Also, the book is available on scribd u can download from there.. Also i have this drive link u can get the pdf from here: Introduction to fluid mechanics b. Introduction to. Fluid Mechanics and. Fluid Machines Revised Second Edition. S K SOM Department of Mechanical Engineering Indian Institute of Technology. Som & Biswas 3rd Edition. Uploaded by Aravind. good book. Copyright: © All Rights Reserved. Download as PDF or read online from Scribd. Flag for.
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Title: Introduction to Fluid Mechanics and Fluid Machines Author(s): S.K. Som, Gautam Biswas Publisher: McGraw Hills Edition: Second Pages: PDF Size . Web Course: Fluid Mechanics Prof. Gautam Biswas Prof. S.K. Som. Download PDF. FLUID MECHANICS Science of fluid mechanics M1 C1 L1 · Fluid . solution of introductoin to fluid mechanics and machines(Prof. Som and Prof. Biswas) Chapter 4. 1. Scanned by. CamScanner; 2. Scanned by.
The content in the book has been rearranged and rejuvenated in such a way that each chapter introduces the topic and then familiarises the students with all the associated principles and applications in a systematic manner. This book features the inclusion of a chapter end summary, to help the readers analyze their understanding of each topic.
This book also includes a set of solved examples and challenges readers with exercise problems, that have been strategically chosen to explain the nuances of the basic principles of fluid mechanics.
The book was written by the authors with the aim to help the students analyze and design various flow systems and help them master fluid mechanics and machines.
Introduction and Fundamental Concepts 2. Fluid Statics 3. Kinematics of Fluid 4. Applications of Equations of Motion and Mechanical Energy 6. Principles of Physically Similarity and Dimensional Analysis. Flow of Ideal Fluids 8. Viscous Incompressible Flows 9. For any gaseous substance, a change in pressure is generally associated with a change in volume and a change in temperature simultaneously. A functional relationship between the pressure, volume and temperature at any equilibrium state is known as thermodynamic equation of state for the gas.
However, this equation is also valid for the real gases which are thermodynamically far from their liquid phase. The relationship between the pressure p and the volume V for any process undergone by a gas depends upon the nature of the process. If there is no heat transfer to or from the gas, the process is known as adiabatic.
A frictionless adiabatic process is called an isentropic process and x equals to the ratio of specific heat at constant pressure to that at constant volume.
The Eq. The value of E for air quoted earlier is the isothermal bulk modulus of elasticity at normal atmospheric pressure and hence the value equals to the normal atmospheric pressure. Invoking 2 this relationship into Eq. In other words, if the flow velocity is small as compared to the local acoustic velocity, compressibility of gases can be neglected. Considering a maximum relative change in density of 5 per cent as the criterion of an incompressible flow, the upper limit of Mach number becomes approximately 0.
In case of air at standard pressure and temperature, the acoustic velocity is about Hence a Mach number of 0. The force of attraction between the molecules of a liquid by virtue of which they are bound to each other to remain as one assemblage of particles is known as the force of cohesion.
This property enables the liquid to resist tensile stress. On the other hand, the force of attraction between unlike molecules, i. This force enables two different liquids to adhere to each other or a liquid to adhere to a solid body or surface. Consider a bulk of liquid with a free surface Fig.
A molecule at a point A or B is attracted equally in all directions by the neighbouring molecules. Due to the random motion of the molecules, the forces of cohesion, on an average over a period of time can be considered equal in all directions. Moreover, this force is effective over a minute distance in the order of three to four times the average distance between the adjacent molecules.
Therefore, one can imagine a sphere of influence around those points. A molecule at C, very near to the free surface has a smaller force of attraction acting on it from the direction of the surface because there are fewer molecules within the upper part of its sphere of influence. In other words, a net force acts on the molecule towards the interior of the liquid.
This force has its maximum value when the molecule is actually at the surface, as at D. This net inward force at D depends not only on the attraction of the molecules inside the liquid, but also on the attraction by the molecules of air on the other side of the surface.
