1 0 - FLUID MECHANICS Page Introduction Fluid is a matter in a state which can flow. Liquids, gases, molten metal and tar are examples of fluids. Fluid mechanics is studied in two parts: ( i ) Fluid statics - Study of the forces and pressures acting on stationary fluid. Pascal s law and Archimedes principle and surface tension are discussed in fluid statics. ( ii ) Fluid dynamics - Study of motion of fluid and properties related to it as a result of forces acting on fluid. Bernoulli s theorem and its applications and viscosity of fluid are discussed here. Fluid dynamics is studied in two sections: Hydrodynamics and Aerodynamics. 0. Pressure Pressure is the force acting on a surface per unit area in a direction perpendicular to it. It is a scalar quantity and its SI unit is N / m named pascal ( Pa ) in honour of the French scientist Blasé Pascal. Its dimensional formula is M L - T -. Thus, Pressure, P ( Pa ) = Force, F ( N ). Area, A ( m ) A bigger unit of pressure is bar. bar = 0 5 Pa. atmosphere pressure ( atm ) = Pa or N /m = 760 mm ( 76 cm ) of Hg column. Density: Density is the ratio of mass to volume of an object. It is a scalar quantity and its S I unit is kg /m. Liquids are almost incompressible. Hence, the density of a liquid remains almost constant at a given temperature for small change in the value of pressure. Gases are compressible. Hence, the volume of gas decreases and density increases with increase of pressure. Relative density / Specific density / Specific gravity: Relative density also known as specific density or specific gravity of a given substance is the ratio of its density to the density of water at 77 K ( i.e., 4 C ). It is a dimensionless quantity and hence does not have a unit. Also, Relative ( specific ) density of an object = Mass of the Mass of an object same volume of water at 77 K 0. Pascal s Law A change in pressure applied to an enclosed ( incompressible ) fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel. This statement is known as Pascal s law.
2 0 - FLUID MECHANICS Page Pascal s law is also given as If the effect of gravitation is neglected, the pressure at every point in an incompressible liquid, in equilibrium, is the same. Applications of Pascal s Law: The figure shows the principle of a hydraulic lift used to raise heavy loads. This device has two vertical cylinders of different diameters connected by a horizontal tube. A liquid is filled in this vessel. Airtight pistons having cross-sectional areas A and A ( A < A ) are fitted touching the liquid surface in both the cylinders. According to Pascal s law, in equilibrium, the pressure on liquid in both the arms is the same. Hence, F A = P = P = F A F = F A A Thus, a large force, F, is generated using a small force, F, as the magnitude of F is A times the magnitude of F. Using Pascal s law, devices like hydraulic lift, hydraulic A jack, hydraulic brake and hydraulic press are developed. Pressure due to a fluid column: For liquid of density ρ in a static equilibrium in a container, pressure at all points at the same depth ( or in other words, at the same horizontal layer ) is the same. Consider an imaginary cylindrical volume element of height dy and cross-sectional area A at the depth y from the surface of liquid as shown in the figure. The weight of liquid in this volume element is dw = ρ g A dy
3 0 - FLUID MECHANICS Page If P and P + dp are the pressures on the upper and lower faces of the element, then PA and ( P + dp ) A are the forces acting on them respectively. In equilibrium, PA + dw = ( P + dp ) A PA + ρ g A dy = PA + A dp dp = ρ g dy This equation is valid for any fluid ( liquid or gas ). It shows that the pressure increases with increase in the depth. Here, ρ g is the weight density, i.e., weight per unit volume of the fluid. Its value for water is 9800 N / m. Pressure P at the depth y = h can be obtained by integration as under. P dp = P a h ρ g dy 0 As ρ is independent of pressure and constant for liquid, the above integration gives P - P a = ρ g h P = P a + ρ g h This equation is valid only for incompressible fluid, i.e., liquid and gives the pressure at depth h in a liquid of density ρ. Here, P ( = P a + ρ g h ) is the absolute pressure whereas P - P a ( = ρ g h ) is the gauge pressure also known as the hydrostatic pressure. The pressure at any point in a liquid does not depend on the shape or cross-sectional area of its container. This is known as hydrostatic paradox. When liquid is filled in the containers of different shapes and sizes, joined at the bottom as shown in the figure, the height of liquid columns in all the containers is found to be the same. 0. Buoyancy and Archimedes principal When a body is partially or completely immersed in a liquid, the buoyant force acting on it is equal to the weight of the displaced liquid and it acts in the upward direction at the centre of gravity of the displaced liquid. This statement was given by Archimedes and is known as Archimedes principle. Buoyant force = weight of the displaced liquid ( or any fluid, i.e., liquid or gas ) = decrease in the weight of the immersed body
