Pre-requisite: Knowledge of Engineering Mathematics with Integral Calculus, Differential Calculus and Vector Calculus is Preferable

Course Description & Outline :

Lagrangian and Eulerian description, streamline, streakline and pathline, acceleration of a fluid element, continuity equation, stream-function, rotation and angular deformation, irrotational flow,velocity potential

Reynolds transport theorem - conservation of mass, linear and angular momentum

Continuity equation, Navier-Stokes equations – derivations and some exact solutions

Schedule for Lecture Delivery

Session 1 : 11^{th}-Aug-2015 - 10-12 noon

Session 2 : 12^{th}-Aug-2015 - 10-12 noon

Session 3 : 13^{th}-Aug-2015 - 10-12 noon

Teacher Forum

Kinematics of Fluid Flow

1.1 INTRODUCTION

Kinematics of fluid flow is that branch of fluid mechanics which describes the fluid motion and its consequences without consideration of the nature of forces causing the motion. 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:

(a) The development of methods and techniques for describing and specifying the motions of fluids.

(b) Characterization of different types of motion and associated deformation rates of any fluid element.

(c) The determination of the conditions for the kinematic possibility of fluid motions, i.e., the exploration of the consequences of continuity in the motion.

1.2 FLOW FIELD AND DESCRIPTION OF FLUID MOTION: LAGRANGIAN AND EULERIAN DESCRIPTION

A flow filed is a region in which the flow velocity is defined at each and every point in space, at any instant of time. Usually, the velocity describes the flow. In other words, a flow field is specified by the velocities at different points in the region at different times. 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 ( of the same density as that of fluid) that exists at that point at that instant. In order to obtain a complete picture of the flow, the fluid motion is described by two methods discussed as follows:

(a) Lagrangian Method:

In this method, the fluid motion is described by tracing the kinematic behaviour of each and every individual particles constituting the flow. Identities of the particles are made by specifying their initial position ( spatial location) at a given time. The position of a particle at any other instant of time then becomes a function of its identity and time. This statement can be analytically expressed as

1.3 STREAMLINES, PATHLINES AND STREAKLINES

1.3.1 Streamlines

The analytical description of flow velocities by the Eulerian approach is geometrically depicted through the concept of streamlines. In the Eulerian method, the velocity vector is defined as a function of time and space coordinates. If for a fixed instant of time, a space curve is drawn so that it is tangent everywhere to the velocity vector, then this curve is called a streamline. Therefore, the Eulerian method gives a series of instantaneous streamlines of the state of motion. In other words, a streamline at any instant can be defined as an imaginary curve or line in the flow field so that the tangent to the curve at any point represents the direction of the instantaneous velocity at that point. In an unsteady flow where the velocity vector changes with time, the pattern of streamlines also changes from instant to instant. In a steady flow, the orientation or the pattern of streamlines will be fixed. From the above definition of streamline, it can be written

Equation (1.4) describes the differential equation of a streamlines in a Cartesian frame of reference

A bundle of neighbouring streamlines may be imagined to form a passage throgh which the fluid flows ( Fig. 1.1(a)). This passage ( not necessaarily 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. Since the stream tube is bounded on all sides by streamlines, and by definition, velocity does not exist across a streamline, no fluid may enter or leave a stream tube except through its ends. The entire flow in a flow field may be imagined to be composed of flows through stream tubes arranged in some arbitrary positions.

1.3.2 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. In other words, a pathline is the trajectory of a fluid particle of fixed identity as defined by Eq.(1.1). Therefore a family of path lines represents the trajectories of different particles, say, P_{1}, P_{2}, P_{3}, etc (Fig. 1.2). It can be mentioned in this context that while streamlines are referred to a particular instant of time, the description of path lines inherently involves the variation of time, since a fluid particle takes time to move from one point to another. Two pathlines can intersect with one another or a single path line itself can form a loop. This is quite possible in a sense that, under certain conditions of flow, different particles or even a same particle can arrive at same location at different instants of time. The two stream lines, on the other hand, can never intersect each other since the instantaneous velocity vector at a given location is always a unique. It is evident that path lines are identical to stream lines in a steady flow as the Eulerian and Lagrangian versions become the same.

1.5 TRANSLATION, RATE OF DEFORMATION AND ROTATION

The movement of a fluid element in space has three distinct features, namely translation, rate of deformation and rotation. A fluid motion, in general, consists of these three features simultaneously. Translation and rotation without deformation represent rigid-body displacements which do not induce any strain in the body. Fig. 1.4 shows the picture of a pure translation in absence of rotation and deformation of a fluid element in a three-dimensional flow described by a rectangular Cartesian coordinate system. In absence of deformation and rotation, there will be no change in the length of the sides or in the included angles made by the sides of the fluid element. The sides are displaced parallely. This is possible when the flow velocities ( the x component velocity), ( the y component velocity), and ( the z component velocity) are neither a function of x, y nor of z, in other words, the flow field is totally uniform.

Rate of linear deformation

Rate of angular deformation and rotation of fluid elements

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Integral Forms of Conservation Equations

2.1 REYNOLDS TRANSPORT THEOREM (RTT)

2.1.1 Derivation of Reynolds Transport Theorem

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Differential Form of Conservation Equations

3.1 CONSERVATION OF LINEAR MOMENTUM IN DIFFERENTIAL FORM

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