Stokes Definition and 268 Threads

hey pf! i am studying fluid mechanics and was wondering if any of you are familiar with a flow around some geometry? for example, perhaps a 2-D fluid flowing around a circle? if so please reply, as i am wondering how to model the navier-stokes equations. i'll be happy to post the equations...

hey pf! i am studying fluid mechanics and we are reviewing navier/stokes equations. we have gone over a few problems, but i could definitely use practice on more. do you all have any suggestions that include solutions, not just answers, so if I am stuck i can see how to solve? problems...

Hi, I am new to fluid dynamics and I would really appreciate some help on the subject. When a droplet of liquid (water/blood) is moving through the air in a spherical shape, assuming the only external forces are drag and gravity, what is the range of the diameter that the drop can have so...

What is the least restrictive set of conditions needed to utilize the formula ##\int\limits_{\Omega}\mathrm{d}\alpha=\int\limits_{\partial\Omega} \alpha##?

Homework Statement Use either Stokes' theorem or the divergence theorem to evaluate this integral in the easiest possible way. ∫∫V \cdotndσ over the closed surface of the tin can bounded by x2+y2=9, z = 0, z = 5, if V = 2xyi - y2j + (z + xy)k The bolded letters are vectors...

Homework Statement Calculate the line integral: F = <xz, (xy2 + 2z), (xy + z)> along the curve given by: 1) x = 0, y2 + z2 = 1, z > 0, y: -1 → 1 2) z = 0, x + y = 1, y: 1→0 3) z = 0, x-y = 1, y: 0 → -1 Homework Equations The Attempt at a Solution I don't think the...

I'm back with more questions! :approve: I'm wondering what conditions must a manifold satisfy to be able to use Stokes' Theorem. I understand that it must be orientable, but does it have to necessarily be smooth? I tried to see if it was possible to prove Cauchy's Residue Theorem and Cauchy's...

Homework Statement Let \vec{F}=<xy,5z,4y> Use Stokes' Theorem to evaluate \int_c\vec{F}\cdot d\vec{r} where C is the curve of intersection of the parabolic cylinder z=y^2-x and the circular cylinder x^2+y^2=36 Homework Equations Stokes' Theorem, which says that \int_c\vec{F}\cdot...

Homework Statement I have to control stokes theorem( I have to calculate line-and surface integral. I have a vectorfield a=(3y,xz,yz^2). And surface S is a paraboloid 2z=x^2+y^2. And it is limited by plane z=2. For line integral the line is a circle C: x^2+y^2=4 on the plane z=2. Vector n is...

Can anyone point me to a derivation of the navier stokes equations in polar? I don't see where the single derivative in theta terms are coming from in the first 2 components.

Homework Statement Use the Stokes' Theorem to show that [SIZE="4"]\intf(∇ X A) dS = \int(A X ∇f) dS + \ointf A dl Homework Equations Use vector calculus identities. Hint given : Start with the last integral in the above relation. The Attempt at a Solution To be honest, I really...

Homework Statement Verify Stokes' Theorem F(x,y,z) = (xz,-y,x2y) and S is the surface bounded by the planes x = 0, y = 0, z = 0, and 2x + y + 2z = 8, excluding the part contained in the xz-plane. Homework Equations Stokes' Theorem: ∫∫A∇xF dA = ∫dAF dR The Attempt at a Solution...

Stokes' Theorem says the curve integral of any surface S simply equals the counter-clockwise circulation around its boundary-curve C.How can this be right? Let's say you have a hemisphere surface S with centre in origo, and its shadow on the xy plane. Both surfaces will have C as their boundary...

I have been doing some serious review of fluids in order to prep for some CFD. I have been re-deriving the NS Equations in all of their various forms. Something seems to have cropped up that I have worked myself in circles about. Let's take the momentum equation in Conservative Integral form...

One of stokes relation is that r=r'. What does this mean exactly? Is the phase difference between incident beam and reflected beam on a boundary between 2 mediums of different refractive indices ∏??

Homework Statement Use Stokes' Theorem to evaluate $$\int_{\gamma} y\,dx + z\,dy + x\,dz,$$ where ##\gamma## is the suitably oriented intersection of the surfaces ##x^2 + y^2 + z^2 = a^2## and ## x + y + z = 0## The Attempt at a Solution Stokes' says that this is equal to $$\iint_S...

