Navier-stokes equation (fluid mechanics)

In summary, the conversation discusses the Navier-Stokes equation for viscous fluid flow and its terms, specifically the term v(del squared)u. There is some confusion about the meaning of this term as it applies to a vector field. The conversation also delves into the concept of using the differential operator \nabla^{2} on vector and tensor fields. The main focus is on understanding the convection acceleration term in the Navier-Stokes equation and how it contributes to its non-linearity. The conversation concludes by clarifying the difference between V.delV and V.(delV) and how they both refer to the same concept.
  • #1
alsey42147
22
0
i'm revising for my exams, and i didn't go to many of my fluids lectures, now I'm well confused. in the navier-stokes equation for viscous fluid flow, there is a term:

v(del squared)u

where v is the kinematic viscosity and u is the velocity field of the fluid. at this point in my notes, the lecturer seems to start doing crazy things which don't make sense.

first of all, its (del squared)u, not (del squared)(dot)u. i thought (del squared)u only had any meaning if u is a scalar field, but its not, its a vector field. what does this mean?
 
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  • #2
[itex]\nabla^{2}[/itex] is a differential operator that perfectly well can be applied to a vector.
 
  • #4
V.delV is convection accelaration term in NSE it is the major source for non-linearity of the equation

You can work it out by

(V.del)V or V.(del V) both methods are same
 
  • #5
altruistic said:
V.delV is convection accelaration term in NSE it is the major source for non-linearity of the equation

You can work it out by

(V.del)V or V.(del V) both methods are same

I believe the OP was asking about the viscous dissipation term not the convective term.
 

1. What is the Navier-Stokes equation?

The Navier-Stokes equation is a set of mathematical equations that describe the motion of fluids, such as liquids and gases. It takes into account factors such as fluid density, velocity, and viscosity to determine how a fluid will behave under different conditions.

2. Who is Navier-Stokes and why is the equation named after them?

The Navier-Stokes equation is named after French mathematician and physicist Claude-Louis Navier and Irish mathematician George Gabriel Stokes. Both scientists made significant contributions to fluid mechanics in the 19th century and their work formed the basis for the Navier-Stokes equation.

3. What are the applications of the Navier-Stokes equation?

The Navier-Stokes equation has a wide range of practical applications, including predicting weather patterns, designing airplanes, and studying ocean currents. It is also used in industries such as automotive, aerospace, and marine engineering to optimize the design and efficiency of fluid systems.

4. Is the Navier-Stokes equation solved analytically or numerically?

The Navier-Stokes equation can be solved using both analytical and numerical methods. Analytical solutions involve finding an exact mathematical formula to describe the behavior of a fluid, while numerical solutions use computers to solve the equation using approximations and simulations.

5. Are there any limitations to the Navier-Stokes equation?

While the Navier-Stokes equation is widely used in fluid mechanics, it does have some limitations. It assumes that fluids are continuous and incompressible, and neglects certain factors such as turbulence and surface tension. These limitations can be addressed by using more complex equations or by incorporating experimental data into the calculations.

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