Boundary Definition and 1000 Threads

The equation is Uxx + Uyy = 0 And domain of solution is 0 < x < a, 0 < y < b Boundary conditions: Ux(0,y) = Ux(a,y) = 0 U(x,0) = 1 U(x,b) = 2 What I've done is that I did separation of variables: U(x,y)=X(x)Y(y) Plugging into the equation gives: X''Y + XY'' = 0 Rearranging: X''/X = -Y''/Y = k...

what does the relation between the temperature gradient inside the thermal boundary and thermal boundary layer thickness i mean what will be the temperature gradient ( high or low) when the thermal boundary layer is thick relative to the thin one? Kindly explain mathematically and physically as...

This isn't homework but could be labeled "textbook style" so I'm posting it here. Homework Statement I'm trying to solve \frac{\partial^2 u} {\partial x^2} +\frac{\partial^2 u} {\partial y^2}=0 on the domain x \in [-\infty,\infty], y\in[0,1] with the following mixed boundary conditions...

given the generalized SL conditions Let's say psi_m and psi_n are eigenfunctions of the given y. Its Wronskian is 0 because otherwise the boundary condition doesn't make sense much. However, I wonder if it is possible to have, S={ x | W[psi_m(x) , psi_n(x)] =/= 0 } otherwise...

hi, I m designing a current transformer in ansys. initially i designed a 3D toroidal (split) core. i wanted to get the B-H curves by varying different permabilities of core with different frequencies.the current injection is through the conductor which is in centre of core. but i am facing a...

Hi, PF! Recently, while reading chapter 6 of Incropera's Fundamentals of Heat and Mass Transfer I got into a confusion regarding the velocity boundary layer. The book first states that, as the flow becomes more turbulent, the boundary layer gets thicker, as indicated by both figures attached at...

Is the Blasius solution valid for internal flow in an underdeveloped pipe? Can the function for the boundary layer thickness be used to calculate when the boundary layers will meet in the center of the pipe? I don't see how it could be valid since the assumption in its derivation is that...

I'm modelling a system with a nanosized semiconductor in 1d, inside which I want to find the electrostatic potential. Having found this I am unsure what boundary conditions to put on this, when it is connected to a metal on one side and to vacuum on the other. So far I have put that it is...

Hi there, I'm solving the equation for the transverse vibrations of a Euler-Bernoulli beam fixed at both ends and subject to axial loading. It's a similar problem to that described by Rao on page 355 of his book "Vibration of Continuous Systems" (Google books link), except the example he uses...

Homework Statement Consider a gas (note: treat as ideal) that has phase coexistence between diatomic and monatomic forms at ##T_0## and ##P_0##. Compute the equation for the P,T phase boundary between monatomic and diatomic gases. Homework Equations ## u_v (V,T) = \frac{5}{2} RT -...

As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω. How about if the function is just discontinuous at the boundary? Actually, this function is an electric field. So its tangential component is naturally continuous, but the...

The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##. I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##. The answer (given in eqns 6&7 of...

There is something I have been wondering about with refraction. There have been many explanations of why the light bends. However, it still does not feel intuitive. The question I have is with how light enters the clear object. Is it proven that light indeed enters at an angle that is instantly...

Hello. I'm trying to wrap my head around how Lagrangians work in classical field theory. I have a book that is talking about the gauge invariance of the Lagrangian: \mathscr{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^\mu A_\mu. It shows that we can replace A^\mu with A^\mu+\partial^\mu\chi for...

http://local.eleceng.uct.ac.za/courses/EEE3055F/lecture_notes/2011_old/eee3055f_Ch4_2up.pdf (Page 4.4 )I am having a trouble with understanding why closed loop line integration is 0 at dielectric boundary. As far as I know, closed loop line integration is 0 because electric field is...

Hello everyone. I posted this question in another forum and got no answers so I'll try and re-post it question here. I need to deliver a correct answer to the cited question below to my course responsible teacher. Below is also my own solution and thoughts on the problem but I don't know if I am...

Firstly, my main question boils down to speaking about the initial conditions and boundary conditions. I was given: $$ u(0,y,t) = u(\pi,y,t) = u(x,0,t) = u(x,\pi,t) = 0 $$ but then the initial condition was: $$ u(x,y,0) = 1 $$ Aren't the initial and boundary conditions inconsistent in such...

