Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

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Chapter 10 Laminar Flow

  • Fluid particles move along straight parallel paths in layers or laminae
  •  It occurs at low velocity; Viscosity force predominates inertial force.
  •  Relation between Shear and Pressure Gradients in Laminar Flow
  •  For a steady uniform flow,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • Thus, for a steady uniform laminar flow the pressure gradient in the direction of flow is equal to the shear stress gradient in the normal direction.
  • By using newton’s law of viscosity :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • The differential equation of laminar flow is given by,

Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • Steady Laminar Flow in Circular Pipes (Nagen - Poiseulle flow)
  • In a circular pipe with steady laminar flow, the shear stress t varies linearly along the radius of the pipe as,

Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • The maximum value of strees t0 occurs at r = R (i.e., at the centre of pipe),

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • The negative sign on (p/x) indicates decrease in pressure in the direction of flow. The pressure must decrease  because pressure force is the only means available to compensate for resistance to the flow, the potential  and kinetic energy remain constant.
  • In laminar flow shear stress is entirely due to viscous action. Hence by Newton’s law of viscosity.

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • By combining (i) and (iii) and integrating under proper limits, we get,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

which shows that for laminar flow through circular pipe, velocity of flow varies parabolically and the
surface of velocity distribution is a paraboloid of revolution.

  • The maximum velocity occurs at axis and is given by,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • By combining equal (iv) and (v), the velocity distribution in laminar flow is obtained as,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

  • The corresponding expression for discharge is given by,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

This equation is called Hagen Poiseulle equation.

  • The mean velocity of flow (V) is given by,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE).                  ..............(viii)

Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)            ........ ...(ix)

  • The point where local velocity is equal to mean velocity is given by,

Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

so mean velocity of flow occurs at a radial distance of 0.707 R from the centre of the pipe.

  • The velocity and shear stress distribution are as shown below :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

Laminar Flow Between Parallel Plates
Case 1 : Both plates are at Rest

  • For the laminar flow of fluid between two fixed parallel flat plates located at distance B apart, the velocity

distribution is given by

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)          ...(i)

  • The maximum velocity occurs at y = (B/2) and it is given by,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)           ...(ii)

  • The discharge q per unit width of plate is given by,

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)         ...(iii)

  • The mean velocity of flow is given by

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)       ........(iv)

By comparing (ii) and (iv)

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)           ....(v) 

  • The pressure drop between any two points distance L apart is given by

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)     ...(vi)

  • The distribution of shear stress is given by

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)            ...(vii)

  • The shear stress is maximum at y = 0 and is given by :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)              ...(viii)

  • The velocity and shear stress distribution is as shown below :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

Case 2 : When one Plate Moving And Other at Rest is known as COUETTE Flow

  • The velocity and stress distribution in COUETTE FLOW is shown below :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

Expression For Head Loss
(a) In case of laminar flow through pipes :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

(b) In case of laminar flow through parallel plates :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

(c) In case of open channel flow :

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

(d) The general equation is

 Laminar Flow Notes | Study Civil Engineering SSC JE (Technical) - Civil Engineering (CE)

where,
hf = loss of head in length L
V = mean velocity of flow
D = characteristic dimension representing the geometry of passage.
k = constant, whose value depends upon the shape of passage.

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