What is Stokes: Definition and 293 Discussions

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

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  1. G

    Navier Stokes Equations - General Question

    (This is from the perspective of Geophysical Fluid Dynamics) In the Navier Stokes equations I am confused as to why there is both a pressure term and a gravity term. Is this pressure resulting from differences in densities and temperature differences alone? I would think that the gravity term...
  2. R

    Vector Calculus question Div and Stokes Theorem

    If you start with the two dimensional green's theorem, and you want to extend this three dimensions. F=<P,Q> Closed line integral = Surface Integral of the partials (dP/dx + dQ/dy) da seems to leads the divergence theorem, When the space is extended to three dimensions. On the...
  3. C

    Computing a Line Integral: Stokes' Thm

    Homework Statement Compute the line integral of v = 6i + yz^2j + (3y + z)k along the path (0,0,0) -> (0,1,0) -> (0,0,2) -> (0,0,0). Check your answer using Stokes' Thm Homework Equations The Attempt at a Solution I've tried breaking into three pieces. The first with dx = dz =...
  4. S

    Why Does Changing the Orientation Affect the Results in Stokes' Theorem?

    -------------------------------------------------------------------------------- I am having some issues with this problem... F=( x+y, y+z, z+x) bounded by the plane with vertices at {2,0,0},{0,2,0},{0,0,2} I need to do both sides of stokes thm and I am running into problems when I try...
  5. D

    Discovering Stokes Law to Understanding its Origins and Applicability

    Hi - Is it possible to derive Stokes law or is it an emprirical law.? http://en.wikipedia.org/wiki/Stokes'_law I was thinking of using the Navier-Stokes equations but i don't want to start out if it impossible.. Thx.
  6. M

    Calculating Circulation of Field F w/ Stokes' Theorem

    Homework Statement Use the surface integral in Stokes' theorem to calculate the circulation of field F around the curve C in the indicated direction. (3) F = (y)i + (xz)j + (x^2)k. C: Boundary of the triangle cut from the plane x+y+z=1 by the first octant, counterclockwise as seen...
  7. Simfish

    Stokes Theorem Problem: Evaluating Line Integral with Vector Field

    Homework Statement Let C be the closed curve that goes along straight lines from (0,0,0) to (1,0,1) to (1,1,1) to (0,1,0) and back to (0,0,0). Let F be the vector field F = (x^2 + y^2 + z^2) i + (xy + xz + yz) j + (x + y + z)k. Find \int F \cdot dr By Stokes Theorem, I know that I can...
  8. haushofer

    Variations, Euler-Lagrange, and Stokes

    Hi, I have some questions which I encountered during my thesis-writing, I hope some-one can help me out on this :) First, I have some problems interpreting coordinate-transformations ( "active and passive") and the derivation of the Equations of Motion. We have S = \int L(\phi...
  9. S

    Proving \int curl A.n dS = 0 w/ Stokes Theorem

    (1) using stokes theorem and cutting the surface into 2 parts how can we prove that \int curl A.n dS = 0 assume the surface "S" to be smooth and closed, and "n" is the unit outward normal as usual. (2) How can you prove \int curl A.n dS = 0 using the divergence theorem?
  10. A

    Application of Stokes' Theorem

    Homework Statement Solve the following question by using Stokes' Theorem. (Line integral on C) 2zdx + xdy + 3ydz = ? where C is the ellipse formed by z = x, x^2 + y^2 = 4. Homework Equations The Attempt at a Solution We have the vector A=(2z,x,3y) which is cont...
  11. H

    Navier Stokes, separation steady/non-steady

    Hello, I want, for obscur reasons which would lead us too far to explain, to split my flow into two component, one steady and another one non-steadyv = v_0 + v' I'm looking for a simple equation governing the evolution of this non steady components. The complete momentum equation gives...
  12. haushofer

    Understanding Stokes Theorem and is the variation of the metric a tensor?

    Hi, I have 2 little questions and hope to find some clarity here. It concerns some mathematics. 1) Is the variation of the metric again a tensor? I have the suspicion that it's not, because i would say that it doesn't transform like one. How can i get a sensible expression then for the...
  13. T

    Proving Stokes Theorem: The Intuition and Application

    I was wondering as to how to prove stokes theorem in its general and smexy form.Also what is the intuition behind it(more important) aside from the fact that its a more general form of the other theorems from vector calculus?
  14. D

    Stokes' Theorem - Limits of Integration

    Stokes' Theorem - Limits of Integration - Urgent! Please give a hand :) Homework Statement Assume the vector function \vec{A} = \hat{a}_x \left( 3x^2 y^3 \right) + \hat{a}_y \left( -x^3 y^2 \right) Evaluate \int \left( \nabla \times \vec{A} \right) \cdot d\vec{s} over the triangular...
  15. J

