The Navier-Stokes equations describe fluid flows in continuum mechanics.

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Is the flow of a viscous fluid in free space under no pressure gradient always laminar?

Consider a (Newtonian) incompressible viscous fluid in three spatial dimensions, whose velocity field $\mathbb{v}=\mathbb{v}(x,y,z,t)$ moves according to the Navier-Stokes equations $$\tag{1}\label{...
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What is the velocity in the Navier-Stokes equation?

I have been looking at the Navier-Stokes equation, and can't seem to find anywhere a clear description of what velocity it represents. From what I have read it could be any of the following: The '...
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Energy/density fluid/solid boundary condition

Working on solid/fluid couplings (for atmospheric purposes) I try to better understand how things work at the interface between an elastic medium and a compressible fluid. What I understood from ...
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Non-dimensionalizing incompressible Navier-Stokes

I have a question about non-dimensionalization of the incompressible Navier-Stokes (NS) equations. My understanding is that the purpose of non-dimensionalization is to "collapse" solutions onto one ...
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Inertial Waves - Why neglecting the advecting term?

I'm trying to derive the dispersion relation for Inertial waves. In Cartesian coordinates: Inviscid and incompressible fluid is rotating uniformly with Angular Velocity: $\Omega = (0, 0, \Omega)$ ...
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Quotient Rule in Vector Calculus

Wikipedia gives the quotient rule for (1) the gradient of two scalar fields "$f$" and "$g$" and (2) the divergence of a vector/tensor field and a scalar field "$\boldsymbol{A}$" and "$g$" as $$\nabla ...
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Why is turbulence caused?

In high Reynolds numbers we have turbulent flow. This is because the inertial forces are much greater than the viscous forces. I understand inertial forces to be actually the fictional forces due to ...
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Finding boundary condition of stationary solid body

A fluid flows past a stationary solid body of arbitrary shape. Write down the boundary condition on the fluid velocity $\textbf u$ for an inviscid fluid and for a viscous fluid, at the solid surface. We ...
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Stokes-Einstein Relationship to find time taken to diffuse $x$ distance? [closed]

A molecule has a diffusion coefficient of 0.5 × 10-9 m2s-1. Calculate how long it would take on average for the molecule to diffuse 10 µm. So I have the above ...
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Transverse and longitudinal random forces

I am trying to read following article: http://arxiv.org/pdf/1410.1262v1.pdf According to the equation (2.10) and (2.11), the random force is defined as $ \langle f_i(x) \ f_j(x) \rangle = \delta(t-t'...
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Navier-Stokes Energy Equation

I've been assigned (for homework in a mathematical modelling course) the task of deriving the Navier-Stokes energy equation in one space dimension: Consider a fluid flowing through a cylindrical ...
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Wall Shear Stress

I have the solution of a Navier-Stokes simulation with an incompressible, Newtonian fluid with laminar flow. Now I compute the wall shear stress (vector) as $$\tau_n = \mu (\nabla u) n,$$ where $\mu$ ...
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Mathematical understanding of vortex solitons

I am wondering if anyone has ever come up with a mathematical description of something that (to me, and I am no expert) look like soliton vortexes. The example I can think of is if you create two ...
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Advection Operator shift in scalar product

Can someone help me with advection operator shifts? I can't figure out the rule for the shift inside of a scalar product. The terms $(u,(v\cdot \nabla)\delta v)_\Omega$ and $(u,(\delta v\cdot \nabla)...
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Basis for Derivation of Stokes Friction Law for Spheres

When deriving Stokes law one uses the Navier Stokes equation with the assumptions: low Reynolds number stationary flow in compressible flow leading to this version of the N.S : $$\nabla p = \...
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Navier Stokes : what about angular momentum?

I play with CFD for a while, and suddenly, a transcendantal question raises: :-) Navier Stokes is basically Newton applied on a continuum in Eulerian. For solids, we would consider linear, but also ...
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What fluid dynamics equations are like in zero gravity?

