3. Factory wind tunnels are used to analyze the aerodynamic properties of fast moving objects like bikes, cars, trucks, planes, or spaceships. Large wind turbines combined with pulses of smoke create flow lines (see 0:20 to 0:40 of this video) that can be studied to perfect a design. Let F : R^3 x (0, ∞) → R^3 be the time-dependent velocity vector field of the air flow inside the space R^3 for times (0, ∞). At time t ∈ (0, ∞), the vector F(x, t) is the velocity of a particle in the air at point x ∈ R^3.
• A pathline is the trajectory that individual fluid particles follow. The velocity of the particle will be determined by the velocity vector field at each moment in time.
• A streamline is the trajectory that individual fluid particles would follow if the flow were stable from a fixed moment in time and onwards. The velocity of the particle would be determined by the velocity vector field at that fixed moment in time.
(3a) Which statement ensures that the trace of ̴ : (0, ∞) → R^3 is a pathline of F?
Fill in EXACTLY ONE circle.
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), F(̴(t), t0) = 0
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴(t) = F(̴'(t), t0)
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴'(t) = F(̴(t), t0)
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴'(t) = F(̴'(t), t0)
○ ∀t ∈ (0, ∞), F(̴'(t), t) = 0
○ ∀t ∈ (0, ∞), ̴(t) = F(̴'(t), t)
○ ∀t ∈ (0, ∞), ̴'(t) = F(̴(t), t)
○ ∀t ∈ (0, ∞), ̴'(t) = F(̴'(t), t)
(3b) Which statement ensures that the trace of ̴ : (0, ∞) → R^3 is a streamline of F?
Fill in EXACTLY ONE circle.
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), F(̴(t), t0) = 0
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴(t) = F(̴'(t), t0)
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴'(t) = F(̴(t), t0)
○ ∃t0 ∈ (0, ∞) s.t. ∀t ∈ [t0, ∞), ̴'(t) = F(̴'(t), t0)
○ ∀t ∈ (0, ∞), F(̴'(t), t) = 0
○ ∀t ∈ (0, ∞), ̴(t) = F(̴'(t), t)
○ ∀t ∈ (0, ∞), ̴'(t) = F(̴(t), t)
○ ∀t ∈ (0, ∞), ̴'(t) = F(̴'(t), t)
(3c) A leaf is blowing in the wind. The wind's velocity vector field and the leaf are plotted below at a fixed moment in time. Sketch the streamline (if possible) and the pathline (if possible) of the leaf. Label your curve(s). If it is not possible to sketch one or both of them, explain why not in a single sentence.