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?
 A: 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.
A: 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.
A: 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.
A: If you want to see this in terms of force you can see it as below.
Consider the point when the water has just started falling(take it as a reference cross section), obviously at this point the atmospheric pressure and pressure from inside the cross section in outward direction will be in equilibrium. Also at this moment the water has not reached its max compressibility yet.
As i suppose you already know that as velocity increases pressure decreases (this same phenomenon is used for air particles to fly you know what(in accordance with Bernoulli theorum)). Due to this fact as velocity of falling water increases the outward pressure exerted by it decreases and thus due to atmospheric pressure the cross section decreases until it reaches its max compressibility.
