# Fluid Mechanics. Jim Emery 9/3/ Bernoulli s Law 2. 2 Torricelli s Law 3. 3 Time to Empty a Tank 3. 4 Viscosity 4. 5 The Acceleration 5

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1 Fluid Mechanics Jim Emery 9/3/2010 Contents 1 Bernoulli s Law 2 2 Torricelli s Law 3 3 Time to Empty a Tank 3 4 Viscosity 4 5 The Acceleration 5 6 The Equation of Motion Due to Cauchy 7 7 The Deformation Tensor 7 8 The Navier-Stokes Equation for Incompressible Flow 7 9 Flow in Pipes, Poiseuille Flow 8 10 Flow in River Channels 8 11 Ground Water Flow 8 12 Diffusion 8 13 Dimensional Analysis and the Reynolds Number 8 14 Hydrology 9 1

2 15 Bibliography 9 1 Bernoulli s Law Consider flow in a pipe of variable diameter. Assume the fluid is not compressible, there is zero viscosity and friction, and the internal energy and temperature of the fluid is constant. Consider the volume of fluid between two cross sections of the pipe. At one end let the cross sectional area be A 1, the pressure p 1, the height h 1, and the velocity v 1. At the other end let the cross sectional area be A 2,the pressure p 2, the height h 2,andthevelocityv 2. Suppose the surface boundary of the volume moves positive distance dx 1 in time dt at the first end. Then in effect a volume dv = dx 1 A 1, is moved to the other end with a new velocity v 2 with dv = dx 2 A 2. The work done on the volume V is dw = dx 1 A 1 p 1 dx 2 A 2 p 2 = dv (p 1 p 2 ). This is equal to the change de of the kinetic and potential energy of the transferred volume dv. de = dv ρ 2 (v2 2 v2 1 )+dv gρ(h 2 h 1 ). Hence and so So dw de dv =(p 1 p 2 ) dw de =0, [ ] ρ 2 (v2 2 v1)+gρ(h 2 2 h 1 ) =0. p 1 + ρ 2 v2 1 + gρh 1 = p 2 + ρ 2 v2 2 + gρh 2. That is p gρ + v2 2g + h, is constant anywhere in the pipe. This is Bernoulli s Law. 2

3 2 Torricelli s Law Suppose we have a tank of relatively large diameter and height h. Suppose thereisanexitholeofareaa at the bottom of the tank. We may apply Bernoulli s law where we take both the pressure and velocity at the top of the tank to be zero. And we assume that the pressure at the bottom of the tank, in the exit hole, is also zero. Thus we have h = v2 2g So the exit velocity is This is Torricelli s law. v = 2gh. 3 Time to Empty a Tank From Torricelli s law the change of volume due to flow out of a bottom hole of area A is dv dt = A 2gh, where dv dt might be measured in cubic meters per second. If the cylindrical tank has radius r then So V (h) =hπr 2 dv = πr 2 dh Thus dt = πr2 A 2gh dh. Thus the time to drain the tank is t = πr2 0 A 2g h 0 3 dh h,

4 where h 0 is the full height of the tank. Thus t = πr2 A 2g [2 h] 0 h 0 = 2πr2 A h 0 2g = πr2 2h 0 /g A = A t A h 2h 0 /g, where A t is the cross sectional area of the tank, and A h is the cross sectional area of the hole at the bottom of the tank. This is assuming zero viscosity of the fluid. Notice that if A h = A t the time is 2h 0 /g, which is the time it takes a mass to fall a distance h 0. However, we are using Torricelli s law, whose derivation assummed that the cross sectional area of the tank to be very much larger than the exit hole. 4 Viscosity Newton first introduced the concept of viscosity using a idea analogous to a shear strain, replacing static displacement by velocity. Thus in the simplest case, if a fluid is flowing between two parallel plates, one moving with velocity V and the other fixed, then in a time δt the fluid contacting the upper plate moves a distance Vδt while the fluid attached to the lower plate is at rest. So the fluid is being sheared. If it were a solid this displacement would be accompanied by a shear stress equal to a force per unit area on the upper plate. Assuming that the velocity between the plates varies linearly with height, the shear strain would be du dy δt If the fluid were a solid this shear strain and shear stress would be proportional. In the case of a fluid the force between layers of fluid is determined by 4

5 the change in velocity rather than the displacement. Hence the shear stress τ is proportional to the gradient of the velocity rather than to the gradient of the displacement as is the case in elasticity. Thus the shear stress is τ = µ dv dy where v is the horizontal velocity and y is the distance perpendicular to the flow and to the plates. µ is called the viscosity. Newton first introduced this equation and hence a fluid that obeys this law is called a Newtonian fluid. From this equation the units of viscosity are FT/L 2 where F is force, T is time, and L is distance. So in the SI system the unit of viscosity is Newton second per meter squared, Ns m 2. In the cgs system it is in dyne seconds per square centimeter a unit called the Poise. 5 The Acceleration ds cm 2, Acceleration in fluid mechanics can be confusing. This is because there are two different velocities, a particle velocity attached to a moving particle, or to an infinitesimal volume in the fluid, and a flow velocity at a fixed point in the fluid. As a particle p moves in the fluid with a particle velocity, which we write as v p (t), it may experience an acceleration, as any particle can in mechanics. This is the acceleration of Newton s second law, which is proportional to the force on the particle. However, at the same time we may consider the flow velocity at a fixed point x in space, which we call the field velocity, which we write as v(t, x). As time passes this velocity will be the velocity of different particles as they pass this fixed point. We consider this to be the fluid velocity at that point. This fluid velocity, the field velocity, is written as v(t, x). The rate of change of this field velocity at the fixed point x is not the acceleration of a particle that happens to be at x at time t. However, acceleration 5

