[M.W]

Hi everyone,

I am working on derivation of the Reynolds thin film theory, i got a question about the assumptions or simplify of the derivation process. I have been reading some literatrure about the derivation process and known it should start from integration of:
∂τ/∂z=∂P/∂x (i)
∂τ/∂z=∂P/∂y (ii)
(for clearly, the axis of the thin film thickness is z, the film is spreading in x&y, the velocitys are written as u,v,w respectively)
My dumb question is: how could we have these eq.?

I know the Reynolds's eq is a special form of N.S. eq, so I start from the N.S. eq in x&y-axis：(body force ρa_x&ρa_y is neglected)
ρDu/Dt=-∂P/∂x+∂τ_xx/∂x+∂τ_yx/∂y+∂τ_zx/∂z (iii)
ρDv/Dt=-∂P/∂y+∂τ_xy/∂x+∂τ_yy/∂y+∂τ_zy/∂z (iV)
If I want to have eq(i)and(ii) from eq(iii)and (iV), the following conclusions are made:
a) material derivative of velocity (LHS) equals 0;
b) ∂τ_xx/∂x=∂τ_yx/∂y=∂τ_xy/∂x=∂τ_yy/∂y=0;

∂τ_zx/∂z and ∂τ_zy/∂z are caused by the different velocity through z, which is comprehensible for me. I dont understand what assumptions or simplify are made to achieve the a) and b).
For example, the ∂τ_yx/∂y=0 means there is no shear stress applied on XZ surface of the fluid element, is that means the u is constent through the y-axis?

Hopefully someone could help me out, thank you in advance!

[FMDenaro]

Originally Posted by M.W

Hi everyone,

I am working on derivation of the Reynolds thin film theory, i got a question about the assumptions or simplify of the derivation process. I have been reading some literatrure about the derivation process and known it should start from integration of:
∂τ/∂z=∂P/∂x (i)
∂τ/∂z=∂P/∂y (ii)
(for clearly, the axis of the thin film thickness is z, the film is spreading in x&y, the velocitys are written as u,v,w respectively)
My dumb question is: how could we have these eq.?

I know the Reynolds's eq is a special form of N.S. eq, so I start from the N.S. eq in x&y-axis：(body force ρa_x&ρa_y is neglected)
ρDu/Dt=-∂P/∂x+∂τ_xx/∂x+∂τ_yx/∂y+∂τ_zx/∂z (iii)
ρDv/Dt=-∂P/∂y+∂τ_xy/∂x+∂τ_yy/∂y+∂τ_zy/∂z (iV)
If I want to have eq(i)and(ii) from eq(iii)and (iV), the following conclusions are made:
a) material derivative of velocity (LHS) equals 0;
b) ∂τ_xx/∂x=∂τ_yx/∂y=∂τ_xy/∂x=∂τ_yy/∂y=0;

∂τ_zx/∂z and ∂τ_zy/∂z are caused by the different velocity through z, which is comprehensible for me. I dont understand what assumptions or simplify are made to achieve the a) and b).
For example, the ∂τ_yx/∂y=0 means there is no shear stress applied on XZ surface of the fluid element, is that means the u is constent through the y-axis?

Hopefully someone could help me out, thank you in advance!

Why did not you introduce the newtonian closure law and see the velocity gradients components?

[LuckyTran]

a) is the low Reynolds number assumption
b) is the thin film assumption. The film thickness in z is much smaller than the dimension in x and y.


Note that you don't formally equate these terms to zero, but order of magnitude analysis yields that they are much smaller than the remaining terms.