Fluid dynamics is a branch of physics that studies the motion of liquids or gases, which involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc.

learn more… | top users | synonyms

0
votes
0answers
9 views

Cauchy Equations and Navier Stokes

I'm attempting to take the Navier Stokes Equation and coming up with an expression that will allow me to numerically determine the velocity of non-Newtonian fluid flow. The text I'm using is Cengel ...
0
votes
1answer
29 views

Euler equations

What's the relationship between the incompressible, free surface euler equations and the euler equations? Are the latter just the former when the free surface is identically zero?
1
vote
0answers
17 views

Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
0
votes
0answers
14 views

Importance of Initial Guess in the numerical solution to the following fluid flow problem

Greetings Stackexchange community. Forgive me if the question is repetitive and/or answered before. I am currently working on a simple fluid flow problem, 'Heated laminar vertical Jet'/Brand and Lahey ...
1
vote
0answers
31 views

Stokes Equation

I came across the Stokes equation expressed in following form: I am trying to expand to check if it is correct but having hard time evaluating it. Can anyone give some hint on how can i expand it ...
0
votes
1answer
30 views

well-posedness of a mathematical model

what is the meaning of Well-posedness of a mathematical model of a physical phenomena for example stokes equation in fluid dynamics ? what is the necessity to prove that a model is well-posed? how ...
0
votes
1answer
40 views

What should be the exponent on the parameter to get the same solution?

Regarding the Boussinesq equations of motion: I was reading a paper which stated the following: Shouldn't the $\alpha$ before the $t$ be $\alpha^{-1}$ instead? Could be a typo or maybe I am not ...
2
votes
2answers
131 views

Prove an identity using differential calculus to a problem connected to fluids

Euler's equation for a incompressible inviscid fluid is $\displaystyle \frac{\partial \textbf{v}_t}{\partial t}+(\textbf{v}_t \cdot \nabla)\textbf{v}_t=-\nabla p_t$ where ...
1
vote
1answer
36 views

$L^2$ regularity of a convolution with Newtonian potential

I am reading Vorticity and incompressible flow (Bertozzi, Majda) and on page 71-72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
1
vote
3answers
71 views

Showing instablity of differential equation.

Assume differential equation $$ x'=2x+y+x \cos t-y \sin t $$ $$ y'=-x+2y-x\cos t+y \sin t $$ Show that solution $(x(t),y(t))=(0,0)$ is unstable. Is there a non-trival solution such that ...
0
votes
1answer
13 views

$\int_C (\alpha x, -\alpha y) . dr = 0$ where C is the unit circle

Circulation is given by $$\int_C u . dr$$ I want to show that the circulation around the unit circle is $0$ for $u = (\alpha x, \alpha y)$. Ie. $$\int_C (\alpha x, -\alpha y) . dr = 0$$ How would ...
1
vote
0answers
24 views

What is the connection (if any) between Knot Theory and Fluid Dynamics?

I've heard there is a connection to physics, but I'm unsure about any specific connection to fluid dynamics. Thanks!
1
vote
1answer
64 views

Fluid Mechanics for Low Reynolds Number

I've tried to answer this question but I seem to get a really weird (and suggestively incorrect) answer. The question is: "Fluid is injected radially and slowly into a circle, radius R, with the ...
0
votes
0answers
42 views

Calculating force per unit width

Question: A line source of strength $2πm$ is located a distance $a$ above a horizontal plate. Find the force per unit width on the plate, ignoring gravity and taking the pressure below the plate to be ...
2
votes
0answers
26 views

Objectivity or frame invariant

I am not sure it is maths or physics question. A stress tensor of a nematic liquid crystal is given as follows $$ T_{ij}=-P\delta_{ij}-n_{k,i}n_{k,j}+\widetilde{T_{ij}} $$ where ...
0
votes
0answers
65 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
2
votes
1answer
58 views

Undergraduate Project Suggestions

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
1
vote
1answer
37 views

differential inequality involving the square of the function

It is written in a book, (Bertozzi- Majda, vorticity and incompressible flow page 106) that given a differential inequality of the following type: $ \frac{d}{dt}\|u^{\epsilon}(t)\| \leq ...
0
votes
1answer
48 views

Fluid Dynamical problems

I'm just going through Batchelor's book on Fluid Dynamics, and I'm not too sure about a couple of questions in "Exercises for Chapter 4", which are the following: Exercise 3 for Chapter 4 "A long ...
0
votes
0answers
24 views

What would global irregularity of the Navier-Stokes Equations do?

Suppose Terrance Tao's hints at showing finite-time blowup for the true Navier-Stokes Equations prevailed, and the Navier-Stokes Problem was solved negatively (no existence and uniqueness). What good ...
1
vote
0answers
24 views

What is the “Enstrophy Miracle”?

I read that one of the main differences between the establishing global regularity for the Navier-Stokes equations for viscous and incompressible flow in 2D in 3D is that in 2D there is something ...
0
votes
0answers
56 views

Would a Counterexample to Navier-Stokes Problem be Sufficient?

