Questions tagged [navier-stokes]

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

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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 ...
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Stressing Over Stress Tensor Symmetry in the Navier-Stokes Equations

How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations: Behavior of a Vorticity Influenced ...
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Derivation of Darcy's law from Stokes equation

On the Wikipedia entry of Darcy's law, a derivation of Darcy's law from Stokes equation is provided. The derivation starts at the Stokes equation, which reads: $$ \mu \nabla^2 u_i + \rho g_i - \...
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Introducing stream function with given velocity equation

Bit information about the problem We are dealing with the slide coating process - where basically a polymer is being put onto a slot, which is moving in the $x$-direction with velocity $v_0$. The ...
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(Fluid Dynamics) Euler's equation including gravity

In fluid dynamics, we can write down the Euler's equation as $\dfrac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v} \cdot \mathbf{\text{grad}} ) \mathbf{v} = - \dfrac{\mathbf{\text{grad}} \; p}{\rho}...
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Stokes flow between 2 spheres

The problem is given like this: Given two rotating spheres with constant angular velocity $\Omega_1$ and $\Omega_2$ around the vertical axis and pressure on the borders of spheres is $p_1$ and $p_2$...
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navier stokes equation with no boundary condition

if we may consider the unsteady motion of flat plate in infinite fluid and make Navier stoke equation into one-dimensional heat equation can we use only initial condition and one derivative condition ...
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Derive mechanical pressure from strain energy density

Similarly to thermodynamic pressure, when we have: $$P_{therm} = (\frac{\partial U}{\partial V})_{S}$$ Can we define the mechanical pressure for a fluid as: $$P_{mech} = \frac{\partial \psi}{\...
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Navier stokes equation - two dimensional simplified [closed]

The general Navier Stokes Equation is $\dfrac{D\vec{v}}{D t}= \dfrac{d\vec{v}}{d t}+ \vec{v} .\nabla \vec{v} = \vec{g} - \dfrac{1}{\rho} \nabla p + \nu \nabla^2 \vec{v}$ The above equation can be ...
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What are the assumptions of the Navier-Stokes equations?

I wanted to model a real life problem using the Navier-Stokes equations and was wondering what the assumptions made by the same are so that I could better relate my entities with a 'fluid' and make or ...
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What is the meaning of pressure in the Navier-Stokes equation?

I have a hard time to wrap my head around pressure in the Navier-Stokes equation! It may sounds ridiculous but still I cannot understand the true meaning of pressure in the Navier-Stokes equation. Let'...
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Non-dimensionalization of Navier-Stokes equations multiphase flows

I am currently dealing with multiphase flows and have to use the non-dimensional form of the Navier-Stokes equations (NSE). In the scientific literature I found various formulations (and almost no ...
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Coupling Navier-Stokes and stochastic models for particle tracking in micro-scale free convection?

I have been using a commercially available software to simulate laminar free convection in a specific small domain (let's use channel w/ heated lower wall as an example). The scale is approx 50-100 ...
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Derive total energy balance equation from Chapman-Enskog analysis of lattice Boltzmann equation

I'm interested to derive the total energy balance from Chapman-Enskog analysis of lattice Boltzmann equation (LBE). I know, I should go to the second moment of LBE (zeroth moment gives mass ...
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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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Divergence of Cauchy Stress Tensor [closed]

On the wikipedia page for the Cauchy Momementum Equation, it's stated that the equation can be written as $$\rho \frac{D\,\textbf{v}}{D\,t} = \nabla \cdot \sigma + \textbf{f}$$ Where $\sigma$ is ...
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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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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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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 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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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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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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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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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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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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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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Is there a nice way to write Navier-Stokes equations in exterior calculus

I'm considering to study some high-dimensional Navier-Stokes equations. One problem is to do write the viscous equation for vorticity, helicity and other conserved quantities. I think it might be ...
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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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Flow field generated by a sphere in fluid

What is a flow field generated by a swimming sphere in the fluid? (in low reynolds regime)? What is the flow field generated by a swimming cylinder in this regime? It seems that the question is not ...
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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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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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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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Existence and uniqueness of solutions to $\nabla_a T^{ab}=0$ in general (or special) relativity

The equation in the title of this question can be a relativistic analogue of the Navier-Stokes equation (in the sense that, in the low-velocity limit, it reduces to Euler's equation when $T^{ab}$ is ...
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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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About turbulence modeling

I have some questions about this paper: Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids. R. J. Becker. Phys. Rev. Lett. 58 no. 14 (1987), pp. 1419-1422. After reading the ...
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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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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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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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Physical Meaning of Divergence of Convective Velocity Term

When taking the divergence of the convective velocity term, I get the following: \begin{align} \nabla\cdot\left[\mathbf u\cdot\nabla\mathbf u\right]&=\frac{\partial}{\partial x_i}\left[u_j\frac{\...
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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 ...