The substance on the other side may be in general, any gas, immiscible liquid or solid. Hence, work is done on each molecule arriving at the surface against the action of. Air D Free surface C. Liquid Fig. Thus mechanical work is performed in creating a free surface or in increasing the area of the surface. Therefore, a surface requires mechanical energy for its formation and the existence of a free surface implies the presence of stored mechanical energy known as free surface energy.
Any system tries to attend the condition of stable equilibrium with its potential energy as minimum. Thus a quantity of liquid will adjust its shape until its surface area and consequently its free surface energy is a minimum. For example, a drop of liquid free from all other forces, takes a permanent spherical shape, since for a given volume, the sphere is the geometrical shape having the minimum surface area. Free surface energy necessarily implies the existence of a tensile force in the surface and the surface, in fact, is in a stretched condition due to this force.
If an imaginary line is drawn on the surface, the liquid molecules on both sides will pull the linear element in both the directions and this line will be subjected to a state of tensile force. The magnitude of surface tension is defined as the tensile force acting across such short and straight elemental line divided by the length of the line.
Surface tension is a binary property of the liquid and gas or two liquids which are in contact with each other and define the interface. It decreases slightly with increasing temperature. It is due to surface tension that a curved liquid interface in equilibrium results in a greater pressure at the concave side of the surface than that at its convex side. Consider an elemental curved liquid surface Fig. The surface is assumed to be curved on both the sides with radii of curvature as r1 and r2 and with the length of the surfaces subtending angles of dq1 and dq2 respectively at the centre of curvature as shown in Fig.
Let the surface be subjected to the uniform pressure pi and po at its concave and convex sides respectively acting perpendicular to the elemental surface.
The surface tension forces across the boundary lines of the surface appear to be the external.
Considering the equilibrium of this small elemental surface, a force balance in the direction perpendicular to the surface results. If the liquid surface coexists with another immiscible fluid, usually gas, on both the sides, the surface tension force appears on both the concave and convex interfaces and the net surface tension force on the surface will be twice as that described by Eq.
Hence the equation for pressure difference in this case becomes. Special Cases For a spherical liquid drop, the Eq. When a liquid is in contact with a solid, if the forces of adhesion between the molecules of the liquid and the solid are greater than the forces of cohesion among the liquid molecules themselves, the liquid molecules crowd towards the solid surface. The area of contact between the liquid and solid increases and the liquid thus wets the solid surface. The reverse phenomenon takes place when the force of cohesion is greater than the force of adhesion.
These adhesion and cohesion properties result in the phenomenon of capillarity by which a liquid either rises or falls in a tube dipped into the liquid depending upon whether the force of adhesion is more than that of cohesion or not Fig. The angle q, as shown in Fig. The value of q may be different from zero in practice where cleanliness of a high order is seldom found. Since h varies inversely with D as found from Eq.
The rate of evaporation depends upon the molecular energy of the liquid which in turn depends upon the type of liquid and its temperature. The vapour molecules exert a partial pressure in the space above the liquid, known as vapour pressure.
If the space above the liquid is confined Fig. Eventually an equilibrium condition will evolve when the rate at which the number of vapour molecules striking back the liquid surface and condensing is just equal to the rate at which they leave from the surface.
The space above the liquid then becomes saturated with vapour. The vapour pressure of a given liquid is a function of temperature only and is equal to the saturation pressure for boiling corresponding to that temperature.
Hence, the vapour pressure increases with the increase in temperature. Therefore the phenomenon of boiling of a liquid is closely related to the vapour pressure. In fact, when the vapour pressure of a liquid becomes equal to the total pressure impressed on its surface, the liquid starts boiling. Solids can resist tangential stress at static. In the continuum approach, properties of a system can be expressed as continuous functions of space and time.
The shear stress at a point in a moving fluid is directly proportional to the rate of shear strain. The constant of proportionality m is known as coefficient of viscosity or simply the viscosity.
