# Why does fluid flow through an orifice under pressure not depend on fluid viscosity?

The equation for fluid volume flow rate through a orifice is

$$Q = CA\sqrt{\frac{2\Delta P}{d}}$$

$$C$$ - discharge coefficient

$$\Delta P$$ - Change in pressure

$$d$$ - density of the fluid

$$A$$ - Area of the orifice

I was surprised to see that the flow rate does not depend on the viscosity of the fluid. Why is viscosity not a factor? Is the above equation only valid for some cases of fluid flows where viscosity does not affect the flow?

Is the fluid viscosity taken into account in the discharge coefficient?

• How do you calculate the pressure drop term?
– user207455
Commented Jul 24, 2019 at 21:54
• @SolarMike The equation is for fluid volume flow for a given pressure difference. Commented Jul 24, 2019 at 22:24
• What is your discharge coefficient? Check out a Borda mouthpiece...
– user207455
Commented Jul 24, 2019 at 22:25
• It does depend on viscosity but this effect is lumped into the discharge coefficient. For an ideal orifice (i.e. without viscous dissipation), $C = 1$. Commented Jul 25, 2019 at 13:05
• @nluigi Can you add that as a answer, so I can accept it? Commented Jul 25, 2019 at 18:57

This equation is only valid in an ideal situation i.e. when there is no viscosity between the layers of the fluid.

For dynamic viscosity mu in fluids,

mu=Fy/Au,

where mu is the dynamic viscosity, F is the applied force, y is separation distance, A is area of each plate & u is the fluid speed.

So,

muu=Fy/A muu*A=Fy

Therefore, A=Fy/mu*u

Substituting the value of A for the area of the orifice in the equation for fluid flow through an orifice, we can get an additional viscosity dependence part on the same equation.

EDIT: This does depend on viscosity but this effect is lumped into the discharge coefficient. For an ideal orifice (i.e. without viscous dissipation), C=1. (AS SUGGESTED BY @NLUIGI

It reinforces my previous statement.