Some Properties of Laminar Boundary Layers on Curved Surfaces

1968 ◽  
Vol 90 (2) ◽  
pp. 301-312 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Fourth Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Curved Surfaces ◽  
Two Dimensional ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Similar Solutions

Equations have been developed [1] which describe the flow in a steady, two-dimensional, incompressible, laminar boundary layer on a curved surface. The method of “similar solutions” yields a fourth-order, nonlinear, ordinary differential equation which may be solved on a digital computer. Account is taken of the effects of both surface curvature and displacement thickness, and in this paper attention is given to the influence of these effects on other boundary-layer properties up to and including the position of flow separation.

A Note on the Accuracy of Similar Solutions of the Equations for Laminar Boundary Layers on Curved Surfaces

1969 ◽  
Vol 73 (699) ◽  
pp. 226-228 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Stokes Equations ◽  
Single Equation ◽  
Navier Stokes ◽  
Constant Density ◽  
Curved Surfaces ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Laminar Boundary Layers ◽  
Density Flow ◽  
Similar Solutions

Summary To describe steady, two-dimensional, constant-density flow in a laminar boundary layer on a curved surface, a single equation may be derived from the complete Navier-Stokes equations, with no approximations being necessary. The conditions under which similar solutions are attainable are discussed, and the validity of some previous calculations is upheld.

scholarly journals Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
A. A. Hemeda ◽  
E. E. Eladdad
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Reynolds Number ◽  
Fractional Order ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Axisymmetric Flow ◽  
Nonlinear Ordinary Differential Equation ◽  
Absolute Residual

An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically.

Laminar Boundary Layers and Their Separation From Curved Surfaces

1965 ◽  
Vol 87 (2) ◽  
pp. 483-493 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Stokes Equations ◽  
Equation Of Motion ◽  
Navier Stokes ◽  
Curved Surfaces ◽  
Integration Technique ◽  
Navier Stokes Equations ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Order Of Magnitude ◽  
Similar Solutions

The equation of motion in a laminar boundary layer on curved surfaces has been developed from the Navier-Stokes equations. After an order-of-magnitude analysis terms of highest and next order have been retained and a single ordinary differential equation of motion derived by the method of “similar solutions.” An iteration procedure has been used to determine effects of displacement thickness and surface curvature up to and including separation. The equation of motion has been solved by a digital computer using the Runge-Kutta step-by-step integration technique.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Effects of Curvature on Laminar Boundary Layers in Sink-Type Flows

1969 ◽  
Vol 91 (3) ◽  
pp. 353-358 ◽  
Author(s):  
W. A. Gustafson ◽  
I. Pelech
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Momentum Equation ◽  
Similarity Solutions ◽  
Momentum Thickness ◽  
Two Dimensional ◽  
Thickness Increase ◽  
Laminar Boundary Layers ◽  
Converging Channel ◽  
Special Case

The two-dimensional, incompressible laminar boundary layer on a strongly curved wall in a converging channel is investigated for the special case of potential velocity inversely proportional to the distance along the wall. Similarity solutions of the momentum equation are obtained by two different methods and the differences between the methods are discussed. The numerical results show that displacement and momentum thickness increase linearly with curvature while skin friction decreases linearly.

Design of evolutionary cubic spline intelligent solver for nonlinear Painlevé-I transcendent

2021 ◽  
Author(s):  
Sharafat Ali ◽  
Iftikhar Ahmad ◽  
Muhammad Asif Zahoor Raja ◽  
Siraj ul Islam Ahmad ◽  
Muhammad Shoaib
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Genetic Algorithms ◽  
Statistical Analysis ◽  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Process Variation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Derivative ◽  
Search Technique

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Riccardo Barsotti
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Nonlinear Ordinary Differential Equation ◽  
Experimental Results ◽  
Limit Case ◽  
Reasonable Approximation ◽  
Constituent Material ◽  
Actual Solution ◽  
Wrinkled Membrane

The axisymmetric inflation problem for a wrinkled membrane is solved by means of a simple nonlinear ordinary differential equation. The solution is illustrated in full details. Both the free and constrained cases are addressed, in the limit case where the membrane is fully wrinkled. In the constrained inflation problem, no slippage is allowed between the membrane and the constraining surfaces. It is shown that an actual membrane can in no way reach the fully wrinkled configuration during free inflation, regardless of the membrane's initial configuration and constituent material. The fully wrinkled solution is compared to some finite element results obtained by means of an expressly developed iterative–incremental procedure. When the values of the inflating pressure and length of the meridian lie within a suitable applicability range, the fully wrinkled solution may represent a reasonable approximation of the actual solution. A comparison with some numerical and experimental results available in the literature is illustrated.

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