4. Fluid MechanicsSavchenko Formulas, chapter 4 of 14, 12 formulas
Sections follow the book, and within a section the most used formulas come first. An italic problem number means the formula appears in its statement. Rest the cursor on a number to see the statement.
$\displaystyle F = pS\ \text{(сила на плоскую стенку, }p\text{ в её центре)}$$\displaystyle p(r) = p_0 + \frac{\rho\omega^2 r^2}{2}\ \text{(вращающаяся жидкость)}$
p0
pressure at the surface, usually atmospheric
h
depth
$\rho$
density of the liquid
Pressure in a liquid at rest grows with depth by $\rho g$ per metre and is the same at all points of one level, whatever the shape of the vessel. The force on a wall is the pressure at its centroid times the area, and on a curved wall it goes by the projection. In a rotating vessel the pressure grows from the axis as $\rho\omega^2 r^2/2$ and the surface is a paraboloid.
Pressure applied to a liquid is transmitted equally to all its points, so the forces on two pistons are as their areas. The gain in force is paid by a loss in displacement, the work is the same. Pressure on the walls of a vessel of any shape acts normal to them.
$\displaystyle \rho_{\text{тела}} V g = \rho_{\text{ж}} g V_{\text{погр}}\ \text{(плавание)}$$\displaystyle N = (\rho - \rho_{\text{ж}})gV\ \text{(вес в жидкости)}$
$\rho_{\text{ж}}$
density of the liquid
$V_{\text{погр}}$
submerged volume
FA
buoyant force, applied at the centroid of the displaced liquid
The buoyant force equals the weight of the displaced liquid, because the pressure on a submerged body is the same as on the liquid that would fill its place. A floating body displaces its own weight of liquid, a sinking one presses on the bottom with the difference of weight and buoyancy. In an accelerating liquid $g$ is replaced by the effective one.
$\displaystyle Q = S\sqrt{2gh}\ \text{(расход через отверстие)}$$\displaystyle x = 2\sqrt{h(H - h)}\ \text{(дальность струи из бака)}$
h
depth of the hole below the free surface
v
speed of the outflow
A jet leaves a hole at the speed a body would gain falling from the liquid's level, Bernoulli's equation between the surface and the hole at the same atmospheric pressure. The level is taken as fixed when the hole is small.
An incompressible liquid does not pile up, so the same volume passes any cross-section of a pipe each second, and the flow is faster where the pipe is narrow. For a spherically diverging flow the speed falls as $1/r^2$.
$$p + \frac{\rho v^2}{2} + \rho g h = \text{const}$$
$\displaystyle v = \sqrt{\frac{2\Delta p}{\rho}}$$\displaystyle \Delta p = \frac{\rho v^2}{2}\ \text{(скоростной напор)}$
p
pressure in the flow
v
flow speed there
h
height of the point
$\rho$
density
Along a streamline of an ideal incompressible liquid the sum of pressure, kinetic energy per unit volume and $\rho g h$ is constant, energy conservation for a fluid. Where the speed is higher the pressure is lower, hence lift, the atomiser and speed measurement from a pressure difference.
$\displaystyle \vec F = \dot m(\vec v_{\text{вых}} - \vec v_{\text{вх}}) + \sum p\vec S$$\displaystyle F = \rho S v^2\ \text{(струя гасится о стенку)}$
$\rho S v$
mass passing per second
p
pressure in the pipe
S
cross-section
A pipe or a wall receives a force equal to the change of the flow's momentum per second, plus the pressure forces on the inlet and outlet sections. Hence the reaction of an outflowing jet, $\rho S v^2 = 2\rho g h S$, twice the hydrostatic force on a closed hole, and the force on a pipe bend.
$\displaystyle \bar v = \frac{R^2\Delta p}{8\eta l}$$\displaystyle Q = \frac{\rho g\sin\alpha}{3\eta}h^3\ \text{(плёнка по наклонной плоскости, на единицу ширины)}$
Q
volume flow rate
R, l
radius and length of the pipe
$\Delta p$
pressure drop over the length $l$
$\eta$
viscosity
In a pipe a viscous liquid flows with a parabolic profile, and the flow rate is proportional to the fourth power of the radius and to the pressure drop. A tube half as thick passes 16 times less at the same drop. Gravity can supply the drop too, $\Delta p = \rho g l\sin\alpha$.
$\displaystyle F = 6\pi\eta r v\ \text{(шар, формула Стокса)}$$\displaystyle v(y) = v_0\frac{y}{h}\ \text{(течение между пластинами)}$
$\eta$
viscosity of the liquid
$dv/dy$
velocity gradient across the flow
S
area of the layer
Neighbouring layers of liquid moving at different speeds drag each other with a force proportional to the viscosity, the area and the velocity gradient. At a wall the liquid is at rest. Hence the linear velocity profile between moving plates and the parabolic one in a pipe or down an incline.
$\displaystyle \sigma = \frac{F}{l} = \frac{A}{\Delta S}$$\displaystyle F = 2\sigma l\ \text{(плёнка с двумя поверхностями)}$
$\sigma$
surface tension
l
length of the boundary along which the force acts
$\Delta S$
change of surface area
A liquid's surface behaves as a stretched film, the force on a boundary is proportional to its length and tangent to the surface. The same quantity is the energy per unit area, and the work to create a surface is $\sigma\Delta S$. A soap film has two surfaces.
$\displaystyle \Delta p = \frac{4\sigma}{R}\ \text{(мыльный пузырь, две поверхности)}$$\displaystyle \Delta p = \frac{\sigma}{R}\ \text{(цилиндрическая поверхность)}$
R1, R2
principal radii of curvature of the surface
$\sigma$
surface tension
$\Delta p$
excess pressure on the concave side
Under a convex surface the pressure exceeds the outside one by $\sigma$ times the curvature. In a water drop it is $2\sigma/R$, in a soap bubble twice that for its two surfaces, and under a concave meniscus the pressure is below atmospheric. A small bubble pushes harder than a large one, so connected bubbles pump air into the larger.
$\displaystyle h = \frac{2\sigma}{\rho g d}\ \text{(между двумя пластинами на расстоянии }d\text{, полное смачивание)}$
r
radius of the capillary
$\theta$
contact angle
$\sigma$
surface tension
$\rho$
density of the liquid
A wetting liquid rises in a thin tube until the column's weight $\rho g h\pi r^2$ balances the tension force around the rim $2\pi r\sigma\cos\theta$, which is also the Laplace pressure under the concave meniscus. A non-wetting liquid is depressed by the same height. Between plates the meniscus radius is twice as large and the rise half.