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1. In the aerodynamic field, we like to know in which direction the flow is going, since in this way all the physical phenomena involved in it are investigated

2. In the aerodynamic field, we like to know in which direction the flow is going, since in this way all the physical phenomena involved in it are investigated

   • Pathlines , streamlines , and streaklines of a flow
     • Pathlines
       • the line traced by a given particle as it flows from one point to another
         • its respective equation
           • dẊ/ dt (t) = up(Ẋp(t),t) Ẋp(to)=Ẋpo
             • P: indicates that we are following the motion of a fluid particle
             • Ẋp: the curve is parallel to the velocity vector
             • t :the speed evaluated in a certain time
     • streamlines
       • is a curve whose tangent at any point is in the direction of the velocity vector at that point
         • its respective equation
           • ds X V = 0
             • ds: a directed element of the streamline
             • V:is the speed parallel to the current line
             • Expanding it in Cartesian coordinates
               • ds = dx(i) + dy(j) + dz(k)
               • V= ui + vj + wk
                 • Making Cross Product
                   • 0=i(w dy - v dz) + j(u dz- w dx) + k(v dx- u dy)
                     • decomposing by each component
                       • w dy - v dz=0 u dz - w dx=0 v dx - udy=0
     • streaklines
       • a line created by particles in a flow that have previosly passed through a common point
         • its respective equation
           • dẊstr/ dt = up(Ẋstr,t) Ẋstr(t=τp)=Ẋpo
             • Ẋstr :speed of a particle at location and time
     • is the distinction between path lines and stream lines
   • Angular Velocity , Vorticity , and Strain
     • element orientation and its shape change as it moves along a streamline
     • Angular Velocity
       • he average of the angular velocities of the lines are taken from one point to another point
         • Terms that compose it
           • • ωz=1/2(dθ1/dt + dθ2/dt)
               • ω : angular velocity
               • dθ:number of angular changes
               • n three-dimensional terms
                 • ω=ωi + ωj + ωk
                   • the angular velocity adds all its components in the velocity field
     • Vorticity
       • they are the most powerful quantities in theoretical aerodynamics to quantify the amount of rotation and distortion.
         • Terms that compose it
           • • v△t
               • Distance in y direction that A moves during time increment △t
             • (v + au/ax dx) △t
               • Distance in y direction that C moves during time increment △t
             • (v + au/ax dx) △t - V△T
               • Net displacement in y direction of C relative to A
       • ξ=2ω
     • Strain

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