Lecture 24 Surface tension viscous flow thermodynamics Surface tension surface energy The atoms at the surface of a solid or liquid are not happy Their bonding is less ideal than the bonding of atoms in the interior of the material For this reason they try to minimize the amount of surface energy that they have to waste In a liquid which is falling through a gas the smallest surface energy occurs when the fluid forms a spherical drop e g a raindrop However when a liquid drop is placed on a solid surface it may form different shapes depending on whether the solid surface attracts or repels the liquid Surface energy has units J m2 ie energy per unit area However the discussion of surface energy effects are often in terms of surface tension The concept of surface tension is really useful for doing calculations but is actually a bit misleading conceptually First note that surface tension has units of N m and that these units are the same as J m2 so where ever we discuss surface tension we can reinterpret the discussion in terms of surface energy which is usually more meaningful physically Due to surface tension a light object placed on the surface of the liquid may deform the surface of the liquid instead of becoming submerged The reason for this is that if the surface of the liquid is broken it creates new surfaces and this is energetically unfavorable For calculation purposes the surface of a liquid acts like an elastic membrane until it breaks and object falls into the liquid There are many insects which can walk on water using this effect This is also the reason that the water feels hard when diving from a high board Measuring surface tension The surface tension of a liquid is the maximum restoring force that can be provided by the surface before it breaks A common experimental procedure is to slowly lift a small ring out of a liquid and measure the maximum force required Ft weight F where F is the force due to surface tension We then find Fmax 2L 1 where Fmax is the maximum of the excess force F From measurements such as this the surface tension of water at 200 C is found to be 0 059N m 1 0 059J m2 Actually the best way to think of this physically is that it is the energy cost of forming a water air interface An interesting application is an insect walking on water Consider an insect of mass m 2 10 5 kg with feet of radius r 1 5 10 4 m walking on water Find the contact angle The weight of the insect is mg and this must be supported by the upward force of surface tension so that 6 2 r Cos mg 2 Solving yields 620 How much weight can the insect put on before it falls in the water The maximum weight that can be supported is 6 2 r 3 4 10 5 kg Equilibrium contact angle As remarked above a liquid drop in a gas will form a sphere to minimize its surface areas Solids are different and form a crystal surface which reflects the crystal structure and directional bonding inside a material We are not concerned with the shapes of crystals here When a liquid drop is placed on a surface it may wet the surface or it may ball up into drops The case where it balls up is typical of water on glass or on wax If the liquid molecules have a favorable atomic bonding with the substrate then the liquid wets the surface while if the bonding is less favorable drops form The wetting angle is the contact angle between the drop and the solid substrate and it is related to the surface bonding energies between the liquid and the substrate as well as the surface tension of the liquid and that of the solid Capillary forces If a liquid has a contact angle with a thin tube it produces an associated vertical force on the liquid the capillary force which is given by F 2 r Cos 3 If the wetting angle is less that 900 this causes the fluid to rise in a thin tube while if the angle is greater than 900 the fluid in the capillary drops The real reason for this effect is the surface energy between the fluid and the walls of the capillary If this energy is favorable the fluid wants to make more contact with the capillary surfaces which induces the rise in the capillary 2 tube The height to which a fluid rises or falls is determined by a balance of the capillary force with the force of gravity F mg V g r 2 h g 4 Solving for h we find that h 2 Cos gr 5 Viscous flow transport phenomena Shearing a liquid The force required to shear a liquid gives a measure of its viscosity F Av d 6 The units of are N s m2 Viscosity has its own unit the poise where 1 poise 0 1N s m2 7 Note 1cp 1centipoise 10 2 poise 10 3 N s m2 Flow due to pressure or concentration differences Note that the Bernoulli equation ignores viscosity as we did not include the energy dissipated in the Work Energy theorem Viscosity is due to dissipation and needs a different treatment It is very difficult to calculate fluid flow in the presence of viscosity However we shall look at a couple of simple but really important cases If there is a pressure difference between two places in a fluid the fluid begins to flow The larger the pressure difference the faster the flow but it also depends on how viscous the fluid is e g is it honey which has high viscosity or is it water which has low viscosity Finding the flow pattern of a viscous fluid is generally a hard problem however flow through a cylindrical pipe of radius R and length L is quite simple and was calculated by Poiseuille 1797 1869 who found that F low Rate Due to P ressure Gradient 3 R4 P1 P2 V t 8 L 8 Note that it is a lot easier to push flow through a wide pipe than through a narrow one Another way to induce flow in a viscous fluid is by having a difference in concentration of a chemical species We will discuss this in more detail when we do thermodynamics For now we just learn about the effect The most important example is when the salt concentration is different on two sides of the cell membrane then water and a small amount of salt crosses the membrane until the salt concentrations are equal This is called osmosis and can be deadly as if too much water leaves a cell it shrinks and can die while if too much water goes into a cell it can expand and explode In general the flow rate is given by Fick s law Dif f usive F low Rate M C2 C1 DA t L 9 where D is …
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