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Fluid Dynamics operators

42 operators in the fluid_dynamics category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
FL1Continuity equation: conservation of mass for a fluid expressed as density flux divergence.\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0
FL10Vorticity as the curl of the velocity field, measuring local fluid rotation.\vec{\omega} = \nabla \times \vec{v}
FL2Bernoulli's equation: conservation of energy along a streamline in an ideal fluid.P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2
FL3Navier-Stokes equations: momentum conservation for a viscous, incompressible Newtonian fluid.\rho\left(\frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}\right) = -\nabla P + \mu\nabla^2\vec{v} + \rho\vec{g}
FL4Reynolds number: dimensionless ratio of inertial to viscous forces predicting flow regime.Re = \frac{\rho v L}{\mu}
FL5Drag force on a body in a fluid: proportional to dynamic pressure, drag coefficient, and area.F_D = \frac{1}{2}\rho v^2 C_D A
FL6Lift force on an airfoil: proportional to dynamic pressure, lift coefficient, and wing area.F_L = \frac{1}{2}\rho v^2 C_L A
FL7Volumetric flow rate: cross-sectional area times mean flow velocity.Q = A v
FL8Hagen-Poiseuille law: pressure drop for laminar flow through a cylindrical pipe.\Delta P = \frac{8\mu L Q}{\pi r^4}
FL9Mach number: ratio of flow speed to local speed of sound in compressible flow.Ma = \frac{v}{c_s}
HYD1Water balance equation: precipitation equals evapotranspiration plus runoff plus storage change.P = ET + R + \Delta S
HYD10Penman-Monteith equation for reference evapotranspiration combining energy balance and aerodynamics.ET_0 = \frac{0.408\Delta(R_n-G) + \gamma\frac{900}{T+273}u_2(e_s-e_a)}{\Delta + \gamma(1+0.34u_2)}
HYD11Crop coefficient: ratio of crop evapotranspiration to reference evapotranspiration.K_c = \frac{ET_c}{ET_0}
HYD12Horton infiltration model: exponential approach from initial to final infiltration capacity.I = f_0 + (f_c - f_0)e^{-kt}
HYD13Richards equation: unsaturated flow through porous media combining Darcy's law and continuity.\frac{\partial\theta}{\partial t} = \frac{\partial}{\partial z}\left[K(\theta)\left(\frac{\partial\psi}{\partial z} + 1\right)\right]
HYD14Capillary pressure in soil: Young-Laplace equation applied to pore water meniscus.\psi = -\frac{2\sigma\cos\alpha}{\rho g r}
HYD15Snyder synthetic unit hydrograph peak discharge estimation.Q_p = \frac{2.08 A^{0.65}}{T_c^{0.31}}P_e^{0.87}
HYD16SCS curve number method for rainfall-runoff modeling from soil and land use data.S = C_n(P - I_a)^2 / (P - I_a + S)
HYD17Saint-Venant continuity equation for unsteady open channel flow.\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = q
HYD18Saint-Venant momentum equation for unsteady open channel flow with friction.\frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}\left(\frac{Q^2}{A}\right) + gA\frac{\partial y}{\partial x} = gA(S_0 - S_f)
HYD19Froude number: ratio of flow speed to shallow water wave speed for open channel flow regime.Fr = \frac{v}{\sqrt{gy}}
HYD2Rational method for peak runoff: flow equals runoff coefficient times intensity times area.Q = CIA
HYD20Specific energy in open channel flow: depth plus velocity head.E = y + \frac{v^2}{2g}
HYD3Manning equation for open channel flow velocity from roughness, hydraulic radius, and slope.Q = \frac{1}{n}AR^{2/3}S^{1/2}
HYD4Darcy's law: groundwater flow velocity proportional to hydraulic gradient and permeability.v = K\frac{dh}{dl}
HYD5Darcy's law for volumetric groundwater flow through a porous medium.Q = KA\frac{dh}{dl}
HYD6Specific storage of an aquifer relating pressure change to volume of water released.S = \rho g(\alpha + n\beta)
HYD7Transmissivity: product of hydraulic conductivity and aquifer thickness.T = Kb
HYD8Theis equation: drawdown in a confined aquifer from a pumping well using the well function.s = \frac{Q}{4\pi T}W(u)
HYD9Theis well function argument: dimensionless parameter combining radius, storativity, transmissivity, and time.u = \frac{r^2 S}{4Tt}
NS0Full Navier-Stokes momentum equation for viscous incompressible flow.∂v/∂t + (v·∇)v = -∇p + ν∇²v + f
OC1Shallow water wave speed: square root of gravity times water depth.c = \sqrt{gH}
RHE1Newtonian viscosity law: shear stress proportional to shear rate via dynamic viscosity.\tau = \eta \dot{\gamma}
RHE10First normal stress difference in non-Newtonian flow, causing the Weissenberg effect.N_1 = \tau_{11} - \tau_{22}
RHE2Power-law (Ostwald-de Waele) model for shear-thinning or shear-thickening fluids.\eta(\dot{\gamma}) = K\dot{\gamma}^{n-1}
RHE3Bingham plastic model: flow only occurs above a yield stress threshold.\tau = \tau_y + \eta_p\dot{\gamma}
RHE4Cross model for viscosity: smooth transition from zero-shear to infinite-shear viscosity.\eta = \eta_\infty + \frac{\eta_0 - \eta_\infty}{1 + (\lambda\dot{\gamma})^2}
RHE5Complex shear modulus: storage modulus plus imaginary loss modulus for viscoelastic materials.G^* = G\prime + iG\prime\prime
RHE6Oldroyd-B viscoelastic model with relaxation and retardation time constants.\tau + \lambda_1\frac{d\tau}{dt} = \eta\left(\dot{\gamma} + \lambda_2\frac{d\dot{\gamma}}{dt}\right)
RHE7Loss tangent: ratio of loss modulus to storage modulus measuring viscoelastic dissipation.\tan\delta = \frac{G\prime\prime}{G\prime}
RHE8Creep compliance function: instantaneous, retarded, and steady-state flow components.J(t) = J_0 + J_1(1-e^{-t/\tau_1}) + t/\eta_0
RHE9Stress relaxation modulus: exponential decay of stress under constant strain.G(t) = G_0 e^{-t/\tau}

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["FL1"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​