What is Viscous flow: Definition and 19 Discussions

In physics, the Navier–Stokes equations () are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.
The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).
The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics.
The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions – i.e. they are infinitely differentiable (or even just bounded) at all points in the domain. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample.

View More On Wikipedia.org
  1. HighPhy

    Double experiment: "Falling ball viscometry" and viscous flow in an horizontal pipe

    TL;DR Summary: Analyze, with the help of @Chestermiller, the importance of viscous interactions in a vertical geometry ("Falling ball viscometry") and in an horizontal viscous flow. The following double experiment was conducted by me a few weeks ago. @Chestermiller and I discussed this...
  2. Ophi-Siren-Kit

    Speed/Velocity and Volume Flow Rate of Viscous Fluids

    In my first attempt, I started off converting the radii of all three sections from centimeters (10, 8, 6) to meters (0.10 , 0.08 , 0.06), then used the VFR=Av formula to find the speed/velocity of section one. VFR== 0.063 m^3/s A== pi*r^2=pi*(10cm)^2=pi*(0.10)^2=pi*0.01 == 0.031415927 VFR/A=v...
  3. WhiteWolf98

    Flow Rate (Q) of Viscous Flow between Two Plates

    I know how to derive the formula, but I have no idea how to actually use this. Where it says ##50~cm## wide, I'm not sure if that's in the x-direction or the z-direction. ##W##, the width in the equation, is in the z-direction. ##\mu## can be calculated, but I'm not sure what the pressure...
  4. WeiShan Ng

    [Fluid Mechanics] Negative sign on the viscous work term

    Homework Statement I am revising on the derivation of the differential equation of energy (White's Fluid Mechanics 7th ed) and I'm having trouble understanding the sign convention used in the viscous work term. The textbook first define an elemental control volume and list out the inlet...
  5. A

    Relation Between Ma and Re for Isentropic Expansion

    Homework Statement Consider the isentropic expansion of air from a fixed given reservoir (i.e. total pressure and temperature). Investigate the behaviour of the value of the Reynolds number of the flow, as a function of the Mach Number M of the expanded flow. For small values of M, the...
  6. JTC

    Bernoulli to explain Magnus Effect

    Hi I have been reading some internet articles that state the Bernoulli equation does NOT explain the Magnus Effect. The articles state that the effect is due to circulation (Bernoulli requires inviscid flows) Could someone explain the cause of the Magnus effect without reference Bernoulli's...
  7. C

    Does electron gas in metals experience viscosity?

    Does electron gas in metals experience viscosity? Also, does highly charged plasma experience viscosity?
  8. M

    Gravity-induced viscous flow around a cylinder

    Homework Statement A viscous liquid with density and viscosity ##\rho## and ##\mu## respectively is discharged onto the upper surface of a cylinder with radius ##a## at a volume flow rate ##Q##. This is a gravity-driven flow, and it forms a film around the cylinder--see picture. What is the...
  9. K

    Change in gauge pressure using viscous flow through a pipe.

    Homework Statement Water flows at 0.25 L/s through a 9.0-m-long garden hose 2.0 cm in diameter that is lying flat on the ground. The temperature of the water is 20 ∘C. What is the gauge pressure of the water where it enters the hose? Side question: does the velocity of the water flow need to...
  10. B

    Thermofluids problem (Viscous Flow)

    Homework Statement A sphere is moving in water at a depth where the absolute pressure is 124kPa. The maximum velocity on a sphere occurs from the forward stagnation point and is 1.5 times the free stream velocity. Calculate the speed of the sphere at which cavitation will occur. Can anyone...
  11. B

    Thermofluids, Viscous Flow Problem

    Homework Statement Q.2 One type of bearing that can be used to support very large structures is shown below. Here fluid under pressure is forced from the bearing midpoint (A) to the exterior zone B. thus a pressure distribution occurs as shown. For this bearing which is 30cm wide, what...
  12. Feodalherren

    Fluid Mechanics: viscous flow in pipes

    Homework Statement Homework Equations ρ=789, μ=.0012 The Attempt at a Solution From the energy equation we get hf=0.9 We know that hf=f(L/D)(V^2)/(2g) [sorry don't know how to use latex after they removed the bar on the right] Now I can substitute V for Q, but I'm stuck with f and Q as...
  13. I

    Numerical solution to stationary viscous flow with a Darcy term

    Hi, I'm trying to solve the flow profile inside an inhomogeneous porous material between two parallel moving plates (essentially Couette flow with a deviation), and I model my system by the following equations: \nabla^2 \mathbf{u} = p(x,y,z) \mathbf{u}\\ \nabla \cdot \mathbf{u} = 0...
  14. S

    How Is the Boundary Layer Analyzed as Incompressible in Viscous Flow?

    Hello, I'm looking at this problem which states: "A parallel air flow along a semi infinite flat plate has undisturbed parameters as follows: " Then they list the speed, temprature, viscousity and pressure. "Demonstrate that the boundary layer can treated by means of incompressible...
  15. S

    Viscous Flow and Pressure Coefficient

    Homework Statement What effect does a large positive dCp/dX have on the real viscous flow? This is for an airfoil design project using thin airfoil theory + vortex panel method. The Attempt at a Solution As far as I know Viscosity is independent of pressure, so the only thing I can...
  16. S

    Viscous Flow and Pressure Coefficient

    I am doing a project for my aerodynamics class, basically its a simplified version of the thin airfoil theory + vortex paneling method; and I have been asked the following the question: What effect does a large positive dCp/dX (Cp=Pressure Coefficient, x=position) have on the real viscous flow...
  17. J

    Inviscid flow and viscous flow

    viscous flows are always rotational because of shear stress that is exerted on the fluid element due to viscosity. what about the inviscid flows? can they be rotational ? if yes then what are the factors which makes the inviscid flows rotational?
  18. M

    How Do You Calculate the Radius of a Pipe in Fluid Mechanics?

    Homework Statement Water (n = 1.00 10-3 Pa·s) is flowing through a horizontal pipe with a volume flow rate of 0.029 m3/s. As the drawing shows, there are two vertical tubes that project from the pipe. From the data in the drawing, find the radius of the horizontal pipe. Homework...