Question

I have this really hard advanced calculus assignment and these questions are stumping me hard. Asking for full solutions but anything is fine. Of course will give a thumbs up to good responses. I have copy and pasted the explanation for the questions and attached pictures of it in case the format is broken. Thanks

MARK WHICH STATEMENTS BELOW ARE TRUE, USING THE FOLLOWING,

In a classical fluid every molecule making up the fluid is subject to Newton's laws. In a continuum approximation of a large number of these particles we may think in terms of a small element of the continuum, containing those particles, being subject to Newton's laws. Thinking in terms of local conditions, the fluid elements follow a path with a tangent to the path given by the fluid velocity vector field,

v(x,y,z,t)=drdt=dxdti^+dydtj^+dzdtk^

If this were air, we would think of this as wind velocity.

We need the acceleration to employ Newton's second law,

a(x,y,z,t)=dvdt,

which can be expressed explicitly with the chain rule.

We will also use,

ρ,p,g,η,

which represent the mass density, pressure, gravitational acceleration, and dynamic viscosity respectively.

After simplifying the acceleration and multiplying by the mass density, we get the force per unit volume. If we suppose this force originates from pressure differences, gravity, and internal friction then

ρ(v⋅∇)v+ρ∂v∂t=−∇p+ρg+η∇2v,

which is known as the Navier-Stokes equation. If the viscosity is ignored the resulting equation,

ρ(v⋅∇)v+ρ∂v∂t=−∇p+ρg,

gets a different name: the Euler equation.

Fluids can be solenoidal so we define the vorticity,

Ω≡∇×v

To a high approximation fluids also behave as being incompressible, which means that the mass density does not depend on time when only considering the flow.

BEWARE: MARKING A STATEMENT TRUE THAT IS ACTUALLY FALSE RECEIVES A NEGATIVE MARK.

In a classical fluid every molecule making up the fluid is subject to Newton's laws. In a continuum approximation of a large number of these particles we may think in terms of a small element of the continuum, containing those particles, being subject to Newton's laws. Thinking in terms of local conditions, the fluid elements follow a path with a tangent to the path given by the fluid velocity vector field, v(x, y, 2, t) = Fi+ dy: i + k dt If this were air, we would think of this as wind velocity We need the acceleration to employ Newton's second law, a(I,y,z,t) = which can be expressed explicitly with the chain rule. We will also use, P, P, 8, 11 which represent the mass density, pressure, gravitational acceleration, and dynamic viscosity respectively After simplifying the acceleration and multiplying by the mass density, we get the force per unit volume. If we suppose this force originates from pressure differences, gravity, and internal friction then plv.V)vtpot - Vp+p8+nD?v,