Fluid Mechanics for MAP/Fluid Statics

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Definitions
Fluid statics is the study of fluids which are either at rest or in rigid body motion  with respect to a fixed frame of reference. Rigid body motion means that there is no relative velocity between the fluid particles.

In a fluid at rest, there is no shear stress, i. e. fluid does not deform, but fluid sustains normal stresses.

We can apply Newton's second law of motion to evaluate the reaction of the particle to the applied forces.



Force balance in $$ \displaystyle i^{th}$$ direction:

$$\displaystyle F_{i}^{net} = m\cdot a_{i}$$

We can also say,$$ \displaystyle F_{i}^{body} + F_{i}^{surf}= m\cdot a_{i}$$

Force created by pressure is :

$$ \displaystyle F^{surf} = F^{pressure} = -p\cdot \vec{A}$$

$$ \displaystyle \vec{A}$$ is the vector having the surface area as magnitude and surface normal as direction.

Thus,

$$ \displaystyle F^{pressure} = -p\cdot\vec{A} = -p\ A \ \vec{n}$$

Force caused by the pressures opposite to the surface normal.

For a differential fluid element: $$\displaystyle dF_{i}^{body} + dF_{i}^{surf} = dm\cdot a_{i}$$

Remember Taylor Series expansion: $$ \displaystyle F(x+\Delta x) = F + \frac{\partial F}{\partial x} + \frac{1}{2}\frac{\partial ^{2}F}{\partial x^{2}}\Delta x^{2} + \ldots$$

P is the pressure in the center of the fluid element, therefore the pressure on the surface in direction of $$x_i$$ is $$P + \frac{\partial P}{\partial x_{i}}\frac{dx_{i}}{2}$$.

Thus,
 * $$ \displaystyle dF_{2}^{pressure} = \left[P-\frac{\partial P}{\partial x_{2}}\frac{dx_{2}}{2}\right]dx_{1}\ dx_{3} - \left[P + \frac{\partial P}{\partial x_{2}}\frac{dx_{2}}{2}\right]dx_{1}\ dx_{3}$$:$$\displaystyle =-\frac{\partial P}{\partial x_{2}}dx_{1}dx_{2}dx_{3} = -\frac{\partial P}{\partial x_{2}}dV$$

$$ \displaystyle dF_{i}^{pressure} = -\frac{\partial P}{\partial x_{i}}dV$$

$$ \displaystyle dF_{i}^{body} = dm\ g_{i}$$

Thus,

$$ \displaystyle -\frac{\partial P}{\partial x_{i}}dV + dm\ g_{i} = dm\ a_{i}$$ or,$$ \displaystyle -\frac{\partial P}{\partial x_{i}}dV + \rho\ dV\ g_{i} = \rho\ a_{i}$$

or,$$ \displaystyle -\frac{\partial P}{\partial x_{i}} + \rho\ g_{i} = \rho a_{i}$$

or,$$ \displaystyle -\frac{\partial P}{\partial x_{1}} = \rho a_{1}\ ; \ -\frac{\partial P}{\partial x_{2}} = \rho a_{2}\ ; \ -\frac{\partial P}{\partial x_{3}} - \rho g = \rho a_{3}$$

for $$ \displaystyle a_{i}=0$$

$$ \displaystyle \frac{\partial P}{\partial x_{1}} = 0 \ ; \ \frac{\partial P}{\partial x_{2}} = 0 \ ; \ \frac{\partial P}{\partial x_{3}} = -\rho g$$

Pressure changes only in $$ \displaystyle x_{3}$$ direction.

Pressure variation in an incompressible and static fluid


$$ \displaystyle \frac{\partial P}{\partial x_{3}} = -\rho g \ ; \ x_{3} = z \rightarrow \frac{\partial P}{\partial z} = - \rho g $$ is constant since $$ \displaystyle \rho$$ and $$ \displaystyle g$$ are constants.

$$ \displaystyle \int^{p_{2}}_{p_{1}}{dP} = -\int^{z_{2}}_{z_{1}}{\rho g dz}$$

$$ \displaystyle p_{2} - p_{1} = -\rho g\ (z_{2}-z_{1})$$

If we take $$ \displaystyle p_{2}$$ at the surface, then:

$$ \displaystyle p_{atm}-p_{1} = -\rho g\ (z_{2}-z_{1})$$

$$ \displaystyle p_{1} = p_{atm} + \rho gh\ ; \ \ h = z_{2}-z_{1}$$

h is measured from the surface.

Other related topics
Buoyancy and calculation of forces on the submerged surfaces are topics related to fluid statics.