    Stokes' Theorem: Evaluating a Surface Integral on a Hemisphere

    Here's my problem: Take u=(x^3)+(y^3)+(z^2) and v=x+y+z and evaluate the surface integral double integral of grad(u) x grad(v) ndS where x is the cross product and between the cross product and the ndS there should be a dot product sign. The region S is the hemisphere x^2+y^2+z^2=1 with z...
  16. J

    Evolution of pressure in navier stokes

    Hello, I haven't studied PDEs much yet, but checked out what the Navier Stokes equations are. I think I understood meaning of the terms in Navier Stokes equations, and what is their purpose in defining the time evolution of velocity of the fluid, but I couldn't see any conditions for the...
  17. J

    Stokes Theorem for Surface S: Parametrization, Flux and Integral

    For the surface S (helicoid or spiral ramp) swept out by the line segment joining the point (2t, cost, sint) to (2t,0,0) where 0 is less than or equal to t less than or equal to pi. (a) Find a parametrisation for this surface S and of the boundary A of this surface. I can only guess that...
  18. S

    Using Navier Stokes for rigid body with constant angular speed

    I've seen several examples of using Navier Stokes in a rotating container where gravity is purely in the Z direction. These solutions generally used cylindircal coordinate systems. I wanted to attempt this problem where the gravity vector does not point purely in the Z direction. (ie...
  19. I

    Fundamental Theorems for Vector Fields

    Please check my work for the following problem: Homework Statement By subsituting A(r) = c \phi(r) in Gauss's and Stokes theorems, where c is an arbitrary constant vector, find these two other "fundamental theorems": a) \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds b) - \int_{S} \nabla...
  20. M

    Exploring Circulation and Flux with Stokes' Theorem

    Just a couple quick conceptual questions about Stokes' Theorem (maybe this belongs in the non-homework math forum?). Does Stokes' theorem say anything about circulation in a field for which the curl is zero? I would think that all it says is that there is no net circulation. Also, if F is a...
  21. T

    What Steps Are Involved in Deriving Stokes Law?

    Would anybody derive stokes law for me or show me how to do it?
  22. S

    Test Stokes' Theorem for the function

    Test Stokes' Theorem for the function \vec{v} = (xy) \hat{x} + (2yz) \hat{y} + (3yz) \hat{z} / for the triangular shaded region \int_{S} (\grad \times v) \dot da = \oint_{P} v \bullet dl for the left hand side \int_{0}^{2} \int_{0}^{2} (-2y \hat{x} - 3z \hat{y} - x \hat{z})...
  23. S

    Help stokes theorem - integral problem

    Hello all, http://img244.imageshack.us/img244/218/picture8ce5.png I am completely new to this stokes theorem bussiness..what i have got so far is the nabla x F part, but i am unsure of how to find N (the unit normal field i think its called). any suggestions people? i get that nabla...
  24. D

    Solving Stokes Theorem Problem: F(x,y,z)

    Hi, i can't seem to figure out how stokes theorem works. I've run through a lot of examples but i still am not having any luck. Anyway, some advice on a particular problem would be greatly appreciated. The problem is: F(x,y,z) = <2y,3z,-2x>. The surface is the part of the unit...
  25. U

    Solving Vector Field Problem: Computing Curl F and Finding Potential Function f

    For the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f. F(x,y,z)=-3xi-2yj+k f(x,y,z)=? I'm not sure what the problem is asking. calcualting curl F needs integration and a boundary. I don't know why they ask for...
  26. D

    Who discovered the general case of Stokes' Theorem?

    I know that th three dimensional case was discovered by William Thompson, but who discovered the general case?
  27. F

    Stokes Thm - don't undestand this question fully

    Stokes Thm - I don't undestand this question fully Question: In Stoke's law, let v_1 = -y and v_2 = 0 to show that the area of S equals the line integral -\int_C y\,\,\,dx . Find the area of an ellipse ( x = a \cos t , y = b \sin t , x^2/a^2+y^2/b^2 = 1 , 0 \leq t \leq 2\pi ). It's...
  28. I

    Intensity of stokes and anti-stokes lines?

    According to Raman effect, the intensity is directly proportional to the 4th power of the wavelength. Then how come stokes lines, which have higher wavelengths than anti-stokes lines, are more intense than the anti-stokes lines?
  29. S

    Stokes Law & Parachutes: Explained

    How is stokes law related to parachutes?
  30. S

    Can Navier Stokes equations explain pressure on a stationary body's surface?

