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. J

    Evaluating Line Integrals Using Stokes' Theorem

    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...
  2. Y

    How can Stoke's theorem be applied to vector fields?

    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...
  3. M

    How to Use Stokes' Theorem to Solve a Sphere Integration Problem?

    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...
  4. M

    Applicable domain of Stokes Flow

    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...
  5. T

    Exploring the Magnetic Field: Applying Maxwell's Equation and Stokes' Theorem

    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...
  6. H

    Navier stokes and pressure on objects

    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...
  7. K

    Incompressible Navier Stokes - Short Question

    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
  8. R

    Stokes theorem and line integral

    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...
  9. F

    How Do You Apply Stokes' Theorem to Evaluate a Line Integral?

    Homework Statement Use Stokes's Theorem to evaluate \int F · dr In this case, C (the curve) is oriented counterclockwise as viewed from above. Homework Equations F(x,y,z) = xyzi + yj + zk, x2 + y2 ≤ a2 S: the first-octant portion of z = x2 over x2 + y2 = a2 The Attempt at a...
  10. G

    Stokes' Theorem on Intersection of Cone and Cylinder

    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...
  11. S

    Stokes' Theorem: Evaluating a Contour Integral for a Given Surface

    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...
  12. Saladsamurai

    Navier Stokes EquationQuestion about the Diff EQ

    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...
  13. Y

    Verifying Stokes' Theorem for a Hemispherical Cap

    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 =...
  14. M

    Stokes law - Settling velocity and rate

    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...
  15. B

    Projectile motion or stokes law

    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...
  16. A

    Verifying Stokes' Theorem: Am I Doing Anything Wrong?

    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...
  17. S

    Stokes Theorem in cylindrical coordinates

    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...
  18. V

    Understanding Stokes' Theorem: A Tutorial

    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...
  19. I

    Proving Total Current through Insulating Wire Using Spherical Coordinates

    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...
  20. J

    The Statistics of Navier Stokes Equations

    I am curious about what insight people might have as to the statistics of Navier Stokes equation. I thought of the following way someone might try to calculate these. 1) Choose a bais (Basis A) 2) Pick a discrete number of points to constrain the solution of stokes equation. 3) Find the...
  21. S

    Stokes' theorem and unit vector

    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...
  22. J

    Why is \hat{a} \bullet \vec{ds} = ds? Explaining Stokes Theorem.

    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...
  23. H

    Questions about deriving the naviers stokes equations

    Hello, I read some fluidmechanics and there was something I didn't understand. The shear stress in a Newtonian fluid is tau=viscosity*dV/dy, (no need to be dy, but dx and dz also can do.) A shear component called tau(xx) came up, I have two questions about this component: 1. Shear is...
  24. C

    Solving Basic Stokes Theorem Homework on Ellipse

    Homework Statement Use the surface integral in stokes theorem to find circulation of field F around the curve C. F=x^2i+2xj+z^2k C: the ellipse 4x^2+y^2=4 in the xy plane, counterclockwise when viewed from above Homework Equations stokes theroem: cirlulation=double integral of nabla...
  25. R

    Calculating Surface Integral with Stokes' Theorem on a Cube?

    Homework Statement I have to use stokes' theorem and calculate the surface integral, where the function F = <xy,2yz,3zx> and the surface is the cube bounded by the points (2,0,0), (0,2,0),(0,0,2),(0,2,2),(2,0,2),(2,2,0),(2,2,2). The back side of the cube is open. [/B] Homework...
  26. T

    Surface integral, grad, and stokes theorem

    Hi I'm practicing for my exam but I totally suck at the vector fields stuff. I have three questions: 1. Compute the surface integral \int_{}^{} F \cdot dS F vector is=(x,y,z) dS is the area differential Calculate the integral over a hemispherical cap defined by x ^{2}+y ^{2}+z...
  27. T

    Evaluating Surface Integral with Stokes Theorem

    use the stokes theorem to evaluate the surface integral [curl F dot dS] where F=(x^2+y^2; x; 2xyz) and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane. I don't see however how to take F dot dr. What is this closed curve dr bounding...
  28. J

    Use Stokes' Theorem to Calculate F on Triangle 1,0,0...0,1,0...0,0,1

    Homework Statement Use stokes theorem F = xyi + yzj + zxk on triangle 1,0,0,,,,,,,0,1,0,,,,,,0,0,1 Homework Equations The Attempt at a Solution First i found the curl F curl F = -yi - zj - xk Then i found the equation of the plane for the triangle z = g(xy) = 1 -...
  29. J

    How Do Stokes' and Divergence Theorems Apply to a Cube's Surface Integral?

