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.
The question is a problem from Leithold's calculus book. I didn't understand the (x = 5 \cos(t)). Shouldn't it be (x = 2 cos(t))? I'm referring to item b.
i tried this way. i don't know what is wrong.
Calculate surface integral ## \displaystyle\iint\limits_S curl F \cdot dS ## where S is the surface, oriented outward in below given figure and F = [ z,2xy,x+y].
How can we answer this question?
##curl([x^2z, 3x , -y^3],[x,y,z]) =[-3y^2 ,x^2,3]##
The unit normal vector to the surface ##z(x,y)=x^2+y^2## is ##n= \frac{-2xi -2yj +k}{\sqrt{1+4x^2 +4y^2}}##
##[-3y^2,x^2,3]\cdot n= \frac{-6x^2y +6xy^2}{\sqrt{1+4x^2 + 4y^2}}##
Since ##\Sigma## can be parametrized as ##r(x,y) = xi + yj +(x^2...
The last formula is what I was going for, since it arises as the momentum flux in fluid dynamics, but in the process I came across the rest of these formulas which I’m not sure about.
The second equation is missing a minus sign (I meant to put [dA X grad(f)]).
Are they correct? Do they have...
Basically surface B is a cylinder, stretching in the y direction.
Surface C is a plane, going 45 degrees across the x-y plane.
Drawing this visually it's self evident that the normal vector is
$$(1, 1, 0)/\sqrt 2$$
Using stokes we can integrate over the surface instead of the line.
$$\int A(r)...
I parameterize surface A as:
$$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$
Then I get y from surface B:
$$y = 2 - x = 2 - 2cos t$$
$$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$
Now I'm asked to integral over the surface, not solve the line integral.
So I create a new function to cover the...
Greetings
the solution is the following which I understand
I do understand why the current orientation of the Path is positive regarding to stocks (the surface should remain to the left) but I don´t understand why the current N vector of the surface is positive regarding stockes theorem...
From Stokes' theorem: ##\int_{C}^{}\vec F\cdot d\vec r=\iint_{S}^{}curl\vec F\cdot d\vec S=\iint_{D}^{}curl\vec F\cdot(\vec r_u \times \vec r_v)dA ##
To get to the latter surface integral, I started by parametrizing the triangular surface in ##uv## coordinates as:
$$\vec r=<1-u-v,u,v>, 0\leq...
From Stokes we know that ##\iint_{\textbf{S}}^{}curl \textbf{F}\cdot d\textbf{S}=\int_{C}^{}\textbf{F}\cdot d\textbf{r}##.
Now, we can calculate the surface integral of the curl of F by calculating the line integral of F over the curve C.
The latter ends up being 0(I calculated it parametrizing...
So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force.
$$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$
$$\nabla \cdot \textbf{u}=0$$
We can apply the...
https://www.researchgate.net/publication/301874096_Emergent_behavior_in_active_colloids/link/5730bb3608ae08415e6a7c0a/download (expression 9 on this document derivation). I understand the need for substitution etc into the integral. What puzzles me is how the integral equals what it does. If...
Here is my answer to this question:
Stokes shift is the difference in wavelength between positions of the band maxima of the excitation and emission spectra of the same electronic transition.
When Stokes shift is large, it means there is more energy loss, which is not favorable regarding...
Hi,
So my goal is to compute the integral of the "curl" of the vector field ##A_i(x_i)## over a 2-dimensional surface. Following a physics book that I am reading, I introduce the antisymmetric 2-nd rank tensor ##\Omega_{ij}##, defined as:
$$\Omega_{ij} = \frac {\partial A_i}{\partial x_j} -...
Given surface ##S## in ##\mathbb{R}^3##:
$$
z = 5-x^2-y^2, 1<z<4
$$
For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...
It is more or less a generic problem of stokes theorem:
##\int_{\gamma} F dr##, where ##F = (-y/(x²+y²) + z,x/(x²+y²),ln(2+z^10))## and gamma is the intersection of ##z=y^2, x^2 + y^2 = 9## oriented in such way that its projection in xy is traveled clockwise.
So, i decided to apply stokes...
in the limit as Re→0 , we can neglect the material derivate of v ( Dv/Dt =0 ) but why in books they always make the gravity effects equal to 0?
i can't understand and no one really explains this stuff
Hi,
I was just working on a homework problem where the first part is about proving some formula related to Stokes' Theorem. If we have a vector \vec a = U \vec b , where \vec b is a constant vector, then we can get from Stokes' theorem to the following:
\iint_S U \vec{dS} = \iiint_V \nabla...
I've tried a few ways of solving this, both directly and by using Stokes' Theorem. I may be messing up what the surface is in the first place
F= r x (i + j+ k) = (y-z, z-x, x-y)
Idea 1: Solve directly. So ∇ x F = (-2,-2,-2). I was unsure on which surface I could use for the normal vector...
Stokes theorem relates a closed line integral to surface integrals on any arbitrary surface bounded by the same curve. Gauss theorem relates a closed surface integral to the volume integral within a unique volume bounded by the same surface. What causes this asymmetry in these 2 theorems, in the...
In Stokes' theorem, the closed line integral of f=the surface integral of curl f on ANY surface bounded by the same curve. But in Gauss' theorem, the surface integral of f on a surface=the volume integral of div f on a unique volume bounded by the surface. A surface can only enclose 1 volume...
Hi,
My question pertains to the question in the image attached.
My current method:
Part (a) of the question was to state what Stokes' theorem was, so I am assuming that this part is using Stokes' Theorem in some way, but I fail to see all the steps.
I noted that \nabla \times \vec F = \nabla...
Summary:: This question is about a Stokes' Theorem question that I saw on Khan Academy and I am trying to attempt to solve it a different way.