I don't know if this is a proper question. I am not so familiar with fluids. I am just curious about what Navier-Stokes equations for fluids will look like in zero gravity. Are they stay the same? If ...
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Incompressible Navier-Stokes boundary conditions

Let's say I have a unit cube $\Omega\in[0,1]^2$ where the inflow is on the left and outflow on the right, at the top and bottom boundary I have no-slip $u_1 = u_2 = 0$. At the inflow I prescribe ...
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Why Velocity is referred as momentum? [closed]

In many text books the velocity is referred to as a linear momentum which is being convected. For example the table in the following page My old conception is that the momentum or more precisely ...
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Numerical model for a Air core Vortex, why it's still so limited? [closed]

The Question can be Formally presented as follows; How is the Numerical Model (CFD, Navier Stokes) of fluid dynamics for Vortexes at in it's limits? At this Publication from 2013 at page 48 is said;...
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The change in time of a concentration in a fluid can be described by Reynolds' theorem. Is that the whole story?

Let $d\in\left\{2,3\right\}$ and $\Omega_t\subseteq\mathbb R^d$ be the bounded set occupied by a fluid at time $t\ge 0$. Moreover, let $\eta_t:\Omega_t\to[0,\infty)$ be the concentration of imaginary ...
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Is a creeping flow with $u_r \sim \ln r$ physically possible?

I was wondering if it is possible to have a 2D cylindrical flow where the radial velocity scales with $ln (r)$. I understand that a flow with $u_r \sim 1/r$ corresponds to a line source or sink. Also $...
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assumptions about sound waves

When deriving the sound wave equation: $${1 \over c^2} {\partial^2 p' \over \partial t^2 }= \Delta^2 p' $$ by linearizing the Euler equation: $$\rho {d v \over dt }= - \nabla p $$ and the continuity ...
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Does this dimensioneless quantity have a name?

When studying creeping flows, a common choice for a characteristic pressure scale is $$p_0 = \frac{\mu_0 U_0}{L_0},$$ where $\mu_0$ is a reference dynamic viscosity, $U_0$ is a reference velocity and $...
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Fluid dynamics equations, number of variables and number of equations

Continuity and Navier-Stokes equation for fluid are, \begin{eqnarray} \frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \mathbf{u}) &=& 0 \\ \rho\left(\frac{\partial \mathbf{u}}{\partial t} ...
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Reference values for viscosity and density in incompressible NSE

I come from a pure mathematics background, so I have very limited physics knowledge. I'm currently working out the non-dimensional form for the Navier-Stokes equations and have some questions. Where ...
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Navier-Stokes: equation or equations [closed]

In textbooks and papers, you see both forms: the Navier-Stokes equation and the Navier-Stokes equations. Which one is correct and why?
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What does $\mathbb{R}^3$ and $\mathbb{T}^3$ look physically for the Navier-Stokes equation?

What does the Navier-Stokes equation solution according to the Clay Math Institute look like in real life? As in how do you visualize $\mathbb{R}^3$ and $\mathbb{T}^3$ without the math? I actually ...
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Is there a formulation for (self-)accelerating fluid flow through permeable medium?

I have a permeable system where is an accelerating fluid flow. Imagine a sponge that is squeezed. The fluid starts at rest, accelerates and flows out from the sponge. How to calculate the fluid speed? ...
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Stokes law in 2-dimensions

Stokes' law states that force on slow moving sphere (i.e. $Re\ll1$) in liquid is $$ F_d = 6 \pi \mu R V $$ In two dimensions we are in trouble (flow around disk in 2d or around cylinder in 3d), ...
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Visco-elastic fluid | stress/strain relationship

I'm working on a moving visco-elastic fluid with a absorption law (against frequency) that can be represented by a Zener model (Gaussian quality factor). I try to make a numerical modeling of it (...
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Maximum pressure in a die-less wire drawing apparatus

I ran across this patent: http://www.google.com/patents/US4549421 and was interested by the idea of reducing the cross sectional area of a wire with only shear stress caused by the wire being pulled ...
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How does the dissolution of salt affect the solution density?

