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Where formula_18 is the energy density of the system, with formula_10 being the pressure, and formula_20 being the four-velocity of the system.
Expanding out the sums and equations, we have, (using formula_21 as the material derivative)
Then, picking formula_23 to observe the behavior of the velocity itself, we see that the equations of motion become
Note that taking the non-relativistic limit, we have formula_25. This says that the of the system is dominated by the rest energy of the fluid in question.
In this limit, we have formula_26 and formula_27, and can see that we return the Euler Equation of formula_28.
In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:
We prove this by looking at formula_30 and then multiplying each side by formula_31. Upon doing this, and noting that formula_32, we have formula_33. Relabeling the indices formula_34 as formula_35 shows that the two completely cancel. This cancellation is the expected result of contracting a temporal tensor with a spatial tensor.
Where we have implicitly defined that formula_37.
Then, let's note the fact that formula_40 and formula_41. Note that the second identity follows from the first. Under these simplifications, we find that
We have two cancellations, and are thus left with
The Starling equation describes the net flow of fluid across a semipermeable membrane. It is named after Ernest Starling. It describes the balance between capillary pressure, interstitial pressure, and osmotic pressure. The classic Starling equation has in recent years been revised. The Starling principle of fluid exchange is key to understanding how plasma fluid (solvent) within the bloodstream (intravascular fluid) moves to the space outside the bloodstream (extravascular space).
Transendothelial fluid exchange occurs predominantly in the capillaries, and is a process of plasma ultrafiltration across a semi-permeable membrane. It is now appreciated that the ultrafilter is the glycocalyx of the plasma membrane of the endothelium, whose interpolymer spaces function as a system of small pores, radius circa 5 nm. Where the endothelial glycocalyx overlies an inter endothelial cell cleft, the plasma ultrafiltrate may pass to the interstitial space. Some continuous capillaries may feature fenestrations that provide an additional subglycocalyx pathway for solvent and small solutes. Discontinuous capillaries as found in sinusoidal tissues of bone marrow, liver and spleen have little or no filter function.
The rate at which fluid is filtered across vascular endothelium (transendothelial filtration) is determined by the sum of two outward forces, capillary pressure (formula_1) and interstitial protein osmotic pressure (formula_2), and two absorptive forces, plasma protein osmotic pressure (formula_3) and interstitial pressure (formula_4). The Starling equation describes these forces in mathematical terms. It is one of the Kedem–Katchalski equations which bring nonsteady state thermodynamics to the theory of osmotic pressure across membranes that are at least partly permeable to the solute responsible for the osmotic pressure difference. The second Kedem–Katchalsky equation explains the trans endothelial transport of solutes, formula_5.
The classic Starling equation reads as follows:
By convention, outward force is defined as positive, and inward force is defined as negative. If Jv is positive, solvent is leaving the capillary (filtration). If negative, solvent is entering the capillary (absorption).
Applying the classic Starling equation, it had long been taught that continuous capillaries filter out fluid in their arteriolar section and reabsorb most of it in their venular section, as shown by the diagram.
However, empirical evidence shows that, in most tissues, the flux of the intraluminal fluid of capillaries is continuous and, primarily, effluent. Efflux occurs along the whole length of a capillary. Fluid filtered to the space outside a capillary is mostly returned to the circulation via lymph nodes and the thoracic duct.
A mechanism for this phenomenon is the Michel-Weinbaum model, in honour of two scientists who, independently, described the filtration function of the glycocalyx. Briefly, the colloid osmotic pressure πi of the interstitial fluid has been found to have no effect on Jv and the colloid osmotic pressure difference that opposes filtration is now known to be π'p minus the subglycocalyx π, which is close to zero while there is adequate filtration to flush interstitial proteins out of the interendothelial cleft. Consequently, Jv is much less than previously calculated, and the unopposed diffusion of interstitial proteins to the subglycocalyx space if and when filtration falls wipes out the colloid osmotic pressure difference necessary for reabsorption of fluid to the capillary.
The revised Starling equation is compatible with the steady-state Starling principle:
Pressures are often measured in millimetres of mercury (mmHg), and the filtration coefficient in millilitres per minute per millimetre of mercury (ml·min−1·mmHg−1).
In some texts the product of hydraulic conductivity and surface area is called the filtration co-efficient Kfc.
Staverman's reflection coefficient, "σ", is a unitless constant that is specific to the permeability of a membrane to a given solute.
The Starling equation, written without "σ", describes the flow of a solvent across a membrane that is impermeable to the solutes contained within the solution.
"σn" corrects for the partial permeability of a semipermeable membrane to a solute "n".