4 Proof of Archimedes Principle: Suppose a solid of height h, uniform cross-sectional area A and volume V is immersed in a liquid of density ρ. Net force on the vertical surface of the solid is zero. The pressure, P, on top of the solid in the downward direction and, P, on the bottom of the solid in the upward direction are respectively given by 0 - FLUID MECHANICS Page 4 P = P a + ρ g h and P = P a + ρ g h P - P = ρ g ( h - h ) = ρ g h ( Q h - h = h ) Hence, the resultant ( buoyant ) force acting in the upward direction is F b = ρ g h A = ρ g V ( Q h A = V ) = weight of the displaced liquid Although the above result has been proved for a symmetrical solid of uniform height and cross-sectional area, it is valid for a solid of any shape. With the help of Archimedes principle, specific density of solid or liquid, volume of a body of irregular shape, the constituents of the mixture and its proportion in an alloy may be known. Law of floatation: When a body is partially or completely immersed in a liquid, two forces act on it. ( i ) Weight of the body, W = ρ s g V s and ( ii ) Buoyant force, F b = ρ f g V f Here, ρ s and V s are the density and volume of the solid body, while ρ f and V f are the density and volume of the displaced liquid respectively. When the body is fully immersed in the liquid, V s = V f ( provided the solid body is not hollow or has no cavities ). If W > F b, the body sinks. If W = F b, the body remains in equilibrium at any depth. Submarine works on this principle. If W < F b, the body floats in liquid and if its partially immersed volume is V p, then the buoyant force F b = ρ f g V p balances the weight W of the body. ρ W = F b which implies ρ s g V s = ρ f g V p. s V p =. ρ f Vs Thus, When the weight of a body is equal to the weight of the liquid displaced by a part of the body immersed in it, the body floats on the surface of the liquid. This statement is known as the law of floatation which is used in designing a steamer.
5 0 - FLUID MECHANICS Page 5 Fluid Dynamics: 0.4 Characteristics of fluid flow ( ) Steady flow: In a steady flow of fluid, the velocity of the fluid at each point remains constant with time. Every particle of the fluid passing through a given point will have the same velocity. Let the particles of fluid, P, Q and R have velocities v P, v Q and v R respectively which may all be different. But these velocities do not change with time and all particles of the fluid in its flow passing through these points will have these velocities at all times. Such a condition is achieved only at low speeds, e.g., a gently flowing stream. ( ) Unsteady flow: In an unsteady flow of fluid, the velocity of the fluid at a given point keeps on changing with time as in motion of water during ebb and tide. ( ) Turbulent flow: In turbulent flow, the velocity of fluid changes erratically from point to point and from time to time as in waterfalls, breaking of the sea waves. ( 4 ) Irrotational flow: In irrotational flow, no element of fluid has net angular velocity. A small paddle wheel placed in such a flow will move without rotating. ( 5 ) Rotational flow: In rotational flow, an element of fluid at each point has net angular velocity about that point. A paddle wheel kept in such a flow has turbulent motion while rotating. Rotational flow includes vortex motion such as whirlpools, the air thrown out of exhaust fans, etc. ( 6 ) Incompressible flow: In incompressible flow, the density of fluid remains constant with time everywhere. Generally, liquids and sometimes even a highly compressed gas flow incompressibly. Flow of air relative to the wings of an aeroplane flying below sonic velocity is incompressible. ( 7 ) Compressible flow: In compressible flow, density of fluid changes with position and time. ( 8 ) Non-viscous flow: In non-viscous flow, fluid with small co-efficient of viscosity flows readily. Normally, the flow of water is non-viscous. ( 9 ) Viscous flow: In viscous flow, fluid having large co-efficient of viscosity cannot flow readily. Castor oil, tar have viscous flow. 0.5 Streamlines, Tube of flow and Equation of continuity The path of motion of a fluid particle is called a line of flow. In a steady flow, velocity of each particle arriving at a point on this path remains constant with time. Hence, every particle reaching this point moves in the same direction with the same speed. However, when this particle moving on the flow line reaches a different point, its velocity may be different.