For a Newtonian fluid, viscosity is a function of temperature only. With an increase in temperature, the viscosity of a liquid decreases, while that of a gas increases. For a non-Newtonian fluid, the viscosity depends not only on temperature but also on the deformation rate of the fluid. When the flow velocity is equal to or less than 0. The interplay of these two intermolecular forces explains the phenomena of surface tension and capillary rise or depression.
A free surface of the liquid is always under stretched condition implying the existence of a tensile force on the surface. The magnitude of this force per unit length of an imaginary line drawn along the liquid surface is known as the surface tension coefficient or simply the surface tension.
Surface tension is a binary property of liquid and gas and bears an inverse relationship with temperature. It is due to the surface tension that a curved liquid interface, in equilibrium, results in a greater pressure at the concave side than that at its convex side. The pressure difference DP is given by DP.
A liquid wets a solid surface and results in a capillary rise Hr r K 1 2 when the forces of cohesion between the liquid molecules are lower than the forces of adhesion between the molecules of liquid and solid in contact. Non-wettability of solid surfaces and capillary depression are exhibited by the liquids for which forces of cohesion are more than the forces of adhesion.
Example 1. For the flow of such a fluid over a flat solid surface, the velocity at a point 75 mm away from the surface is 1. Calculate the shear stresses at the solid boundary, at points 25 mm, 50 mm, and 75 mm away from the boundary surface.
Assume i a linear velocity distribution and ii a parabolic velocity distribution with the vertex at the point 75 mm away from the surface where the velocity is 1.
Solution Consider a two dimensional cartesian coordinate system with the velocity of fluid V as abscisa and the normal distance Y from the surface as the ordinate with the origin O at the solid surface Fig. O V Fig. According to Eq. It is observed that the shear stress decreases as the velocity gradient decreases with the distance y from the plate and becomes zero where the velocity gradient is zero.
Both the cylinders are 0. Determine the viscosity of the liquid which fills the space between the cylinders if a torque of 0. Now according to Eq.
Hence, 0. Determine the isothermal bulk modulus of elasticity and compressibility of water at 1, 10 and atmospheric pressure. It is found from the above example that the bulk modulus of elasticity or compressibility of water is almost independent of pressure. The air is compressed to 0.
Boundary conditions: What is the height h at the wall? Solution The curved interface is plane in other direction. Hence the pressure difference across the interface can be written according to Eq.
The value of A and B are found out using he boundary conditions as follows: Let d be the diameter of the smaller drops. What is the capillary rise of water in the annulus if s is the surface tension of water in contact with air?
To and W in Eq. The inner and outer radii of the annulus are ri and r0 respectively. Find the work required in splitting up the drop. From conservation of mass d3 0.
Assume s for pure water to be 0. Solution An increase in the surface area out of a given mass takes place when a bigger drop splits up into a number of smaller drops. According to the kinetic theory of gas. Assuming an isothermal process. If the annular space is filled with a lubricating oil having a viscosity of 0. The clearance. It is desired to test the vessel at bar by pumping water into it. The estimated variation in the change of the empty volume of the container due to pressurisation to bar is 6 per cent.
Ignoring the edge effects. Neglect the inertia of the piston. Determine the lubricant thickness assuming linear velocity distribution. Calculate the mass of water to be pumped into the vessel to attain the desired pressure level. The viscosity of lubricant is 0. It has been discussed in Section 1. The shear force is zero for any fluid element at rest and hence the only surface force to a fluid element. Body Forces These forces act throughout the body of the fluid element and are distributed over the entire mass or volume of the element.
Surface Forces They include all forces exerted on the fluid element by its surroundings through direct contact at the surface. Therefore these forces appear only at the surface of a fluid element.
Fluid Statics 2. The ratios of these forces and the elemental area in the limit of the area tending to zero are called the normal or shear stresses respectively. Though surface forces are considered as external forces acting on the free body of a fluid element a fluid element in isolation from its neighbouring fluid. Since a fluid at rest. These forces are generally caused by external agencies such as gravitation. Body forces are usually expressed per unit mass of the element or medium upon which the forces act.