Where "σ" is close to 1, the plasma membrane is less permeable to the denotated species (for example, larger molecules such as albumin and other plasma proteins), which may flow across the endothelial lining, from higher to lower concentrations, more slowly, while allowing water and smaller solutes through the glycocalyx filter to the extravascular space.
Following are typically quoted values for the variables in the classic Starling equation:
It is reasoned that some albumin escapes from the capillaries and enters the interstitial fluid where it would produce a flow of water equivalent to that produced by a hydrostatic pressure of +3 mmHg. Thus, the difference in protein concentration would produce a flow of fluid into the vessel at the venous end equivalent to 28 − 3 = 25 mmHg of hydrostatic pressure. The total oncotic pressure present at the venous end could be considered as +25 mmHg.
In the beginning (arteriolar end) of a capillary, there is a net driving force (formula_28) outwards from the capillary of +9 mmHg. In the end (venular end), on the other hand, there is a net driving force of −8 mmHg.
Assuming that the net driving force declines linearly, then there is a mean net driving force outwards from the capillary as a whole, which also results in that more fluid exits a capillary than re-enters it. The lymphatic system drains this excess.
J. Rodney Levick argues in his textbook that the interstitial force is often underestimated, and measurements used to populate the revised Starling equation show the absorbing forces to be consistently less than capillary or venular pressures.
Glomerular capillaries have a continuous glycocalyx layer in health and the total transendothelial filtration rate of solvent (formula_7) to the renal tubules is normally around 125 ml/ min (about 180 litres/ day). Glomerular capillary formula_7 is more familiarly known as the glomerular filtration rate (GFR). In the rest of the body's capillaries, formula_7 is typically 5 ml/ min (around 8 litres/ day), and the fluid is returned to the circulation "via" afferent and efferent lymphatics.
The Starling equation can describe the movement of fluid from pulmonary capillaries to the alveolar air space.
The principles behind the equation are useful for explaining physiological phenomena in capillaries, such as the formation of edema.
Woodcock and Woodcock showed in 2012 that the revised Starling equation (steady-state Starling principle) provides scientific explanations for clinical observations concerning intravenous fluid therapy.
The Starling equation is named for the British physiologist Ernest Starling, who is also recognised for the Frank–Starling law of the heart. Starling can be credited with identifying that the "absorption of isotonic salt solutions (from the extravascular space) by the blood vessels is determined by this osmotic pressure of the serum proteins" in 1896.
In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.
The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting bathymetric changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.
A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972. Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.
In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.
For monochromatic waves according to linear theory—with the free surface elevation given as formula_1 and the waves propagating on a fluid layer of mean water depth formula_2—the mild-slope equation is:
The phase and group speed depend on the dispersion relation, and are derived from Airy wave theory as:
For a given angular frequency formula_7, the wavenumber formula_12 has to be solved from the dispersion equation, which relates these two quantities to the water depth formula_20.
the mild slope equation can be cast in the form of an inhomogeneous Helmholtz equation:
In spatially coherent fields of propagating waves, it is useful to split the complex amplitude formula_4 in its amplitude and phase, both real valued:
This transforms the mild-slope equation in the following set of equations (apart from locations for which formula_30 is singular):
The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy formula_32 is transported in the formula_33-direction normal to the wave crests (in this case of pure wave motion without mean currents). The effective group speed formula_40 is different from the group speed formula_41
The first equation states that the effective wavenumber formula_33 is irrotational, a direct consequence of the fact it is the derivative of the wave phase formula_43, a scalar field. The second equation is the eikonal equation. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with formula_44 the splitting into amplitude formula_45 and phase formula_43 leads to consistent-varying and meaningful fields of formula_45 and formula_33. Otherwise, "κ"2 can even become negative. When the diffraction effects are totally neglected, the effective wavenumber "κ" is equal to formula_12, and the geometric optics approximation for wave refraction can be used.
When formula_50 is used in the mild-slope equation, the result is, apart from a factor formula_51:
Now both the real part and the imaginary part of this equation have to be equal to zero:
The effective wavenumber vector formula_33 is "defined" as the gradient of the wave phase:
Note that formula_33 is an irrotational field, since the curl of the gradient is zero:
Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by formula_45:
The first equation directly leads to the eikonal equation above for formula_61, while the second gives:
which—by noting that formula_63 in which the angular frequency formula_7 is a constant for time-harmonic motion—leads to the wave-energy conservation equation.
The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. These assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using potential flow theory.
Luke's Lagrangian formulation gives a variational formulation for non-linear surface gravity waves.
