Motion of Fluid Particles, An Essential Need of Humans…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Kinematics of Viscous Fluid Flows

The Convection Theorem Suppose that S t is a region of fluid particles and let p(x,t) be a scalar function. The volume integral of p(x,t) has capability to generation convection in fluid. Generates kinematic properties to the fluid field.

Description of a Fluid Flow

Lagrangian description: Picture a fluid flow where each fluid particle caries its own properties such as density, momentum, etc. The procedure of describing the entire flow by recording the detailed histories of each fluid particle is the Lagrangian description. The particle properties density, velocity, pressure,... can be mathematically represented as follows:  p (t), v p (t), p p (t),.. The position of Any particle is completely defined in terms of a position vector which is a function of time and initial position.

Lagrangian Description of Flow

The Material Derivative Let scalar property  is identified to a certain fluid parcel, e.g. temperature or density. Suppose that, as the parcel moves, this property is varying with time. This fact is denoted by Since this means that the time derivative is taken with particle label fixed, i.e. taken as we move with the fluid particle in question. Such a scalar is called as material. A material is the one attached to a fluid particle. Further, suppose that, as the parcel moves, this property is invariant in time. This fact is denoted by the equation

Material Derivatives to Define Kinematic Properties

Practical Use of Lagrangian Description Flow through Francis Turbine

Parts of A Francis Turbine

Runner inlet (Φ 0.870m) Guide vane outlet for designα) (Φ 0.913m) Closed Position Max. Opening Position

Water from spiral casing Water particle

Parts of A Francis Turbine

Engineering Use of Lagrangian Description The Lagrangian description is simple to understand. Conservation of mass and Newton’s laws directly apply directly to each fluid particle. However, it is computationally expensive to keep track of the trajectories of all the fluid particles in a flow. The Lagrangian description is used only in Extreme cases of numerical simulations other particles carried by the fluid paricles.

Lagrangian Description to Control Sand erosion in the guide vanes