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.

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Solving a second Order ODE with a complex coeffcient

I have been asked to derive a solution for F(y) in the form F(y) = A cos ky + B sin ky + C , where k = (1 + I)/δ , δ = sqrt(2ν/ω) and A,B and C are constants to be found. The ODE I have found is F''+(...
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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 ...
Sanket Biswas's user avatar
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Approximate solution to $\epsilon y'' + x^2 y' - \lambda y = 0$ near $x=0$

I would like to approximate the solution of $$\epsilon y''(x)+ x^2 y'(x) - \lambda y(x) = 0 $$ with boundary conditions $y'(0)=0$ and $y(\infty)=0$. The parameter $\epsilon$ is small. To approach this ...
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Small parameter expansion of solution to ODE

I'm working with an ODE of the form \begin{equation} \begin{split} C'(z) &= (z^2 + \epsilon^2)C'' \\ C(a) &= b \\ C(1) &= 1 \\ \end{split} \end{equation} where $0 < a,b < 1$, and $\...
Mike D's user avatar
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Spotting distinguished limits from Robin boundary conditions

I am currently working on a problem involving solving PDEs and using boundary layers. In this problem, $f(x,y)$ is the main dependent variable, $f(x,y)=f_R (X_R ,y)$ and $f(x,y)=f_B(x,Y_B)$ where $X_R$...
epsilonD3LT4's user avatar
2 votes
1 answer
303 views

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} \...
FreeMind's user avatar
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How to solve the singularly perturbed problem below

There is a singularly perturbed problem: \begin{align} &\epsilon u''(x) + u(x) - u^2(x)=0,\quad x<0<1,\\ & u(0)=1, \quad u(1)=0. \end{align} where $\epsilon \ll 1$ is a tiny parameter. ...
ZR Tang's user avatar
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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 ...
Tanmay Agrawal's user avatar
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2 answers
301 views

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 ...
mosx's user avatar
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WKB for non-homogeneous ODE

Consider the ODE $$\epsilon^2 y'' + \epsilon x y' - y = -1, \; y(0) = 0, \; y(1) = 3$$ I've seen the WKB method applied to homogeneous (linear) ODEs, but here we have the $-1$ term. I could perhaps do ...
hirotaFan's user avatar
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Inner and outer expansions

I'd really appreciate some help with the following question. Find inner and outer expansions, correct up to and including terms of O(ε), for the function $$ f(x;ε) = \frac{e^{-\frac{x}{ε}}}{x} + \...
Stack123's user avatar
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How does the analysis of a single perturbation problem change when $\epsilon$ changes sign?

Just for reference I am looking at the problem $\epsilon y''(x)+xy'(x)+y(x)=0$ with $y(1)=0$ and $y(2)=1$ for which I have found the leading-order outer and inner solutions and composite solution. ...
user3709's user avatar
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Help understanding how to input/simplify a certain partial derivative in order to solve

The Problem I have the following information from an article. I am not seeking help with the engineering side of this, but am more looking for information on how to write out / solve a portion of a ...
user881818's user avatar
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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 $\...
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Integral representation of the pressure of the Stokes flow

I'm currently reading these three books: S. Kim, S. J. Karrila - Microhydrodynamics: Principles and Selected Applications O. A. Ladyzhenskaia -The Mathematical Theory of Viscous Incompressible Flow ...
Cicciopassss's user avatar
3 votes
2 answers
260 views

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 $\varepsilon \rightarrow 0^+$, ...
glowstonetrees's user avatar
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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 ...
Eric Hester's user avatar
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Matched Asymptotic Expansion- Boundary Layer Problem

$\epsilon y''=e^{\epsilon y'}+y$ for $0<x<1$ where $y(0)=1$ and $y(1)=-1$. Compute the first term matched asymptotic expansion for the equation. The outer expansion expansion gives me $y(x)=-...
Anon's user avatar
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How to find inner solution in the method of matched asymptotic expansions

Consider the equation of the form $\epsilon y^{\prime \prime}+a y^{\prime}=0$ on $x\in[0,1]$ with $a\in\mathbb{R}$, $0<\epsilon\ll1$,$y(0) =\alpha$, and $y(1)=\beta$. Show that if $a >0$ then ...
Thinkpad's user avatar
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Matched Asymptotic Expansion- Boundary Layer

In this problem $\epsilon y''+(x+\frac{1}{2})y'+y=0$ for $0 < x <1 $ $y(0)=2, y(1)=3$ In the outer expansion, $y\approx y_0(x)+\epsilon y_1(x)+\cdots$ I found the $O(1)$ problem to be: $(...
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$y_t+-\epsilon .y_{xx}+ M.y_x=0\, ;(x,t) \in (0,1)\times(0,T)$ Boundary layers

