Bernoulli's equation

Euler's equation is given as  $$dp =  - \rho vdv$$ This equation applies to an incompressible and in-viscid flow where $\rho  = \,{\rm{constant}}$. In a streamline, in between two points 1 and 2, the above Euler equation can be integrated as  $$\int\limits_{{p_1}}^{{p_2}} {dp =  - \rho \int\limits_{{v_1}}^{{v_2}} {vdv} }$$ $${p_2} - {p_1} =  - \rho \left( {\frac{{v_2^2}}{2} - \frac{{v_1^2}}{2}} \right)$$ $$\Rightarrow {p_1} + \frac{1}{2}\rho v_1^2 = {p_2} + \frac{1}{2}\rho v_2^2$$ This can be written as $$p + \frac{1}{2}\rho {v^2} = {\rm{constant}}$$ for a streamline.For an rotational flow the value of constant is changing from streamline to another.For irrotational flow,the constant is same for all streamlines and  $$p + \frac{1}{2}\rho {v^2} = {\rm{constant}}$$ throughout the flow. Physical significance of Bernoulli's equation is that when the velocity increases, the pressure decreases  and when the velocity decreases, the pressure increases.

Example: Calculate the velocity at a point on the airfoil, where the pressure is${\rm{0}}{\rm{.4 \times 1}}{{\rm{0}}^{\rm{5}}}{\rm{N/}}{{\rm{m}}^{\rm{2}}}$. The free stream velocity is ${\rm{70m/s}}$ in a standard sea level conditions.

Solution: At sea level conditions ${{\rm{\rho }}_\infty }{\rm{ = 1}}{\rm{.23kg/}}{{\rm{m}}^{\rm{3}}}$ and  ${p_\infty }{\rm{ = 1}}{\rm{.01}} \times {\rm{1}}{{\rm{0}}^5}{\rm{N/}}{{\rm{m}}^{\rm{2}}}$ $${p_\infty } + \frac{1}{2}\rho v_\infty ^2 = p + \frac{1}{2}\rho {v^2}$$ $$v = \sqrt {\frac{{2\left( {{p_\infty } - p} \right)}}{\rho } + v_\infty ^2}$$ $$v = \sqrt {\frac{{2\left( {1.01 - 0.4} \right) \times {{10}^5}}}{{1.23}} + {{\left( {70} \right)}^2}}$$ $$\Rightarrow v = 322.63\,{\rm{m/s}}{\rm{.}}$$ Example: In an inviscid, incompressible flow of air along a streamline, the air density is 0.002377 ${\rm{slug/f}}{{\rm{t}}^{\rm{3}}}$. At a point on the streamline pressure and velocity are 2000 ${\rm{lb/f}}{{\rm{t}}^{\rm{2}}}$ and 15 ft/s, respectively. Downstream at other point on the streamline the velocity is 150 ft/s. What is the pressure at this point.
From, Bernoulli's equation $${p_1} + \frac{1}{2}\rho v_1^2 = {p_2} + \frac{1}{2}\rho v_2^2$$ $${p_2} = {p_1} + \frac{1}{2}\rho \left( {v_1^2 - v_2^2} \right)$$ $${p_2} = 2116 + \frac{1}{2}\left( {0.002377} \right)\left[ {{{\left( {15} \right)}^2} - {{\left( {150} \right)}^2}} \right]$$ $$= 2089.526\,\,lb/f{t^2}$$