The problem is as follows:
Problem: Let \vec{F} = \begin{pmatrix} -y^2 \\ x \\ z^2 \end{pmatrix} . Evaluate \oint \vec F \cdot d \vec {r} over the...
Hi PF!
I'm running a CFD software that non-dimensionalizes the NS equations. The problem I'm simulating is a 2D channel flow: relaxation oscillations of an interface between two viscous fluids, shown here. I'm trying to see what they are non-dimensionalizing time with, which is evidently just...
Stokes' Law gives us the value fo viscous force when a spherical body is under motion inside a fluid.
##F_{viscous} = 6\pi\eta av## (where ##a## is the radius of the spherical body and ##v## is the velocity with which it is moving)
What is the reason for the Viscous drag to depend upon the...
Stokes' Theorem states that:
$$\int (\nabla \times \mathbf v) \cdot d \mathbf a = \oint \mathbf v \cdot d \mathbf l$$ Now, if for a specific situation, I can work out the RHS and it's equal to zero, does it necessarily mean that ##\nabla \times \mathbf v = 0##? I mean all that tells me is that...
Hello everyone,
Attached is the homework problem (FluidHmk.PNG) as well as the attempt (Attempted 1 and 2).
Just wanted to know if this is method to approach the problem
Thanks in advance.
I want to check Stokes' theorem for the following exercise:
Consider the vector field ##\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k##.
A closed curve ##C## lies in the plane ##x + y + z = 3##, oriented counterclockwise. The parametric representation of this curve is defined as...
Just trying to derive the Navier-Stokes equation.
(1)The velocity at any point in space of an infinitesimal fluid element is v(x,y,z,t)
(2) acceleration ##\frac{dv}{dt}=\frac{\partial v}{\partial t}+\sum_i\frac{\partial v_i}{\partial x_i}{\dot x_i}##
##a=\frac{dv}{dt}=\frac{\partial v}{\partial...
Choking mass flow seems to reflect the fact that fluid momentum density has a maximum value (in stationary conditions) equal to ##\rho_* c_*## where ##\rho_*## is the critical mass density and ##c_*## is the critical velocity, which is closely related to the speed of sound (see...
Homework Statement
Use Stokes' Theorem to evaluate ∫cF ⋅ dr, where F(x, y, z) = x2zi + xy2j + z2k and C is the curve of the intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9 oriented counterclockwise as viewed from above.
Homework Equations
Stoke's Theorem:
∫cF ⋅ dr = ∫s...
Homework Statement
With the stokes' theorem transform the integral ## \iint_\sigma \vec{\nabla}\times\vec{F}\cdot\vec{\mathrm{d}S} ## into a line integral and calculate.
## \vec{F}(x,y,z) = y\hat{i} -x^2\hat{j} +5\hat{k}##
##\sigma(u,v) = (u, v, 1-u^2)##
## v\geq0##, ##u\geq0##...
Homework Statement
[/B]
Homework Equations
Navier strokes theorem
The Attempt at a Solution
May I ask why would there suddenly a "h" in the highlighted part?
"h" wasnt existed in the previous steps, which C2=0 shouldn't add height of the liquid as a constant in the formula...
thanks
I have been spending an embarrassing amount of time, trying to figure out what these two theorems are actually telling me.
As I understand it, it is suppose to tell me, what the "curl" around a boundary is. However there are several examples I can find, where this doesn't make sense. My...
Homework Statement
Homework Equations
Stokes Theorem
The Attempt at a Solution
I'm having a tough time "cancelling" out integrals from both sides of an equation (if possible). On the right hand of the equation, we know since it is a closed curve, that Stoke's Theorem applies and we can...
Hi, in first attachment/picture you can see the generalized navier stokes equation in general form. In order to linearize these equation we use Beam Warming method and for the linearization process we deploy JACOBİAN MATRİX as in the second attachment/picture. But on my own I can ONLY obtain the...
An article in Quanta Magazine discusses the math behind the Navier Stokes equations, why they are so difficult to solve and whether they truly represent fluid flow:
https://www.quantamagazine.org/what-makes-the-hardest-equations-in-physics-so-difficult-20180116/
Hey! :o
I want to calculate $\int_{\sigma}\left (-y^3dx+x^3dy-^3dz\right )$ using the fomula of Stokes, when $\sigma$ is the curve that is defined by the relations $x^2+y^2=1$ and $x+y+z=1$.
Is the curve not closed? Because we have an integral of the form $\int_{\sigma}$ and not of the form...
Hello! I am reading this paper and on page 9 it defines the De Rham's period as ##\int_C \omega = <C,\omega>##, where C is a cycle and ##\omega## is a closed one form i.e. ##d\omega = 0##. The author says that ##<C,\omega>:\Omega^p(M) \times C_p(M) \to R##. I am a bit confused by this, as...
Homework Statement
The goal is to verify Stoke's Theorem. I've uploaded the image showing the problem and diagram. I'd like to get a double check on my work as I work on part b.
Homework Equations
Curl in cartesian coords and vector E.
Integral of E dot dl = Integral of (Curl of E) dot dS
The...
I'm studying the Navier Stokes equations right now, and I've heard that those set of equations are invalid in some situations (like almost any mathematical formulation for a physics problem). I would like to know in which situations I cannot apply the NS equations, and what is the common...
I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...
Homework Statement
Verify Stokes' theorem
∫c F • t ds = ∫∫s n ∇ × F dS
in each of the following cases:
(a) F=i z2 + j y2
C, the square of side 1 lying in the x,z-plane and directed as shown
S, the five squares S1, S2, S3, S4, S5 as shown in the figure.
(b) F = iy + jz + kx
C, the three...
Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it.
I honestly will never use the higher dimensional version but I still want to see a full proof...