1. Continuum Mechanics (MTH487)
Lecture 14
Instructor
Dr. Junaid Anjum

2. Recap …. kinematics of deformation and motion
material and spatial coordinates
Reference P
Current P

3. Recap ….
Langrangian and Eulerian description

4. Aims and Objectives examples displacement field
the material derivative

5. Problem 1: The motion of a continuous medium is specified by the component equations
Show that the Jacobian determinant J does not vanish, and solve for the inverse equations
Calculate the velocity and acceleration components in terms of the material coordinates.
Using the inverse equations developed in part (a), express the velocity and acceleration components in terms of spatial coordinates.

7. Problem 2: Let the motion of continuum be given in component form by the equations
Show that and solve for the inverse equations.
Determine the velocity and acceleration
at time s for the particle which was at point when s.
at time s for the particle which was at point when

9. The displacement field
A displacement is the change in the position of the particle as it transits from the reference configuration to the current configuration
In component form, the displacement vector is written as
Reference P
Current p
displacement of the particle starting at
displacement that the particle now at has undergone

10. The displacement field …
Problem 3: Let the motion equations be given in components form by
Obtain the displacement field in both materials and spatial descriptions.

12. The material derivative
: any physical or kinematic property of a continuum body
The material derivative of is the time rate of change of for a specific collection of particles (one or more) of the continuum body. This derivative can be thought of as the rate at which changes when measured by an observer attached to, and travelling with, the particle, or group of particles. We will use or superpositioned dot to represent the material derivative.
For material description …
For spatial description …
local rate of change
convective rate of change

13. The material derivative
For spatial description …
the material derivative operator for properties expressed in the spatial description

14. The material derivative …
Problem 4: Let a certain motion of continuum be given by the component equations,
and let the temperature field of the body be given by the spatial description,
Determine the velocity field in spatial form, and using that, compute the material derivative of the temperature field.

16. The material derivative …
Problem 5: Verify the spatial velocity components determined in Problem 4 by applying the material derivative definition to the displacement components in spatial form

18. The material derivative …
Problem 6: Cilia are motile cells that are responsible for locomotion of single-cell bodies, hearing, and moving fluid in the body. The cilia beat in a manner that propels the surrounding fluid. The motion of cilia is complex. A model for cilia motion is to enclose the tips by a flexible envelope that moves in a fashion similar to the tips of the cilia. This is reasonable given that in many biological processes fluid motions are slow. Two possible descriptions of the envelope motion are (T. J. Lardner and W. J. Shack (1972) Cilia transport, Bulletin of Mathematical Biophysics, 34 (3) )
Case 1: Case 2:
Here is the position of the envelope and is an initial position of the particles.
(a) Show that the both descriptions indicate that particles move in an elliptical path centered about
(b) Fluid mechanics has a free-slip condition that requires the velocity of the fluid to be the same as the velocity of the envelope. Determine the velocity of the particles in the two descriptions.

21. Aims and Objectives displacement field the material derivative
For material description …
For spatial description …