Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

Mechanical Engineering SSC JE (Technical)

Mechanical Engineering : Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev

The document Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).
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​​​Chapter 10LAMINAR FLOWFluid 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,
dt/dy = Δp/ΔxThus, 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 :
Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevThe differential equation of laminar flow is given by,
μd2v/dy2 = Δp/Δx
 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,Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevThe maximum value of stress t0 occurs at r = R (i.e., at the walls of the pipe),Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevThe negative sign on 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. Fully developed horizontal pipe flow is merely a balance between pressure and viscous forces–the pressure difference acting on the end of the cylinder of area πr2 and the shear stress acting on the lateral surface of the cylinder of area  2πrl. This force balance can be written asp1πr2−(p1−∆p)πr2−2πr/τ=0 ∴ Δp/l= 2τ/rFor laminar flow of a Newtonian fluid, the shear stress is simply proportional to the velocity gradient,τ=μdu/dy.   In the notation associated with our pipe flow, this becomes       τ=−μdu/dr By combining the above two equations, we obtain       du/dr=−(∆p/2μl)which can be integrated to give the velocity profile:      u=−(∆p/4μl).r2 + C1  where C1 is a constant. Because the fluid is viscous, it sticks to the pipe wall so that 'u=0' at 'r=D/2'. Thus, C1=4R2.(∆p/16μl). Hence, the velocity profile can be written as Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevAnd,Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevwhere Vmax= (ΔpR2/4μl) is the centerline velocity i.e. at the center of the pipe.By definition, the average velocity is the flow rate divided by the cross-sectional area,V=qv/πR2 so that for this flow,Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev The point where local velocity is equal to average velocity is given by,
Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevso mean velocity of flow occurs at a radial distance of 0.707 R from the centre of the pipe. Pressure drops Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevPutting the value of vmax in the above equation,Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevThis above equation is commonly referred to as Hagen-Poiseuille’s law. The velocity and shear stress distribution are as shown below :Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevLaminar Flow Between Parallel Plates 
Case 1 : Both plates are at Rest   Using the Navier-Stokes equations, we can determine the flow between two fixed horizontal, infinite parallel plates.  In order to to this, we will need to describe how the fluid particles move.  For this case, there will be no flow in the y or z direction; v = 0 and w = 0.  As a result, all of the fluid flow will be in the x-direction.  Hence, the resulting continuity equation will be ∂u/∂x=0.  In addition,  u will have no variation in the z-direction for the infinite plates. This means that the stead flow ∂u/∂t = 0 so that u = u(y)  Taking these conditions into account the Navier-Stokes equation will be reduced the following equations.(Eq 1) = -∂p/x+u (∂2u/∂y2)   (Eq 2)  = -∂p∂y - ρg   (Eq 3)  = -∂p∂z​In these equations, gx=0, gy = -g, and gz=0.​Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev  As a result, the y-axis will point up. Next, we will integrate equation 2 and 3 to generate the following equation. In turn, this shows that there is a variation of pressure hydrostatically in the y-direction.Next, equation 1 will be rewritten into the following form and integrated twice.After integrating and determining all the constants,the velocity distribution can be fully derived.Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevIn turn, the resulting velocity profile between the fixed plates is parabolic.Volume Rate of Flow: The volume flow rate ,q, represents the total volume of fluid pass between the two plates.          Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevMean Velocity: In turn, taking in consideration that the pressure gradient is inversely proportional to the viscosity and has a strong dependence on the gap width, the mean velocity can be determined.         Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevMaximum Velocity: Finally, maximum velocity will occur at y = 0 between the two the parallel plates.  As a result, the maximum velocity can be expressed in the following mathematical form.         Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevThe pressure drop between any two points distance L apart is given by      Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev The distribution of shear stress is given by        Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev        The shear stress is maximum at y = 0 and is given by :           t0 = (∂p/∂x)(B/2)             Case 2 : When one Plate Moving And Other at Rest is known as COUETTE  Flow  The velocity distribution in COUETTE FLOW is shown below :Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevConsider two-dimensional incompressible plane(∂/∂z=0) viscous flow between parallel plates a distance 2h apart, as shown in fig.. We assume that the plates are very wide and very long , so the flow is essentially axial, u≠0 but v=w=0. The present case is where the upper plate moves at velocity 'V' but there is no pressure gradient. Now, by applying the continuity equation and integrating, then finding the constants, we get the velocity distribution:   Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev     (for  -h  ≤  y  ≥  +h)Expression For Head Loss
(a) In case of laminar flow through pipes :hf = Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev(b) In case of laminar flow through parallel plates : hf =Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev(c) In case of open channel flow : hf =Chapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRev(d) The general equation isChapter 10 Laminar Flow - Fluid Mechanics, Mechanical Engineering Mechanical Engineering Notes | EduRevwhere, 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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