Let be a particle droped in a 2D flow without speed. The speed of the flow is initialy on the x direction and then the velocity change (direction and magnitude). Is there ananlytical solution for this problem? Where can I find some references about this subject?

So,initially the location of the particle is known, and the relative velocity is also known and is in x-direction. From this and the particle diameter, and Reynolds number you can compute the drag force acting on the particle. ( you need this formula for drag coefficient on a sphere at different Reynolds number , and you need to do some searching to look for this formula on your own. not me!) Then you can use Newton's 2nd law to compute the particle motion for a small time step delta t. You do this for the motion in x-direction and the motion in y-direction. ( the force in y-direction is the gravitational force.) F=m*a, m is the mass of the particle, and F is the component of force in either x- or y-directions, a is the acceleration=dV / dt. Now you have dV=a*dt, you can pick a small dt to compute your dV, that is the change in velocity at the end of dt. The velocity is then V(t+dt)=V(t) + dV. At this point the particle should move to a new location defined by x(t+dt)=x(t) + V,in x-direction * dt, y(t+dt)=y(t) + V,in y-direction * dt. Since the particle will have both V,x and V,y velocity components, make sure to find the relative velocity and the direction so that the proper drag force can be computed ( this is also the direction of the relative velocity). In y-direction, the force acting on the particle is m*g + F,y . m*g is the gravitational force which is roughly constant and F,y is the component of the darg force in y-direction which depends on the relative velocity at time t and t=dt and so on..

Here are some publications that cover the problem (I assume you are interested in the behavior of small particles): 1-Tchen,C.M. Mean Value and Correlation Problems Connected with the motion of Small particles suspended in a turbulent flow, Diss. TH Delft, 1947 2-Millikan,R.A. The general law of a small spherical body through a gas, Phys Review, Sec.Ser.,Vol. 22,No.1, 1923 3-Allen,M.D., Raabe,O.G. Re-evaluation of Millikans Oil drop datafor the motion of small particles in air, J.Aerosol Sciences, Vol13,No.6,1982 4-Kurz,R., Experimentelle und theoretische Untersuchungen an gleichfoermig und ungleichfoermig geteilten Turbinengittern, Diss UBw Hamburg, 1991.

Hope this helps!