Navier-Stokes Equation

the Navier-Stokes equation but

the so called particle image velocimetry is suitable for visualizing the velocity field of an instationary flow the method is

based on the idea of putting fine particles into a fluid and

comparing successive images with each

other then the velocity of each

particle is determined with the cover distance and the time passed In the experimental set up of the method 1

uses a pulsed laser to create shop video images of the moving particles 2 successive images

are processed so that the velocity can be determined with the time passed and the covered distance here delta x 1 can illustrate

exemplarily the procedure with the Sean experiment in image 1 observes a particle at the position r 1 at the time t 1 after the time delta t 1 sees on the success of image that the particle has covered the distance that R R 1 1 obtains the velocity at the position r 1 which can be illustrated with the vector to obtain an image of the total velocity field the procedure must be repeated for many particles several times

finally 1 obtains a vector map of the flow velocity field that

varies in space and time to

show the spatial dependence of the velocity field each point of the flow is

colored according to the absolute value of the velocity 1 observes that the absolute value of the velocity varies in space this means that the gradient of the

square of you is generally not 0 that In the next step 1 compares the

directions of the velocity it's 2 different positions are 1 and R 2 1 sees also that the

direction varies in space the the the further the velocity varies in time at the position r 2 for example the partial time differentiation is not 0 according to the instationary flow here is the

velocity in a vector diagram at equidistant points in time the the

important observations can be compared with the theoretical description of a flow by the Navier-Stokes equation to set up the equation of motion for an infinitesimal volume element of a viscous fluid 1st 1 starts with a Newtonian axiom on the left side of the equation different forces are part of the total force F the the pressure force the gravitational force and the viscous force the the total time differentiation of the velocity on the right side called substantial acceleration can be splitted into Adams the partial time differentiation and for the time dependence of the velocity u at a certain position the and the term for the spatial variation which can be stated again in 2 parts the the 1st depends on the absolute value of the velocity and the 2nd depends on the direction all parts of the substantial

acceleration play a role in the show and turbulent flow