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Questions tagged [boundary-layer]

Use this tag for questions related to boundary-layer theory, which refers to asymptotic approximations of solutions of boundary value problems for differential equations containing a small parameter in front of the highest derivative in sub-regions where there is a substantial effect from terms containing the highest derivatives on the solution.

Boundary-layer theory refers to asymptotic approximations of solutions of boundary value problems for differential equations containing a small parameter in front of the highest derivative (singular problems) in sub-regions where there is a substantial effect from terms containing the highest derivatives on the solution.

Boundary-layer phenomena arise in narrow zones near the parts of the boundary on which there is a difference in the numbers of boundary conditions for the initial problem and the degenerate one (with the small parameter taking the value zero) as well as near surfaces of discontinuity of the solution of the degenerate problem.

41 questions
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Why don't these ODEs produce the same result?

I am relatively new to differential equations, and the following problem is confusing me. Consider, for example, the ODE $x'+x=0$ such that $x(0)=1$. This has solution $x(t)=e^{-t}$. But consider an $\epsilon>0$ and the ODE. $$\epsilon x'' + x' + x…
user818261
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1 answer

Integration in cylindrical coordinate system

Context: I am trying to derive an equation given in a Journal of Fluid Mechanics paper (2.2). It deals with the analysis of an axisymmetric turbulent wake where cylindrical coordinate system has been used (which to me is a little hard to understand…
Tanmay Agrawal
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Question about Navier Stokes Equation and boundary layers

I'm trying to understand the Navier Stokes Equation by solving a fluid dynamics problem. Not too sure if this is an appropriate place for my question. I have a thin layer of paint with constant thickness $h$ spread evenly on a very long and wide…
mathnoob
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ODE with nested boundary layers

Problem: Consider the equation $$\varepsilon^3 \frac{d^2y}{dx^2} + 2x^3 \frac{dy}{dx} - 4\varepsilon y = 2x^3 \qquad \qquad y(0) = a \;, \; y(1)=b$$ in the limit as $\varepsilon \rightarrow 0^+$, where $0
glowstonetrees
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Thickness of the Boundary Layer

Given an ODE $$\epsilon y''+2xy'=x \cos(x)$$ with boundary condition $y(\pm {\pi \over 2})=2$ Where is the boundary layer and what is the thickness of it?
SamC
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non-homogeneous laplace equation with mixed boundary condition

Consider this problem $$ \begin{cases} -\Delta u=10 \hspace{6mm} \mbox{in} \hspace{6mm} \Omega \\ u=0 \hspace{6mm}\mbox{on}\hspace{6mm}\Gamma_d \\ \frac{\partial u}{\partial n}=-\sqrt{4x^2+64y^2} \hspace{3mm} \mbox{on} \hspace{3mm}…
FreeMind
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WKB for a boundary value problem with two layers

The equation $\epsilon y''-x^4y'-y=0,\ y(0)=y(1)=1$, can be solved by boundary layer analysis and turns out it has two layers of size $O(\epsilon) $ and $O(\sqrt{\epsilon})$ at $1,0$ accordingly. Is possible to solve this with WKB method? How do I…
mosx
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Reference request: Vector calculus using signed distance coordinates for boundary layers near curved surfaces

I am looking for references which give vector calculus expressions in boundary layers around curved surfaces. My application is fluid dynamics, so I want to be able to write the Navier-Stokes equations in these coordinates $$ \partial_t u + u\cdot…
Eric Hester
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Boundary layer in time

Consider the initial value problem $\varepsilon x'' + x' + tx = 0$ where $x(0) = 0$ and $x'(0) = 1$. I'm solving this problem using a matched asymptotic expansion. First, I let $$x(t, \varepsilon) = \varepsilon x_1(t) + \varepsilon^2 x_2(t) +…
Patrick Lewis
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Boundary layer ODE, is my solution okay?

The original question is from here, I am trying to work out a full solution following the hint in the answer and the comments. Given the ODE $$y' - (y-1)^2 -\epsilon\frac{y^2}{x^2} = 0, \quad y(1) = 1,$$ we would like to find a uniform…
Xiao
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1 answer

Are boundary conditions required between subdomains of 1D PDE?

I am using finite difference software to solve across 1D line from x=0 to x=1 (left Domain), and x=1 to x=2 (middle/central domain) and x=2 to x=3 (right domain). The only difference between domains is that the central domain posses a source…
SPIL
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Construct leading order approximation without given specific function

The Question: $$\varepsilon y''+f(x)y'+y=0 \qquad y(-1)=0 \qquad y(1)=1$$ where $0<\varepsilon \ll 1$ and $f$ is a given smooth function that is strictly positive with $f(1)=f(-1)=1$. (i) Determine the location of the boundary layer (ii) Obtain…
glowstonetrees
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How to find boundary layer

The question tell me that there is a boundary layer at $x=0$ for this differential equation $$\epsilon y''+y y'-y=0, 0
SamC
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Uniform solution for transition layer

Background Looking for an approximated uniform solution for: $$\epsilon y''+xy'+xy=0, \quad y(-1)=e, \quad y(1)=\frac{2}{e}$$ I found the following Outer solutions: $$y^{outer}_{left}(x)=e^{-x},\quad y^{outer}_{right}(x)=2e^{-x}$$ Matching…
Michael
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Leading order matching of $\epsilon x^py'' + y' + y = 0$

Question: The function $y(x)$ satisfies $$\epsilon x^py'' + y' + y = 0,$$ in $x\in [0,1]$, where $p<1$, subject to the boundary conditions $y(0) = 0$ and $y(1)=1$. Find the rescaling for the boundary layer near $x = 0$, and obtain the leading order…
Sanket Biswas
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