Who Causes the Deformation/Rotation inViscous Fluid Flow

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Presentation transcript:

Who Causes the Deformation/Rotation inViscous Fluid Flow P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Scalar Product of  and a Tensor ……

Cartesian Fluid Element

Cartesian Fluid Element

Fluid element under a general three-dimensional stress condition. The first index i refers to the axis, to which the fluid element surface is perpendicular. The second index j indicates the direction of the force component.

Generation of Extensive Nature to Force

The Culprit

Force per Unit Area Divide the individual components of the above force vectors by their corresponding area of the fluid element side. The results of these divisions exhibit the components of a second order stress tensor.

Force per unit area : An Important Tensor in Viscous Fluid Flows The Stress tensor

General Stress Condition

Stresses on Two Opposite Faces

Force Balance on A Plane Force Balance on Plane 1 in x1 direction: Force Balance on Plane 1 in x2 direction: Force Balance on Plane 1 in x1 direction:

The Resultant Force on A Plane The resulting Force on Plane 1: The resulting Force on Plane 2: The resulting Force on Plane 3:

Force Balance per Unit Volume The total resulting forces acting on the entire surface of the element are obtained by adding the nine components. Defining the volume element , we divide the results by and obtain the resulting force vector that is acting on the volume element.

Transpose : A Natural Thinking

Stress is the Mother of Force The stress is A tensor It can be easily shown that The above expression is a scalar differentiation of the second order stress tensor and is called the divergence of the tensor field. We conclude that the force acting on the surface of a fluid element is the divergence of its stress tensor. The stress tensor is usually divided into its normal and shear stress parts.

Stress Tensor in an Incompressible Viscous Fluid Flow How do they act on fluid elements entering/ Leaving a Control Volume (Viscous Fluid Flow)????

Format of Examinations Minor Examinations: each of one hour duration. Major Examination: Two hours duration. All the examinations are open – hand written notes examinations. All kinds of your own hand written notes are allowed to be used during examinations. (Class notes, Tutorial notes, etc.) No printed or photocopied material is allowed. Printed copies of my lectures are also not allowed.

Weightage of examinations A Grand total of 200 points will be considered. Following distribution is sued for individual evaluations: Minor-1 30 Minor – 2 40 Major 70 Assignment 40 Course Project 20

Outcome of Viscous Fluid Flow Analysis

The Art of Using Principle of Conservation The conservation laws in integral form are, strictly speaking, valid for closed systems. In fluid flows, the mass flow continuously crosses the system boundary. To apply the conservation laws to open systems, special mathematical tools are essential. A treatment for the volume integral of an arbitrary field quantity f(x,t) demands a special engineering law.