#### Derivation of a Pressure Field and Fluid at Rest: Screencasts

Derives the equation for pressure within a fluid with no shear stresses using a differential element.

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Derives the equation for a fluid at rest by simplifying the pressure field derivation.

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##### Important Equations:

A Taylor Series expansion (neglecting higher order terms) and Newton’s second law are essential for the derivation of a pressure field and fluid at rest.

$\sum _{n=0} ^{\infty} \frac {f^{n} (a)} {n!} (x-a)^{n} = f(a) + \frac {f'(a)} {1!} (x-a) + …$

where $$a$$ is a point in the $$x$$ direction, $$f(a)$$ is the function of interest evaluated at $$a$$, and $$f'(a)$$ is the derivative of $$f(x)$$ with respect to $$x$$ and evaluate at point $$a$$. $\sum \vec{F} = m \vec{a}$Where $$\vec{F}$$ is a force vector, $$\vec{a}$$ is an acceleration vector, and $$m$$ is mass.

Using a fluid element, the surface and body forces can be used to derive the equation of motion for a fluid with no shear stresses.

$-\nabla P -\gamma \hat{k} = \rho \hat{a}$

where $$\nabla P$$ is the gradient of pressue, $$\gamma$$ is specific weight of the fluid, $$\hat{k}$$ is the unit vector in the z direction, $$\rho$$ is the density of the fluid, and $$\hat{a}$$ is an acceleration vector.

When a fluid is at rest, the acceleration vector is zero. Thus, the only non-zero component is in the z direction.

$\frac{dP}{dz}=-\gamma$