CBSE Class 11-science Answered

Assuming that the velocity of flow of a liquid (v) through a narrow tube depends on the radius of the tube(r) , density of the liquid(D) and coefficient of viscosity(n),find the expression for velocity.

Asked by | 24 May, 2012, 12:23: AM

Expert Answer

In fluid dynamics, the HagenPoiseuille equation is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe. The assumptions of the equation are that the flow is laminar, viscous and incompressible and the flow is through a constant circular cross-section that is substantially longer than its diameter. It is also assumed that there is no acceleration of liquid in the pipe. The equation is also known as the HagenPoiseuille law, Poiseuille law and Poiseuille equation.

The flow of fluid through a pipe of uniform (circular) cross-section is known as HagenPoiseuille flow. The equations governing the HagenPoiseuille flow can be derived directly from the NavierStokes equations in cylindrical coordinates by making the following set of assumptions:

The flow is steady ( $\text{[math]}$ ).

The radial and swirl components of the fluid velocity are zero ( $\text{[math]}$ ).

The flow is axisymmetric ( $\text{[math]}$ ) and fully developed ( $\text{[math]}$ ).

Then the second of the three NavierStokes momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to $\text{[math]}$ , i.e., the pressure $\text{[math]}$ is a function of the axial coordinate $\text{[math]}$ only. The third momentum equation reduces to:

$\text{[math]}$ where $\text{[math]}$ is the dynamic viscosity of the fluid.
The solution is
$\text{[math]}$

Since $\text{[math]}$ needs to be finite at $\text{[math]}$ , $\text{[math]}$ . The no slip boundary condition at the pipe wall requires that $\text{[math]}$ at $\text{[math]}$ (radius of the pipe), which yields

$\text{[math]}$

Thus we have finally the following parabolic velocity profile:

$\text{[math]}$

The maximum velocity occurs at the pipe centerline ( $\text{[math]}$ ):

$\text{[math]}$

The average velocity can be obtained by integrating over the pipe cross section:

$\text{[math]}$