Stokes Definition and 268 Threads

Hi, If the viscosity in the NS equations was a variable, what extra equation is used to solve the NS equations? Thanks.

If we consider a sphere oscillates in viscous fluid with frequency w, then sphere has velocity u=u_0*e^{-iwt} In Laudau's book, he defined the velocity of fluid is: v=e^{iwt}*F where F is a vector with only spatial variable involved. The boundary condition then becomes u=v at |x|=R, where R is...

I was wondering, how you break down dS to something with dA? I know that dS is equal to ndS. The n is equal to grad f / (magnitude of grad f) and the dS is equal to the same as the magnitude of grad f right? So is the formula the same as double integral of region D (curl F * grad f) dA?

Homework Statement for the vector field E=x(xy)-y(x^2 +2y^2) find E.dl along the contour find (nabla)xE along the surface x=0 and x=1 y=0 and y=1 Homework Equations The Attempt at a Solution i tried the second question (nabla)xE over the surface by finding the...

Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...

this Q want to check Stokes' theorem ? for http://latex.codecogs.com/gif.latex?F=(x^2,xy,-z^2) and surface http://latex.codecogs.com/gif.latex?x^2+y^2+z^2=1 and http://latex.codecogs.com/gif.latex?z\geqslant%200 i should equal http://latex.codecogs.com/gif.latex?\oint%20Mdx+ndy+pdz with...

So I'm going over Rudin's chapter on differential forms in his Principles of Mathematical Analysis and I'm looking at Example 10.36 which gives the 1 form \eta = \frac{xdy-ydx}{x^2+y^2} on the set \mathbb{R}^2-{0} and then parametrizes the circle \gamma(t)=(rcos(t),rsin(t)) for fixed r>0 and...

Not really a homework problem, just me wondering about this: why is there a problem here? Say you want to use the divergence theorem in conjunction with Stokes' theorem. So, from Stokes' you know: Line integral (F*T ds)= Surface integral (curl(F)*n)dS. And you know that Surface...

I don't think these three: {Determinism, Stokes' Theorem, Relativity Theory}, are compatible. The notion of determinism, as applied to spacetime physics, means that if we know everything on an R3 spacelike hypersurface at time ta, we can predict what will be will be the state of things on an...

Given c(t) = [cos t, sin t, 2 + sin (t/2)] where t \epsilon [0, 2pi] and F(x,y,z) = (2-y + x2, x + sin y, \sqrt{}z4+1) --- Find \intF.dS over c(0, 2pi). I've no idea how to do this... any help would be awesome! Thanks!

Homework Statement An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates. The two plates move in opposite directions with constant velocities U1 and U2. The pressure gradient in the x-direction is zero and the only body force is due to the fluid weight. Use...

Homework Statement See figure attached for problem statement Homework Equations The Attempt at a Solution See figure attached for my attempt. I found this problem to be a little long and drawn out, which leads me to believe I made a mistake somewhere. Is this the case? Or...

Homework Statement Use Stoke's theorem to evaluate the line integral \oint y^{3}zdx - x^{3}zdy + 4dz where C is the curve of intersection of the paraboloid z = 2 + x^{2} + y^{2} and the plane z=5, directed clockwise as viewed from the point (0,0,7). Homework Equations The...

Hey!, I was repeating for myself a course I had from a earlier year, fluid mechanics. I looked at the derivation of the navier stokes equations, and there is one term that does not give meaning to me. Take a look at the x-momentum equation here...

Homework Statement The flow due to translation of a sphere in a Newtonian fluid at rest is given by the following streamfunction, ψ(r,Θ) = (1/4)Ua2(3r/a - a/r)sin2Θ which consists of a stokeslet and a potential dipole. If the contribution of the dipole is less than 1%...

I'm having trouble using Stokes' theorem in order to find a simple formula for the area found within a two dimensional simplex. I know the formula, but I'm interested in the derivation. For simplicity, I've been working with a unit triangle with vertices at the coordinates (0,0), (0,1), and...

Let's assume that I have a surface defined parametrically by a vector \mathbf{\ r}(r,\theta) Is it acceptable to simplify the Stokes theorum surface integral to: \iint\limits_D\,\nabla \times f \cdot\!(r_r\times\!r_\theta) \,\, \!r \mathrm{d}r\,\mathrm{d}\theta Where r_r and r_theta are...