For the case of a horizontally unbounded domain with a constant density formula_36, a free fluid surface at formula_66 and a fixed sea bed at formula_67 Luke's variational principle formula_68 uses the Lagrangian
where formula_70 is the horizontal Lagrangian density, given by:
where formula_72 is the velocity potential, with the flow velocity components being formula_73 formula_74 and formula_75 in the formula_76, formula_77 and formula_78 directions, respectively.
Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.
Taking the variations of formula_79 with respect to the potential formula_72 and surface elevation formula_81 leads to the Laplace equation for formula_82 in the fluid interior, as well as all the boundary conditions both on the free surface formula_66 as at the bed at formula_84
In case of linear wave theory, the vertical integral in the Lagrangian density formula_70 is split into a part from the bed formula_86 to the mean surface at formula_87 and a second part from formula_88 to the free surface formula_89. Using a Taylor series expansion for the second integral around the mean free-surface elevation formula_87 and only retaining quadratic terms in formula_82 and formula_92 the Lagrangian density formula_93 for linear wave motion becomes
The term formula_95 in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the Euler–Lagrange equations, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to formula_96 in the potential energy.
The waves propagate in the horizontal formula_6 plane, while the structure of the potential formula_82 is not wave-like in the vertical formula_78-direction. This suggests the use of the following assumption on the form of the potential formula_100
Here formula_104 is the velocity potential at the mean free-surface level formula_103 Next, the mild-slope assumption is made, in that the vertical shape function formula_106 changes slowly in the formula_6-plane, and horizontal derivatives of formula_106 can be neglected in the flow velocity. So:
The Euler–Lagrange equations for this Lagrangian density formula_93 are, with formula_114 representing either formula_115 or formula_116
Now formula_118 is first taken equal to formula_115 and then to formula_120
As a result, the evolution equations for the wave motion become:
with ∇ the horizontal gradient operator: ∇ ≡(∂/∂"x" ∂/∂"y")T where T denotes the transpose.
The next step is to choose the shape function formula_106 and to determine formula_123 and formula_124
Vertical shape function from Airy wave theory.
Since the objective is the description of waves over mildly sloping beds, the shape function formula_125 is chosen according to Airy wave theory. This is the linear theory of waves propagating in constant depth formula_126 The form of the shape function is:
with formula_128 now in general not a constant, but chosen to vary with formula_76 and formula_77 according to the local depth formula_2 and the linear dispersion relation:
Here formula_133 a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals formula_123 and formula_135 become:
The following time-dependent equations give the evolution of the free-surface elevation formula_81 and free-surface potential formula_138
From the two evolution equations, one of the variables formula_115 or formula_141 can be eliminated, to obtain the time-dependent form of the mild-slope equation:
and the corresponding equation for the free-surface potential is identical, with formula_141 replaced by formula_144 The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around formula_145
Consider monochromatic waves with complex amplitude formula_4 and angular frequency formula_147
with formula_7 and formula_133 chosen equal to each other, formula_151 Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:
Applicability and validity of the mild-slope equation.
The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3. However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.
The vorticity equation of fluid dynamics describes evolution of the vorticity of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity).
where is the material derivative operator, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.
The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid.
In the case of incompressible (i.e. low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation
where is the kinematic viscosity and is the Laplace operator.
Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to
Alternately, in case of incompressible, inviscid fluid with conservative body forces,
For a brief review of additional cases and simplifications, see also. For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.
The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains
Now, vorticity is defined as the curl of the flow velocity vector. Taking the curl of momentum equation yields the desired equation.
The following identities are useful in derivation of the equation:
The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol :
In the atmospheric sciences, the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is
Here, is the polar () component of the vorticity, is the atmospheric density, , , and w are the components of wind velocity, and is the 2-dimensional (i.e. horizontal-component-only) del.
The Vogel-Fulcher-Tammann equation, also known as Vogel-Fulcher-Tammann-Hesse equation or Vogel-Fulcher equation (abbreviated: VFT equation), is used to describe the viscosity of liquids as a function of temperature, and especially its strongly temperature dependent variation in the supercooled regime, upon approaching the glass transition. In this regime the viscosity of certain liquids can increase by up to 13 orders of magnitude within a relatively narrow temperature interval.
where formula_2 and formula_3 are empirical material-dependent parameters, and formula_4 is also an empirical fitting parameter, and typically lies about 50 °C below the glass transition temperature. These three parameters are normally used as adjustable parameters to fit the VFT equation to experimental data of specific systems.
The VFT equation is named after Hans Vogel, Gordon Scott Fulcher (1884–1971) and Gustav Tammann (1861–1938).
In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.
The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours.