[Brandani]

Hello!

I am trying to figure out this set of equations: (please see attached screenshot)

My question is what does the subscript k represent?

Why does term dUk/dxk = 0 for incompressible fluid?


Thank you

[LuckyTran]

This is Einstein / tensor notation, a particular kind of index notation.

Originally Posted by Brandani

My question is what does the subscript k represent?

Why does term dUk/dxk = 0 for incompressible fluid?


Thank you

i,j,k are different indexes and k means not i or j.


U is the velocity vector with components u,v,w: e.g. U=(u,v,w). The position vector x also has components (x,y,z).


dUk/dxk means du/dx + dv/dy + dw/dz


du/dx + dv/dy + dw/dz=0 is the definition for a flow to be incompressible. A substance/fluid which is incompressible will automatically fulfill this constraint on the velocity field. You can prove this property from the continuity equation by expanding the derivative, collecting some terms, and then setting the material derivative of density to be 0. You can look in the Conservation of Mass section of the wikipedia article for some hints on how to prove this.

[FMDenaro]

The previous answer is what you were looking for. I can just add that repeated subscript stands for summation over the index. Therefore, assuming x1,x2,x3 and u1,u2,u3 the components along the three Cartesian direction you have



dui/dxi = Sum (i=1,2,3) dui/dxi



To understand that dui/dxi =0 for flows where the density is assumed constant both in time and space, just consider the density equation in your figure setting rho=constant.

[Brandani]

Thank you!