Homework Statement From Calculus:Concepts and Contexts 4th Edition by James Stewart. Pg.965 #13 Verify that Stokes' Theorem is true for the given vector field F and surface S F(x,y,z)= -yi+xj-2k S is the cone z^2= x^2+y^2, o<=z<=4, oriented downwards Homework Equations The Attempt at a...

Hi, I am doing physics coursework on finding viscosity of fluids by dropping a marble into fluids, finding terminal velocity, then using stoke's law to find viscosity. (using density of fluid, sphere, sphere diameter etc). I have completed all the practical, now just the write up However ... I...

Homework Statement Derive an expression for frictional force acting on a spherical objects of radius R moving with velocity V in a continuous viscous fluid of fluid's viscosity η . Homework Equations please do not use dimension analysis to prove. The Attempt at a Solution

Do you agree that the following identity is true: \int_S (\nabla_\mu X^\mu) \Omega = \int_{\partial S} X \invneg \lrcorner \Omega where \Omega is volume form and X\invneg \lrcorner \Omega is contraction of volume form with vector X.

I find in a homework an alternative Stokes theorem tha i wasn't knew before. I wold like to know that it is really true. Any can give me a proof please? It is: Int (line) dℓ′× A = Int (surface)dS′×∇′× A

I am not able to understand the following situations. In stokes' experiment a tiny lead shot falls freely under gravity in a highly viscous column of liquid. When the viscous force becomes equal to the net weight of the lead shot it is said that the shot moves down with a constant...

Homework Statement Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...

Here's where I try to explain Stokes' theorem in my own words and you tell me if I'm right / what I need to clarify on. Essentially, it's a method to compute a line integral around a closed curve in three dimensions, with a given vector field F, without having to parametrize this field and...

Homework Statement Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...

Hi everyone, I have to prove this problem but I have no idea how to approach this problem. I tried something but it seems not working... Suppose F is a vector field in R3 whose components have continuous partial derivatives. (So F satisfies the hypotheses of Stoke's Theorem.) (a)...

Homework Statement Verify Stokes Theorem ∬(∇xF).N dA where surface S is the paraboloid z = 0.5(x^2 + y^2) bound by the plane z=2, Cis its boundary, and the vector field F = 3yi - xzj + yzk. The Attempt at a Solution I had found (∇xF) = (z+x)i + (-z-3)k r = [u, v, 0.5(u^2 + v^2)]...

Homework Statement magnetic field is azimuthal B(r) = B(p,z) \phi current density J(r) = Jp(p,z) p + Jz(p,z) z = p*exp[-p] p + (p-2)*z*exp[-p] z use stokes theorem to find B-filed induced by current everywhere in space Homework Equations stokes -...

Homework Statement Evaluate \int\int Curl F\cdot dS where F=<z,x,y> (NOTE: the vector in my post preview is showing me the wrong one despite me trying to correct it, the right one is F=<z,x,y>) and S is the surface z=2-\sqrt{x^2 +y^2} above z=0. Homework Equations I used Stokes'...

Homework Statement Evaluate the line integral I = (x2z + yzexy) dx + xzexy dy + exy dz where C is the arc of the ellipse r(t) = (cost,sint,2−sint) for 0 <= t <= PI. [Hint: Do not compute this integral directly. Find a suitable surface S such that C is a part of the boundary ∂S and use...

Homework Statement \nabla \times f \vec{v} = f (\nabla \times \vec{v}) + ( \nabla f) \times \vec{v} Use with Stoke's theorem \oint _C \vec{A} . \vec{dr} = \int \int _S (\nabla \times \vec{A}) . \vec{dS} to show that \oint _c f \vec{dr} = \int \int _S \vec{dS} \times \nabla...

Homework Statement Use stokes' theorem to find I = \int\int (\nabla x F) n dS where D is the part of the sphere x^2 + y^2 + (z-2)^2 = 8 that lies above the xy plane, and F=ycos(3xz^2)i + x^3e^[-yz]j - e^[zsinxy]kAttempt at solution: I want to use the line integral \int F dr to solve this. I...

Hi, I'm working on a calculation of flow through a rectangular duct and I'm assuming I'm in the Stokes' flow regime (Re<<1), but I also want to experiment on this system and I was wondering if anyone knows until what Re-number Stokes' flow is still a good approximation (and how good, i.e. in...

