In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. The liquid or gas in the boundary layer tends to cling to the surface.
The boundary layer around a human is heated by the human, so it is warmer than the surrounding air. A breeze disrupts the boundary layer, and hair and clothing protect it, making the human feel cooler or warmer. On an aircraft wing, the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow. In the Earth's atmosphere, the atmospheric boundary layer is the air layer (~ 1 km) near the ground. It is affected by the surface; day-night heat flows caused by the sun heating the ground, moisture, or momentum transfer to or from the surface.
Homework Statement
Determine all the solutions, if any, to the given boundary value problem by first finding a general solution to the differential equation:
y" + y = 0 ; 0<x<2π
y(0)=0 , y(2π)=1
The attempt at a solution
So the general solution is given by: y = c1sin(x) +...
Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<
Homework Statement
How do you I calculate the Laminar Boundary Layer Y in meters with Blasius Equation?
I have an expression for U (m/s) and u/U and u(m/s) and also eta=0.1, 0.2...5.2. I am wondering how one could calculate the y laminar boundary layer (m), the y buffer boundary layer (m)...
Hi, All:
Let S g,2 be the orientable genus-g surface with two boundary components, and let C be a
simple-closed curve in S g,2 .
If C is homologically non-trivial (i.e., C does not bound a subsurface of Sg,2), and C
intersects one of the boundary components
, must C also...
suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression
\int\nablag(r)dv=M\int\hat{n}\delta(r-rs)dv=M\hat{n}\intd\deltav
where...
Hello everyone and greetings from my internship!
It's weekend and I'm struggling with my numerical solution of a 1+1 wave equation.
Now, since I'm eventually going to simulate a black hole ( :D ) I need a one-side open grid - using advection equation as my boundary condition on the end of my...
Hi, I was reading Lemon's Perfect Form and it talked about "natural boundary conditions". But I don't understand exactly how one determines them. It seems to me that one imposes some random condition then deduce stuff from it...?!
Advanced thanks for any enlightenment!
I am trying to set up for an experiment, and I need to know the time dependence of the temperature of the front surface of an assembly of plates. The assembly has a heater on one side and is exposed to a gaseous environment (of constant known pressure and temperature) on the other. I am...
Homework Statement
I have been self studying Spivak's Calculus on Manifolds, and in chapter 1, section 2 (Subsets of Euclidean Space) there's a problem in which you have to find the interior, exterior and boundary points of the set
U=\{x\in R^n : |x|\leq 1\}.
While it is evident that...
Suppose I have a mass M_0 (here denoted with lowercase zero because of previous discussions on relativistic mass), and I have a gravitational field \phi which can under make a shift of 180^o between a negative plane and a positive plane. Assume also that the mass is considered as a charge...
Does anyone know any good references for a moving boundary problem? I am looking at a cylinder of charge being injected into a fluid, the PDE is:
-\nabla^{2}\varphi +a\frac{\partial^{2}\varphi}{\partial t^{2}}+b\frac{\partial\varphi}{\partial t}=0
I want \varphi =\varphi_{0}, a constant on...
Homework Statement
n is given by:
∂2Θ/∂x2=1/α2 ∂Θ/∂t
, where Θ(x, t) is the
temperature as a function of time and position, and α2
is a constant characteristic for the
material through which the heat is flowing.
We have a plate of infinite area and thickness d that has a uniform...
I am trying to model the diffusion of fluorophores in a cell with a source in the middle by solving the appropriate differential equation. I can solve the PDE easily enough, however as I haven't done DE's in a while, I need a refresher on how to apply the appropriate boundary conditions for my...
Something has been bugging as of late: usually, derivatives (ordinary and partial) are defined for interior points. However, I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some...
I am trying to solve the following heat equation ODE:
d^2T/dr^2+1/r*dT/dr=0 (steady state) or
dT/dt=d^2T/dr^2+1/r*dT/dr (transient state)
The problem is simple: a ring with r1<r<r2, T(r1)=T1, T(r2)=T2.
I have searched the analytical solution for this kind of ODEs in polar coordinate...
Hi
I am studying magnetic vector potential from Griffiths book. The eq 5.76 in his book gives
the boundary condition for the magnetic vector potential.
