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

3
votes
0answers
89 views

Water explosion and Navier-Stokes global regularity problem

In this article, Terence Tao is talking about a water explosion thought experiment that can lead to the solution of Navier-­Stokes global regularity problem. Can anybody explain this in more ...
3
votes
0answers
29 views

Presentation of Navier-Stokes eqns

This may be a trivial point, perhaps it's a lack of understanding on my part? When I was first introduced to fluid mechanics I was instructed to write the continuity and (generalized) Navier-Stokes ...
0
votes
0answers
47 views

How much water could be stored?

We have two water storage tanks--- Tank A and Tank B --- on the roof of the upper storey of our two-storey house. The tanks are cylindrical in shape. Each of the two tanks have a circular opening ...
0
votes
1answer
39 views

Significance of 'faces' in Stress tensor components?

I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor. Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress ...
0
votes
1answer
18 views

Math Basis for Partial Fluid Change in Car Transmission

I have a fluid mixing problem with my car and I can't seem to find the answer: I have to change my transmission oil. The transmission has a 7 liter total capacity, but due to the torque converter, I ...
1
vote
0answers
46 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
2
votes
1answer
62 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
0
votes
0answers
7 views

using complex potential to calculate stream function

The answer is $ \psi = Br^{12} sin12 \theta $ I dont know how to get this? say the complex potential equals $ Bz^{n} $ = $ Bre^{i\theta} $ how do i get the above answer? thank you
1
vote
1answer
15 views

Finding streamlines from complex potential.

I'm currently studying a Fluid Dynamics module and mock exam question has me completely stumped, I have been given the complex potential and shown it to be in the form given, however when trying to ...
1
vote
1answer
20 views

fluid dynamics Bernoulli's equation

I really cant seem to prove this, using A1u1 = A2u2 I get (r_a)^2/(r_b)^{2} = 1 using Bernoulli's equation, i cant figure out what the other terms should be? 1/2ro u_a^{2} + rogz1 + pa = ...
0
votes
0answers
55 views

Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
0
votes
0answers
25 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
1
vote
1answer
33 views

Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
0
votes
0answers
24 views

Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
0
votes
0answers
42 views

Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
1
vote
0answers
22 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
1
vote
0answers
11 views

Visco-elastic fluid reference

What is a good book on visco-elastic fluids for self-teaching after one has studied Gurtin's Intro to Continuum mechanics? Thanks!
0
votes
0answers
41 views

Concept of a continuum

I have to explain the concept of a continuum that is used for the description of the dynamic behaviour of the fluids, and to explain how this concept is related on the one side with the laboratory ...
0
votes
0answers
32 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
1
vote
0answers
37 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
0
votes
1answer
30 views

How to find the complex potential for the following flow under certain conditions?

We've used $z=i(Z+4/Z)$ as a conformal mapping to map the exterior of a circle $|Z|=2$ to the exterior of the line segment $(-4i,4i)$. We now want to write the complex potential of the uniform flow ...
4
votes
2answers
45 views

Help solving a first order non-linear differential equation derived from the navier-stokes equation

I am an engineer studying an unsteady-state flow through a pipe. The transient Bernoulli equation of this system, which I picked up from here ...
0
votes
0answers
9 views

Representation of polynomial order in CFD codes

I currently working on a CFD code over a cubic grid. Now, the number of elements used in the simulation is decomposed among the number of processors. Each of those processors (a section of the cube) ...
1
vote
1answer
22 views

Does the Laplace operator include the second derivative with respect to time variable?

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$? We aren't sure whether or not time is included in it or not.
1
vote
2answers
26 views

Is a 3D or 2D Poisson's equation separable or non sperable?

Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable? Thank you
2
votes
1answer
39 views

When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
0
votes
0answers
17 views

Von Karman similarity solution of Navier-Stokes equations

I was wondering if anyone could help me; I've been looking at Von Karman's similarity solutions of the Nav-Stokes equations for a rotating disk in a fluid $u = r \Omega U(n)$ $v = r \Omega V(n)$ ...
0
votes
1answer
128 views

Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
0
votes
0answers
13 views

Linearisation of instability of flames

I am not sure it is a maths or physics question, kinda in between. Large-scale disturbances of a plane flame $x=\eta(y,t)$ are describe by Euler's equation and the continuity equation: $$ ...
0
votes
0answers
19 views

Problem on Majda's Vorticity and Incompressible Flow

I'm reading Majda's book, on page 110, I cannot understand how to get $v\in C_W (0,T;H^{m})$ from the above. And he wrote "$[\phi,v^{\epsilon}]\to[\phi,v]$ uniformly on $[0,T]$ " two times, do the ...
0
votes
0answers
29 views

Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
2
votes
0answers
24 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
2
votes
0answers
19 views

Velocity potential of flow under rigid disk

Determine velocity potential of the flow in this system: Rigid disk of radius R at a heigh h(t) above horizontal plane z=0 with incompressible, inviscid flow between them, and h< The flow is ...
1
vote
0answers
19 views

Linearising equations about a base state.

