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
5 views

Convective operator: Distributive with superposition of vector fields?

I'm unable to find a great deal of information on this. I'm mostly sure that the convective operator over a vector field $A$ acting on a function $f$: $$ (A \cdot \nabla )f$$ is distributive, i.e. ...
1
vote
0answers
19 views

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$?

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$? ($ k > -2$). Using the divergence theorem, I got that the flux is: $\frac{3\pi}{k}(1-(-1)^k)$ and ...
0
votes
2answers
39 views

Solve $\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$

$$\frac {\partial^2 v}{\partial r^2}+\frac 1r \frac {\partial v}{\partial r}-\frac v{r^2} =0$$ i did $$\frac {1}{r}\frac {\partial }{\partial r}\bigg( r \frac {\partial v}{\partial r} \bigg) =\frac ...
1
vote
1answer
40 views

Finding $x$ and $y$ components of Navier Stokes

An incompressible viscous fluid of constant densite and kinematic viscosity occupies the space between porous walls at $y=0$ and $y=d$. The steady two dimensional flow is subject to a constant ...
1
vote
1answer
86 views

Integrate second order DE once

Given the vorticity equation $$\frac{D \omega}{Dt}=(\omega \cdot \nabla)\textbf{u}+ν\nabla^2ω$$ and $\textbf{u} = (−αr/2,v(r),αz) $ in cylindrical polars where alpha is positive constant. Find ...
0
votes
0answers
23 views

solution of small perturbation in fluid dynamics

Suppose we have a 2D flow in a narrow gap as shown in the picture The flow is governed by Navier-Stokes equation $\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - ...
1
vote
1answer
38 views

Stream function and Vorticity relationship with Streamlines

For a two-dimensional flow, the velocity is given by $\textbf u= (u(x,y,t),v(x,y,t),0)$. Define the stream function $ψ (x,y,t)$. Evaluate $(\textbf u·∇) ψ$ and deduce a relationship between the stream ...
2
votes
1answer
18 views

Derivative of the stress tensor

Let $\partial u_i/\partial x_i=0$ then given that $$\sigma_{ij} = -p\delta_{ij}+\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$ Show that $$\frac{\partial ...
0
votes
1answer
46 views

Finding pressure using Bernoulli's Theorem

The inviscid irrotational flow around a circular cylinder of radius $a$ is described by the complex potential $$w =Uz+ \frac{Ua^2}{z}$$ where $U$ is a positive constant. I found $\psi=U \cos \theta ...
1
vote
1answer
64 views

Finding stagnation points and stream function

Sorry for the lack of latex. The question I want to ask would need all this info and it would take very long to write it. (a) Irrotational flow means $\nabla \times \textbf u =0$ so we can define ...
1
vote
1answer
32 views

Using Euler's equation and vector identity

An unsteady incompressible inviscid fluid flow satisfies the continuity equation $∇·\textbf u = 0$ and Euler’s equation $$\frac{∂ \textbf u }{∂ t} +(\textbf u·∇)\textbf u = − \frac1ρ ∇p$$ where $\textbf ...
0
votes
0answers
17 views

Weak form of PDE for 2-phase-flow

I want to simulate the flow of 2 incompressible and immiscible fluids in porous media. It is assumed that the conditions are satisfied for the Darcy equation to be valid. The combination of mass ...
0
votes
1answer
31 views

Solve $f'' - \frac{iw}{\upsilon}f=0$

We are given $u=u(y,t)$ and $$\frac{\partial u }{\partial t}= \upsilon \frac{\partial ^2 u }{\partial y^2}$$where $\upsilon$ is the viscosity. Look for solutions of the form $u=Re\{Ue^{iwt}f(y)\}$ ...
0
votes
0answers
31 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge ...
1
vote
1answer
21 views

Viscous fluid boundary condition

Consider an incompressible viscous fluid of kinematic viscosity $ν$ , dynamic viscosity $µ$ and density $ρ$ . A viscous boundary layer is located over a solid surface at $y = 0$ and $x > 0$. The flow ...
1
vote
1answer
22 views

What does $p_b \propto ρ^n _b $ mean

$p_b \propto ρ^n _b $, in fluid mechanics, where $p_b$ is the pressure inside a bubble and $\rho$ is the density. What does that symbol looking like alpha mean?
1
vote
1answer
24 views

Manipulating vorticity equation

We have $\omega = (0,0, \xi(x,y,t))$ and $\textbf u =(u(x,y,t),v(x,y,t),0)$ and that $$\frac{\partial \xi}{\partial t} +u \frac{\partial \xi}{\partial x} +v\frac{\partial \xi}{\partial y}=0$$ is a ...
0
votes
2answers
20 views

${\partial\over{\partial x_j}}\left(\partial x_i\over\partial t\right)\ne{\partial\over{\partial t}}\left(\partial x_i\over\partial x_j\right)$?