Hi. I am new here so I please let me know if I should post this in another forum. I have been struggling for a while with the following homework problem: "State the boundary layer equations for incompressible flow over a solid, weakly curved boundary of a Newtonian fluid. What approximations...

The Euler-Lagrange equations give a necessary condition for the action be extremal given some lagrangian which depends on some function to be varied over. The basic form assumes fixed endpoints for the function to be varied over, but we can extend to cases in which one or both endpoints are free...

I was reading this: http://www.creatis.insa-lyon.fr/~dsarrut/bib/Archive/others/phys/www.mas.ncl.ac.uk/%257Esbrooks/book/nish.mit.edu/2006/Textbook/Nodes/chap06/node29.html Under the first figure it states "Figure 6.20: The boundary layer at a stagnation point on an airfoil has a constant...

I am trying to solve the differential equation ##\frac{d^{2}y}{dr^2}+(\frac{1}{r}+1)y=0## with the boundary conditions ##y(r) \rightarrow r \frac{dy}{dr}(0)## as ##r \rightarrow 0## and ##y(r) \rightarrow \sin(kr+\delta)## as ##r \rightarrow \infty##. I know that the shooting method is the...

Hi! I can't understand how to implement boundary conditions in a 2D axisymmetrical model. How should be the value of pressure, x-velocity and y-velocity at the axis of symmetry? Thank you!

I have come across the paper attached in which a 1D fluid piston is modeled. I have question on the boundary conditions (BCs) of the system. Essentially, the problem consists of a fluid chamber in contact with a spring (a mass -spring system). ALE is used to move the mesh. I am not certain...

I want to model the diffusion-controlled combustion of a small carbon particle. The system I want to model is similar to this one However, I'm not going to use the stagnant gas film model as shown in the figure, since I lack data for the film thickness, and I want to evaluate the problem...

As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω. How about if the function is just discontinuous at the boundary? For example, I intend to make a volume integration of a product of electric fields, the field function is...

Homework Statement Consider the boundary value problem \begin{equation} u''(t)=-4u+3sin(t),u(0)=1,u(2)=2sin(4)+sin(2)+cos(4) \end{equation} Homework Equations Derive the linear system that arise when discretizating this problem using \begin{equation} u''(t)=\frac{u(t-h)-2u(t)+u(t+h)}{h^2}...

Hello, Can anyone help me find the boundary conditions of the below given beam please. Its a clamped-free beam but the overhanging sectiona and the mass makes it confusing. Actually I am puzzled about finding the initial conditions.

Hi, so I have this question: A wave pulse on a string has the dimensions shown in the figure (Figure 1) at t = 0. The wave speed is 40 cm/s. a) Draw the total wave on the string at t=15ms, 20ms 25ms, 30ms, 35ms, 40ms and 45ms. b) repeat part (a) for the case in which the end of the string is...

Hello, I am simulating an experiment I did in the lab where we had air flow over a tray of water to determine the mass transfer coefficient scaling with velocity, as well as boundary layer thickness scaling with velocity. Now I am using COMSOL to simulate the experiment, and here is the...

Beginning with the Schrodinger equation for N particles in one dimension interacting via a δ-function potential ##(-\sum_{1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(x_i-x_j))\psi=E\psi## The boundary condition equivalent to the ##\delta## function potential is...

Hello, I'm doing an experiment where I will be blowing warm air parallel to a stagnant water surface, and I will investigate the scaling of air velocity with mass transfer coefficient. I am trying to find some kind of scaling of the concentration boundary layer thickness with air velocity, and...

Hello! (Wave)Consider the boundary value problem $\left\{\begin{matrix} - \epsilon u''+u'=1 &, x \in [0,1] \\ u(0)=u(1)=0 & \end{matrix}\right.$ where $\epsilon$ is a positive given constant. I have to express a finite difference method for its numerical solution. How can we know whether it...

Hello, PF! Recently, while reading chapter 10 (microscopic energy balances) of the second edition of BSL, I found a minor discrepancy which is confusing me, especially when considering the mathematical analogies of heat and mass transfer. In section 10.1, the authors introduce Newton's law of...

