List of Equations

Susann Beier

List of Equations

Name	SI Unit	Equation	Number

Macrofluidics
Density	Kg/m³	$\rho = \frac{M}{V}$	2.1
Shear stress	N/m²	$\tau = \mu \frac{du}{dy}$	2.2
Shear force	N	$F = \tau A = \mu A \frac{du}{dy}$	2.3
Kinematic viscosity	m²/s	$v = \frac{\mu}{\rho}$	2.4
Bulk modulus	Pa	$k = \rho \frac{dp}{d\rho}$	2.5
Surface tension	N/m²	$\sigma_s = \frac{F}{L}$	2.6
Hydraulic diameter of a duct	m	$D_h = \frac{4a^2}{4a} = a$	2.8
Bernoulli equation	–	$P + \frac{1}{2}\rho v^2 + pgh = const.$	2.9

Microfluidics
Conservation of mass	–	$\frac{\delta p}{\delta t} + \nabla(pv) = 0$	3.1
Conservation of momen-tum	–	$\p\frac{\delta V}{\delta t} + p(V \cdot \nabla)V = -\nabla p + \eta \nabla ^2 V + pg$	3.2
Euler Equation	–	$\p\frac{\delta V}{\delta t} + p(V \cdot \nabla)V = -\nabla p + \eta \nabla ^2 V + pg$	3.3
Stokes flow	–	$\nabla p = \eta \nabla ^2 V$	3.4

Microfluidics
Casson model	$\sqrt{m Pa}$	$\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\upeta \dot{\gamma}}}$	4.1
Blood wave speed (1)	m/s	$c = \frac{r_i}{2 \beta p}$	4.2
Blood wave speed (2)	m/s	$c = \sqrt{\frac{Eh}{4pr_i}}$	4.3
Flow separation	–	$\frac{\delta ^2 u}{\delta y^2} = \frac{1}{\mu} \frac{\delta p}{\delta x}$	4.4
Navier stokes equation	–	$\frac{\delta(\rho u)}{\delta t} + \bigtriangledown (\rho u \overrightarrow{V} = - \frac{\delta p}{\delta x} + \frac{\delta \tau_{xx}}{\delta x} + \frac{\delta \tau_{yx}}{\delta y} + \frac{\delta \tau_{zx}}{\delta z} + \rho f_x$ $\frac{\delta(\rho v)}{\delta t} + \bigtriangledown (\rho v \overrightarrow{V} = - \frac{\delta p}{\delta y} + \frac{\delta \tau_{xy}}{\delta x} + \frac{\delta \tau_{yy}}{\delta y} + \frac{\delta \tau_{zy}}{\delta z} + \rho f_y$ $\frac{\delta(\rho w)}{\delta t} + \bigtriangledown (\rho w \overrightarrow{V} = - \frac{\delta p}{\delta z} + \frac{\delta \tau_{xz}}{\delta x} + \frac{\delta \tau_{yz}}{\delta y} + \frac{\delta \tau_{zz}}{\delta z} + \rho f_z$	7.1
Laser fre-quency	1/s	$\omega = \frac{2v sin. \frac{\theta}{2}}{\lambda}$	8.1

Dimensionless numbers
Reynolds number	–	$Re = \frac{v \cdot D}{v} = \frac{\rho \cdot v \cdot D}{\mu}$	2.7
Dean number	–	$De = \frac{\rho D v}{\mu} \sqrt{\frac{D}{2R_c}} = Re\sqrt{\frac{D}{2R_c}}$	3.5
Peclet number	–	$Pe = \frac{U \cdot L}{D}$	3.6
Womersley number	–	$\alpha = L(\frac{\omega \rho}{\mu})^{\frac{1}{2}}$	4.5
Strouhal number	–	$St = \frac{fd}{v}$	Table 7.1

Cavitation number	–	$Ca = \frac{p - p_v}{\frac{1}{2} \rho v^2}$	Table 7.1
Prandtl number	–	$Pr = \frac{\mu C_p}{k}$	Table 7.1
Weber number	–	$We = \frac{\rho v^2 L}{\sigma}$	Table 7.1
Capillary number	–	$C = \frac{\mu v}{\sigma}$	Table 7.1

List of Equations

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