A fluid has constant density ρ and velocity v = zi + (y2 + 2y)j − 2xk.
(a) Calculate the particle paths (field-lines).
(b) Calculate the mass flux through the boundary of the region x2 + y2 ≤ 4 and 0 ≤ z ≤ (4−y) by direct calculation and by using the Divergence Theorem.
To calculate the particle paths (field-lines) for the given fluid velocity vector, we need to solve the following differential equation:
dx/dt = v_x
dy/dt = v_y
dz/dt = v_z
where v_x, v_y, and v_z are the x, y, and z components of the velocity vector v.
To solve this system of differential equations, we'll integrate each component individually.
(a) Particle paths (field-lines):
Integrating the x-component:
dx/dt = v_x
Integrating both sides with respect to t:
∫dx = ∫v_x dt
x = ∫v_x dt
Since v_x = 0, we have x = constant.
Integrating the y-component:
dy/dt = v_y
Integrating both sides with respect to t:
∫dy = ∫v_y dt
y = ∫v_y dt
Since v_y = y^2 + 2y, we can integrate:
y = ∫(y^2 + 2y) dt
y = ∫(y^2 + 2y) dt
To solve this integral, we need to know the initial conditions (starting point) for y. Without those initial conditions, we cannot fully determine the particle paths.
Similarly, for the z-component:
dz/dt = v_z
Integrating both sides with respect to t:
∫dz = ∫v_z dt
z = ∫v_z dt
Since v_z = -2x, we can integrate:
z = ∫(-2x) dt
Again, without the initial conditions for x, we cannot determine the particle paths completely.
(b) To calculate the mass flux through the boundary of the region x^2 + y^2 ≤ 4 and 0 ≤ z ≤ (4-y), we can use the Divergence Theorem.
The Divergence Theorem states:
∬(F . dA) = ∭(div F) dV
Where F is a vector field, dA is an outward-facing vector element of the surface, and dV is an infinitesimal volume element.
In this case, the vector field F is the fluid velocity vector v = zi + (y^2 + 2y)j - 2xk.
The divergence of F (div F) can be calculated as follows:
div F = (∂v_x/∂x) + (∂v_y/∂y) + (∂v_z/∂z)
To calculate the mass flux through the given boundary, we need to integrate div F over the enclosed volume. However, we need the complete form of the velocity vector v and the boundary conditions for x and y to perform the calculation.
Without the specific values for x, y, and the initial conditions, we cannot provide a direct calculation using the Divergence Theorem.