$ \boldsymbol x = f(\boldsymbol X,t)$ is the position of a particle in an instant of time $\boldsymbol X$ is the initial position $t$ time $\boldsymbol u$ velocity In my opnion $f$ is continuos... ...
2
votes
1answer
46 views

Finding pdes of velocity component and pressure

An incompressible viscous fluid of constant density $ρ$ and kinematic viscosity $ν$ occupies the space above a solid boundary at $y = 0$ in two-dimensional Cartesian coordinates $(x,y)$. For time $t ...
-1
votes
1answer
25 views

Finding an expression for velocity [closed]

Consider an annulus formed by two circular cylinders, with one cylinder inside the other. The inner cylinder has radius $a$ and the outer cylinder has radius $b$. The cylinders have a common axis, and ...
2
votes
1answer
37 views

Taking curl of Euler equation

Consider an inviscid incompressible flow. Euler’s equation can be written as $$\frac{\partial \textbf u}{\partial t} + \textbf ω × \textbf u = −\textbf∇\bigg( \frac pρ + \frac 12 \textbf u^2 + V ...
2
votes
0answers
13 views

Show that $δ_{KL}$ is a Cartesian tensor

By using the definition of the Kronecker delta $δ_{KL}$, show that $δ_{KL}$ is a Cartesian tensor, that is $δ'_{MN} = L_{MK}L_{NL}δ_{KL}$ under the rotation $X_K = L_{MK}X'_M$. Solution: Using the ...
1
vote
1answer
34 views

Numerical Method for KdV travelling waves

Can someone please direct me to the best NUMERICAL method or some references for solving $$-cu_x + uu_x + u_{xxx} = 0$$ with periodic boundary conditions. This governs travelling waves of the KdV ...
1
vote
1answer
28 views

Find a limit for Doublet Stream function

In fluid Mechanics, The superimposed stream function of point source and sink is: $\psi=-\frac{Qcos\theta_1}{4\pi}+\frac{Qcos\theta_2}{4\pi}$ Graphical image of the function and for a sink - ...
0
votes
1answer
22 views

Finding streamlines

Find the streamlines, particle paths and streaklines when $$u=xe^{2t-z}, \, \, \, v=ye^{2t-z}, \, \, \, w=ze^{2t-z}$$ What is the track of the particle passing through $(1,1,0)$ at time $t = 0$? To ...
-1
votes
0answers
19 views

integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
4
votes
0answers
31 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
1
vote
1answer
28 views

Why is the magnitude of the curl of a vectorfield twice the angular velocity?

(if V is a vectorfield describing the velocity of a fluid or body, and $x\in R^3$) I agree that it should be when you look at the calculation, but intuitively speeking... If $\nabla \times V(x)= ...
0
votes
0answers
9 views

Derivation of the Quadrilateral Source Element used in Fluid Potentials

Background: Potential method is used to solve linear fluid dynamics problems still to this date, which is based off reducing the Navier-Stokes equation to an inviscid, irrotational, incompressible ...
1
vote
0answers
18 views

Flux of the amount of buffalo entering a square kilometer per minute

So Eq.(13) is just $\int F.n dS = \int \int div(F) dA$ So I did $div(F) = \nabla . F = y+1$, then I did $\int_2^3 \int_2^3 y + 1 dx dy = \int_2^3 y + 1 dy = .5(3^2) + 3 - (.5(2^2) + 2) = 3.5$ This ...
0
votes
0answers
14 views

Rankine ovals are oval shaped

I'm studying Rankine oval and I have a question. How do you prove the equation for its zero streamline curve is in fact oval shaped? Is there some family of oval shaped curves including Rankine Ovals? ...
1
vote
0answers
35 views

Dynamics of fluid

While reading on Wikipedia about the partial differential equations (https://en.wikipedia.org/wiki/Partial_differential_equation), I wondered how dynamics for the fluid occur in an ...
6
votes
0answers
92 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
0
votes
0answers
5 views

Confusion regarding phase velocities

I am revising a waves module and a past exam paper asks to calculate the horizontal components of the phase velocity (the horizontal wavevector is (k,l)). Immediately I believe that this is ...
0
votes
0answers
16 views

how to formulate an equation for this?(modified 3D sphere)

I was reading about a situation, where a sphere is close to a solid plane boundary. whose radius is 'a' and whose centre is 'a+b' away from the solid plane boundary. so they have a function h(r,t) ...
0
votes
0answers
18 views

How did they arrive at those equations?

