Begin2.DVI

If dρdt = 0,then the fluid is an incompressible fluid, and the velocity field is solenoidal.

If the fluid flow is also irrotational, then V is derivable from a potential function

Φcalled the velocity potential of the fluid flow. The potential function must be a

solution of Laplace’s equation. The field lines associated with the velocity field V

produce a family of curves which are termed streamlines.

Heat Conduction

In the basic equations describing heat conduction in materials, the following

assumptions and terminology are employed:

1. Let T =T(x, y, z, t) denote the temperature (◦C) at a point (x, y, z )within the

material at time t.

Heat flow within the material is denoted by the vector qhaving units [q] = cm J (^2) ·sec

3. Heat flows from regions of higher temperature to regions of lower temperature

and the direction of heat flow is in the direction of the greatest rate of change

of the temperature. Expressing this as a mathematics statement, we write

q=−kgrad T, (9 .94)

where kis a proportionality constant having units of cm ·secJ·◦C and is called the

thermal conductivity of the material. Since the gradient of temperature points

in the direction of increasing temperature, the negative sign in the relation (9.94)

indicates that heat is flowing in the direction of decreasing temperature.

4. The symbol c is used to denote the specific heat of the material which is a

measure of the heat capacity per unit mass of material. The specific heat cis

measured in units gJ◦C.

5. The symbol ρis used to denote the density of the material [cmg 3 ].

6. The total amount of heat in an arbitrary volume Vbounded by a closed surface

Sis given by

H=

∫∫∫

V

cρT dV, (9 .95)

where His in joules.

If an imaginary closed surface Senclosing a volume Vis placed within a body in

which heat is flowing, then the heat flux across this surface is given by the integral

∫∫

S

q dS=