    Homework Statement Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem Homework Equations Divergence theorem: ∫∫∫∇.FdV = ∫∫∇.ndS Stokes...
  30. A

    Stokes Theorem Problem: Surface Integral on Ellipse with Curl and Normal Vector

    Homework Statement F = xi + x3y2j + zk; C the boundary of the semi-ellispoid z = (4 - 4x2 - y2)1/2 in the plane z = 0Homework Equations (don't know how to write integrals on here, sorry) double integral (curl F) . n dsThe Attempt at a Solution curl F = 3y2x2k n = k curl F . n = 3y2x2 So I...
  31. S

    Using Stokes' Theorem for Hemispherical Surface Area Calculation

    Homework Statement Calculate \int \int _{S}\nabla \times \overline{F} \cdot \hat{N}dS where \overline{F} = 3y\hat{i} - 2xz\hat{j} + (x^{2}-y^{2})\hat{k} and S is a hemispherical surface x2 + y2 + z2 = a2, z ≥ 0 and \hat{N} is a normal of the surface outwards. Can you use Stokes' theorem...
  32. B

    Stokes' Theorem for Line Integrals on Closed Curves: A Problem Solution

    Homework Statement Please help me to check whether I did the right working for this problem. Thanks. The numerical answer is correct but I'm not very sure if the working is correct also. Find \int y dx + z dy + x dz over the closed curve C which is the intersection of the surfaces whose...
  33. L

    Investigating Stokes Law: Different Formulas, Different Results?

    Hello all. I'm investigating a little bit about stokes law in order to understand the settling velocity of falling particles and on the net i encountered with 2 different formulas and i simply can't find the reason why they are different. every formula gives me a different answer. The 2...
  34. B

    Stokes' Theorem ( Surface Integral )

    [SOLVED] Stokes' Theorem ( Surface Integral ) Homework Statement Use stokes' theorem to find the value of the surface integral \int\int (curl f) dot n) dS over the surface S: Let S by the part of the plane z=y+1 above the disk x^2+y^2<=1, and let f=(2z,-x,x). Homework Equations...
  35. J

    COMSOL doing periodic boundary conditions in navier stokes

    Please can anyone tell me how to set this up? I know how to do the required settings in the Physics/Period Conditions. However, to fully implement it, I'm also required to choose boundary conditions in the 2D incompressible navier stokes solver (e.g. wall, inlet, outflow, open boundary...
  36. S

    Understanding Stokes Theorem: Solving Boundary Curve Dilemmas in Vector Calculus

    Homework Statement This is a question about stokes theorem in general, not about a specific problem. Directly from lecture: "If S has no boundry (eg. if S is the boundry of a solid region) then \int\int_{S}Curl(\stackrel{\rightarrow}{F})\bullet ds = 0 " because apparently "no boundry C...
  37. T

    Stokes' Theorem and Maxwell's Equations

    Homework Statement Faraday’s Law can be written as: \oint_P \vec{E} \cdot \vec{dl} = -\frac{d}{dt}\Phi Where \Phi is the magnetic flux. Use Stokes’ theorem to obtain the equvilant Maxwell equation (i.e. Faraday’s Law in differential form). Homework Equations Stokes' Law...
  38. T

    Stokes theorem under covariant derivaties?

    in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this... So, is it true that: \int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x if so, how do I make sense of the d^{n-1}x term? would...
  39. S

    Solve Stokes Equation Help: Homework Statement

    Homework Statement let F be vector field: \[\vec F = \cos (xyz)\hat j + (\cos (xyz) - 2x)\hat k\] let L be the the curve that intersects between the cylinder \[(x - 1)^2 + (y - 2)^2 = 4 \] and the plane y+z=3/2 calculate: \[\left| {\int {\vec Fd\vec r} } \right|\] Homework Equations...
  40. S

