Final answer:
To determine the value of k for a physically possible incompressible fluid flow, the divergence of the velocity field must equal zero. Through the application of the divergence theorem to the given velocity components, the value k = 0 is the only solution that satisfies the equation for all values of x, y, and z. Option b is the correct answer.
Step-by-step explanation:
The student's question involves the topic of fluid mechanics, specifically the continuity equation for an incompressible fluid flow. Firstly, one of the fundamental principles for an incompressible flow is that the flow rate, or volumetric flow rate Q, must remain constant throughout the fluid's path. According to the continuity equation, this can be expressed as Q = A * u, where A represents the cross-sectional area of the fluid flow, and u is the average velocity.
To ensure a physically possible flow for the given velocity components of the fluid, one needs to ensure mass conservation or flow rate constancy. The divergence of the velocity field (u, v, w) should equal to zero, which is a mathematical expression for the conservation of mass in fluid mechanics for an incompressible fluid. The divergence of a vector field (u, v, w) is given by the sum of partial derivatives of each component: ∇ ⋅; ℓ; = ∂u/∂x + ∂v/∂y + ∂w/∂z.
Substituting the given components into the divergence we get:
∂u/∂x = k(y + z)
∂v/∂y = k(x + z)
∂w/∂z = -k(x + y) - 2z
The sum of these derivatives must be zero, so:
k(y + z) + k(x + z) - k(x + y) - 2z = 0
Simplifying the expression:
κ(x + y + 2z) - 2z = 0
κ(x + y + 2z) = 2z
Because the terms x + y + 2z and 2z are independent of each other and can vary independently, the only way for this equality to hold for all values of x, y, and z is if k equals zero. Thus, k = 0 is the value that makes the flow physically possible.
In conclusion, the correct option is b. k = 0.