Suppose you have a container of water as a solvent and you a certain amount of salt as a solute sitting at the bottom of the container that has yet to start dissolving. Supposing temperature and ...
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Why is the Reynolds number “the way it is?” Why is its order the way it is?

Why is the Reynolds number “the way it is?” Why is its order the way it is? I'm not sure if this is an appropriate question for this context, but I would like more intuition on this matter and so I'...
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What is this form of mass conservation equation?

I found the following equation of conservation of mass (continuity) in "Computational Fluid Dynamics Vol.III" by Hoffmann: $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho u)+ \...
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How does the mathematical definition of drag reduce to Stokes or form drag?

I know that for the flow of flow of a Navier-Stokes fluid in a domain, once the velocity $\mathbf{v}$ and pressure p are known, the drag over a solid object with boundary $\partial R$ is given by ...
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Reynolds Average Navier Stokes equations and turbulence scale

To obtain the time average of an unsteady term like $\frac{\partial u_{i}}{\partial t}$ by definition we perform the following: \begin{align} \overline{\frac{\partial u_{i}}{\partial t}} &= \...
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Origin of pressure gradient in Navier-Stokes integral

I am not understanding the origin of the second term on RHS of momentum conservation equation (cf. the Wiki page), $$ \frac{\partial}{\partial t}\int_V\rho\mathbf u\,dV=-\oint_S\left(\rho\mathbf u\...
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Velocity profile of a viscously damped wave

For a test case, I want to determine the velocity profile of a viscously damped standing wave. By linearizing the density ($\rho=\rho_0+\rho'$) and velocity ($ux=ux'$), the continuity and Navier-...
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No diffusion term in conservation of mass in Navier-Stokes equations?

I have followed derivations of the Navier-Stokes equations and I can see how the various terms arise in the "main equation", the momentum conservation equation. However I don't understand why the ...
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Does a whirlpool(vortex in water) continue in air(vortex in air),and when does a vortex stop?

First part: The question is both about the continuity of the water vortex(whirlpool) to vortex in air in time and in space. About continuity in time,does the vortex of the water slowly produce a ...
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Why can't the Navier Stokes equations be derived from first principle physics?

At the 109th UCLA Faculty Research lecture, Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles [...
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Value of Stokes constant for air-water vapour interaction

I wish to estimate the Stokes force between air and water vapour. Where can I find a reference for the corresponding "Stokes constant"? Assume we have a composition of water vapour with air. I ...
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Specific form of Stokes's differential equation

Coming from a chemical background, I have next to no knowledge of the (as it seems to me) complex field of fluid dynamics, so bear with me here. I'm reading a paper written by seismologist Norman ...
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Incompressible Navier-Stokes pressure solve in simulations

I am a complete newcomer when it comes to fluid simulations. I'm currently working through some tutorials to understand the idea of of the discretized Navier-Stokes equations for numerical simulations....
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Solve unsteady state Bernoulli equation for flow in a pipe

I am an engineer studying an unsteady-state flow through a pipe. The Pipeline has been cleanly cut into two halves, without deforming the cylindrical form of the pipe, exposing the contents to ...
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Exact solution to a 2D/3D Poiseuille flow in a channel

Hi everyone and thank you in advance for any help. I am struggling to find an analytical solution to either a 2D or 3D Poiseuille flow in a rectangular duct. All I can find is 1D example. Can someone ...
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Why is the solution to the Blasius boundary layer problem self-similar?

In every course or textbook that I encountered so far, the authors transform the Navier-Stokes equations of the Blasius boundary layer problem into the Blasius ODE. The problem with many of those ...
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Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ...
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Fluid flow: Force acting on the fluid and the Navier-Stokes equation

Consider a one dimensional fluid flow in a rectangular tube. Typical streams are the poiseuille streams. Consider the case in wich we apply a force on the fluid. The Navier-Stokes equation (for ...