Questions tagged [navier-stokes]

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

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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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Why does Anderson ignore a derivative of a normal viscous stress?

I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. In part 15.6, he said: Consider a steady two-dimensional, viscous, compressible flow. The x-momentum equation for such a ...
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How can a system of hoses each defined in their own non-inertial reference frame transmit energy information?

Rotating pipes <- image Since energy is frame dependent, I can't find a relationship that makes energy invariant at the hose connection points in respect to each local frame in the general case. ...
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a flow through a pipe submeged in a moving fluid

There is a classic problem by which a fluid having a uniform velocity profile enters a tube. The pressure at the outlet is atmospheric, and the dimensions of that tube (diameter and length) are known. ...
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Can someone explain the Navier Stokes equations? [duplicate]

I am doing a research work on the Navier stokes equations but I don't really understand them, as I am a high school student. Could someone explain them to me in an easy way but kind of specific? Thank ...
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Incompressible Navier-Stokes equation in Fourier Space

I'm trying to derive the incompressible Navier-Stokes equation in Fourier space, given in e.g. Kraichnan & Montgomery (1980), Rep. Prog. Phys. 43 547 (PDF) (but probably countless other places), ...
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Original Derivations of Euler Equations or Navier-Stokes Equations

I've seen the derivations for both viscous and inviscid momentum balance in fluid flow in courses, but I'm curious as to where the original derivation for the equations now referred to Euler equations ...
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Stokes's law proportionality to radius

Is there a logical explanation why Stokes's drag $F_d=6\pi R \eta v$ is proportional to the radius, $R$ of the sphere? Naively I would have expected that it is proportional to the cross section, i....
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Bulk viscosity in jump conditions for shock waves

One of the jump conditions for shock waves can be derived from the Navier-Stokes equation and it takes the form (see Shu, Gas dynamics, p.212) $\dfrac{d}{dx}\left(\rho u^2 + P - \frac{4}{3}\mu \frac{...
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Incompressible Fluid Flow around sphere (Stokes)

Stokes solved this 1851. I have a question regarding the derivation. Following Batchelor the equations to be solved are \begin{align} \nabla \left( \frac{p - p_0}{\mu} \right) = \nabla^2 \vec{u} = -\...
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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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Friction between 2 fluids in Navier Stokes

I have a system with 2 different fluids, and I want to write a friction/viscosity term between them in their Navier-Stokes equation. Should it be in the form $M(v_{1}-v_{2})$ with $M$ a friction ...
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Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration?

The incompressible Navier-Stokes equations widely used in hydrodynamics don't have the gravitational acceleration. $$ \begin{align} \frac{\partial u_i}{\partial x_i} & = 0, \\ \frac{\partial u_i}{\...
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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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Complex solutions to the Navier-Stokes equation

I had recently ended up with a case where on solving the Laplace equation (for a fluid under certain conditions), the radial dependence turned out to be complex (in general). In such cases, do we work ...
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How to prove that Navier-Stokes equations are Lorentz invariant?

I've been hearing recently that the Navier-Stokes (NS) equations are invariant under a Lorentz transformation, so I tried to prove it just changing terms of transformed velocity instead of the ...
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Mass conservation equation

We have a small drop spreading on a completely wetting solid substrate. The drop shape is h(s,t). The coordinate system is cylindrical (s,z). The velocity fields are: u(s,z) in the s direction and w(s,...
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A Hurschel-Bulkley fluid flows through a conical nozzle

I got an physical problem in paste extrusion that I am unable to solve: A pasty material flows through a conical nozzle. The material follows Hurschel-Bulkley flow model (H-B model): $\tau=\tau_y+...
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Physical intuition for Incompressible Navier Stokes second term

I'm trying to get an intuitive understanding of the Incompressible Navier Stokes equation (as for me thats the only way I can use it effectively and avoid the rote learning method): $\rho(\frac{\...
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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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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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Incompressible 2D Navier-Stokes equation

I am trying to solve for and simulate the vorticity numerically (finite difference method), however there's one part I was hoping to get some help with. I need to find the fluid velocity $\mathbf{u}$ ...
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2D Uniform Flow Inclined Plane - Reynolds Averaging: Leads to no Turbulence?

We're looking at a fully developed flow along an inclined plate, the $x$ coordinate is along the plate and the $z$ coordinate is perpendicular to it. In 2D, uniform flow I end up with the continuity ...
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Interpretation of the time scale $L^2/\nu$

During the scaling of the Navier-Stokes equations it is often made use the viscous time scale $L^2/\nu$, where $L$ is the characteristic length and $\nu$ the kinematic viscosity. What is the physical ...
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Cascade vs Inverse Cascade: Energy flow in 2D and 3D turbulence

We know that vortices tend to coalesce in 2D(inverse cascade) as opposed to 3D where they 'break down' into smaller vortices(cascade) ie. energy flows fro larger to smaller scales in 3D and opposite ...
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Boundary Layer Physics and Bernoulli Principle