6 0 - FLUID MECHANICS Page 6 But this different velocity also remains constant with respect to time. The path so formed is called a streamline and such a flow is called a streamline flow. In unsteady flow, flow lines can be defined, but they are not streamlines as the velocity at a point on the flow line may not remain constant with time. Streamlines do not intersect each other, because if they do then two tangents can be drawn at the point of intersection and the particle may move in the direction of any tangent which is not possible. Tube of flow: The tubular region made up of a bundle of streamlines passing through the boundary of any surface is called a tube of flow. The tube of flow is surrounded by a wall made of streamlines. As the streamlines do not intersect, a particle of fluid cannot cross this wall. Hence the tube behaves somewhat like a pipe of the same shape. Equation of continuity: In a tube of flow shown in the figure, the velocity of a particle can be different at different points, but is parallel to the tube wall. In a non-viscous flow, all particles in a given crosssection have the same velocity. Let the velocity of the fluid at cross-section P, of area A, and at crosssection Q, of area A, be v and v respectively. Let ρ and ρ represent density of the fluid at P and Q respectively. Then, as the fluid can not pass through the wall and can neither be created or destroyed, the mass flow rate ( also called mass flux ) at P and Q will be equal and is given by dm = ρ A v = ρ A v dt This equation is known as the law of conservation of mass in fluid dynamics. For liquids, which are almost incompressible, ρ = ρ. A v = A v ( ) which implies Av = constant ( ) or, v A Equations ( ) and ( ) are known as the equations of continuity in liquid flow. The product of area of cross-section, A and the velocity of the fluid, v at this cross section, i.e., Av, is known as the volume flow rate or the volume flux. Thus, velocity of liquid is larger in narrower cross-section and vice versa. In the narrower cross-section of the tube, the streamlines are closer thus increasing the liquid velocity. Thus, widely spaced streamlines indicate regions of low speed and closely spaced streamlines indicate regions of high speed.
7 0 - FLUID MECHANICS Page Bernoulli s equation Bernoulli s equation is derived in fluid dynamics with the help of work-energy theorem. The figure shows a streamline flow of a non-viscous liquid which is steady and irrotational through a hypothetical pipe or a flow tube. One end of the pipe is horizontal at a height y above some reference level and has uniform cross-section A upto some length. The pipe gradually widens and rises and becomes horizontal at the other end which is at a height y from the reference level and has uniform crosssection A. Now consider the portion of fluid shown by shaded area as the system. Suppose the system of fluid gets displaced from the position shown in figure ( a ) to that in figure ( b ) in a small time interval. ( ) The work done on the system, W, due to the pressure P and hence by the force, P A, during displacement l at the left end is W = force displacement = P A l ( ) The work done on the system, W, due to the pressure P and hence by the force, P A, during displacement l at the left end is W = force displacement = - P A l [ negative sign is because of force, P A, and displacement, l, being in opposite directions. ] ( ) The work done on the system, W, by the gravitational force, mg, is W = force displacement = - mg ( y - y ) the total work done on the system, [ negative sign is due to force, mg, and displacement, ( y - y ), being in opposite directions. ] W = W + W + W = change in K. E. of the system [ By work-energy theorem ] = mv - mv P A l - P A l - mg ( y - y ) = mv - mv ( )
8 0 - FLUID MECHANICS Page 8 As the liquid is incompressible, the mass flow rate at both the ends must be the same. ρ A l = ρ A l = m = ρ V A l = A l = m = V ( ) ρ From equations ( ) and ( ), P m m - P ρ ρ - mg ( y - y ) = P - P - ρ g ( y - y ) = P + P + ρ v ρ v mv ρ v - ρ v + ρ g y = P + ρ v + ρ g y - mv + ρ g y = constant ( for every point on a streamline ) This equation is known as Bernoulli s equation applicable only to steady, irrotational, incompressible and non-viscous streamline flow. Dividing the above equation by ρ g, P ρ g + v g + y = constant Each term in the above equation has the dimension of length and hence every term is known as head. The first term is called pressure head, the second velocity head and the third is known as the elevation head. Applications of Bernoulli s equation ( ) Venturie meter ( principle ) The figure shows the venturie meter used to measure the flow rate of liquid in a pipe. One end of a manometer is connected to the broad end of the venturie meter and the other end to the throat. The cross-sectional area of the broad end and velocity and pressure of liquid there are A, v and P respectively. The crosssectional area of the throat and velocity and pressure of liquid there are a, v and P respectively. The densities of the liquid flowing through the venturie meter and of the manometer liquid ( mercury ) are ρ and ρ respectively.
