# How does the cross section of a stream of falling water decrease

Consider a continuous stream of water flowing down from a tap. Since the water flow is continuous, by equation of continuity, the cross sectional area of the stream decreases. But what makes the water flow sideways, ie, which force is responsible for decreasing the area of cross-section?

• For the principle of continuity shouldn't the initial momentum of the liquid remain constant? But when the water falls down because of gravity, there is a force acting on it – N.S.JOHN Jan 23 '16 at 10:11

## 3 Answers

For an incompressible viscous fluid, the equation of continuity (mass conservation) for an axisymmetric deformation is given by$$\frac{1}{r}\frac{\partial (ur)}{\partial r}+\frac{\partial w}{\partial z}=0$$ where u is the radial velocity, w is the axial velocity, r is the radial coordinate, and z is the axial coordinate. The key components of the stress tensor for axisymmetric stretching of a cylinder of viscous fluid are given by: $$\sigma_r=-p+2\eta \frac{\partial u}{\partial r}$$ $$\sigma_z=-p+2\eta \frac{\partial w}{\partial z}$$ where p is a parameter with units of pressure that must be determined from the boundary conditions, $\eta$ is the fluid viscosity, and the sigmas are the stresses in the radial and axial directions, respectively.

From the continuity equation, it follows that if dw/dz is the (constant) axial rate of deformation for the cylinder, the radial velocity can be integrated to give:$$u=-\frac{r}{2}\frac{dw}{dz}$$ This equation indicates that, as the fluid is stretched axially, it contracts radially. The stress in the radial direction is equal to zero, so, the parameter p is given by:$$p=2\eta \frac{du}{dr}=-\eta \frac{dw}{dz}$$ If we substitute this into the axial stress equation, we obtain:$$\sigma_z=3\eta\frac{dw}{dz}$$ This is a well-know equation which indicates that the so-called elongational viscosity of a fluid is 3 times its shear viscosity.

In summary, according to this development, even through the overall stress in the radial direction is zero, the cylinder of viscous fluid still contracts radially when stretched axially in order to satisfy the continuity equation (provided that the fluid does not break up). The viscous compressive contribution to the radial stress holds the cylinder together. This all is totally analogous to the Poisson effect in stretching a solid (as a result of elastic forces).

Of course, in the case of molten polymers, the contribution of surface tension is typically negligible, unless the cylinder is of tiny diameter.

• Can you please write an answer which does not make use of tensors(if it is possible)? I and many other high-school students have not yet studied tensors but this example is often shown in books as an application of continuity equation. – Apoorv Potnis Nov 11 '17 at 5:27

For the sake of the argument, assume that the stream of water is 10 cm tall. The water drops at the bottom of the stream have been falling for a longer period of time than the water drops 1 cm above them, and this is true for all the water drops in the falling stream. Because of this, the water drops at the bottom of the stream are moving at a higher velocity than the water drops 1 cm above them, because the water drops that are lower in the stream have been accelerating for a longer time. Since the flow rate is constant, the stream must get thinner in cross-sectional area as it falls, in order to satisfy the continuity equation.

• Your answer does not address which force in responsible for the contraction. You only prove that continuity is satisfied, but that is not really what is asked here. – Bernhard Jan 23 '16 at 22:45
• The acceleration due to gravity causes the thinning. When the stream gets thin enough, surface tension breaks the stream up into drops. – David White Jan 24 '16 at 14:49

It is due to a matter of fact that same mass and hence same volume of water has to flow through a given cross section.This is termed as law of continuity in fluid mechanics. It is a known fact that velocity of a falling object increases with height, and hence to satisfy the law of continuity the cross section has to decrease.(to keep the volume constant).And so higher the velocity of fluid flow lesser the cross section becomes.

• Your answer does not address which force is responsible for the contraction. – Bernhard Jan 23 '16 at 22:44