary condition also should be approximated in the fourth-order. Amounts of

Comparison of

2nd-order boundary | 4th-order boundary | |
---|---|---|

0 | 1.2321389 | 1.2325876 |

0.05 | 1.2941257 | 1.2946467 |

0.1 | 1.3689191 | 1.3695389 |

0.2 | 1.5863305 | 1.5873276 |

0.3 | 2.1078014 | 2.1108185 |

Comparison of

2nd-order boundary | 4th-order boundary | |
---|---|---|

0 | 0.3319149780 | 0.3320573425 |

0.1 | 0.2969615173 | 0.2970753479 |

0.2 | 0.2682281494 | 0.2683211517 |

0.3 | 0.2443289184 | 0.2444062805 |

0.4 | 0.2243995666 | 0.2244654083 |

0.5 | 0.2068298721 | 0.2068865966 |

0.6 | 0.1923221588 | 0.1923710250 |

0.7 | 0.1803870391 | 0.1804304504 |

0.8 | 0.1697863769 | 0.1698255157 |

0.9 | 0.1598538398 | 0.1598894882 |

1 | 0.1505471420 | 0.1505795288 |

Comparison of

2nd-order boundary | 4th-order boundary | |
---|---|---|

0 | −0.44364105 | −0.44349197 |

0.2 | −0.45671180 | −0.45658203 |

0.4 | −0.47542854 | −0.47529945 |

0.6 | −0.50540725 | −0.50525421 |

0.7 | −0.53035644 | −0.530166503 |

It is obvious that the amounts of

In this study, the fourth-order predictor-corrector method is employed for solving the three viscoelastic fluid flow problems. Equations of motion for these problems, stagnation-point flow, Blasius flow, and Sakiadis flow, are obtained using Cauchy equation of motion, with assumption of boundary layer flow and turned into nondimensional forms. The discretized equations on the approach of the predictor-corrector method are solved, and the results are compared with the results of last studies, that is, clarified the high accuracy of the method. Also it has been shown that as elasticity increases, the stress on the solid boundary increases in stagnation-flow problem and decreases in Blasius and Sakiadis flows. In addition, it is shown that the results of the last study about Sakiadis flow by Sadeghy and Sharifi [