In this article we will discuss about:- 1. Laminar Incompressible Flow in a Circular Pipe 2. Laminar Incompressible Flow in a Pipe of Varying Diameter 3. Two-Dimensional Laminar Flow between Two Parallel Plates. [with solved examples]
Fig. 11.6 shows a horizontal circular pipe of radius R conveying a viscous incompressible fluid of viscosity μ. Consider sections 1-1 and 2-2, l units apart.
Let p1 and p2 be the pressure intensities at these sections.
In this case the velocity of the fluid is different at different distances from the axis of the pipe.
Let the velocity be u at a radius r.
∴ Velocity gradient = – du/dr
(The negative sign indicates that as r increases u decreases.)
∴ Shear stress intensity = μ x velocity gradient = – μ(du/dr)
Total resistance to the motion of a cylinder of fluid of radius r and length l
= Shear stress intensity x Surface area of cylinder
= – μ (du/dr)2πrl
Consider sections 1-1 and 2-2 of a pipe whose diameter varies uniformly from D1 at section 1-1 to D2 at section 2-2. Let the sections be l apart. The pipe conveys a liquid of viscosity μ. Let the pressure intensities at these sections be p1 and p2. Consider a section XX at a distance x from section 1-1.
Consider a steady laminar flow between two parallel plates 2t apart. Let b be the width of each plate. Let the width of each plate be very large compared with the distance between the plates. In such a case we can make the approximation that the flow is two-dimensional.
All liquid particles in one level parallel to the plates will have the same velocity. The liquid flows in layers parallel to the plates. The velocity of the liquid particles in any such layer is dependent on its distance from the centre. Consider sections 1-1 and 2-2, l units apart. Let pt and p2 be the pressure intensities at these sections.
Consider the central rectangular block of liquid of height 2y, width b between the two sections.
Shear stress intensity at the upper and lower faces of this liquid block = μ x velocity gradient