Introduction To Navier-Stokes Equations
The Navier-Stokes equations describe the motion of Newtonian fluids, They can describe a variety of phenomena. They may be used to model the weather, ocean currents, water flow in a pipe, and airflow around a wing.
The first equation expresses the conservation of mass while the second expresses the conservation of momentum for fluids, you know how Newton's second law of motion F=ma describes the motion of objects, the second equation is nothing but the second law of motion but for fluids, it considers the forces that act on the fluid and hence determine how the fluid evolves with time, these forces are pressure gradient, viscosity and any external force exerted on the fluid ( the Navier-Stokes equations do not account for all the forces that act on the fluid they neglect force due to turbulence and compressibility)
Let’s learn what each term means
Conservation Of Mass
This equation represents the conservation of mass, it says that the divergence of the velocity vector field is zero, the divergence of a vector field is a measure of how much a point acts as a source or a sink of the field, and since it’s impossible for mass to appear out of nowhere and visa versa divergence of velocity is zero.
This condition is also the same as saying that the fluid is incompressible, the condition for incompressible flow is that the density remains constant within a small volume, mathematically this means that the material derivative(explained below) of density is zero, the material derivative considers the change of density with respect to time and space, so the condition that
is not sufficient because density could be changing as the fluid moves from one point to another.
The Material Derivative
The material derivative DV/Dt describes the acceleration of the fluid, you can have acceleration in two ways either the velocity of the fluid changing as time passes, or velocity changing as you move from one location to another, the material derivative accounts for both of these changes to give a total acceleration of the fluid.
Which can also be written as
where velocity gradient is the expresses the difference in velocity from one point in the fluid to another.
Take as an example the vector field shown below as the fluid moves down it accelerates due to say gravity so we have acceleration as we go from top to bottom while if you observe a fixed point in this field as time passes the velocity of the fluid at that point remains constant so the change in velocity with time will equal zero while the velocity gradient(change in velocity with as you change position) will be non-zer0.
Pressure Gradient
pressure gradient means the pressure difference, so when pressure differs from one region to another a pressure gradient forms and in fact, it’s that pressure gradient that drives the fluid to move.
A balloon for instance when punched air will flow outside it because of the pressure gradient between the balloon and its surrounding.
Viscosity
Viscosity can be conceptualized as quantifying the internal frictional force that arises between adjacent layers of fluid that are in relative motion. In liquids, viscosity arises generally due to cohesive forces between the molecules of the liquid while in gases it’s due to momentum diffusion between the partials of gas.
This squared upside-down triangle(called nabla) expresses the Laplacian of the velocity field, the Laplacian measures how the vector field (the velocity of the fluid) will diffuse or get spread out over time so that it becomes evenly distributed all over the fluid, This is the typical behavior of a dissipative (i.e., “viscous”) fluid.
And finally, external forces are exerted on the fluid which is usually gravity.
Resources
- https://en.wikipedia.org/wiki/Momentum_diffusion
- https://andrew.gibiansky.com/blog/physics/fluid-dynamics-the-navier-stokes-equations/
- https://en.wikipedia.org/wiki/Incompressible_flow
- https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives/divergence-and-curl-articles/a/divergence
