# 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.

38 questions
Filter by
Sorted by
Tagged with
55 views

### 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''+(...
1 vote
44 views

### 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 ...
• 3,935
1 vote
76 views

### 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 ...
• 1,442
118 views

1 vote
33 views

### 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 ...
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 $\varepsilon \rightarrow 0^+$, ...
• 4,915
32 views

### 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 ...
195 views

• 361
86 views

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 ...
• 1,280
1 vote
869 views

### 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 ...
• 53
1 vote
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 ...
1 vote
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 ...
• 323
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 ...
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 ...
• 109