Slide 1 : 1 CHAPTER - FLUID MECHANICS
PART – 2
HYDRODYNAMICS aroragaurav.wordpress.com
STREAMLINE, LAMINAR AND TURBULENT FLOW : aroragaurav.wordpress.com 2 STREAMLINE, LAMINAR AND TURBULENT FLOW (A) Streamline flow Streamline flow of a liquid is that flow in which every particle of the liquid follows exactly the path of its preceding particle and has the same velocity in magnitude and direction as that of its preceding particle while crossing through that point. In a streamline flow, no two streamlines can cross each other. The greater is the crowding of stream lines at a place, the greater is the velocity of liquid particles at that place and vice-versa. Important properties of stream lines
Slide 3 : aroragaurav.wordpress.com 3 (B) Laminar flow A flow, in which the liquid moves in layers is called a laminar flow. The velocity of liquid flow is always less than the critical velocity of the liquid. In general, laminar flow is a streamline flow. (C) Turbulent flow When a liquid moves with a velocity greater than its critical velocity, the motion of the particles of liquid becomes disorderly or irregular. Such a flow is called turbulent flow. Critical Velocity The critical velocity is that velocity of liquid flow, upto which its flow is streamlined and above which its flow becomes turbulent.
Slide 4 : aroragaurav.wordpress.com 4 REYNOLD NUMBER Reynold number is a pure number which determines the nature of flow of liquid through a pipe. NR = h is the coefficient of viscosity of the liquid, r is the density of liquid NR is a constant called Reynold number. If the value of Reynold number lies between 0 to 2000, the flow of liquid is stream line or laminar. For values of NR above 3000, the flow of liquid is turbulent and for values of N between 2000 to 3000
Slide 5 : aroragaurav.wordpress.com 5 EQUATION OF CONTINUITY Consider a non-viscous liquid in streamline flow through a tube AB of varying cross-section. Let a1, a2 = area of cross-section of the tube at A and B respectively. v1, v2 = velocity of flow of liquid at A and B respectively, r1, r2 = density of liquid at A and B respectively. \ Volume of liquid entering per second at A = a1v1 Mass of liquid entering per second at A = a1v1r1 Similarly, mass of liquid leaving per second at B = a2v2r2
Slide 6 : aroragaurav.wordpress.com 6 Mass of liquid entering per second at A
= mass of liquid leaving per second at B
Or a1 v1 r1 = a2 v2 r2 If the liquid is incompressible,
Then r1 = r2 a1 v1 = a2 v2 a v = constant This is known as equation of continuity.
Slide 7 : aroragaurav.wordpress.com 7 ENERGY OF A LIQUID A liquid in motion may possess three types of energy. Pressure energy : It is the energy possessed by a liquid by virtue of its pressure. It is the measure of work done in pushing the liquid against pressure without imparting any velocity to it. Pressure energy per unit volume of the P Potential energy : It is the energy possessed by liquid by virtue of its height or position above the surface of earth or any reference level taken as zero level. Potential energy per unit volume of liquid = = r gh Kinetic energy : It is the energy possessed by a liquid by virtue of its motion or velocity. K.E. per unit volume of the liquid r v2
Slide 8 : aroragaurav.wordpress.com 8 BERNOULLI’S THEOREM Bernoulli’s theorem states that for the streamline flow of an ideal liquid, the total energy (the sum of the pressure energy, potential energy and kinetic energy) per unit mass remains constant at every cross-section throughout the flow. An ideal liquid is one which is perfectly incompressible and nonviscous. Proof : Consider a tube AB of varying cross-section through which an ideal liquid is in streamline flow.
Slide 9 : aroragaurav.wordpress.com 9 Let
P1 = pressure applied on the liquid at A
P2 = pressure at the end B, a1, a2 = area of cross-section of the tube
at A and B respectively, h1, h2 = mean height of section A and B from the ground or a reference level. v1, v2 = normal velocity of the liquid flow at section A and B respectively r = density of the ideal liquid flowing through the tube. By equation of continuity is a1 v1 = a2 v2 = m/r = V (say) a1 > a2 \ v2 > v1
Slide 10 : aroragaurav.wordpress.com 10 Force on the liquid at section A = P1 a1 Work done/second on the liquid at
section A = P1 a1 × v1 = P1V Similarly, Work done/second by the liquid at section B = P2 a2 × v2 = P2V Net work done/second on the liquid = P1V – P2V Change in potential energy/sec of the liquid from A to B = mgh2 – mgh1 Change in kinetic energy/sec of the liquid from A to B = mv22 – mv12 Work done/sec by the pressure energy
= change in P.E./sec + change in K.E./sec
Slide 11 : aroragaurav.wordpress.com 11 Dividing throughout by m, we get P1V – P2V = (mgh2 – mgh1) + P1V + mgh1 + m v12 = P2V + mgh2 + mv22 Hence
Slide 12 : aroragaurav.wordpress.com 12 Limitations of Bernoulli’s Theorem While deriving the Bernoulli’s equation, it is assumed that velocity of every particle of liquid across any cross-section of tube is uniform. Practically it is not correct. Infact, the particles of the liquid in the inner most (i.e. central layer) have maximum velocity and those on a layer in contact with the tube have least velocity. Therefore, we should take the mean velocity of the liquid. The viscous drag of the liquid which comes into play when the liquid is in motion has not been taken into account. While deriving the above equation, it is assumed that there is no loss of energy when liquid is in motion. Infact some K.E. is converted into heat and is lost.