Questions tagged [navier-stokes]

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

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What is the physical explanation as to why the kinematic boundary condition must hold at the free surface of a wave?

The kinematic boundary condition at the surface of a water wave is given as: "a particle on the free surface remains there always". This is then written as the material derivative of the free surface ...
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How can I derive the stress tensor for a Newtonian fluid in more physical terms?

The question is quite fundamental and more on a beginner's level (not sure if good in this high-level-forum, but I try): I have big problems in understanding the stress tensor for Newtonian fluids in ...
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130 views

Velocity field to a permeability field using poisson pressure equation

I have a velocity field and I want to get a pressure field. In my experiment we're controlling the pressure at the inlet and the outlet. I have Dirichlet boundary conditions at the inlet and outlet ...
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Governing equations vs Transport equations

I posted it in computational-science SE site, and it was suggested I shift it here. This is a basic question. But I did not find any explanations for this. How are governing equations, like mass, ...
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Viscous Flow past a cylinder

I'm trying to solve the incompressible, viscous and small Reynolds number flow past a cylinder. At the surface of the cylinder ($r=R$) the velocity is zero and at infinity it is $v_0 {\vec e}_y$ where ...
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How do I calculate the change of pressure on a fluid moving on block?

Lets assume block motion for a fluid. From Navier-Stokes equation we got $\vec \nabla p = \rho (\vec g - \vec a)$ Lets say s is the direction where pressure has the steepest increase. How do you ...
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59 views

Deriving the Integral Form of the Navier Stokes equation

I'm trying to follow the book Turbulence by Davidson. Currently I'm having trouble in converting the differential NS equation to its integral form but I cannot see clearly how the Divergence theorem ...
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623 views

Inversion of a metric

I am currently reading a paper by Bredberg $et.al$ arXiv:1101.2451 titled "From Navier-Stokes to Einstein". In this paper, the authors have considered a metric of the form \begin{eqnarray}ds^2_{p+2} = ...
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Why shear stress is assumed constant in the inner layer

In the derivation of the log-law and the viscous sub-layer velocity profiles, it is customary to assume that the shear stress is constant and equal to the wall shear stress. Is there any physical or ...
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How should energy loss in a hydraulic jump be calculated?

The energy loss in a hydraulic jump is still calculated with the old equation of Bresse from year 1860; (I.e., equation 7 in this paper from 2017) $$ \frac{\Delta E}{E_1} = \frac{(\sqrt{1+8Fr^2}-3)^3}...
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What are the limitations of this form of the Navier-Stokes equation?

$$ \frac{∂u}{∂t}+u\frac{∂u}{∂x}+v\frac{∂u}{∂y}+w\frac{∂u}{∂z}=\frac{−1}{ρ}\frac{∂P}{∂x}+gx+ν(\frac{∂^2u}{∂x^2}+\frac{∂^2u}{∂y^2}+\frac{∂^2u}{∂z^2}) $$ Why would someone use a form of this equation ...
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How do speed and density change in a turbulent flow?

Suppose a fluid passes from having laminar flow, to having a turbulent flow (like when passing after an object). How do fluid speed and fluid density change after that?
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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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Governing equations and boundary conditions for a steady-state compressible viscous flow in an axisymmetric annular orifice

I'm trying to simulate a 2D axisymmetric model of steady-state compressible viscous flow using Mathematica, but I get some errors. There is a chance that I'm making some mistakes with the governing ...
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What is the turbulent Navier Stokes equation for cylindrical coordinates?

I am looking for turbulent Navier Stokes equation for cylindrical coordinates. I know that RANS (Reynolds Averaged Navier Stokes) eq. is the solution, I understood the point of it but only for ...
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Navier-Stokes : divergence or covariant derivative of a tensor : 1 vector result?