Consider a shallow-water system with mean depth H, where the base state consists of the flow (u,v)=($u_{0},$0), with a sloped water surface $\eta_{0}$(x,y) = - $\gamma y$, where u$_{0}$ and $\gamma$ ...
0
votes
0answers
14 views

Vorticity of a rigid body.

Consider a fluid in solid body roation about the z-axis with angular speed $\varOmega$ Derive an expression for the velocity field (u(x,y), v(x,y)) and show the vorticity field is the same at every ...
1
vote
0answers
29 views

The free surface of the wave is a material surface

If we define the free surface by: $F(x,y,t)=y-h(x,t)=0$ Then for this to be a material surface $\frac{DF}{Dt}=0$ on $y=h(x,t)$ However on $y=h(x,t)$, $F=0$, so doesn't this just imply ...
0
votes
1answer
14 views

Particle paths and standing waves

Here $x_0$ is the $x$ coordinate of a point in $x-y$ space Here $x_0$ is the $x$ coordinate of a point in $x-y$ space. I understand where the nodes and crests are on the figure. However, I don't ...
0
votes
0answers
20 views

Elementary Fluid Dynamics help!

I'm revising for my Fluids exams next month and I'm trying to understand a few definitions, and maybe grasp a physical interpretation of what exactly they are. I click on 'velocity field' on ...
0
votes
1answer
22 views

Mass conservation

I am trying to prove that $$\frac{d}{dt}\int_{a(t)}^{b(t)} \rho(x, t)g(x, t)dx = \int_{a(t)}^{b(t)} \rho(x, t)\frac{D}{Dt}g(x, t)dx$$ I have tried to evaluate the integral using Liebniz' rule, so ...
0
votes
1answer
24 views

Conservation of norms by the 2-d euler vorticity equation

In the book of Filho Lopes, Weak solutions for the equations of incompressible and inviscid Fuid dynamics. Page 59 They want to prove the following: Take $w^{\epsilon}_0$ a ...
1
vote
2answers
47 views

N2 diffusion through a vertical fluid column

Trying to figure out the mathematical model that might correlate to laboratory results. I have a cylindrical pressure vessel (picture a can) with height, h, and radius, r. It is filled with distilled ...
1
vote
1answer
45 views

Circulation of a Flow Field

Given the velocity components for a flow $$ u = 16x^2+y, \hspace{10pt} v = 10, \hspace{10pt} w = yz^2 $$ and a rectangular region $R$ in the $xy$-plane formed by the points $(0,0)$, $(10,0)$, ...
0
votes
1answer
27 views

Wall boundary condition

Why is it that at $y=0$ (at the wall), we have $v=0$ (vertical component of velocity)? Obviously $v$ cannot be negative there as there is no flow through the wall, however how do fluid particles ...
5
votes
2answers
96 views

What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
1
vote
0answers
21 views

A gronwall inequality

In Majda/Bertozzi book, Incompressible flows etc.. p.118,he uses Gronwall theorem on the following inequality: $$|\nabla v(.,t)|_{L^{\infty}} \leq C\left( 1 + \int_0^t|v(.,s|_{L^{\infty}}ds ...
-1
votes
1answer
45 views

Continuum hypothesis confusion (fluid dynamics)

For the quantity $\rho(x,t)$ with the continuum hypothesis am I taking the average value of the density at each point in the small volume surrounding the point $x$ or am I taking the average density ...
1
vote
1answer
17 views

One dimensional flow slowly changing cross sectional area

I am rather confused by what's written in the green box. If $\frac{\partial A}{\partial x}$ was not $<<1$ would this mean that the velocity now has a vertical component and is hence not ...
1
vote
1answer
43 views

Is the Mass flow rate (Mass flux) a scalar quantity?

Wikipedia states that mass flow rate is a scalar quantity, however Mass Flow Rate= Density x Cross Sectional Area x Velocity and velocity is a vector quantity, so this would imply Mass Flow Rate is ...
0
votes
0answers
103 views

How is Euler fluids equation considered unsolved?

Apart from the Navier-Stokes equation, the Euler equation is described by Clay Math Inst. as unsolved or not well understood. My question is, is there a special case of Euler fluids equation that they ...
0
votes
0answers
15 views

Is $ \textbf{u} = y e_x − \sin x e_ y + b e_z$ a solution of the unforced incompressible Euler equations with $D = \mathbb{R}^3?

Hint; compute $∇ × (u · ∇u)$ and use to solve the problem. I dont even know how to start this problem. May you help me solve this problem?