Show that the Stokes equation: ∇ = ( ∂/∂x,∂/∂y,∂/∂z) ρv ̇=-∇p+η∇^2 v can be written as: ρv ̇= - ∂ p / ∂x_1 + η ∂^2 v_1 / ∂x^2 whch is the fluid motion equation. η∇^2 v = ∇p ∇ . v = 0 (dot product is zero) Ω = ∇ ⋀ v (curl of velocity vector) can be written as follows for an incompressible fluid, where Ω is vorticity. ρv ̇=-∇p-η∇ ⋀ Ω -∇p=-∂p/∂x_i ⋀ is the cross product η is the dynamic of shear viscosity, viscosity is a measure of a fluid's resistance to shear deformation. [Newtonian fluids] where v is velocity vector, where v = (v_i , v_j , v_k) where i is in the x direction, j in the y direction and k in the z direction. ρ is mass density p is pressure incompressibility is when: ∂v_i/∂x_i + ∂v_j/∂x_j + ∂v_k/∂x_k = 0

State Fundamental theorem of Gradient, Divergence and Stokes.

Show that divergence of a curl is zero

In R3 filled with an incompressible fluid, we say that the z-axis is a line source ofstrength m if every interval ∆z along that axis emits fluid at volume rate dV /dt = 2πm ∆z. The fluidthen spreads out symmetrically in all directions perpendicular to the z-axis. Show that the velocityfield of the flow isv(x, y) = mx2 + y2 (xi + yj).

Consider this equation: du/dx+dv/dy+dw/dz=0.Which of the following statements is incorrect with regards to this equation?Group of answer choicesThis applies for incompressible flowsThis applies for steady flowsThis applies for certain types of three-dimensional flowsThis equation is derived from the integral conservation of momentum equation

If F(x,y,z)𝐹(𝑥,𝑦,𝑧) is a vector field such that curl(F)≠0(𝐹)≠0 then:a.there will be a scalar field f𝑓 such that F=∇(f)𝐹=∇(𝑓).b.F𝐹 is a conservative vector field.c.F𝐹 is irrotational.d.none of the choices are true.