9 0 - FLUID MECHANICS Page 9 Using Bernoulli s theorem at points and which are at the same elevation, P + ρ v = P + ρ v ρ ( v - v P - P = ) = ( ρ - ρ ) gh ( from the manometer reading ) ( ρ - ρ ) gh = ρ ( v - v ) = A ρ v - v ( using the continuity equation, Av = av ) a = A ρ v - a v = a ( ρ' ρ ( A - ρ ) gh - a ) Using this value of v, the volume flow rate, v A, can be calculated. The velocity of liquid is low at the broad end and high at the throat of the venturie tube. In an automobile carburetor, air flows through a venturie tube reducing its pressure at the throat where the fuel is sucked in and mixed with proper amount of air for satisfactory combustion. In a spray pump, the piston pushes the air which comes out of a hole with high velocity. This reduces the pressure near the hole and sucks the liquid to be sprayed through a capillary. ( ) The change in pressure with depth: The expression for hydrostatic pressure in a stationary liquid can be obtained as a special case of Bernoulli s equation taking v = v = 0. Taking point at a depth of h from the surface of liquid and point on the surface where pressure is atmospheric pressure, P a, and noting that h = y - y, we get P = P a + ρgh. ( ) Dynamic lift and Swing bowling: A body in liquid experiences buoyant force which is known as static lift. When the body is in motion with respect to a fluid, it also experiences another force known as dynamic lift. The figure shows a ball moving horizontally in air without spinning. The velocity of air at points and is the same. So, by Bernoulli s equation, the pressures at and would also be the same. Hence the dynamic lift on the ball is zero.
10 Now consider a ball which is moving horizontally with velocity v and also spinning about the horizontal axis passing through its centre and perpendicular to the plane of the figure with its peripheral velocity v. When the surface of the ball is rough, it drags some air along with it. This results in increase in velocity of streamline flow of air at point to v + v and decrease at point to v - v. Hence pressure of air at point is less than the pressure at point which gives dynamic lift to the ball in the upwards direction. Again, if the ball was spun about a vertical axis lying in the plane of the figure making it rotate in the horizontal plane, it may deviate towards the off or leg stump. This is the reason for swing of the ball in fast bowling. ( 4 ) Aerofoils: The figure shows an aerofoil which is a solid shaped to provide an upward vertical force when it moves horizontally through air and hence it can float in air. The aeroplane wings are shaped like an aerofoil. Air has streamline flow about the wings when the angle between the wing and the direction of motion, called an angle of attack, is small. The crowded streamlines above the wings results in high velocity and low pressure, while the sparse streamlines below the wings indicate low velocity and high pressure. This difference in pressure results in dynamic upward lift which helps the aeroplane in motion to float in air. 0.7 Viscosity An appropriate external force is required to maintain the flow of fluids. A property of fluid responsible for this is called viscosity of the fluid. 0 - FLUID MECHANICS Page 0 Consider the steady flow of liquid on some horizontal stationary surface as shown in the figure. The layer of the liquid in contact with the surface remain stuck to it due to adhesive force and has zero velocity. The velocity of layer gradually increases on moving upwards from the surface and is the largest at the top. In a steady flow, different layers slide over each other without getting mixed which is called laminar flow. In a laminar flow, the relative velocity between the consecutive layers of fluid results in tangential force at the surfaces of the layers known as viscous force and the property of the fluid causing it is known as viscosity. To maintain the flow, some minimum external force has to be applied to balance the viscous force.