A fluid element. The component along the normal to the area is called the normal force. These stresses are denoted by a scalar quantity p Fig. Fluid Statics ' can develop neither shear stress nor tensile stress. D z are infinitesimal. With conventional notation of the positive sign for the tensile stress.
Substituting the values of D A cos a. It concludes that the normal stresses at any point in a fluid at rest are directed towards the point from all directions and are of equal magnitude. Considering gravity as the only source of external body force. Consider a fluid element of given mass at rest which ocupies a volume V bounded by the surface S Fig.
The fundamental equation of fluid statics. In accordance with Gauss divergence theorem. If gravity is con- sidered to be the only external body force acting on the fluid.
If we consider an expanse of fluid with a free surface. Hence the Eq. Fluid Statics! The pressure p0 at free surface is the local atmospheric pressure. From Eqs 2. With the help of Eq. For the standard atmosphere.
If the fluid is a perfect gas at rest at constant temperature. Constant Temperature Solution Isothermal Fluid The equation of state for a compressible system generally relates its density to its pressure and temperature. When a pressure is expressed as a diference between its value and the local atmospheric pressure.
Pressure is usually ex- pressed with reference to either absolute zero pressure a complete vacuum or local atmospheric pressure. Fluid Statics!! The absolute pressure is the pressure expressed as a difference between its value and the absolute zero pressure.
Experimental evidence of the temperature variation with altitude in differ- ent layers of the atmosphere is shown in Fig. For a liquid without a free surface in a closed pipe. The direct proportional- ity between gauge pressure and the height h for a fluid of constant density enables the pressure to be simply visualized in terms of the vertical height.
Gauge pressure Absolute Vacuum pressure pressure Local Absolute pressure atmospheric pressure Absolute zero complete vacuum Fig. The height h is termed as pressure head corresponding to pressure p. If such a piezometer tube of suffi- cient length were closed at the top and the space above the liquid surface were a perfect vacuum.
This almost vacuum condition above the mercury in the barometer is known as Torricellian vacuum. Then we get from Eq. The pressure at A equal to that at B Fig.
One of its ends is connected to a pipe or a container having a fluid A whose pressure is to be measured while the other end is open to atmosphere. This principle is used in the well- known mercury barometer to determine Torricellian the local atmospheric pressure. If water was used instead of mercury. B A perfect vacuum at the top of the tube Fig. For measuring very small gauge pressures of liquids.
The lower. The pressures at two points P and Q Fig. Hence it becomes. This fluid is called the manometric fluid. When the pressure of the fluid in the container is lower than the atmospheric pressure. Then equating the pressures at P and Q in terms of the heights of the fluids above those points. Sometimes it is desired to express this difference of pressure in terms of the difference of heads height of the working fluid at equilibrium.
If the working fluid is a gas. Applying the principle of hydrostatics at P and Q we have. For example. For this purpose. One limb is usually made very much greater in cross-section than the other. When a pressure difference is applied across the manometer. Air is used as the manometric fluid. If the level of the surface in the wider limb is assumed constant. From continuity of gauge liquid. The typical arrangement is shown in Fig.
But we consider the case as if the surface A shown in Fig. Let p denote the gauge pressure on an elemental area dA. In fact. The individual forces distributed over an area give rise to a resultant force. The determination of the magnitude and the line of action of the resultant force is of practical interest to engineers.
The point of action of the resultant force on the plane surface is called the centre of pressure cp. Equating the moment of the resultant force about the x axis to the summation of the moments of the component forces.
Let xp and yp be the distances of the centre of pressure from the y and x axes respectively. By applying the theorem of parallel axis. Equation 2. An arbitrary submerged curved surface is shown in Fig.
This is obvious because of the typical variation of hydrostatic pressure with the depth from the free surface. Eqs 2. With the help of Eqs 2.
