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Distributed Parameters. ECX 5241. Academic year 2003. Prepared by D.A.Mangala Abeysekara. ASSIGNMENT NO.1. ASSIGNMENT NO.2. ASSIGNMENT NO.3. ASSIGNMENT NO.4. ASSIGNMENT NO.1. Symmetry in the physical universe Coordinates systems Operators in vector calculus The theorem of gauss

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    1. Distributed Parameters ECX 5241 Academic year 2003 Prepared by D.A.Mangala Abeysekara


    3. ASSIGNMENT NO.1 • Symmetry in the physical universe • Coordinates systems • Operators in vector calculus • The theorem of gauss • The theorem of Stokes

    4. Symmetry in the physical universe • Most of the problems in the physical universe exhibit the spherical or cylindrical symmetry. • Examples • the gravity field of the Earth is to the first order spherical symmetric.

    5. Waves excited by a stone thrown into water are usually cylindrically symmetric • An earthquake excites a tsunami in the ocean The Cartesian coordinates is usually not very convenient to use for study such a problems

    6. The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. There are two coordinates systems used for study the symmetric system.

    7. Spherical Coordinates • Cylindrical Coordinates Relationship between the Cartesian coordinates and spherical coordinates

    8. Relationship between the Cartesian coordinates and cylindrical coordinates

    9. Operators in vector calculus  There are several operators can be identified in the vector calculus. They are, ·Gradient (f ) ·Divergent(.v ) ·Curl (v )

    10. Gradient • Lets consider the particle, which moved according to the function f from point A to point B.(f Is a function of x and y) • Grad f =

    11. The divergence of a vector field vx dz dy dx Outward flux through the right hand surface perpendicular through the x-axis vx (x+dx, y, z) dydz

    12. The flux through the left hand surface perpendicular through the x- axis – vx (x, y, z) dydz The the total outward flux through the two surface vx (x+dx, y, z) dydz - vx (x, y, z) dydz = (vx/x)dxdydz

    13. (.v)= d/dv The divergence of a vector field is the outward flux of the vector field per unit volume

    14. The curl of a vector field Curl V=v

    15. Physical meaning of the curl operator The component of curl v in a certain direction is the closed line integral of v along a closed path perpendicular to this direction, normalized per unit surface area.

    16. The theorem of gauss The theorem of Stokes

    17. Example The magnetic field induced by a straight current I B The Maxwell equation for the curl

    18. (Using Stoke theorem) Solving this equation,

    19. Reference A Guided Tour Of Mathematical Methods For The Physical Science By Role Sineder Cambridge University Press

    20. THE END Thank you for your attention

    21. ASSIGNMENT NO.2 • Classification of PDE • Acoustic sound propagation in gas

    22. A partial differential equation of the form is said to be ØElliptic if 4AC-B2>0 Ø  Parabolic if 4AC-B2=0 ØHyperbolic if 4AC-B2<0

    23. Consider the partial differential equation, where A,B and C are constants. Define variables, and where ,, and  are constants. Using chain rule we can show that ,

    24. If we can select ,, and  such that,  and Then and the general solution is ,

    25. hyperbolic Equations It can be shown that, if the equation is hyperbolic if B2>4AC, =2A =2A Satisfies this condition. Hence hyperbolic equations have two characteristic given by,

    26. Parabolic Equation It can be shown that the equation is parabolic, if 4AC-B2=0 The solution of the quadric equation, is If =2A, and =-B, the coefficient of , that is,

    27. The partial differential equation is reduce to, The solution is, Where p and q are arbitrary functions.

    28. The parabolic equations have only one characteristic given by, 2Ax-Bt = constant

    29. Elliptic Equation Elliptic equations have no characteristics. However, the transformation, Reduce the partial differential equation to,

    30. Let consider the cubic element of the fluid (gas). Y X Z

    31. net force in the positive x direction is similarly the net force in the positive y and z directions are

    32. The net vector force on the cubical element is therefore where Using Newton’s second low

    33. Let write the divergence of the gradient of pressure The incremental pressure and the accompanying dilation are linearly related through the bulk modules B,

    34. By eliminating . where The left side of equation is the three dimensional laplacian of the pressure p

    35. THE END Thank you for your attention

    36. ASSIGNMENT NO.3 • Potential of the velocity field in fluid

    37. IRROTTIONAL FLOW Lets consider the two point in the fluid with distance of dr z B dr A y x

    38. E B dz A D dy dx C Les conceder the two points of A and B as a diagonal of the rectangular pipe. We can write the velocity of the point c can be given in the term of the velocity of point A.

    39. This equation can be written in three dimensional form

    40. z E D A y C x Hence the velocity at the point c can be represent in the term of velocity of the point A

    41. A C Like in the point c, the velocities at the other two are also can be written relative to the point A z E D y x

    42. Now conceder particle C. it is clear that is the rate of elongation of line segment AC. We can write the elongation rate per unit original length,as . This is known as the normal stain Hence,

    43. Lets investigate the rate of the angular change of the sides of the rectangular pipe. The average rate of rotation about the z axis of the orthogonal line segments AC and AD is The rate of change of the angle CAD (a right angle at time t ) becomes

    44. Thus we know that the time rate of change of the shear anglexy so that Similarly,

    45. Accordingly, we have available to describe the deformation rate of the rectangular parallel piped the strain rate terms which we now set forth as follows: =strain rate tensor

    46. Now lets consider the rigid body rotation. Thus, the expression Is actually more than just the average rotation of line segments dx and dy about the z axis . It represents for a deformable medium what we maybe consider as the rigid_body angular velocity z about the z axis. This is, Similarly,

    47. Thus, Hence, At this time ,we define irrotational flow as those for which =0 at each point in the flow.

    48. For irrotational flow, we require that Thus it become clear that another criterion for irrotationality, and the one we will use,is Curl V= 0

    49. The velocity potential If velocity components at all points in a region of flow can be expressed as continuous partial derivatives of a scalar function (x,y,z,t) thusly In general we can write