I don't understand very well the following definition concerning Navier-Stokes equation : where $\vec{u}\otimes\vec{v}$ is a tensor (2,0), isn't it ? This is not scalar since $\vec{u}\,\vec{v}^{T}...
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Characteristics of the Navier-Stokes equations as a set of PDE's

I am not entirely sure if I should ask this question here or not, but here goes: can anyone suggest any reference (book, article, etc.) about the Navier-Stokes equations from a mathematical point view?...
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How to classify your fluid is a transitioning from a liquid to gas from numerical computation

I saw at one time that if the kinetic energy/potential energy of the gas was approaching 1 then the gas is becoming a liquid. I can't find the reference where I found that though (it was on stack ...
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303 views

Pressure generation due to opposing flows

If two fluids are flowing at unequal velocities towards each other in a circular pipe, will a pressure be generated at the intersection? If yes, what will the direction of this pressure generated and ...
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199 views

Energy dissipation for force-free, incompressible Navier-Stokes equation with tangent boundary conditions on bounded domains

I consider the Navier-Stokes equation for uniformly incompressible, force-free, Newtonian fluids with constant viscosity. The equations describing the situation are: $\partial_tv-v\times \text{curl}(...
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312 views

Relationship between flow and pressure gradient in a one dimensional compressible fluid

Consider a one dimensional model (tube with diamter $D$) with a compressible viscose ($\mu$) fluid (e.g. air). further assumptions are conduction and radiation are negligible gravity's effect is ...
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How do you interpret this symbol within the following Navier-Stokes equations?

I was reading this fluid simulation paper(http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/ns.pdf) by Jos Stam and encountered these Navier-Stokes equations. 1. 2. Now, I know that ...
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Assumptions for 2d simplification of Navier-Stokes flow

There are many cases where Navier-Stokes flow is simplified to a two-dimensional problem to reduce the costs for a numerical simulation, e.g., flow around an airfoil, channel flow or pipe flow. To ...
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Does the “O” in this google doodle for Olga Ladyzhenskaya have anything to do with her work, or is it completely fanciful? [closed]

Google Doodle celebrates mathematician Olga Ladyzhenskaya She was famous for fluid dynamics and partial differential equations, both of which are beyond my pay grade. And she worked on the Navier-...
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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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743 views

Zero velocity divergence for incompressible flow is derived from conservation of energy equation or conservation of mass equation?

I'm a bit confused about incompressible flow definition. In many textbooks or scientific articles, they simply claim that the incompressibility condition for Navier-Stokes equation is: $\nabla \cdot \...
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Navier-Stokes - Reynolds decomposition of energy equation

I am trying to apply the Reynolds decomposition to the Navier-Stokes equations for incompressible flows. At the moment I am doing that for the energy equation following the book Viscous Fluid Flow by ...
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Variational Navier-Stokes: where to find study material “for dummies”?

I have worked with the Navier Stokes equations before but I'm a physicist. I was talking to a mathematician and they use a complete different notation and I am very lost. First of all, I use the ...
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168 views

Species Source Terms on Navier-Stokes Equation (Reacting Flow)

I have a quick question: if i decide to model two reactants (for example kerosene+oxygene) and they have their own NS equations separately (own velocity, density, temperature). For this case, should ...
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First Order Approximation of the Navier-Stokes Equation: Order of Magnitude of the Gradients of First-Order Fields

I am currently working on a project in acoustics and I am studying first and second-order approximations to the Navier-Stokes equation. I have been reading the book 'Theoretical Microfluidics' by ...
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Bulk and dynamic viscosity in the atmosphere

I'm studying the physics of the atmosphere but I'm struggling with the matter of viscosity (Navier-Stokes equation) for gravito-acoustic waves. From Landau-Lifschitz : $$ (T)_{ij} = -p\delta_{ij} + \...
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Some confusions about Navier-Stokes equations

I just started working on the Navier-Stokes equations. I consider the following paper Seibold A compact and fast Matlab code solving the incompressible Navier-Stokes equations on rectangular domains (...
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Is Stokes equation a reduction of Navier-Stokes equations?