11 0 - FLUID MECHANICS Page Velocity gradient: In a laminar flow, the difference in velocity between two layers of liquid per unit perpendicular distance, in the direction perpendicular to the direction of flow, is called velocity gradient. As shown in the figure ( previous page ), the velocity of the layers at distances x and x + x from the surface in the direction perpendicular to the layers as well as perpendicular to the flow, are v and v + v respectively. average velocity gradient of the layers lying within distance x = and for a given layer, velocity gradient = The unit of velocity gradient is s -. x lim 0 v x = v x dv dx Co-efficient of viscosity: The viscous force between two adjacent layers of a laminar flow at a given temperature, F area ( A ) between the two adjacent layers and F dv F A dx or, dv F = η A, dx dv dx where η is a constant known as the co-efficient of viscosity of the fluid. Its magnitude depends on the type of the fluid and its temperature. It s C. G. S. unit is dyne-s-cm - and is called poise and MKS unit is N-s-m -. Its dimensional formula is M L - T -. dv Taking A = unit and = unit in the above equation, η = F. dx Thus, the co-efficient of viscosity can also be defined as the viscous force acting per unit surface area of contact and per unit velocity gradient between two adjacent layers in a laminar flow of a fluid. Note that the co-efficient of viscosity of liquids decrease with increase in temperature, while that of gases increase with the increase in temperature. At normal temperature, blood has higher co-efficient of viscosity than that of water and the relative co-efficient of viscosity of blood, η blood / η water in the temperature range of 0 C to 7 C remains almost constant. 0.8 Stokes Law The resistive force ( viscous force ) on a small, smooth, spherical, solid body of radius r, moving with velocity v through a viscous medium, of large dimensions, having co-efficient of viscosity η is given by F v = 6 π η r v. This equation is called Stokes Law which can be verified using dimensional analysis.
12 0 - FLUID MECHANICS Page Free fall of a sphere in a fluid and its terminal velocity: Suppose a small, smooth, solid sphere of radius r of material having density ρ falls freely in a fluid of density ρ ( < ρ ) and o co-efficient of viscosity η as shown in the figure. The figure shows the forces acting on the sphere at three different instants. The forces acting on the sphere at an instant when its velocity is v are: ( ) its weight, F = mg = 4 π r ρ g ( downwards ), ( ) the buoyant force by the fluid, 4 F = m o g = π r ρ g ( upwards ), o where m o = mass of fluid displaced, and ( ) the resistive force opposing the motion as given by Stokes law, F v = 6 π η r v ( upwards ) the resultant force acting on the sphere is F = F - F - F v = 4 r 4 π ρ g - r ρ g o π - 6 π η r v ( ) This is the equation of motion of the sphere in the viscous medium. v = 0 at t = 0 and hence F v = 0. F = 4 r 4 π ρ g - π r ρ g = 4 π r g ( o ρ - ρ ) ( ) o If the acceleration of the sphere is a 0 at t = 0, F = m a 0 = 4 π r ρ a o ( ) Comparing equations ( ) and ( ), 4 4 π r ρ ao = π r g ( ( ρ - ρ ) ρ - ρ ) a o g o o = ρ ( 4 ) Initially, before the sphere starts its motion in the fluid, its velocity is zero and there is no resistive force acting on it. It accelerates downwards and its velocity increases. The upward resistive force on the sphere also increases with the velocity. This reduces the net downward force on the sphere. As a result, the velocity of the sphere increases while its acceleration decreases. At a definite high velocity, the resultant force on the sphere becomes zero and thereafter the sphere continues to move with the uniform velocity, v t, known as the terminal velocity. At v = v t, the net downward force becomes zero.
13 0 - FLUID MECHANICS Page F = F - F - F v = r g v t = ( ρ - ρ ) 9 η o 4 r 4 π ρ g - π r ρ g - 6 π η r v = 0 o This equation is used to determine the coefficient of viscosity of the fluid experimentally. A bubble of air in a liquid rises up due to lower density of air as compared to water and reaches an upward terminal velocity. 0.9 Reynolds number The type of fluid flow in a given pipe depends on ( i ) the coefficient of viscosity ( η ) of the fluid, ( ii ) the density ( ρ ) of the fluid, ( iii ) average velocity ( v ) of the fluid and ( iv ) the diameter ( D ) of the fluid and is given by the magnitude of the dimensionless number called Reynolds number, N R formed by the combination of these four physical quantities as under. Reynolds number, N R = ρ v D η For N R < 000, the flow is streamline, N R > 000, the flow is turbulent and for 000 < N R < 000, the flow is unstable and its type keeps on changing. If η = 0 ( that is fluid is non-viscous ), N R tends to infinity. Hence the flow of non-viscous fluids can never be streamline, for all values of v > Surface tension Cohesive force - inter-molecular attractive force between molecules of the same matter. Adhesive force - attractive force between molecules of different matters. Range of inter-molecular force ( r o ) - the maximum distance upto which the attractive force between two molecules is significant. Sphere of molecular action - an imaginary sphere of radius r o with molecule at the centre.