One of Maxwell's equations says that \nabla\cdot\vec{B}{=0} where B is any magnetic field. Then using the divergence theore, we find \int\int_S \vec{B}\cdot\hat{n}dS=\int\int\int_V \nabla\cdot\vec{B}dV=0 . Because B has zero divergence, there must exist a vector function, say A...

Hi every one, I am having a few problems with some research I am doing. I put this in the PDE section as it seams related, but it is for a specific application and I am not sure that it wouldn't be better suited to the mechanical engineering section. I am wanting to find the pressure...

Hello! The incompressible Navier Stokes equation consists of the two equations and Why can't i insert the 2nd one into the first one so that the advection term drops out?! \nabla\cdotv = v\cdot\nabla = 0 => (v\cdot\nabla)\cdotv = 0

Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...

Homework Statement Note: the bullets in the equations are dot products, the X are cross products Evaluate: [over curve c]\oint( F \bullet dr ) where F = < exp(x^2), x + sin(y^2) , z> and C is the curve formed by the intersection of the cone: z = \sqrt{(x^2 + y^2)} and the...

Homework Statement Let: \vec{F}(x,y,z) = (2z^{2},6x,0), and S be the square: 0\leq x\leq1, 0\leq y\leq1, z=1. a) Evaluate the surface integral (directly): \int\int_{S}(curl \vec{F})\cdot\vec{n} dA b) Apply Stokes' Theorem and determine the integral by evaluating the corresponding...

Hello! :smile: I am going over an example in my fluid mechanics text and I am confused about a few lines. My question is more about the math then the fluid mechanics. In fact, I doubt you need to understand the FM at all; if you understand Diff eqs, you can probably answer my question. I am...

Homework Statement Suppose we want to verify Stokes' theorem for a vector field F = <y, -x, 2z + 3> (in cartesian basis vectors), where the surface is the hemispherical cap +sqrt(a^2 - x^2 - y^2) The Attempt at a Solution Why is it that if I substitute spherical coordinates x =...

Homework Statement 1. A fly ash (ρ =1.8 g/mL) aerosol consists of particles averaging 13 μm in diameter and with a concentration of 800μg/m3. Use the average diameter to calculate the settling velocity (cm/s) and settling rate (μg/m/s) of the particles in air. The Stokes-Cunningham slip...

Its a while since I've done any motion calcs so I'm after some guidance. I am vertically dropping a range of materials (size 8-20 mm) into a horizontal air stream in a pipe (pipe diammeter d , ~0.3m) The horizontal air velocity in the pipe is 10 m/s The particle bulk density ranges from...

https://nrich.maths.org/discus/messages/27/147417.jpg For the above problem, I simply take the curl of F and then take the cross product of it with the normal to the plane and integrate the whole thing with respect to the surface bounded by the plane. Now, my solution is as followed with...

Homework Statement A vector field A is in cylindrical coordinates is given. A circle S of radius ρ is defined. The line integral \intA∙dl and the surface integral \int∇×A.dS are different. Homework Equations Field: A = ρcos(φ/2)uρ+ρ2 sin(φ/4) uφ+(1+z)uz (1) The Attempt at...

Homework Statement Through Stokes' Theorem, I am given a formula and vector (see attached document), where V is a vector, and S is the right-circular cylinder (including the endcaps) which is bounded by (x^2) + (y^2) = 9, z=0, and z = 5. Homework Equations See attached document...

i am trying to solve this problem which states that J(p) = (I/pi) p^2 e^-p^2 in z direction is the current density flowing in the vicinity of insulating wire. pi = pie in standard spherical polar coordinates. J is the current density. I need to prove that the total current...

Homework Statement Use Stokes' theorem to show that \oint\ \hat{t}*ds = 0 Integration is done closed curve C and \hat{t} is a unit tangent vector to the curve C Homework Equations Stokes' theorem \oint F* \hat{t}*ds = \int\int \hat{n}*curl(F)*ds The Attempt at a Solution Ok...

For stokes theorem, can someone tell me why \hat{a} \bullet \vec{ds} = ds? My notes say it's because they are parallel, but I'm not sure what that means. Also to get things clear, Stokes theorem is the generalized equation of Green's theorem. The purpose of Stokes theorem is to provide a...