Consider an elemental area dA at a depth z from the surface of the liquid. The components of the force dF in x. The components of the surface element d A projected on yz.
Fluid Statics "! In the similar fashion. If zp and yp are taken to be the coordinates of the point of action of Fx on the projected area Ax on yz plane. We can conclude from Eqs 2.
The vertical component of the hydrostatic thrust on the surface in this case acts upward and is equal. The hydrostatic forces on the Fz surface can then be calculated by considering the surface as a submerged Fig.
In some instances Fig. If a free surface does not exist in practice. From Eq. The vertical forces acting on the two ends of such a prism of cross-section dAz Fig. Consider a solid body of arbitrary shape completely submerged in a homogene- ous liquid as shown in Fig.
Fluid Statics " 2. Hydrostatic pressure forces act on the entire surface of the body. To calculate the vertical component of the resultant hydrostatic force. The line of action of the force FB can be found by taking moment of the force with respect to z-axis. For a floating body in static equilibrium and in the absence of any other external force. Depending upon the relative locations of G and B.
In general. It is found from Eq. Thus the net weight of the submerged body. The buoyant force of a partially immersed body. A body is said to be in stable equilibrium. Therefore the buoyant force depends upon the density of the fluid and the submerged volume of the body. In neutral equilibrium. This principle states that the buoyant force on a submerged body is equal to the weight of liquid displaced by the body. On the other hand. When a body is submerged in a liquid. Consider a submerged body in equilibrium whose centre of gravity is located below the centre of buoyancy Fig.
If the body is tilted slightly in any direction. This phenomenon was discovered by Archimedes and is known as the Archimedes principle. When the centre of gravity G and centre of buoyancy B coincides. Figure 2. The force of buoyancy FB is equal to the weight of the body W with the centre of gravity G being above the centre of buoyancy in the same vertical line.
The centre of gravity G remains. This is because. As a result. When M coincides with G. During the movement. For small values of q. Therefore the centre of buoyancy i.
Suppose that for the boat. If M were below G. For the body shown in Fig. M is above G. The new centres of gravity and buoyancy are therefore again vertically in line. As the total weight of the body remains unaltered so does the volume immersed. This may be done simply by considering the shape of the hull. This is so if the planes of flotation for the equilibrium and displaced positions intersect along.
The section on the left. Therefore it is desirable to establish a relation between the metacentric height and the geometrical shape and dimensions of a body so that one can determine the position of metacentre beforehand and then construct the boat or the ship accordingly. It is well understood that the metacentric height serves as the criterion of stability for a floating body.
This may be determined from a graph of nominal values of GM calculated from Eq. It is assumed that there is no overall vertical movement. Let a weight P be moved transversely across the deck which was initially horizontal so that the boat heels through a small angle q and comes to rest at this new position of equilibrium.
The position of the body after a small angular displacement is shown in Fig. Since the point M corresponds to the metacentre for small angles of heel only. The coordinate axes are chosen through O as origin.
OY is perpendicular to the plane of Fig 2. OY lies in the original plane of flotation Fig. The total immersed volume is considered to be made up of elements each underneath an area dA in the plane of flotation as shown in Figs 2. The centre of buoyancy by definition is the centroid of the immersed volume the liquid being assumed homogeneous. The x coordinate xB of the centre of buoyancy may therefore be determined by taking moments of elemental volumes about the yz plane as.
For typical sections of the boat. If the sides are not vertical at the water-line. For rolling movement of a ship. For this reason. The value of BM for a ship is always affected by a change of loading whereby the immersed volume alters. Hence the stability of the body is reduced. Thus not only does the centre of buoyancy B move. Since the torque equals to mass moment of inertia i. The metacentric height of ocean-going vessel is usually of the order of 0. An increase in the metacentric height results in a better stability but reduces the period of roll.