How to find value of integration constant?I know with the help of boundary conditions,but How boundary conditions help in finding integration constant?

Homework Statement Under what condition on the constant ##c## and ##c'## are the boundary conditions ##f(b) = cf(a)## and ##f'(b)=c'f'(a)## self-adjoint for the operator ##L(f) = (rf')'+pf## on ##[a,b]##? (Assume that ##r,p## are real.) Homework Equations The boundary conditions are...

Hi, just want to confirm that with the eigenfunction boundary condition $ p(x) v^*(x)u'(x)|_{x=a} = 0 $, the order of (solutions) v, u doesn't matter? I ask because a problem like this had one solution = a constant, so making that the u solution makes $ p(x) v^*(x)u'(x) = 0 $ no matter the...

I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is $$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4...

I've forgotten a lot of field theory so I've been rereading it in a couple of electric field theory textbooks. What seems like a simple problem falls between the cracks. I hope some readers can help - it will be appreciated. My application seems simple (solution will require numerical FEA but...

I am working with a fixed fixed bar with a distributed axial load to the right as w(x)=CX/L. I am having a hard time determining the force boundary conditions. I know that U(0)=0 and U(L)=0. However, I need to come up with something in regards to U'(Value). Any help would be appreciated.

There is a general topic of boundary constructions, which means how to adjoin idealized points in a sensible way to a given spacetime. There is a menagerie of these methods, including the g-boundary (Geroch), b-boundary (Schmidt), c-boundary (Geroch, Kronheimer, and Penrose) and a-boundary...

Hi, I have two coupled differential equations d^2 phi(z)/dz^2=lambda*phi(z)*(phi(z)^2+psi(z)^2-sigma^2) d^2 psi(z)/dz^2=lambda*psi(z)*(phi(z)^2+psi(z)^2-sigma^2+epsilon/lambda) where lambda, epsilon and sigma are arbitrary constants. The equation subject to the bellow boundary conditions...

I'm upgrading a non-PFC (power factor corrected) offline power supply design to include PFC for European deployment. The total output power is less than 25W, and the two output windings are around 20V. I'm familiar with boost-flyback topologies for isolated PFC supplies, but that seems to...

Consider the heat equation dT/dt - aΔT + v⋅∇T = S where S is a source term dependent of the radiation intensity I and the temperature T. The fluid velocity v is prescribed. We also consider the radiative transfer equation describing the radiative intensity I(x,ω,t) where ω is the ray direction...

Somehow I can't relate two things and confused over this. What I understand when someone say that some spacetime has conformal boundary it means that I can write the metric conformally to some other metric where the coordinates are finite ..So it has boundary. Now I just read something on Ads...

Hey this isn't so much a homework problem but one I have just had an exam over. I have absolutely no idea how to calculate it and in all past papers/tutorial questions and the notes, makes no mention of the sort of problem. I'm not bothered over the exact answer, just how you go about it...

Lumley, turbulence textbook on boundary layers, introduction pages: "The turbulent eddies transfer momentum deficit away from the surface". Can anyone explain what this means, specifically what is momentum deficit? In my mind, the word "deficit" means a shortage of something, so how can one...

Hi PF! I'm using a finite differencing scheme to solve the following $$h_t = h h_{zz} + 2h_z^2$$ where the subscripts denote partial derivatives. The difficulty I'm facing is the boundary conditions are dynamic, and move with time ##t##. This makes choosing a ##\Delta z## very difficult and...

I'm trying to work up some examples to help me understand this concept. Would the periodic flow on a solid torus be transverse to it's boundary?

Hey! :o Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\partial{u}}{\partial{y}}\right ) -(1+x^2+y^4)u=f(x,y), x^2+y^2<1 \\ u(x, y)=g(x,y)...

Solve the boundary value problem: $\left\{ \begin{array}{lcl} y''&=&0,\hspace{1.0mm} 1<x<2\\ y(1)&=&0\\ y(3)+y'(3)&=&0 \end{array} \right. $ For the problem, I first calculate the eigenvalues and after check the roots and finally find the eigenvectors. Is correct this? Help me please :).