Question

A liquid is allowed to flow into a tube of truncated cone shape. Identify the correct statement from the following.
A. The speed is high at the wider end and low at the narrow end
B. The speed is low at the wider end and high at the narrow end
C. The speed is same at both ends in a stream line flow
D. The liquid flows with uniform velocity in the tube

Answer 1

Hint: We can apply the theory of conservation of mass to solve this problem quantitatively. In infinitesimal time dt, we shall calculate the mass that enters the tube and the mass that exits it. Where we can use the continuity theorem to solve this problem to get the desired solution.

Complete answer:
The continuity equation states that the product of the pipe's cross-sectional area and fluid speed at every point all along the tube is always constant. This product is equivalent to the flow rate or volume flow per second. According to the equation, it can be written as
\[R = Av\]= Constant
Where,
\[R\]is the rate of flow of volume,
\[A\]Is the area of flow,
\[v\] is the velocity of the flow.

As we know, the continuity theorem is valid.
\[\therefore {A_1}{v_1}\rho = {A_2}{v_2}\rho \]

Let’s take the density uniform throughout
\[\therefore {A_1}{v_1} = {A_2}{v_2}\]
By the above equation, we can derive the relation between the area of flow and velocity of flow. Area of flow is inversely proportional to velocity of the flow.

Greater the velocity leads to smaller area. Therefore, the second option is correct, that is the speed is low at the wider end and high at the narrow end.

Note: There are various assumptions we need to make before the continuity equation: The tube should have a single entrance and exit. The flow should not be compressed. The fluid flow should be uniform. The passing fluid should have viscosity. It should be noted that this formula is only applicable when the liquid has a homogeneous density. It implies that it must be constant.