The time period i. In practice. Manometers are devices in which columns of a suitable liquid are used to measure the difference in pressure between two points or between a certain point and the atmosphere. A simple U-tube manometer is modified as inclined tube manometer. The pressure expressed as the difference between its value and the local atmospheric pressure is known as gauge pressure.
The scalar magnitude of the stress is known as hydrostatic or thermodynamic pressure. The oscillation of the body results in a flow of the liquid around it and this flow has been disregarded here. If q is small. For an incompressible fluid.
The minus sign in the RHS of Eq. If the cargo is placed further from the centre- line. In cargo vessels the metacentric height and the period of roll are adjusted by changing the position of the cargo. Body forces act over the entire volume of the fluid element and are caused by external agencies. Solved Examples Example 2. M coinciding with G or lying below G refers to the situation of neutral and unstable equilibrium respectively. For stable equilibrium of floating bodies.
The vertical component of hydrostatic force on a submerged curved surface is equal to the weight of the liquid volume vertically above the submerged surface to the level of the free surface of liquid and acts through the centre of gravity of the liquid in that volume. The force acts in a direction perpendicular to the surface and its point of action. The buoyant force on a submerged or floating body is equal to the weight of liquid displaced by the body and acts vertically upward through the centroid of displaced volume known as centre of buoyancy.
Solution a For an incompressible f luid. A submerged body will be in stable. Metacentre of a floating body is defined as the point of intersection of the centre line of cross-section containing the centre of gravity and centre of buoyancy with the vertical line through new centre of buoyancy due to any small angular displacement of the body. Therefore Eq. The elevations EL of the interfaces. For column E: Since the pressure at H is below the atmospheric pressure.
Pressure at EL8. Hence the elevation at N is Pressure at EL From the principle of hydrostatics. The reservoir and tube diameters of the manometer are 50 mm and 5 mm respectively. What will be the percentage error in measuring pS if the reservoir deflection is neglected. Calculate the difference in pressure between Sections A and B.
The deflection of mercury in the manometer is 0. Substituting for h from Eq. From continuity of the fluid in both the limbs. Find the magnitude and point of action of the resultant hydrostatic force on the entire area.
The triangle is isosceles of 3 m high and 4 m wide. A vertical rectangular area of 2 m high is attached to the 4 m base of the triangle and is acted upon by water. Solution The submerged area under oil and water is shown in Fig. For the triangular area. To find the points of action of the forces F1 and F2 on this line. Let G be the centre of gravity of the submerged part of the board Fig.
Find the depth of water h at the instant when the water is just ready to tip the flash board. Find h as a function of q for equilibrium of the gate. Solution Let us consider. Determine the horizontal and vertical components of hydrostatic force on the cylinder. The horizontal component of the hydrostatic force on surface ACB Fig. Determine the components of net hydrostatic force on the gate exerted by water. The gate is 3 m wide. For the equilibrium condition shown in Fig.
It is 0. Fluid Statics 65 left of BC. What is the ratio of a and b for this condition? Water a Steel block b Mercury Fig. Example 2. Alternative method: Consider an elemental area dA of the gate subtending a small angle dq at 0 Fig. The cube is submerged with half of it being in oil and. Under the condition of floating equilibrium as shown in Fig.
The horizontal and vertical components. T T 75 mm Oil Water mm Fig. For equilibrium of the cube. Find the tension in the string if the specific gravity of oil is 0. Solution Let the cube float with h as the submerged depth as shown in Fig. If the specific gravity of the cube material is Sc. G and M be the centre of buoyancy. Solution Let B. Show that the equilibrium is stable and the metacentric height is 3 r FG r.
The distance of G along this line from the base of the hemisphere can be found by taking moments of elemental circular strips Fig. Let V be the submerged volume.
Then from equilibrium under floating condition. If the specific gravity of the cone is S. For the equilibrium. Fluid Statics 69 In a similar way. If rh is the radius of cross-section of the hemisphere at water line. The upper limit of l would be decided from the consideration of stable equilibrium angular stability of the cylinder.
