Navier Stoke’s problem “Water Flow in a Pipe”

In nature, we know water flows from higher elevation to lower elevation. We have seen this from the river flowing from the up north from Himalayas to down towards terai. This is caused by the difference in the elevation which creates a potential gradient.

But for a pipe which is laid out horizontally, the potential gradient is zero (as the start and end of the pipe are at same level), in order to convey water in it, a pressure gradient is needed. We need to supply energy in one end to convey it to the other end. The difference in the pressure from one end to the other end will govern the speed of the water flowing. Higher the pressure difference, faster will the water flow in the pipe.

Let us analyze this mathematically,

Here P1 is the pressure at start point of the pipe, and P2 is pressure at the end of the pipe, and distance be these two points be L.

The blue color profile in the pipe is the velocity profile. If the water is flowing full in the pipe, at the boundary, due to the roughness (which causes opposing friction force) of the pipe, the velocity of the water is zero, while at the center of the pipe, there is nothing to stop the water to flow so the velocity profile will be maximum. From experimentation the velocity flow profile is a parabolic one.

To understand the pipe flow, let us now

As we are flowing water horizontally (x-direction) specific, water cannot move to other direction due to the circular enclosure of the pipe. So the y and z component of the velocity of the water is zero. Thus,

For our case, taking water as in incompressible fluid, the continuity equation is:

Navier-Stoke’s Equation

Reynold number is defined to differentiate the classes of flow. Water flow is classified into laminar, transient and turbulent flow by the state of intermixing of fluid while propagating from one end to other.

Laminar flow is the one where there won’t be intermixing of water and velocity profile we assumed above will holds together. This is numerically defined with Reyonlds number stated below as:


And D is the internal diameter of the pipe.

For Re<1, the flow will have laminar flow (no intermixing/ turbulence)

Navier-Stoke’s (NS) Equation

Here,

From Newton’s Second Law of Motion,


Navier-Stoke’s equations are a force balance per unit volume.

This leaves the question, what accelerates the fluid.

From the right hand side of the equation, the driving force is caused by the three terms, pressure gradient (first term in NS equation),  acceleration due to gravity (second term in NS equation), and the fluid flow (third term in NS equation , fluid accelerates in direction of increasing velocity gradient).

So while solving any FLUID MECHANICS SOLUTIONS, use the above equations and simplify the problems like:

NAVIER- STROKE EQUATIONS + CONTUINITY EQUATION + BOUNDARY CONDITIONS

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