I was reading an article about pertubation in advection-transport equations, nad so they have defined the following equation with the perturbation ($\epsilon). $ $$y_t+-\epsilon .y_{xx}+ M.y_x=0\, ;(...
BrianTag's user avatar
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1 answer
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Leading Order Approximation of $\varepsilon y'' + (1-2x)y' + x^2y = 0$ when there are two boundary layers

Hi so I need to find the leading order approximation to the solution for $$ \varepsilon y'' + (1-2x)y' + x^2y = 0 $$ $$ y(0)=-1, y(1)=1 $$ by doing the lead term setting $\varepsilon = 0$ we can ...
wjmccann's user avatar
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2 votes
4 answers
221 views

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) = \...
Patrick Lewis's user avatar
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53 views

Struggling with Proof of Prandtl's Boundary Layer Equations

Would someone with knowledge in fluid mechanics please help me in understanding this man's argument for why dp/dy, the pressure gradient in the boundary layer, must equal zero? I would greatly ...
Partey5's user avatar
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1 answer
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Matching expansion of an ODE: $\epsilon y'' + xy' + y = 0$

I am trying to solve a boundary layer problem using matched expansion $$\epsilon y'' + xy' + y = 0$$ where the boundary condition is $$y(0) = 1, y(1) = 1$$ and $x\in (0,1)$. So far, I have the outer ...
TurbPhys's user avatar
1 vote
1 answer
718 views

Obtain the leading order uniform approximation of the solution

Obtain the leading order uniform approximation of the solution to $ \epsilon y′′-x^2y′-y=0$. The boundary conditions are $y(0)=y(1)=1$. Since $a(x)<0$ the boundary layer is at $x=1$. The outer ...
maria1991's user avatar
1 vote
1 answer
119 views

Boundary layer type with initial value problem

Consider the initial value problem $\sqrt{\epsilon} \, u'' + u' - u = e^{2t}$ , with $u(0)=1$, and $u'(0)=1/\sqrt{\epsilon}$. I am trying to use a matched asymptotic expansion to find the leading ...
user569959's user avatar
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1 answer
201 views

Conditions necessary for a boundary layer to exist

Determine values of $a$ for which the problem: $\epsilon y^{''} + y^{'}+ae^y=0,$ $ y(0)=y(1)=0$ has a solution with a boundary layer structure. I am familiar with the procedure for tackling this ...
John Simpleton's user avatar
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2 answers
2k views

Navier Stokes Equation

The general Navier Stokes Equation is $\dfrac{D\vec{v}}{D t}= \dfrac{d\vec{v}}{d t}+ \vec{v} .\nabla \vec{v} = \vec{g} - \dfrac{1}{\rho} \nabla p + \nu \nabla^2 \vec{v}$ The above equation can be ...
Raptor's user avatar
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2 votes
0 answers
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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) = ...
Xiao's user avatar
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2 votes
1 answer
190 views

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 ...
SPIL's user avatar
  • 131
2 votes
1 answer
248 views

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) ...
glowstonetrees's user avatar
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Asymptotic inner expansion

Not sure if I should directly ask in the comment section of that thread or start a new thread like I have now. But anyway, I happen to come across this thread on MSE: Asymptotic Inner and Outer ...
glowstonetrees's user avatar
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1 answer
72 views

values for which there is a boundary layer for BVP

Consider $$ \epsilon y'' + x^{\alpha} y' + y = 0 , \; \; \; \; \epsilon \to 0^+ $$ with $y(0)=y(1)=1$. For what value of $\alpha$ there is boundary layer at $x=0$? What is the thickness of the ...
user avatar
0 votes
2 answers
159 views

how to find the thickness of the boundary layer near $0$? [closed]

Given the BVP $$ \begin{cases} \varepsilon y'' + \sqrt{x} y' - y =0 \\ y(0)=0 \\ y(1) = \mathrm{e}^2 \end{cases} $$ How do we find the thickness of the boundary layer near $x = 0$ ?
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1 answer
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Uniformly valid solution to boundary layer problem

If there is a boundary layer at $x=0$ and I have found the outer solutions $y^{left}_{out}$ and $y^{right}_{out}$, and the inner solution $y_{in}$. Than how can I put them together to get a uniformly ...
SamC's user avatar
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3 votes
1 answer
256 views

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's user avatar
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2 votes
2 answers
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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 $$\epsilon y''+y y'-y=0, 0<x<1$$ where $y(0)=0, y(1)=3$. My question is how do we know the boundary ...
SamC's user avatar
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