Boundary Definition and 900 Threads

Hi, If the dirichlet boundary condition is being applied, what does it tell us?

Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at r=R ?

Hi everybody, I’m trying to calculate the shape of a boundary line f(x) between two mediums that collimates rays from a point light source. This requires the rays to hit the boundary line under a certain angle, so I calculated the slope m(φ) of the boundary line for a ray with polar angle φ (φ...

Hello! (Wave) We consider an elliptic operator $L$ in the space $\Omega$ with $c(x) \leq 0$. We suppose that $\partial{\Omega}=S_1 \cup S_2$. What can we say about the solution of the following problem? $$Lu=0 \text{ in } \Omega \\ u|_{S_1}=0...

Homework Statement I am given the following figure: These are converging rays that appear to be going to a point F convert to a plane wave upon hitting the boundary between n2 and n1, and I am asked to find the equation for the boundary between n1 and n2 that perfectly accomplishes this...

Greetings, I hope this is the right place to ask. I have been working with modeling of thermal conductance, G [W/K] of semi-conductor nanowires as a function of temperature. To start, thermal conductivity of a nanowire is modeled using BTE (Boltzmann Transport Equation). Then the conductance...

What is a homogeneous boundary condition? Or, more explicitly, what would make a boundary condition inhomogeneous Many thanks :)

Hi PF! So after scaling Navier-Stokes for a flow over a flat plate we ultimately arrive at ##f f'' + f''' = 0## subject to ##f(0)=0##, ##f'(0)=0##, and ##f'(\infty) = 1## where independent variable is ##\eta##. The source I was reading is trying to reduce this BVP to an IVP. Thus they suggest...

I was wondering how a boundary layer would be dissipative of momentum if it was under the influence of a positive heat gradient. I understand that the reason that we don't see the boundary pressure equal the stagnation pressure is that the boundary is dissipative (so excess pressure above...

When compared to laminar flows, the fluid "sticks" with the solid surface longer in case of turbulent flows. For example, the angle of separation for flow over a circular cylinder is 80 degrees for laminar flows, and 140 degrees for turbulent flows. What is the reason?

Any good places to go to get a better layman understanding of the no boundary proposal other than Hawking books? Id like to see what criticisms there are of the model, how has it evolved over time and is there any chance for experimental probes of it.

Homework Statement Uxx - SU = A ; 0<x<1 Boundary conditions : Ux(0) = 0 U(1) = 0 The Attempt at a Solution I tried to set a new variable W = u + A, I can get rid of the A in the main equation and U(1) becomes = 1. If I set U= C*esqrt(S)x into the equation, its a trivial solution because of...

Homework Statement in the formula of shear stress τ = (V)(Q) / It , Q=Ay = first moment of inertia of area, the area can be located above(or bottom) at the point of interest) when the chosen point is at the wall(boundary) , why shear stress = 0? Homework EquationsThe Attempt at a Solution When...

Hi, I was recently following an example shown in this link and just had a couple questions: http://www.scientificpython.net/pyblog/solving-the-2d-wave-equation-and-making-a-video-of-the-solution I believe I understand the steps, but was just not quite understanding the justification. In the...

What I mean by this question is the following: If, just for example, we define the surface region of a star as that where the matter undergoes a phase transition from plasma to radiation, then that boundary has an objective physical meaning (let's not bother with the fact that the transition is...

Dear all, I made a cad of floating structure (the frame only), figure attached below. It should be located in the water and moored, so it can't go anywhere. The CFD simulation was done. So I have fluid force on the structure. Now I want to do mechanical analysis of the structure by apply the...

Hello everyone, The boundary condition : P=0, z=ζ is very common when studying irrotational flows. When cast with the Bernoulli equation, it gives rise to the famous dynamic boundary conditionn, which is much more convenient : ∂tφ+½(∇φ)2+gζ=0, z=ζ But what happens if the motion is rotational ...

Can there be a bounded space without a boundary without embedding in a higher spatial dimension? This seems to be the kind of question I get stuck on when the big bang comes up. Thanks

The rate of working of the Reynolds Stress can be written as: where ui is the fluctuating velocity and Ūi is the time-averaged velocity. It is stated in the textbook that, if we integrate the above equation over a closed volume V, the divergence term on the left integrates to zero since τRij...

Hi all! I have to calculate the natural frequency of the system. Any idea of boundary conditions of this case? There is beam supported by two springs on the left side.

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have \psi(x) = A\sin(kx) + B\cos(kx) with boundary conditions \psi(x) = \psi(x+L) In the fixed boundary case, we had \psi(0) = 0 which meant B=0 and...

Hello! (Wave) I want to check if the following boundary value problem has a solution $\left\{\begin{matrix} -u_{xx}-4u=\sin {2x}, x \in (0,\pi)\\ u(0)=u(\pi)=0 \end{matrix}\right.$ I have thought the following: We consider the corresponding homogeneous equation $-u_{xx}-4u=0$. The...

How can something have a definitive edge if space can always be more granular?

In calculus of variations, extremizing functionals is usually done with Dirichlet boundary conditions. But how will the calculations go on if Neumann boundary conditions are given? Can someone give a reference where this is discussed thoroughly? I searched but found nothing! Thanks

I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0 where f(x) and g(x) are positive functions. I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this. I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...

Homework Statement Sorry for the dull question. Problem is as shown/attached Homework Equations The waves in part ii) are traveling in a HIL dielectric of permittivity ##\epsilon_{r}## from ##0 <z<d## and then hit an ideal metal boundary at ##z=d##. The Attempt at a Solution I figure this...

Homework Statement Homework Equations 3. The Attempt at a Solution [/B] I know dV=1/C∫idt and that we integrate the voltage from V to V0. What I don't get are the boundary conditions for t - How do we get what we get in the parenthesis? My closest assumption is that the t/T values refer to the...

does anyone know what the boundary condition is for a closing valve using the wave equation pde?

Suppose we are solving a diffusion equation. ##\frac{\partial}{\partial t} T = k\frac{\partial^2}{\partial x^2} T## On the domain ##0 < x < L## Subject to the conditions ##T(x,0) = f(x) ## and ##T = 0 ## at the end points. My question is: Suppose we solve this with some integration scheme...

I have a simple question about Lewkowycz and Maldacena's paper http://arxiv.org/abs/1304.4926v2'][/PLAIN] http://arxiv.org/abs/1304.4926v2 In section 2, they consider the scalar field in BTZ background ground and require boundary condition of the scalar field, $\phi \sim e^{i\tau}$ . This...

I have created a matrix with a class called Lattice. The lattice is filled with objects of type 'Dipole' which is created with another class. The problem I am having is with boundary conditions when I look for a neighbour. e.g If i pick a dipole on the top row, I want its 'above' neighbour to be...

Homework Statement Find the Green's function $G(t,\tau)$ that satisfies $$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$ under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$ Then...

I have a question about the following scenario involving a flow separation issue in a pipe expansion The angle of the expansion is 30* - doubling the diameter from 1D to 2D We can consider this flow fully developed with a Reynolds of 5000+ Associated with this expansion is a head loss...

I can't seem to find an explicit or analytical solution to a boundary value problem and thought I might ask those more knowledgeable on the subject than me. If t is an independent variable and m(t) and n(t) are two dependent variables with the following 8 constraints: a) m' =0 @T=0 and...

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...

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, 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...

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...

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...

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!