Exercises 1

Question 1

Determine the dependence of the pressure on the coordinate z in a stationary fluid of constant
density under a uniform gravitational field.

Answer 1

p(z)=p(z0)-ρg(z-z0)

Question 2

Write the x component of the vectors (u.∇)u and u(∇.u).

Answer 2

1. (u · ∇)u_x = u_x * (∂u_x/∂x) + u_y * (∂u_x/∂y) + u_z * (∂u_x/∂z)
2. u_x * ( ∂u_x/∂x + ∂u_y/∂y + ∂u_z/∂z )

Question 3

Given the function u(x,y,z,t), what is the difference between its total derivative and its
convective or Newtonian derivative?

Answer 3

the total derivative (or material/substantial derivative, (Du/Dt)) measures the change of (u) for an observer moving with the flow (following the (x,y,z) positions that change with time), accounting for both explicit time change and spatial convection; while the convective derivative is a term within that, representing the spatial variation’s effect on a moving particle, often called the advective or Lagrangian derivative, contrasting with the Eulerian (partial) derivative which only sees the explicit time change at a fixed point. In short, (Du/Dt=\partial u/\partial t+(u\cdot \nabla )u), where ((\mathbf{u}\cdot \nabla )u) is the convective part

Question 4

Why is Equation (3) insufficient to be solved in the absence of additional information on the
elastic properties of the fluid? What is the simplest model available for these elastic properties,
and why is it generally quite accurate in the case of liquids.

Answer 4

There is 4 unknowns and only 3 equations provided for the velocity components. Therefore, another equation is needed that reflects the elastic property of the fluid. The simplest model available is incompressibility, which makes pressure calculable.

Question 5

Why does the stress at a point in the middle of a fluid (away from any physical surface)
depend on a certain orientation vector n, and what does this n mean (or where does it come from
in the absence of a real surface)?

Answer 5

Because internal forces act as surface forces and they can vary with the orientation of the surface. The unit vector n speicifies that orientation to an imagined infinitesimal surface passing through the point, even if no physical surface is present.

Question 6

What general principle fixes how the stress depends on n?

Answer 6

Cauchy’s principle of local action and balance of linear momentum (i.e., the requirement that surface forces on an infinitesimal control volume balance so that the net force equals the inertial force) fixes how the stress depends on the normal vector n.

Question 7

What is the difference between the stress (a vector) and the stress tensor?

Answer 7

• Stress vector is the force per unit area acting on a specific surface which normal is n.
• Stress tensor is a matrix that represents the complete stress state at a point, describing force per unit area on any internal surface