14 0 - FLUID MECHANICS Page 4 Consider molecules P, Q and R of a liquid along with their spheres of molecular action as shown in the figure. The sphere of action of molecule P is completely immersed in the liquid. As P is acted upon by equal forces of attraction from all sides, the resultant force on it is zero keeping it in equilibrium. All such molecules at a depth more than r o will be in similar situation. Molecule Q is at a depth less than r o. Part FOEF of its sphere of action is outside the liquid and contains the molecules of air and liquid vapour. As the density of molecules in this part is less than that in GNHG part which contains only liquid molecules, the resultant force due to these two parts is in the downward direction. Molecules in CDHG and CDEF parts being equal, the resultant force on Q due to molecules in these regions is zero. Thus, there is a net downward force on molecule Q. A layer of thickness r o below the free surface AB of the liquid is called the surface of liquid. As we move upwards in this layer, the downward resultant force keeps on increasing and becomes maximum on the molecules like R on the free surface AB. Thus the molecules in this surface layer will have tendency to go inside the body of the liquid. As some of the molecules go down, density below the surface of liquid increases and it decreases on moving upwards in the surface. This results in decreased inter-molecular distance below the surface and more within the surface. As a result, the molecules lying in it experience force of tension parallel to the surface known as surface tension. The force exerted by the molecules lying on one side of an imaginary line of unit length, on the molecules lying on the other side of the line, which is perpendicular to the line and parallel to the surface is defined as the surface tension ( T ) of the liquid. The S I unit of surface tension is Nm - and its dimensional formula is M L 0 T -. The force of surface tension is parallel to the surface, while the resultant cohesive force on the molecules is perpendicular to the surface and towards inside the liquid. There are molecules on either side of an imaginary line in the middle of the surface exerting forces which are equal in magnitude and opposite in direction. Hence, the effect of surface tension is not felt. At the edge of the surface, there are molecules only on one side and hence the surface tension manifests here parallel to the surface and perpendicular to the surface towards the middle of the surface. Surface tension in context of potential energy: Work has to be done to bring a molecule from a point below the surface to a point on top of the surface against the downward force acting on it. Hence, when such a molecule reaches the surface it acquires potential energy. Now a system has a tendency to remain in the state of minimum potential energy. Thus molecules in the surface of a liquid has a tendency to reduce its potential energy. So the surface of the liquid has a tendency to contract and minimize its area.
15 0 - FLUID MECHANICS Page 5 The molecules reaching the surface do not occupy place between the molecules already present in the surface, but generate a new surface. This means that the surface gets expanded. The whole surface of a liquid can be considered to have been generated this way. Thus, the molecules in the surface of a liquid possess potential energy equal to the work done on them in bringing them to the surface. Hence surface tension can also be defined as the potential energy stored in the surface of the liquid per unit area. By this definition, its unit is Jm - which is the same as Nm -. Surface tension of a liquid depends on the type of the liquid and its temperature. It decreases with temperature and becomes zero at a critical temperature. It also depends upon the type of the medium which the liquid is in contact with. Surface - energy: Surface tension, T, is the amount of energy to be given to the surface to increase its area by unity at constant temperature. But the temperature of the surface decreases as it expands. Hence, heat has to be supplied to the surface from outside to maintain its temperature. Total surface energy per unit area = Potential energy ( due to surface tension ) + Heat energy. Thus, at any temperature the value of surface energy is more than surface tension. Both the surface tension and surface energy decrease with temperature and become zero at the critical temperature. The figure shows the graphs of surface tension and surface energy versus temperature for water both of which become zero at the critical temperature of water which is 74 C. Now, consider a rectangular frame ABCD on which a wire PQ can be slided over the sides AD and BC by a light string tied to PQ as shown in the figure.