Given the problem statement for the Navier-Stokes Existence & Smoothness problem from the Clay Institute website, wouldn't one need to show only one counterexample to the conjecture of global ...
1
vote
1answer
115 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
0
votes
1answer
39 views

Mass Continuity Equation for Fluid - Running Into a Problem

I'm running into a problem when trying to show the mass continuity equation for a fluid, which says $$\frac{\partial \rho}{\partial t} + \left(\nabla \cdot \rho \textbf{u}\right) = 0$$ Where ...
0
votes
0answers
26 views

Lubrication Theory: Quick Question!

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
2
votes
0answers
27 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
0
votes
0answers
23 views

Convected 2nd order tensor in component form

I have a convected second order tensor that I'd like to write in component form. $\frac{D\mathbf{a}}{D t} = \frac{\partial \mathbf{a}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{a}$, where ...
3
votes
1answer
40 views

If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
-1
votes
1answer
123 views

Fluid Dynamics Problem

A vertical circular stack 100 ft high converges uniformly from a diameter of 20 ft at the bottom to 16 ft at the top. Coal gas with a unit weight of 0.030 pcf enters the bottom of the stack with a ...
0
votes
0answers
41 views

Why is the Galerkin-Method not optimal for non-self-adjoint equations

often i read phrases that explain the bad behavior of standard Galerkin-FEM for convection dominated problems by the equations beeing non-self-adjoint. Examples: Zienkiewicz, The Finite Element ...
0
votes
1answer
58 views

Velocity of fluid in presence of Sphere

I am struggling with the following problem: A rigid sphere of radius $a$ is placed in a stream of fluid whose velocity in the undisturbed state is $V$. Determine the velocity of fluid at any point of ...
1
vote
1answer
46 views

Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
2
votes
2answers
48 views

Given the velocity field $\ (u,v) = UL( \frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}$), how can we find the Legrangian coordinates?

I'm currently taking Graduate Fluid Dynamics and I'm using Stephen Childress' book, An intro to theoretical fluid mechanics. I've been stuck on one problem for quite some time and, so I'd like to see ...
0
votes
1answer
56 views

Notation regarding the continuity equation for conservation of mass

I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$ (Actually there should be an ellipse ...
0
votes
0answers
99 views

Joukowski transformation of streamlines around cilinder in mathematica

I have problem transforming streamlines around cilinder, which is in fact simple circle that rotates, to a airfoil. It is done using Joukowski transformation $z = z+\frac{c}{z}$. The circle transforms ...
0
votes
1answer
35 views

Inviscid Shallow Water Equation

Aside from wikipedia where might I find a fairly comprehensive, yet simple to read, piece of literature on the inviscid shallow water equation? Can you recommend any texts? I don't want literature ...
1
vote
1answer
64 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
0
votes
0answers
43 views

viscous burgers equation physical meaning

The viscous Burgers' equation : $$ q_{t} + qq_{x} = vq_{xx}, \:\:\: \mbox{where} \:\:\: v > 0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion. What is this equation for? (in ...
1
vote
0answers
66 views

Finite Difference Discretization of Darcy's law and solving with Picard method

I am trying to discretize Darcy's Law using finite differences and then solving the resulting linear system of equations with the Picard method. So far only in 1D and the steady-state (no time ...
3
votes
0answers
88 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
0
votes
1answer
68 views

Normal mode solutions

Why does the solution choose those highlighted in green. Don't you normally have to pick the more general $\displaystyle\eta=\hat\eta (e^{ik(x-ct)}+$complex conjugate$)$ and then consider $\hat\eta ...
0
votes
1answer
42 views

Linear Water Waves

When solving , $\displaystyle {\partial^2 \tilde\phi\over\partial x^2}+{\partial^2 \tilde\phi\over\partial y^2}=0$ Why is it that $-h<y<0$ as opposed to ...
0
votes
1answer
47 views

Surface tension

Consider deep water gravity waves but assume now that there are two fluids separated by the interface at $y=\eta(x,t)$ and suppose that the upper fluid extends to $y\to\infty$. Let the fluid in the ...
0
votes
1answer
33 views

Finite element convergence rates for mixed problems

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh. ...
0
votes
0answers
65 views

Sketching the shock path

I can do the vast majority of this question except the bit underlined in green at the bottom. I don't really understand what it is asking. By 'some large t' does it mean 'at large values of t'. To get ...
3
votes
1answer
241 views

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
1
vote
0answers
40 views

Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case? Recall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm ...
0
votes
1answer
40 views

Proving that pressure at a point does not depend on orientation

In a) the solution states that $dS_1=$cos$\theta dS_2$, in other words it considers the surface area to be equivalent to the length of a line, in order to use basic trigonometry. I understand we are ...
0
votes
0answers
38 views

How to design Boundary condition for Euler equations (CFD)?

I'm developing on the calculation of the euler equations using the finte volume method. As you may know each cell is calculated by the incoming and outgoing flux. That means I need in a 1D System the ...
1
vote
1answer
38 views

Special form of the Divergence theorem

It states in my notes relating to the derivation of the euler equations that a 'special form' of the divergence theorem is: $\iint{\phi\hat{n}}\space{dS}=\iiint\nabla\phi\space{dV}$ with $\phi$ a ...