\frac{\partial \vec{A_2} }{\partial n}- \frac{\partial \vec{A_1} }{\partial n}=-\mu_o \vec{K}
where n is the vector perpendicular to the...
Hey guys, I'm having a conceptual problem implementing the Crank-Nicolson scheme to a PDE with nonlinear boundary conditions.
The problem is the following:
u_t + u_{xxxx} = 0,
u(0,t) = 1,\quad u_x(0,t) = 0, \quad u_{xx}(1,t) = 0,
u_t(1,t) - u_{xxx}(1,t) = f\bigl(u(1,t)\bigr).
Taking m...
Hey guys,
I'm confused as to what happens regarding the amplitude when a pulse moves across mediums.
1. Let's say a pulse is moving from a string of low density to a string of high density:
I know that the reflected amplitude will be less than the incident's, and also the transmitted...
Homework Statement
True or false:
Let S be any set in R2. The boundary of S is the set of points contained in S which are not in the interior of S.
Homework Equations
The Attempt at a Solution
Common sense tells me true. I don't really understand it though, if S is an open set...
Homework Statement
Hi guys, I'm having trouble understanding the finite potential well, in particular the boundary conditions
The well under scrutiny has potential
V(x)= 0 for |x|<a
and
V(x)=V_0 for >a
Homework Equations
\frac{d^2\psi}{dx^2}=-\sqrt{\frac{2mE}{\hbar^2}}\psi=-\alpha^2\psi...
Homework Statement
y^{(4)}+\lambda y=0
y(0)=y'(0)=0
y(L)=y'(L)=0
Homework Equations
The hint says...
let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution
Listening to the hint, I got
r=\pm\mu With multiplicity 2 of each. So that means..
y=c_1 e^{\mu t}+c_2te^{\mu...
I wrote a program that uses the FEM to approximate a solution to the heat conduction equation. I was lazy and wanted to test it, so I only allowed Neumann boundary conditions (I will program in the Dirichlet conditions and the source terms later).
When I input low values for the heat flux, I...
I am disappointed. This take on determinism comes from the Stanford Encyclopedia:
"Determinism: The world is governed by (or is under the sway of) determinism if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law. The...
Okay, so I've read Max Born's book twice and believe I now have a reasonable, high-level understanding of general relativity. The thing I'm now trying to tease out is Mach's principle. One thought experiment that I've been falling asleep thinking about is how the presence of distant masses...
This is an example shown in "Introduction to Electrodynamics" by Griffiths. Page 226 example 5.8.
Given a sheet of current K on the xy-plane where current traveling in +ve x direction. Find the magnetic field.
I am confused on the way the book justify the z direction of B is zero.
The...
Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.).
I said: ∇•(u∇u)=u∆u+∇u2, since ∆u = 0, we have ∇•(u∇u)=∇u2
∫...
Can't seem to work this out,
any solutions would be greatly appreciated!
Thanks in advance!
Solve the boundary-value problem
Uxx + Uyy + U = 0 , 0<x<1,0<y<1
U(0,y) = 0 , Ux(a,y)= f(y)
U(x,0) = 0 , Uy(x,1)= sin(3*pi*x)
I am using Gaussian elimination to solve the airy stress function, but I am having difficulty implementing boundary conditions.
A good synopsis on the problem of identifying boundary conditions is given here (section 5.2.1):
http://solidmechanics.org/text/Chapter5_2/Chapter5_2.htm
Given that...
Greetings, everyone!
The problem below is actually a task on Numerical Methods. But I have difficulties making a mathematical model.
Homework Statement
Let us have a longitudinally homogeneous system of a pipe of radius R and a propeller of nearly the same radius inside it (we shall...
Hi,
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...
Hi,
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...
Homework Statement
Consider
\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} subject to
u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x). Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity)...
So, I do not think I did this properly, but if f(-x)=-f(x), then u(-x,0)=-u(x,0), and if g(-x)=-g(x), then ut(-x,0)=-ut(x,0).
According to D`Alambert`s formula,
u(x,t)=[f(x+t)+f(x-t)]/2 + 0.5∫g(s)ds (from x-t to x+t)
so, u(0,t)=[f(t)+f(-t)]/2 + 0.5∫g(s)ds (from -t to t)
f is odd, and so is...