The following Stokes problem: $$\begin{cases}-\nu\Delta u+\nabla p=f&,\textrm{in }\Omega\\ \nabla\cdot u=0&, \textrm{in } \Omega\end{cases}$$ is a reduction of the Navier--Stokes equations? ...
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By which physical mechanism does the continuity equation in fluid mechanics work?

The Navier-Stokes equations consist of the momentum equation and the continuity equation. Consider the incompressible versions for the purpose of this question. Continuity is always talked about as a ...
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Confused about Navier-Stokes equation

Just look at the L.H.S of the compressible navier-stokes equation from wiki $$\rho(\partial_t \vec{u}+\vec{u}\cdot\nabla\vec{u})=...$$ How can I add a vector $\partial_t \vec{u}$ and a scalar $\vec{...
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Turbulence Model on Unsteady Navier Stokes

I am asking you if the Unsteady (Time-Dependant) Navier-Stokes Equation is able to predict accurately the Flow Turbulence? I know that the RANS (with different Turbulence Models like Spalart–Allmaras, ...
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Is Navier stokes a turbulence model?

Is Navier stokes a turbulence model? If yes, what is the use of k-omega model.. if no, what does the Navier stokes equation got to do with the turbulence models ..? I am very new to fluid dynamics... ...
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Index notation with Navier-Stokes equations

This is an index-notation question rather then the NS one: For incompressible flow and Newtonian fluid, the continuity equation is denoted with: $$\frac{\partial u_i}{\partial x_i} = 0, $$ which ...
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Converting this navier stokes solution into a incompressible solution?

I have an equation for a non-viscous compressible fluid with density, pressure and velocity given by: $$ \begin{align} \rho(x, y, z) &= \frac{3B}{a^2 + x^2 + y^2 + z^2} \\ p(x, y, z) &= \frac{...
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Are there any cases where Stokes law does not apply in viscous fluids?

A friend of mine and I are conducting an experiment to find the relationship between terminal velocity and radius of a sphere (i.e trying to confirm Stokes law). We are using spheres ranging from 1....
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How would you model a single point outputting fluid in all directions whilst enclosed in a sphere?

The idea is this: There is a point at the center of a sphere. This point is releasing a fluid (say water) in all directions towards the edges of the sphere. The fluid can collide with the sphere's ...
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724 views

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 = \eta \...
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Combined Poiseuille-Couette flow

I stumbled upon this exercise in James Fay "Fluid Mechanics" book, which I'm using to learn fluid dynamics by my own, and I am struggling a bit with it, any help will be appreciated: The figure ...
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Flow up an inclined plane (laminar flow)

While learning fluid-dynamics, I stumbled across this problem that requires me to calculate the volumetric flow rate Q of a thin layer of oil leaking from a small hole in a barge, as depicted in the ...
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What is the interpretation of $(\textbf{u} \cdot \nabla)\textbf{u}$ in Stokes flow?

In very low Reynolds number flows, the Navier-Stokes equations can be reduced to the Stokes equation which is given by \begin{gather} \mu \nabla^2 \textbf{u} = \nabla p \end{gather} I am looking at ...
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Should non-relativistic Navier Stokes Equations be modified so that they become pseudo-Lorentz invariant?

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 ...
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Is it possible to define background independent fluid dynamics equations?

Imagine a lake, and you measured the distance from each molecule to it's neighbours for molecules to within say 3 molecular radii. Taking this data you could reconstruct the positions of the ...
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Difference between Eulerian and Lagrangian formulation of Fluid Dynamics

Difference between Eulerian and Lagrangian formulation of Fluid Dynamics. I am completely new to fluid mechanics. Until now definition $F = ma$ was sufficient for me to solve any rigid body problems ...
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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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What are the Lorenz Equations used for?

In fluid dynamics I have come across two sets of equations, the Navier-Stokes equation and the Lorenz equations. From what I have read the Navier-Stokes equations always holds. So why do we need the ...