Question

1. The equation of continuity of a pure fluid is given as др at + V.pv = 0 a) In performing a derivation in fluid mechanics, you find you need an expression for Dinp Dt Show how this can be obtained from the equation of continuity above.

Answer 1

Given

Continuity equation

$\frac{\partial \rho}{\partial t}+\triangledown .\rho v=0$

Where

$\\ v=v_x i+v_yj+v_zk\ \ \ i,j,k\ are \ unit \ vector \ in \ x \ y \ z\ direction \\ \triangledown .\rho v=\frac{\partial \rho v_x }{\partial x}+\frac{\partial \rho v_y }{\partial y}+\frac{\partial \rho v_z }{\partial z} \\Applying\ product \ rule \\ \triangledown .\rho v=v.\triangledown \rho+\rho \triangledown .v=v_x\frac{\partial \rho }{\partial x}+ v_y\frac{\partial \rho }{\partial y}+v_z \frac{\partial \rho }{\partial z}+\rho (\frac{\partial v_x }{\partial x}+\frac{\partial v_y }{\partial y}+\frac{\partial v_z }{\partial z})$

Applying in the continuity equation

+ Фуду + Vi dr др др + V: др at дz v. Vp- at др +v. Vр + pv.v = 0 Eq1 at The total derivative of Dp др др Dt Applying in Eq1 Dp + pv.v = 0 Dt Dp = -pv. Eq2 Dt

We need to find

alnp Dinp ᎠᏆ . Applying chain rule Dp 1 DA Dt ap Dt Dt Using Eq2 Dinp * -pb.v = -1.0 Dt P *

So the solution is

$\\ \frac{\mathrm{D} ln \rho }{\mathrm{D} t}=-\triangledown .v=-(\frac{\partial v_x }{\partial x}+\frac{\partial v_y }{\partial y}+\frac{\partial v_z }{\partial z})$