I was taught that the relative velocity of all fluid particles directly in contact with the boundary in a fluid flow is zero. In other words, that the relative velocity of fluid particles in the ...
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Navier-Stokes eqs. correspond to $F=m*a$

I have the (typical) Navier--Stokes system for incompressible fluid: $$div(u)=0$$ $$\rho(u_t+u\cdot\nabla u)=-\nabla p+div(\nu\nabla u)+\rho g$$ In a paper that I'm reading says that the term $$u_t+...
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momentum balance equation for low reynolds number flow

This is a flow in a 2-D microchannel (h/L<<1). Low Reynolds number flow. The mass - conservation equation states $$\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}=0$$ ...
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Exact solution of micropolar fluid for Poiseuille flow

I am new here, so i wondered if you could help me with a sytem of ordinary equations with constant coefficients. This model of equations refer to the well known Navier Stokes equations of fluid ...
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Second-order covariant derivative in index notation [closed]

So I'm having problems finding the second order covariant derivitive in index notation. My teacher said to just find the covariant derivative of a covariant derivative, so I first started with finding ...
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What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?

I am studying the Navier-Stokes equations and I have the equation in the form: $$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
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Viscosity and energy balance

When solving Navier Stokes equations for viscous fluid over rigid surface, the viscous term in the momentum equation accounts for the momentum transfer between the fluid and surface in the near wall ...
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Will a proof of the existence and smoothness of the Navier-Stokes equations contribute to a GUT? [closed]

Note: throughout the course of the question, the word "describe" will often be used to suggest that a mathematical equation can "describe" what it happening in a physical context. One of the long ...
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Fluid simulation - why gradient of pressure is the term $\delta q$ in the projection step?

I'm learning stable fluids by reading Stam's paper. In the paper, it says, according to Helmholtz-Hodge Decomposition, I vector field $w$ could be decomposed to a divergence-free field and a curl-...
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What does $ \vert \partial^{\alpha} v_o(x) \vert $ mean in the Navier-Stokes initial velocity condition?

The initial condition $\displaystyle \mathbf{v}_0(x)$ is assumed to be a smooth and divergence-free function such that, for every multi-index $\displaystyle \alpha$ and any $\displaystyle K>0$, ...
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What is the importance for the pressure/velocity at infinity be constant in fluid dynamics?

I am studying fluid dynamics on my own and it is commom to see this assumption. I am asking this here because I didn't find any satisfactory answer. For example, I am studying a problem that is as ...
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Rationale behind the linearised Navier-Stokes equation

Some applications of fluid dynamics consider the linearised Navier-stokes equation where the advection term $(\vec{u}\cdot\vec{\nabla})\vec{u}$ is dropped. I am trying to build a convincing argument ...
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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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How were the Navier-Stokes equations found in the first place if we can't solve them?

I was reading up on the Clay Institute's Millenium prizes in mathematics. And I noticed the Navier-Stokes equations were described as minimally understood. As far as I was taught in physics a few ...
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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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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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Validity of the Navier Stokes equations for turbulent flows

The derivation of the Navier-Stokes equation presupposes that the pressure, $p$, and velocity, $u_i$, are infinitely differentiable, so that the forces in each face of the fluid element can be ...
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What causes fluid rotation in the Navier--Stokes equation?

It is known that rotation in the flow results from the viscous terms in the Navier-Stokes (N-S) equation. However, when deriving the N-S equation from the general principle of linear momentum in ...
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Terminal velocity of a settling sphere in 2D vs 3D

How can one analytically calculate the terminal velocity of a settling sphere in 2D? Actually it would be a circular disk. One cannot simple equate boyancy forces minus the drag right? As stated in ...
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Stream function formulation for Navier-Stokes equations in 2D - boundary conditions

I'm particularly interested in 2D incompressible version of Navier-Stokes equations that describe flow in some domain of interest: $$\begin{aligned}\frac{\partial v_x}{\partial t} + \vec{v} \cdot \...
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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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Sound speed on Navier-Stokes/Euler equations

I'd like to simulate strong shock (i.e., Rankine Hugoniot conditions) on inviscid condition, using the non-conservative Euler equations and I don't know if I should use the relation, $$ c^2=\left(\...
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What does it mean that a substance can be smelled from far away?

I thought about this question in the middle of this video. Ok, Thioacetone takes the price for the World's smelliest chemical, I can accept it (why not?), but what about You can smell one drop of ...
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Instantaneity of Stokes Flow

Background Info: A steady-state incompressible newtonian Stokes fluid occupying a domain $\Omega \in \mathbb{R}^2$ has flow velocity $\textbf{u}$ and pressure $p$ given by the following system of ...
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Is there an Okubo-Weiss function for axisymmetric flows

I have recently learned about the Okubo-Weiss criterion in a ($x,y$) and a ($r,\theta$) coordinate system. Is it possible to recast this criterium in an ($z,r$) system, that is for axisymmetric ...