    Consider a stationary body within the flow of some fluid. I want to calculate pressure on the surface of the body. From the Navier Stokes (incompressible, stationary, no volume forces) equations, you would get something like: dp/dx=-rho(u du/dx+v du/dy+w du/dz)+eta(d²u/dx²+d²u/dy²+d²u/dz²)...
  31. M

    Solving Stokes' Theorem: Find \int_{\partial S} F \cdot ds

    Here is the problem: S is the ellipsoid x^2+y^2+2z^2=10 and F is a vector field F=(sin(xy),e^x,-yz) Find: \int \int_S ( \nabla \mbox {x} F) \cdot dS So, I know that Stokes' Theorem states that: \int \int_S ( \nabla \mbox {x} F) \cdot dS = \int_{\partial S} F \cdot ds where...
  32. Reshma

    Stokes' theorem over a circular path

    I need complete assistance on this :-) Check the Stokes' theorem using the function \vec v =ay\hat x + bx\hat y (a and b are constants) for the circular path of radius R, centered at the origin of the xy plane. As usual Stokes' theorem suggests: \int_s {(\nabla\times \vec v).d\vec a =...
  33. Reshma

    Stokes' theorem over a tetrahedron

    Check the Stokes' theorem for the function \vec v = y\hat z Here it is over a tetrahedron. Stokes' theorem suggests: \int_s {(\nabla\times \vec v).d\vec a = \oint_p\vec v.d\vec r For the right hand side I computed the line integral from (a,0,0)--->(0,2a,0)--->(0,0,a)--->(a,0,0); which...
  34. S

    Compressible Navier Stokes in cylinder coordinates

    Hello, I need the Navier Stokes equations for compressible flow (Newtonian fluid would be ok) in cylinder coordinates. Can anybody help? Thanks
  35. Cyrus

    How Is Stokes Theorem Applied to Partial Derivatives?

    I can't figure this out, help me NOW! :-p Just kidding. So anyways, here's the question: Part of stokes theorem has the following in it: \frac {\partial } {\partial x} ( Q + R \frac{\partial z}{\partial y} ) Which is written as: \frac { \partial Q }{\partial x} + \frac {...
  36. W

    Using Stokes' Theorem to Show F(r) is Conservative

    Hi I've got this question that I've been stuck on a while now.. I am sure its really obvious but i can't see to get it: Q: with the help of stokes's theorem, show that F(r) is conservative provided that nabla X F = 0. nabla X F is the same as curl F? Cheers.
  37. K

    Directionality in Stokes Theorem for Volumes

    I'm not sure if this post should go here or in the Calc setion, but I figure more knowledgeable people browse this form. This question is relating to 'directionality' of doing closed loop integrals. If you have some 2D wire structure, let's image it looks like a square wave, or a square well...
  38. B

    Explanation of Stokes' Theorem

    Would anyone be willing to explain Stokes' Theorem to me? I have managed to grasp the concepts of grad, div, curl, and what the text calls "green's theorem", but I cannot seem to grasp the geometric meaning of "stokes theorem." I've been trying to put the theorem together based on the...
  39. G

    Solving Vector Field Flux with Gauss & Stokes

    Hi folks, I'm working on the following problem... Show that the flux of the vector field \nabla \times A through a closed surface is zero. Use both Gauss and Stokes. Where can I begin? Thanks...
  40. M

    Stokes theorem is easy to prove

    i thought stokes theorem (green's thm) was hard after reading it in spivak, who calls it trivial nonetheless. however lang showed it is indeed trivial in his analysis I. the same proof occurs in courant. I.e. the point is that the theorem is easy for a rectangle, where it follows...
  41. M

    When can Stokes' law be used for motion in a liquid?

    Hello, Could anyone tell me what assumptions are made about conditions in the derivation of Stokes' law ( F = 6(pi)(eta)rv )? Also, how is the Ladenburg correction for motion in a fluid derived from/related to this? I have searched high and low on the net and in libraries, but I'm not coming up...
  42. S

    Navier Stokes: Spherical form

    Hi, I'm trying to understand how to convert the cartesian form of the N-S equation to cylinderical/spherical form. Rather than re-derive the equation for spherical/cylindrical systems, I am trying to directly convert the cartesian PDE. I'm ok with converting the d/dx and d2/dx2 terms. What...
  43. B

    Understanding the Vector Laplacian in the Navier Stokes Equations

    I recently came across the vector version of the Navier Stokes equations for fluid flow. \displaystyle{\frac{\partial \mathbf{u}}{\partial \mathbf{t}}} + ( \mathbf{u} \cdot \bigtriangledown) \mathbf{u} = v \bigtriangleup \mathbf{u} - grad \ p Ok, all is well until \bigtriangleup. I know...
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