This is a problem from an old final exam in my Calc 3 class. My book is very bad at having examples for these types of problems, and my instructor only went over one or two. Help would be much appreciated. Homework Statement Verify that the Stokes' theorem is true for the vector field...

I am wondering why z = -sin(t) and not sin(t)

Homework Statement Hey guys, I'm having trouble finding the n in stokes theorem. For example, F(x,y,z)= z2i+2xj-y3; C is the circle x2 + y2=1 in the xy-plane with counterclockwise orientation looking down the positive z-axis. ∫∫CurlF*n I know the curl is -3y2i+2zj+2k The...

Homework Statement Let S be the surface defined by y=10 -x^2 -z^2 with y≥1, oriented with rightward-pointing normal. Let F=(2xyz+5z)i+ e^x Cos(yz) j +x^2 y k Determine ∫∫s ∇×F dS (Hint: you will need an indirect approach) Homework Equations Stokes Theorem ∫∫s ∇×F dS The...

Homework Statement I am some trouble understanding the surfaces required to integrate over in the following questions. I can tackle them, I just don't understand some terminology. Q1) A circle C is cut on the surface of the sphere ##x^2 +y^2 +z^2 = 25## by the plane ##z=3##. The direction...

Homework Statement Homework Equations I'm guessing Stoke's Theorem? However, I'm not sure how to apply it exactly.. The Attempt at a Solution Looking at Stoke's Theorem, I'm still not sure how to apply it. I'm really just lost as to where to begin; is there even a \grad F to take? I know...

Homework Statement Let X^{i} be a vector field in Minkowski space R^{4}_{1}. We define the integral of this vector over a 3-dimensional hypersurface as the integral of the 3-form X^{i}dS_{i}. where dS_{i}=\frac{1}{6}\sqrt{|g|}ε_{jkli} dx_{j}\lambda dx_{k}\lambda dx_{l}(don't know how to type...

I have a simple question but I'm putting down the whole derivation as it is relevant. There is a point that I don't understand, or seems wrong to me. This is a derivation of Group Velocity followed by simplifying(approximating it) for long wavelength waves in shallow water. This appears in a...

Hi, I'm trying to calculate some line integral with both Gauss' and Stokes' theorems, but for some strange reasons I get different results. Since the solution with Stokes' theorem seems to be somewhat easy I doubt that this question was meant to be solved by Gauss' theorem but I still want to...

Does anyone know of a program that can give a good approximation of fluid flow based on the Navier Stokes equations? I know there are FEA programs out there that do linear flow, like in pipes, but what I'm looking for is general flow, for applications that aren't constrained to a pipe. Does such...

Homework Statement Verify stokes theorem where F(xyz) = -yi+xj-2k and s is the cone z^2 = x^2 + y^2 , 0≤ Z ≤ 4 oriented downwards Homework Equations \oint_{c} F.dr = \int\int_{s} (curlF).dS The Attempt at a Solution Firstly the image of the widest part of the cone on the xy plane is the...

Homework Statement Use Stokes Theorem to evaluate the integral\oint_{C} F.dr where F(x,y,z) = e^{-x} i + e^x j + e^z k and C is the boundary of that part of the plane 2x+y+2z=2 in the first octant Homework Equations \oint_{C} F.dr = \int\int curlF . dS The Attempt at a Solution So first...

Homework Statement Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards Homework Equations The Attempt at a...

I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...

Homework Statement Prove that ## \oint_{\partial S} ||\vec{F}||^2 d\vec{F} = -\int\int_S 2 \vec{F}\times d\vec{A} ## Homework Equations Identities: ##\nabla \times (||\vec{F}||^2 \vec{k}) = 2\vec{F} \times \vec{k} ## For ##\vec{k} ## constant i.e. ## \nabla \times \vec{k} = 0 ## Stokes...

We're given x^2+2*y^2=1. so x^2=1-2y^2 now using distance formula d^2=x^2+y^2 since x^2=1-2y^2, substituting it in the distance formula we get: d^2=1-2y^2+y^2=1-y^2; differentiating and then setting the eq to 0 we get; 0=-4y or y=0. now x^2=1-2y^2=1 so x=+-1 so point having...

Homework Statement Verify Stokes' Theorem for F(x,y,z)=(3y,4z,-6x) where S is part of the paraboloid z=9-x2-y2 that lies above the xy-plane, oriented upward. Homework Equations Stokes' Theorem is ∫F*ds=∫∫scurl(F)*ds Where curl(F)=∇*F The Attempt at a Solution I got...