The minimum length corresponds to the situation when the cylinder will just float with its top edge at the free surface Fig. For any length l greater than mm. Find the limits of the length of the wooden portion so that the composite cylinder can float in stable equilibrium in water with its axis vertical. For floating equilibrium of the composite cylinder.
Solution Let l be the length of the wooden piece. Fluid Statics 71 H2 1 or. Fluid Statics 73 -NAH? EIAI 2. What would be the percentage error if the effect of vapour pressure is neglected. Pressure and temperature at sea level are The inverted z 1. The atmospheric pressure at sea level is Barometer reads mm of mercury.
For the given values of heights. Carbon tetrachloride Sp. Water is contained in A and rises to a level of 1. What horizontal force must be applied at B for equilibrium of gate AB? Calculate the magnitude. Fluid Statics 75 2. Assume contact with wall as frictionless. To what depth the log will sink in fresh water with its axis being horizontal? As Water soon as the plug opens it is observed 2m that the plug-float assembly jumps String upward and attains a floating position.
Calculate the metacentric height for Fig. Find the ratio of diameter to length of the cylinder so that it will just float upright in a state 15 m of neutral equilibrium in water. Find the metacentric height of the ship. The centre of buoyancy is 1. Find the volume of the spherical weight if the total mass of the plug and the weight is 5 kg. Can the plug Plug diameter be larger than the float diameter? Find out the maximum 50 mm possible plug diameter.
Fluid Statics 77 2. Explain why. As the water fills Patm up and the level reaches 2 m. Determine the level in the reservoir when the plug closes again. Kinematics of Fluid 3. The magnitude of a vector is a scalar. The magnitude of a scalar a real number will change when the units expressing the scalar are changed..
Therefore the kinematics of fluid is that branch of fluid mechanics which describes the fluid motion and its consequences without consideration of the nature of forces causing the motion.
Vector A quantity which is specified by both magnitude and direction is known to be a vector. Typical scalar quantities are mass. A scalar quantity can be completely specified by a single number representing its magnitude. The basic understanding of the fluid kinematics forms the ground work for the studies on dynamical behaviour of fluid in consideration of the forces accompanying the motion.
The subject has three main aspects: Identities of the particles are made by specifying their initial position spatial location at a given time. Hence S in Eq.
In other words. Vector Field If at every point in a region. Lagrangian Method In this method. The H H velocity V and acceleration a of the fluid particle can be obtained from the material derivatives of the position of the particle with respect to time. Force and velocity fields are the typical examples of vector fields. A fluid mass can be conceived of consisting of a number of fluid particles. Hence the instantaneous velocity at any point in a fluid region is actually the velocity of a particle that exists at that point at that instant.
In order to obtain a complete picture of the flow. Kinematics of Fluid 79 Scalar Field If at every point in a region. The temperature distribution in a rod is an example of a scalar field. The position of a particle at any other instant of time then becomes a function of its identity and time. For the acceleration. Instead it seeks the velocity V and its variation with time t at each H and every location S in the flow field.
The favourable aspect of the method lies in the information about the motion and trajectory of each and every particle of the fluid so that at any time it is possible to trace the history of each fluid particle. Eulerian Method The method due to Leonhard Euler is of greater advantage since it avoids the determination of the movement of each individual fluid particle H in all details.
While in the Lagrangian view. In addition. But the solution of the set of three simultaneous differential equations is generally very difficult. In Eulerian approach. Steady and Unsteady Flows A steady flow is defined as a flow in which the various hydrodynamic parameters and fluid properties at any point do not change with time.
Kinematics of Fluid 81 therefore. This implies that. In the Lagrangian approach. But in steady flow. According to the type of variations. Flow in which any of these parameters changes with time is termed as unsteady flow. To an observer on a bridge. Uniform and Non-uniform Flow When velocity and other hydrodynamic parameters.
Since the examination of steady flow is usually much simpler than that of unsteady flow. The motion of a body or a fluid element is described with respect to a set of coordinates axes.