I was reading this paper and I found the math behind some equations difficult to understand. can someone explain how they got the spatial dependence of the single-droplet velocity field as:(page 6) ...
0
votes
0answers
14 views

Phonon-like acoustic modes in a 1D unconfined crystal

I was studying this paper: http://www.sns.ias.edu/~tlusty/papers/PhysRep2012.pdf In page 21, they say: "To consider small fluctuations of the droplet positions around their lattice points, we define ...
0
votes
0answers
21 views

Find the pressure at a certain section of the pipe

Water flows through a sudden pipe enlargement at 0.35 m^3/s. Upstream the pipe diameter is 0.027 meters and downstream, 0.042 meters. At a point about 15 millimeters upstream of the expansion the ...
2
votes
1answer
89 views

Proof of $-\nabla\times\omega = \nabla^2 U$

What is a proof for $$ -\nabla\times\omega = \nabla^2 U $$ in the scope of fluid mechanics? I'm learning vector calculus for my project and stuck on this seemingly simple proof problem. Detailed ...
0
votes
0answers
23 views

Complex potential question

Consider the flow described by the complex potential $$w = Be^{−imπ}z^{m+1}$$ where $B > 0$ and $m ∈ (0,1]$. (i) Determine the stream function $ψ$ and the potential $φ$ in plane polar coordinates ...
0
votes
1answer
23 views

Calculating the Lagrangian map for a solenoidal vector field

I have two related questions. First what is a Lagrangian map? I've searched online but I have not been able to find an explanation that I understand. Second, for a solenoidal field $u(x,t)$ how do ...
0
votes
0answers
43 views

Solving the Navier-Stokes equations for a known density function

Consider the following form for the Navier-Stokes equations for a compressible fluid in steady-state: $$div(\rho \textbf v)=0$$ $$div(\rho \textbf{vv} )=-k^2 \nabla \rho$$ where k is a constant. If ...
1
vote
1answer
47 views

Milne-Thompson Theorem with a Vortex

I'm doing a problem related with Milne-Thompson theorem which tells that: "A cylinder of radius $a$ is immersed in a counter-clockwise whirlpool, which we model here as a potential vortex of intensity ...
2
votes
2answers
48 views

Curl of a Point Vortex Flow and its Circulation

I have the following 2D vector field $U=(u,v)=\frac{1}{x^2+y^2}(y,-x)$. When taking the curl of this field it returns zero. But when I take the circulation of the field defined as $$\Gamma=\oint_C ...
2
votes
1answer
47 views

Is there a way to relate an axisymmetric 3D flow field to cylindrical planar flow in order to determine the swirl velocity?

I have the following incompressible axisymmetric velocity field. $$u=u_r\hat e_r+u_\theta\hat e_\theta+u_z\hat e_z$$ For the planar analog to this flow (where the swirl velocity $u_\theta=0$) I know ...
2
votes
1answer
32 views

Vorticity Stretching in an Axisymmetric Flow Without Swirl

For an axisymmetric flow with zero swirl the velocity field is $$u=u_r \hat e_r+u_z\hat e_z$$ which results in the following vorticity field $$\omega=\omega_\theta\hat e_\theta$$ Here, ...
1
vote
1answer
19 views

Partial Derivatives and Operator Commutivity

I have an operator $$L\psi=\frac{1}{r^2}\partial_z^2\psi+\frac{1}{r}\partial_r(\frac{1}{r}\partial_r\psi)$$ I am interested in taking $\partial_rL\psi$ and $\partial_zL\psi$. Do the partial ...
0
votes
0answers
19 views

Nonlinear Maximum Principle estimate

Im interested in the in the 2D Boussinesq equations given by $$\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla p+\Lambda^{\alpha}u=\theta e_{2}\\ \nabla\cdot u=0\\ ...
2
votes
3answers
70 views

Why is $(\mathbf{v} \cdot \nabla)\mathbf{v} = (\nabla \times \mathbf{v}) \times \mathbf{v} + \nabla (\frac{1}{2} \mathbf{v}^2)$?

The Convective Derivative or Material Derivative is usually written as $\frac{D}{Dt}=\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla$. According to MathWorld, this equation, multiplied with ...
2
votes
2answers
58 views

Can we get a tensor out of summation of one vector with its transpose?

I don't know much about tensor calculus and here is something I'm trying to figure out. $$T=\mu({\nabla}\vec{V}+{\nabla}\vec{V}^T)$$ T is viscous stress tensor and $\vec{V}$ is the velocity vector. ...