    Calculate this integral using Stokes

    Homework Statement let F be vector field: \[\vec F = \cos (xyz)\hat j + (\cos (xyz) - 2x)\hat k\] let L be the the curve that intersects between the cylinder (x - 1)^2 + (y - 2)^2 = 4 and the plane y+z=3/2 calculate: \[\left| {\int {\vec Fd\vec r} } \right|\] Homework Equations in...
  41. L

    Using Stokes Theorem to $\int_{L}^{} y dx + z dy + x dx$

    Homework Statement Use Stokes Theorem to compute \int_{L}^{} y dx + z dy + x dx where L is the circle x2 + y2 + z2 = a2, x + y + z = 0 The Attempt at a Solution I feel like this problem shouldn't be that hard but I can't get the right answer: (pi)a2/3. I calculated the curl of F as...
  42. A

    Finding Area of L using Stokes Theorem

    Homework Statement Hey. I need to find the circulation of F through out the line L. I know I need to use stokes theorem, the problem is, how do I find the area of L? I mean, I know the intersection line of the sphere and the plot looks like an ellipse on the XY surface, but how do I find...
  43. A

    Struggling with Stokes' Theorem? Need Help Solving?

    Homework Statement I got stuck using the Stokes' theorem, the problem is at the bottom of the pic. I found the Curl of F, and also the normal of the Triangle. As you can see, I ended up with an area integer with 3 variables, how do I solve this? did I do it right? Homework Equations...
  44. D

    Stokes theorem equivalent for cross product line integral

    "Stokes theorem" equivalent for cross product line integral Homework Statement I am aware that the vector path integral of a closed curve under certain conditions is equivalent to the flux of of the curl of the vector field through any surface bound by the closed path. In other words, Stokes...
  45. P

    Stokes' Theorem Verification for Upper Half Sphere with Radius b and z > 0

    Homework Statement A = sin(\phi/z)* a(\phi) I'm having problem verifying Stokes Theorem. I have to verify the theorem over the upper half of the sphere with radius b and the sphere is centered at the origin. The problem also says z > = 0 Could someone help me with this.
  46. O

    Navier Stokes with chemical reaction

    I wasn't sure whether to put this in Aerospace, but decided on physics in the end. 1.) How do you factor a chemical reaction into the solution for the Navier Stokes equations? More precisely, how can you include the affects of a heat absorbing (endothermic), or heat releasing (exothermic)...
  47. B

    Navier Stokes Equations - Helmholtz-Hodge decomposition and pressure

    Hi, I've been doing some work with the NS equations. I've read a few papers by fellow undergrads that imply a relationship between the helmholtz-hodge decomposition and the pressure equation. As far as I can see, they're both separate ways of resolving the problem of keeping the flow...
  48. E

    Stokes and Divergence theorem questions

    Homework Statement Let \vec{F}=xyz\vec{i}+(y^{2}+1)\vec{j}+z^{3}\vec{k} And let S be the surface of the unit cube in the first octant. Evaluate the surface integral: \int\int_{S} \nabla\times \vec{F} \cdot \vec{n} dS using: a) The divergence theorem b) Stoke's theorem c)...
  49. N

    Calculating Line Integrals Using Stokes' Theorem

    [SOLVED] Calculus - Stokes' theorem Homework Statement I have F in Cartesian coordinates (F is a vector): F = (y , x , x*z) and a curve C given by the quarter-circle in the z-plane z=1 (so t : (cos(t) , sin(t) , 1) for t between 0 and Pi/4). I have found the line integral, and it equals...
  50. M

    Stokes Law, Viscosity. (very simple)

    I'm trying to find the viscosity of some glycerol that we dropped various steel balls down using the equation: V = [2r^2 (p – σ) g] / 9η I put in these values: p = 7800 kg m-3 σ = 1200 kg m-3 g = 9.8 m s-2 And ended up with the equation. η = 129360r^2 / 9V My problem is that I...
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