When the exercise tells me to calculate the flux, how do I know when I need to use each of these theorems (Green's, Stokes or Divergence)? Can anyone tell me the difference between them? I'm a LOT confused about this. If anyone knows any good material about this on internet, it'll help me a...

Homework Statement The vector field F is defined in 3-D Cartesian space as F = y(z^2−a^2)i + x(a^2− z^2)j, where i and j are unit vectors in the x and y directions respectively, and a is a real constant. Evaluate the integral  Integral:(∇ ×F)·dS, where S is the open surface of the...

Homework Statement F= xi + x3y2j + zk C is the boundary of the semi-ellipsoid z=√(4-4x2-y2) in the plane z=0 Homework Equations Stokes theorem states: ∫∫(curlF ° n)dS The Attempt at a Solution I found the curl of the F to be 3x2y2k I found that the dot product of CurlF and n =...

Homework Statement This is not actually a homework question, just a question I ran into while studying for my math final. When I am using stokes theorem: ∫∫(curlF ° n)dS I have listed in my notes from lecture that there are time when it is applicable to replace dS with an easier...

Hello, I'm trying out the escript python FEM software package which is so far rather impressive, if for no other reason than the developers have included a Stokes Flow solver. The problem I'm having, however, is that they have formulated the problem in a manner I have not encountered before...

Acording to the non-Abelian stokes thoerem http://arxiv.org/abs/math-ph/0012035 I can transform a path ordered exponential to a surface ordered one. P e\oint\tilde{A}= P e∫F where F is some twisted curvature;F=U-1FU, and U is a path dependet operator.So, I have a system where every element...

Homework Statement Homework Equations The double integral at S of the (curl of)F.n. The Attempt at a Solution We find the the normal vector to the surface S is (2/7, 6/7, -3/7), right or not? We compute the curl of F which is -2y, right? Then, we calculate the dot product and we get 6y/7...

Is a stokes four-vector like (1 1 0 0) being horizontal polarized vector can be treated as a quantum state? If the answer is yes, this state can be used to construct density matrix?

I need to solve 0=u[(d/dr)((1/r)*(d/dr)(r*Vo))] for Vo the prof gets Vo=Co*r/2+C1/r I don't get the same answer as him, does anyone know how to do this?

Hello! :smile: I am doing some review and it has occurred to me that I always confuse myself when I derive the the momentum equation in integral form. So I figure I will try to hammer through it here and ask questions as I go in order to clarify certain points. I know that there are many...

Homework Statement V.Field F(x,y,z)=<x^2 z, xy^2, z^2> where S is part of the plane x+y+z=1 inside cylinder x2 + y2 =9 Homework Equations Line integrals, Stokes Theorem, Parametrizing intersections... The Attempt at a Solution I found the answer to be 81pi/2 using stoke's theorem...

Homework Statement verify Stokes theorem for the given Surface and VECTOR FIELD x2 + y2+z2=4, z≤0 oriented by a downward normal. F=(2y-z)i+(x+y2-z)j+(4y-3x)k Homework Equations ∫∫S Δ χ F dS=∫ ∂SF.ds the triangle is supposed to be upside down. The Attempt at a Solution myΔχF =...

I recently came across the NS millennium problem and I read that uniqueness for the NS equations is unknown. I have two questions. First question, if solutions are found to be non-unique, would the NS equations have to be corrected? Second question, since uniqueness is unknown, if someone...

If the fluorescence is the re-emitting of a photon with a larger wave length due to the transition from a higher energy state to a lower energy state in the case of resonance Raman (where there aren't any virtual states) seems be equal to the fluorescence. Which differences are there?

Is it? If so, can you show how? Thanks :smile:

Hi I was reading Introduction to Fluid Mechanics by Nakayama and Boucher and I got lost in their derivation of the Navier Stokes Theorem. They basically started out with a differential of fluid with dimensions dx, dy, and b. Then they say that the force acting on it F = (F_x, F_y) is F_x...

So I'm self teaching myself Multivariable Calculus from UCBerkeley's Youtube series and an online textbook. I'm up to Stokes Theorem and I'm getting conflicting definitions. UCBerkeley Youtube series says that Stokes Theorem is defined by: \int {(Curl f)} {ds} And then the textbook says that...