Steadiness of flow and uniformity of flow do not necessarily go together. Though minor fluctuations of velocity and other quantities with time occur in reality. He would be comparing the flow with reference axes fixed to the bridge. The latter kind of non- uniformity is always encountered near solid boundaries past which the fluid flows.
Any such parameter will have a unique value in the entire field. This is because all fluids possess viscosity which reduces the relative velocity to zero at a solid boundary no-slip condition as described in Chapter For a steady and uniform flow. For a non-uniform flow. He would compare the water flow with an imaginary set of reference axes in the boat. This is because all movement is relative.
Any of the four combinations as shown in Table 3. Steady non-uniform Flow at constant rate through a duct of non- flow uniform cross-section tapering pipe. The region close to the walls of the duct is however disregarded. Dz and Dt 3. Kinematics of Fluid 83 Table 3. Unsteady uniform flow Flow at varying rates through a long straight pipe of uniform cross-section. The velocity components u. Unsteady non-uniform Flow at varying rates through a duct flow of non-uniform cross-section.
Again the region close to the walls is ignored. Steady uniform flow Flow at constant rate through a duct of uniform cross-section. Dy and Dz in Eqs 3. The first terms in the right hand sides of Eqs 3. Thus we can write. Therefore the terms in the left hand sides of Eqs 3. The last three terms in the right hand side of Eq. Table 3. In a steady and uniform flow. In a uniform flow. In a similar fashion.
Kinematics of Fluid 85 In a steady flow. Existence of the components of acceleration for different types of flow. The acceleration components in a spherical polar coordinate system Fig. This is typically known as centripetal acceleration. Path Lines and Streak Lines Streamlines The analytical description of flow velocities by the Eulerian approach is geometrically depicted through the concept of streamlines.
In the Eulerian method. If for a fixed instant of time. Path Lines Path lines are the outcome of the Lagrangian method in describing fluid flow and show the paths of different fluid particles as a function of time. Since the stream-tube is bounded on all sides by streamlines and. This passage not necessarily circular in cross-section is known as a stream-tube. A stream-tube with a cross-section small enough for the variation of velocity over it to be negligible is sometimes termed as a stream filament.
The entire flow in a flow field may be imagined to be composed of flows through stream-tubes arranged in some arbitrary positions. The above expression therefore represents the differential equation of a streamline.
Then Eq. A bundle of neighbouring streamlines may be imagined to form a passage through which the fluid flows Fig. Kinematics of Fluid 87 Therefore.
From the above definition of streamline. In a steady flow. In a cartesian coordinate H H system. In an unsteady flow where the velocity vector changes with time. If a fluid particle H H S 0 passes through a fixed point S 1 in a course of time t. If dye is injected into a liquid at a fixed point in the flow field.
While a path line refers to the identity of a fluid particle. The two stream lines. Therefore a family of path lines represents the trajectories of different particles. It is evident that path lines are identical to streamlines in a steady flow as the Eulerian and Lagrangian versions become the same. The equation of a streak line at time t can be derived by the Lagrangian method. It can be mentioned in this context that while stream lines are referred to a particular instant of time.
Two path lines can intersect with one another or a single path line itself can form a loop. This line is of particular interest in experi- mental flow visualization. This is quite possible in a sense that. This single space. Let P be a fixed point in space through which particles of different identities pass at different times.
In an unsteady flow. Figure 3.
Let at any instant t1. The fixed point P will also lie on the line. R and S. So one-dimensional flow is that in which all the flow parameters may be expressed as functions of time and one space coordinate only. R and S represent the end points of the trajectories of these three particles at the instant t1. Sometimes simplification is made in the analysis of different fluid flow problems by selecting the coordinate directions so that appreciable variation of the hydro- dynamic parameters take place in only two directions or even in only one.
Kinematics of Fluid 89 This is the final form of the equation of a streak line referred to a fixed point S1. Q and the fixed point P will define the streak line at that instant t1.