I was reading a book on fluids and I didn't understand from where the velocity formulas that were written came from. Judging by the text, it looks like it was derived by taking the material derivative. Also the picture shows an elemental control volume. How can I get the equations (2.69)? Please derive
I know that the material derivative of a quantity Q is given by
$$\frac{DQ}{Dt}=\frac{∂Q}{∂t}+u.∇Q$$
Applying this to the velocity components u and v gives:
$$\frac{Du}{Dt}=\frac{∂u}{∂t}+u\frac{∂u}{∂x}+\frac{v∂u}{∂y}$$
I got it this far but i couldn't find how to get this equation
$$u_{A}=u-\frac{∂u}{∂x}\frac{δy}{2}-\frac{∂u}{∂y}\frac{δy}{2}$$