16 0 - FLUID MECHANICS Page 6 A thin film ABQP is formed by dipping the frame in soap solution and holding the wire PQ in position with a thread. If the string is released, PQ slides inwards and the film contracts. This shows that the surface tension manifests itself on the edge of the surface of the liquid perpendicular to the edge and parallel to the surface. Now displace the wire by x by pulling the string with a force slightly more than the force of surface tension. If T is the surface tension of the solution and l is the length of the wire PQ, then the force acting on the wire is F = T l ( The multiple is because the film has two surfaces. ) work W = Force displacement = T l x = T ( l x ) = T ( A ) work W = T for A = unit Thus, surface tension can also be defined as the work done to increase the area of the surface by unit. Experiments show that the surface tension does not change on expanding the surface. This indicates that the molecules coming to the surface do not occupy place between the already present molecules but create a new surface of their own. 0. Drops and bubbles Free surface of liquids has tendency to minimize its area due to surface tension. Since spherical surface has minimum area for a given volume, drops and bubbles of liquids are always spherical. Consider a bubble of radius R as shown in the figure. The pressures inside and outside are Pi and P o ( Pi > P o ) respectively. The pressure on the concave surface is always more than that on the convex surface. Let the surface tension of the liquid forming the wall of the bubble be T. Suppose, on blowing the bubble, its radius increases from R to R + dr and its surface area increases from S to S + ds. The work done in this process can be calculated in two ways. ( ) Work, W = force displacement = pressure difference area displacement = ( P i - P o ) 4 π R dr ( ) ( ) The surface area of the bubble of radius R is, S = 4 π R the increase in the surface area is, ds = 8 π R dr But the bubble in air has two free surfaces total increase in its area = 8 π R dr = 6 π R dr
17 0 - FLUID MECHANICS Page 7 work, W = surface tension total increase in area = 6 π T R dr ( ) Comparing equations ( ) and ( ), ( P i - P o ) 4 π R dr = 6 π T R dr P i - P o = 4 T R For a bubble formed in a liquid or a liquid drop which has only one free surface, P i - P o = T R 0. Capillarity The phenomenon of rise or fall of a liquid in a capillary, held vertical in a liquid, due to its property of surface tension is called capillarity. When a glass capillary of small bore is held vertical in water, water rises in the capillary and when held in mercury, mercury falls in the capillary as shown in the figure. Also, note that water wets the glass while mercury does not. The meniscus of water in the capillary is concave while the meniscus of mercury is convex. Suppose liquid rises to height h in a capillary of radius r held vertical in the liquid as shown in the figure. The radius of concave meniscus of liquid in the capillary is R. The tangent drawn at a point P, where the surface of meniscus is in contact with wall of the capillary, makes an angle θ with the wall. θ is known as the angle of contact of the liquid with the matter of the capillary. Liquids with θ < 90 rise while those with θ > 90 fall in the capillary. In the figure, OPQ = θ in right-angled OPQ. cos θ = OP = OQ Radius of the capillary ( r ) Radius of the meniscus ( R ) R = r cos θ ( ) The pressure on the concave surface of the meniscus P o > Pi, the pressure on the convex surface.
18 0 - FLUID MECHANICS Page 8 P o - Pi = T R ( because, the liquid has one free surface. ) ( ) Also, for equilibrium, the pressure at point B is the same as at point are at the same horizontal level. A which is P o as both P o - Pi = h ρ g ( ) where ρ = density of the liquid and g = acceleration due to gravity. Comparing equations ( ) and ( ), T R = h ρ g T = R h ρ g = r h ρ g cos θ [ putting the value of R from equation ( ) ] For mercury and glass, θ > 90. Hence, cos θ is negative. Therefore, mercury falls in a glass capillary and its meniscus is convex. Also, Pi > P o. Thus in equation ( ), Pi - P T o = should be taken. As Pi R - P o = h ρ g, the final result remains the same. 0. Detergent and surface tension Stains of grease or oil on the clothes are not removed by water alone as water does not wet the grease or oil. On adding detergent, the broad end of the molecule of detergent as shown in the figure is attracted to the water molecule and the other end is attracted towards the molecule of oil or grease. This forms an interface between water and oil or grease. Thus, on adding detergent to water, the surface tension of the solution is less than that of water. Hence, it wets grease or oil and removes the dirt. Some useful formulae ( ) Velocity, v, of a liquid having co-efficient of viscosity, η, in laminar flow through a tube of radius r and length l across which there is a pressure difference p is given by v = p ( r - x ) 4 η l ( ) Volumetric flow rate, V, of liquid in the above case is given by V = π p r 4 8 η l This equation is called Poiseiulle s Law.
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