# The Equation of Continuity with derivation & Streamlines of fluid flow

Last updated on August 28th, 2021 at 08:51 am

In this post, we will discuss two topics related to fluid dynamics. These topics are **Streamlines** and **the Equation of Continuity with derivation**. This physics tutorial is apt for class 11 physics for ISC, CBSE, IGCSE, and other boards (grade 11 and 12 for international boards).

## Streamlines

The path taken by a fluid particle under a steady (laminar) flow is called a streamline. The velocity of such particle at each point along its path is always tangent to the streamline, as shown in Figure 1.

A set of streamlines like the ones shown in Figure 1 form a *tube of flow. *Note that fluid particles cannot flow into or out of the sides of this tube; if they could, then the streamlines would cross each other.

## Equation of Continuity with derivation

Consider an ideal fluid flowing through a pipe of nonuniform size, as illustrated in Figure 2.

The particles in the fluid move along streamlines in a steady flow.

In a time *t*, the fluid with a speed v1 at the bottom end of the pipe moves a distance Δ*x*1 =*v*1. *t*.

If *A*1 is the cross-sectional area in this region, then the mass of fluid contained in the left shaded region in figure 2 is **m1 = pA1 Δx1 = p A1 v1 t** where p is the (nonchanging) density of the ideal fluid.

Similarly, the fluid that moves with speed v2 through the upper end of the pipe in the time *t *has a mass ** m2 = p A2 Δx2 =p A2 v2 t**.

However, because *mass is conserved *and because the flow is steady, the mass that crosses *A*1 in a time *t *must equal the mass that crosses *A*2 in the time *t*.

That is, m1=m2,

or, **p A1v1t= pA2v2t**

=> this means that **A1 v1=A2 v2 = constant**

This expression **A1 v1= A2 v2 = constant** is called the **equation of continuity**. **It states that the product of the area and the fluid speed at all points along the pipe is a constant for an incompressible fluid.**

This equation tells us that the speed is high where the tube is constricted (small *A*) and low where the tube is wide (large *A*). The product *Av*, which has the dimensions of *volume per unit time*, is called either the ** volume flux or the flow rate**.

The condition **Av=constant** is equivalent to the statement that the volume of fluid that enters one end of a tube in a given time interval equals the volume leaving the other end of the tube in the same time interval if no leaks are present.

## Statements of the Equation of Continuity

1) The Equation of Continuity **states that the product of the area and the fluid speed at all points along the pipe is a constant for an incompressible fluid.**

2) The Equation of Continuity **states that** **the volume of fluid that enters one end of a tube in a given time interval equals the volume leaving the other end of the tube in the same time interval if no leaks are present. **