6 of the particle velocity, sometimes called the material velocity, can be computed by differentiating the field velocity with respect to time. The result is obtained by partial differentiation as a = v t + v v. Indeed, suppose a particle p has coordinates (x p (t),y p (t),z p (t)), at time t. Then the particle velocity at time t is ( dxp (t), dy( pt), dz ) p(t) = v p (t), dt dt dt which has the same value as the field velocity at the point (x p (t),y p (t),z p (t)) at time t. Thatis, v p (t) =v(t, x p (t),y p (t),z p (t)). The two velocities are distinguised as function, being functions of different variables. The x component of particle acceleration is d 2 x p (t) dt 2 = dv px(t) dt = v x t + v x dx p x dt + v x dy p y dt + v x dz p z dt = v x t + v x v p (t). We have written v x for the x component of v. Similarly, dv py (t) dt = v y t + v y v p (t), and dv pz (t) = v z dt t + v z v p (t). The particle velocity has the same value as the field velocity at time t, v p (t) =v(t, x p (t),y p (t),z p (t)), so we can replace v p by v, and then obtain as written above. a = dv p dt = v t + v v, 6

7 6 The Equation of Motion Due to Cauchy ρ dv = ρf + T dt where the left side is the acceleration, f is an external force, and T is the stress tensor. (Serrin, p 136) 7 The Deformation Tensor In the case of elasticity, the strain tensor is defined in terms of the derivatives of the displacement u. Let u 1,u 2,u 3 be the coordinates of a displacement. The strain tensor is defined by e ij =( u i + u j )/2. x j x i See Solkolnikoff, The Mathematical Theory of Elasticity, p25, for comments about alternate notation for the strain coefficients. The strain is a Cartesian tensor, as one sees by looking at the properties of the gradient and the Jacobian. In the usual case the stress tensor is a linear function of the strain tensor. In the case of fluid mechanics we have not just displacement but continuous motions, so the analog of the strain tensor is called the deformation tensor D. It is defined similarly to the strain tensor, replacing the displacement u by the velocity v, d ij =( v i + v j )/2. x j x i 8 The Navier-Stokes Equation for Incompressible Flow In the case where the stress tensor is a linear function of the deformation tensor D, the case of linear viscosity, the Cauchy equation of motion become the Navier-Stokes equation, ρ dv = ρf p + (2µD). dt (Serrin, p 217) 7

8 9 Flow in Pipes, Poiseuille Flow The Navier-Stokes equation is solvable in closed form for only a few special cases. One of these is the steady laminar flow in cylindrical pipes (Munson et. al., page 379). This flow is called Hagan-Poiseuille flow. Poiseuille s law (pwah-zweez) Q = πr4 p, 8µl where Q is the volume rate of flow, R is the pipe radius, p, is the pressure drop, µ is the viscosity, and l is the pipe length. At any cross section the velocity distribution is parabolic. Notice that if the viscosity were zero, then the flow rate would be infinite. This seems a bit disturbing. However, if the viscosity were zero, then there would be no resistance to flow, and the pressure difference would accelerate the velocity to infinity. 10 Flow in River Channels 11 Ground Water Flow 12 Diffusion Various kinds of diffusion processes are governed by a diffusion equation, which is similar to the heat equation. 13 Dimensional Analysis and the Reynolds Number The Reynolds Number is a dimensionless number which determines the boundary between laminar flow and turbulent flow. ρvd µ, where ρ is the density, v is the velocity, d is the pipe diameter (or some other characteristic length in a non-pipe problem), and µ is the viscosity. The 8

9 dimension is (ml 3 )(lt 1 )(l)(l 2 f 1 t 1 ) =(fl 1 t 2 )(l 3 )(lt 1 )(l)(l 2 f 1 t 1 ) = f 0 l 0 t 0, where m is mass, f is force, l is length, and t is time. 14 Hydrology Hydrology is an earth science that studies the movement of water and water vapor on the earth. 15 Bibliography [1]Munson Bruce R, Young Donald F, Okiishi Theodore H, Fundamentals of Fluid Mechanics, John Wily and Sons, [2]Viessman Warren Jr, Lewis Gary L, Knapp John W, Introduction to Hydrology, Harper Collins, Third Edition, [3]Prandtl L., Tietjens O. G., Applied Hydro- and Aeromechanics, Dover reprint, [4]Prandtl L., Tietjens O. G., Fundamentals of Hydro- and Aeromechanics, McGraw-Hill, [5]Serrin James, Mathematical Principles of Classical Fluid Mechanics, insiegfriedflügge (Herausgeber): Handbuch der Physik. Springer- Verlag, [6]Birkhoff Garrett, Hydrodynamics, Dover Edition, 1955, First Edition: Princeton University Press, (Dimensional Analysis, Buckingham Pi Theorem.) (Garrett Birkhoff was a leading mathematician. His father G. E. Birkhoff was considered to be the first great American Mathematician). 9

10 [7]Landau L. D. and Lifshitz E. M., Fluid Mechanics, Volume 6 of Course of Theoretical Physics, Institute of Physical Problems, U.S.S.R Academy of Sciences. Translated from the Russian by Sykes J. B. and Reid W. H., Pergamon Press, [8]Langhaar Henry L., Dimensional Analysis and Theory of Models, John Wiley and Sons, [100]Author, Title, publisher,year. 10

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