I am sometimes just not sure how to go about solving magnetics problems and applying the right boundary conditions. I was hoping for a little advice.
For example in an infinitely long cylinder (along z-axis) with radius a, and a permanent magnetization given by:
\vec{M} =...
Hello,
I'm having trouble getting started on this problem. Here's the question:
[PLAIN]http://img810.imageshack.us/img810/2464/ee323assn3q3.jpg
My issue is in setting up the governing partial differential equation in 3 dimensions. What I've tried so far is setting du/dt equal to the...
I understand application of Snell's law for transition from one medium to another but I have a question regarding this model. When an electromagnetic wave transitions from air into a conductive medium does the wavelength change instantaneously as the theory seems to imply or is there a boundary...
Hi there,
I have completed my analysis in BEM. The results gives me velocity potential in form of
37954305E-04 +1.58625295E-06 +2.10275811E-07 +...
I have real scattering values, imaginary and absolute scattering values in this form.
I am new to the method and from a different...
Homework Statement
A sphere under uniform rotation R, in a simple shear flow, given at infinity by
ui = G(x2 + c)deltai1
The centre of sphere is fixed at x2
Boundary conditions are ui = EijkRjxk on sphere,
and ui = G(x2 + c) at...
Homework Statement
Find the solution of the equation
v''- 4v'+5v=0,such that v=-1 and v'=1 when x=pi=3.14159Homework Equations
...
The Attempt at a Solution
I treat it as a polynomial=>r^2+4r+5=0
=>delta=-4=>r1=2+2i and r2=2-2i
v=e^[x+2](A*cos[2]+B*i*sin[2])
v=-1=e^[pi+2](A*cos[2]+B*i*sin[2])...
Boundary conditions & time domain electromagnetic waves: does classical model fit?
Consider two propagating media: a lossy dielectric medium and a lossless dielectric medium. Thus, the interface that separates them has two tangential components of electric field, one for each medium. One of...
I don't seem to grasp the meaning of boundary conditions for Laplace's equation.
Consider the Lagendre expansion of the potential at x due to a unit charge 1/|x-x'|, where x' is the position of the unit point charge.
To do the expansion, we need to consider a volume in space where the...
Homework Statement
Solve the given boundary value problem or else show that it has no solutions: y'' + 4y = cos x, y'(0) = 0, y'(pi) = 0.
Homework Equations
N/A
The Attempt at a Solution
So I made it all the way through the problem I think, but I am not getting the correct answer...
When solving Schrodinger's eqn for a quantum ring, what would be the boundary conditions?
The solution (polar) should be
Ψ(Φ) = A exp(ikΦ) + B exp(-ikΦ)
And I believe the boundary conditions are
Ψ(0) = Ψ(2pi)
Ψ(0) = A + B
Ψ(2pi) = A exp(ik*2π) + B exp(ik*2π)
and I suppose I can...
Hi Everyone,
Apologies if this is posted in the wrong place.
I am trying to produce numerical solutions for the following initial value partial differential equations, namely the overdamped Schmoluchowski equation
\frac{\partial p(x,t)}{\partial t}=-\frac{\partial}{\partial...
For a parallel polarization EM hitting the conductor boundary in an oblique angle. z axis is perpendicular to the boundary and point into the conductor. y-axis it out of the page which give x pointing up. Let the boundary surface by xy plane. With this:
The direction of the incident is...
Standard electrostatics problem (in spherical polar coords): spherical cavity of radius R in an infinite dielectric of permittivity ε centred at origin of the coord system. Surface charge stuck on to the cavity:
\sigma(\theta) = \sigma_0 \cos (\theta)
Problem is to find the potential in...
Homework Statement
"Solve for t > 0 the one-dimensional wave equation
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
with x > 0, with the use of Fourier transformation.
The boundary condition in x = 0 is u(0,t) = 0.
Assume that the initial values u(x,0) and...
If a PDE has no boundary conditions specified, how does one go about providing a solution--even if this is a general solution?
I'm stuck looking at the separation of variables and other methods which all seem to heavily rely on those boundary conditions and initial conditions.
If anyone...