Motion of Fluid Particles, An Essential Need of Humans……

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Kinematics of Viscous Fluid Flows

The Convection Theorem

• Suppose that St 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

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), vp(t), pp(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

0dt

d

• 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.

0dt

d

• 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

Guide vanesGuide vanesGuide vanesGuide vanesGuide vanesGuide vanes

Runner inlet (Φ 0.870m)

Guide vane outlet for designα) (Φ 0.913m)

ClosedPosition

Max. Opening Position

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanes

R a d i a l v i e wrunner guide vanes and stay vanesR a d i a l v i e wrunner guide vanes and stay vanes

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