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

    Biological Fluid Dynamics - Approximate Navier Stokes Equations Solns

    Homework Statement Good evening. First post on this forum! The problem I wish to state would take too long to write by hand so I thought it best to do so via attachment. The question I am stuck on is part d and, in fact, part e also. Homework Equations All relevant equations are given...
  2. M

    Is the boundary of ##S^{*}## the same as the boundary of ##D##?

    Homework Statement . Let ##F## be the vector field defined by ##F(x,y,z)=(-y,yz^2,x^2z)## and ##S \subset \mathbb R^3## the surface defined as ##S=\{x^2+y^2+z^2=4, z\geq 0\}##, oriented according to the exterior normal vector. Calculate: ##\iint_S (\nabla\times F).dS##. The attempt at a...
  3. S

    Vectors in Stokes and Gauss theorem

    Homework Statement I have a couple of problems With normal vectors (this especially in Stokes theoerm where it get used often). 1) In some tasks they use unit normal vector, in some they use ordinary normal vector, is there any rules on when to use what? couse it seems pretty random, or dose...
  4. M

    Fluid mechanics navier stokes flow around geometry

    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...
  5. M

    Fluid mechanics study navier stokes

    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...
  6. S

    What size range of droplets is ideal for applying Stokes' Law in fluid dynamics?

    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...
  7. Mandelbroth

    General Conditions for Stokes' Theorem

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

    Is my derivation of the Navier Stokes equation using Newton's second law valid?

    hey pf! can you tell me if this derivation sounds reasonable for the navier stokes equation, from Newtons second law into a partial differential equation. i'm really just concerned with one part. specifically, i start the derivation with \Sigma F = ma. I am comfortable with the force term...
  9. L

    Using stokes' or divergence theorem to solve integral

    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...
  10. G

    Stokes theorem, parametrizing composite curves

    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...
  11. Mandelbroth

    Conditions for using Stokes' Theorem

    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...
  12. C

    Finding a surface form the intersection of two surfaces- Stokes' Thrm.

    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...
  13. F

    Flux integral of surface using Stokes theorem

    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...
  14. E

    Deriving navier stokes in polar

    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.
  15. P

    Use Stokes Theorem to show a relationship

    Homework Statement Use the Stokes' Theorem to show that \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 don't know...
  16. I

    Stokes' Theorem for F(x,y,z) over S: Solving for dA on a Bounded Surface

    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...
  17. N

    Is Stokes' Theorem Intuitively Satisfying and Accurate?

    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...
  18. Saladsamurai

    Navier Stokes EQs: Some Insight Please

    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...
  19. A

    Stokes Relation: R=R' Meaning & Phase Difference

    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 ∏??
  20. 0

    Mathematica Reduced navier stokes in mathematica help please

    reduced navier stokes in mathematica urgent help please ok I am modelling airflow in the upper airway for application i obstructive sleep apnoea, but I have hit a brick wall with mathematica. I have a system of 3 differential equations with boundary conditions, and I need to solve to find 3...
  21. C

    Stokes' Thm, intersection of sphere and plane

    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...
  22. S

    Verify that the Stokes' theorem is true for the given vector field

    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...
  23. M

    Understanding Z = -sin(t) in Stokes Theorem: A Simple Explanation

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

    Finding the n in stokes theorem.

    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...
  25. A

    Stokes Theorem;determine double integral

    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...
  26. C

    Some terminology problems involving Stokes' Theorem

    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...
  27. Y

    Surface Integral over a Cone - Stokes?

    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...
  28. R

    A question regarding General Stokes' Equation

    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...
  29. K

    Group Velocity of shallow water Stokes wave derivation seems wrong

    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...
  30. estro

    Calculating line integral using Stokes' and Gauss' theorems

    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...
  31. B

    Weak form of Navier Stokes Equation

    Homework Statement Folks, determine the weak form given Navier Stokes eqns for 2d flow of viscous incompressible fluids ##\displaystyle uu_x+vu_y=-\frac{1}{\rho} P_x+\nu(u_{xx}+u_{yy})## (1) ##\displaystyle uv_x+vv_y=-\frac{1}{\rho} P_y+\nu(v_{xx}+v_{yy})## ##\displaystyle u_x+v_y=0##...
  32. R

    Stokes theorem in a cylindrical co-ordinates, vector field

    Homework Statement given a vector field v[/B=]Kθ/s θ (which is a two dimensional vector field in the direction of the angle, θ with a distance s from the origin) find the curl of the field and verify stokes theorem applies to this field, using a circle of radius R around the origin Homework...
  33. B

    Finding Clear Explanations for Navier Stokes Equations

    Hi I am currently revising for an exam taking place in 3 days. I am finding it difficult understanding Navier Stokes Equations. It won't necessarily involve difficult questions on Navier Stokes Equations, i myself just find it difficult revising from academic books which explain the problems...
  34. F

    Multivalued Navier Stokes Solution

    I was messing around with the Navier-Stokes equations a while ago and I found a time dependent 2D solution. The force I used was periodic, bounded, and smooth. The question I have is with regards to the time functions in the solution. The solution is spatially periodic and has the form: u =...
  35. O

    Exploring Navier Stokes Equations for General Fluid Flow Analysis

    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...
  36. G

    Stokes Theorem cone oriented downwards

    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...
  37. G

    Use Stokes Theorem to evaluate the integral

    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...
  38. G

    Stokes Theorem paraboloid intersecting with cylinder

    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...
  39. T

    Parameterizing Z-Value in Line Integral for Cylinder (Stokes Thm)

    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...
  40. G

    Proving Stokes Theorem w/ Homework Equations

    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...
  41. H

    Trouble with the unit normal vector for stokes theorem

    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...
  42. J

    Verify Stokes' Theorem for F across a paraboloid

    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...
  43. C

    Green's, Stokes and Divergence Theorem

    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...
  44. H

    Stokes theorem over a hemisphere

    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...
  45. M

    Help doing an integral using stokes theorem?

    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 =...
  46. M

    Concept question about stokes theorem?

    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...
  47. C

    Determining the viscosity of glycerol at 20*C using Stokes Law.

    Homework Statement I used a home made falling sphere viscosimeter to obtain: terminal velocity = 0.06216 ms^-1 I used a micrometer to obtain: radius = 0.001835m and, because my value for glycerol viscosity was way off the known value of 1.495 Pa s, I decided to measure the density of the...
  48. T

    Unfamiliar formulation of Stokes Problem

    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...
  49. A

    When is not necesary to do surface ordering in the non-Abelian Stokes theorem?

    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...
  50. H

    Calculating the Surface Integral Using Stokes' Theorem

    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...
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