Navier Stokes thin film on infinite wall

[gfd43tg]

Homework Statement

Consider steady, incompressible, parallel, laminar flow of a film of oil falling down an infinite vertical wall (Figure P-1). The oil film thickness is “h” and gravity acts in the negative Z-direction (downward on the figure). There is no applied pressure driving the flow – the oil falls by gravity alone. Calculate the velocity and pressure fields in the oil film and sketch the normalized velocity profile. Neglect hydrostatic pressure changes in the surrounding air.

Homework Equations

The Attempt at a Solution

Hello,

I wanted to check if I am doing my Navier-Stokes correctly. Also, why is the dP/dz term zero conceptually? Does that have to do with the last statement ''Neglect hydrostatic pressure changes in the surrounding air''?

Also, I do not know how to calculate the ''pressure field''. First of all, is this the pressure profile, just to understand that the names are synonymous. Secondly, how would I go about calculating a pressure profile? From what I recall, all I remember Navier-Stokes doing is calculating velocity profiles.

 

[gfd43tg]

I'm still confused about how to go about solving the pressure profile. How do I do that?

 

[Chestermiller]

Hello,

I wanted to check if I am doing my Navier-Stokes correctly.

Yes. Nice job.

Also, why is the dP/dz term zero conceptually? Does that have to do with the last statement ''Neglect hydrostatic pressure changes in the surrounding air''?

Yes. The air pressure is independent of z at the interface. The normal stress at the interface between the air and the liquid has to be continuous, so, it follows from this that the fluid pressure at the interface is equal to the air pressure. Thus, at the interface at least, the fluid pressure is not varying with z. From the force balance in the x direction, the derivative of pressure with respect to x is also zero. So the pressure is also independent of x. So, throughout the falling film, the fluid pressure is equal to the air pressure.

Also, I do not know how to calculate the ''pressure field''. First of all, is this the pressure profile, just to understand that the names are synonymous.

The pressure field is the pressure expressed as a function of x and z.

Secondly, how would I go about calculating a pressure profile? From what I recall, all I remember Navier-Stokes doing is calculating velocity profiles.

We showed above how we can establish the pressure field for this problem. Basically, you need to consider how the stresses are varying. But, in addition to providing the velocity field, the Navier Stokes equations are also capable of delivering the pressure distribution.

Chet

 

[gfd43tg]

So that means the pressure profile is dp/dx = 0 and dp/dz = 0?

 

[Chestermiller]

So that means the pressure profile is dp/dx = 0 and dp/dz = 0?

Yes.