Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time
A streamline is one that drawn is tangential to the velocity vector at every point in the flow at a given instant and forms a powerful tool in understanding flows. This definition leads to the equation for streamlines.
where u,v, and w are the velocity components in x, y and z directions respectively as sketched.
The flow velocity u of a fluid is a vector field
which gives the velocity of an element of fluid at a position x and time t.
The flow of a fluid is said to be steady if u does not vary with time. That is if
When a fluid is in motion, it must move in such a way that mass is conserved.Consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time). The inflow and outflow are one-dimensional, so that the velocity V and density are constant over the area A.
Now we apply the principle of mass conservation. Since there is no flow through the side walls of the duct, what mass comes in over goes out of , (the flow is steady so that there is no mass accumulation). Over a short time interval
As volume is same so this equation can be written as
This is a statement of the principle of mass conservation for a steady, one-dimensional flow, with one inlet and one outlet. This equation is called the continuity equation for steady one-dimensional flow.