[Paper Review] On a general similarity boundary layer equation

Bernard Brighi, Jean-David Hoernel|ArXiv.org|Jan 16, 2006

Differential Equations and Numerical Methods34 references3 citations

TL;DR

This paper investigates a general third-order nonlinear boundary value problem, $ f''' + f f'' + g(f') = 0 $, with boundary conditions $ f(0) = \alpha $, $ f'(0) = \beta \geq 0 $, $ f'(\infty) = \lambda $, where $ g $ is a given continuous function. Under suitable conditions on $ g $, the authors prove existence and uniqueness of concave or convex solutions, establish asymptotic behavior, and derive nonexistence results, generalizing classical models like Falkner-Skan and free/mixed convection in porous media.

ABSTRACT

In this paper we are concerned with the solutions of the differential equation $f'''+ff''+g(f')=0$ on $[0,\infty)$, satisfying the boundary conditions $f(0)=α$, $f'(0)=β\geq 0$, $f'(\infty)=ł$, and where $g$ is some given continuous function. This general boundary value problem includes the Falkner-Skan case, and can be applied, for example, to free or mixed convection in porous medium, or flow adjacent to stretching walls in the context of boundary layer approximation. Under some assumptions on the function $g$, we prove existence and uniqueness of a concave or a convex solution. We also give some results about nonexistence and asymptotic behaviour of the solution.

Motivation & Objective

• To establish existence and uniqueness of concave or convex solutions for a general third-order nonlinear ODE arising in boundary layer theory.
• To unify and generalize existing models such as Falkner-Skan, free convection, mixed convection, and stretching wall flows under a single analytical framework.
• To analyze the asymptotic behavior of solutions as $ t \to \infty $, particularly the convergence of $ f(t) - (\lambda t + \mu) \to 0 $ under convexity and positivity assumptions.
• To derive sufficient conditions for nonexistence of convex solutions, extending known results for specific physical cases.

Proposed method

• The authors analyze the third-order nonlinear ODE $ f''' + f f'' + g(f') = 0 $ on $[0, \infty)$ with boundary conditions $ f(0) = \alpha $, $ f'(0) = \beta \geq 0 $, $ f'(\infty) = \lambda $.
• They use a transformation $ v(y) = f''(t) $, where $ y = f'(t) $, to reduce the problem to a second-order ODE in terms of $ v $, enabling the study of monotonicity and sign behavior.
• A key technique involves integrating the equation multiplied by $ f'' $, leading to energy-type estimates and the use of anti-derivatives of $ g $ and $ f $ to analyze long-term behavior.
• Uniqueness is proven via contradiction by assuming two distinct solutions and showing they must coincide under strict conditions on $ g $, particularly when $ g > 0 $ on $[\beta, \lambda)$ and $ g(\lambda) = 0 $.
• Nonexistence results are derived by contradiction, assuming a convex solution exists and showing that the resulting integral expression cannot be nonpositive under the given constraints on $ g $.
• The analysis relies on comparison principles and sign analysis of $ g $, particularly focusing on whether $ g $ vanishes at $ \lambda $ and does not vanish in $[\beta, \lambda) $.

Experimental results

Research questions

• RQ1 Under what conditions on $ g $ does the general boundary layer equation admit a unique convex or concave solution?
• RQ2 What is the asymptotic behavior of the solution as $ t \to \infty $, particularly in terms of the difference $ f(t) - (\lambda t + \mu) $?
• RQ3 Can nonexistence of convex solutions be established under specific bounds on $ g $, such as $ g(x) \leq x^2 - \lambda x $?
• RQ4 How do the results generalize known cases such as Falkner-Skan, free convection, and mixed convection in porous media?
• RQ5 What role does the sign and monotonicity of $ g $ play in determining the existence and shape of solutions?

Key findings

• Under the assumption that $ g > 0 $ on $[\beta, \lambda) $ and $ g(\lambda) = 0 $, there exists a unique convex solution to the boundary value problem.
• For such convex solutions, $ \lim_{t \to \infty} \{ f(t) - (\lambda t + \mu) \} = 0 $ for some $ \mu > \alpha $, and $ \lambda t + \alpha \leq f(t) \leq \lambda t + \mu $ for all $ t \geq 0 $.
• A nonexistence result is established: if $ g(x) \leq x^2 - \lambda x $ and $ -\alpha + \max_{x \in [\beta, \lambda]} \{ x^2 - \lambda x - g(x) \} > 0 $, then no convex solution exists.
• The results recover and extend known nonexistence results: for the Falkner-Skan case ($ g(x) = m(1 - x)(1 + x) $, $ \lambda = 1 $), no convex solution exists for $ m \leq -\frac{1}{2} $; for mixed convection ($ g(x) = \frac{2m}{m+1}x(1 - x) $, $ \lambda = 1 $), no convex solution exists for $ -1 < m \leq -\frac{1}{3} $.
• The solution’s concavity or convexity is directly linked to the sign of $ g $, with $ f'' $ vanishing identically if $ g(\mu) = 0 $ and $ f'(t_0) = \mu $, $ f''(t_0) = 0 $ at some point.
• The method confirms that $ f''(t) \to 0 $ as $ t \to \infty $ when $ f $ is of constant sign at infinity, which holds if $ f $ is convex or concave at infinity or if $ l = f'(\infty) \neq 0 $.