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The Magnus effect is the transverse force produced when a spinning body moves relative to a fluid. Rotation changes the flow and circulation around the body; the resulting asymmetric pressure and wake deflect the fluid, with the opposite transverse force on the body. Examples include the curved flight of spinning balls and the lift of rotating-cylinder sails.

A useful quantitative model is steady two-dimensional incompressible flow past a cylinder of radius $a$, with fluid velocity $\mathbf{U}=U\hat{\mathbf{x}}$ far away. Let circulation $\Gamma$ be positive anticlockwise as viewed from $+z$. The potential-flow velocity at the cylinder surface isThe Bernoulli equation outside the boundary layer givesIntegrating the pressure force around the cylinder, per unit span, yieldsThus the Kutta–Joukowski theorem givesThe sign here refers to the fluid velocity relative to the cylinder. For a body moving through otherwise stationary fluid, that relative fluid velocity points opposite to the body's translational velocity.

Real viscosity and the no-slip boundary condition transmit the surface rotation to the fluid and help set the circulation and separation pattern. Pure inviscid flow with only an impermeability condition does not determine $\Gamma$ from the cylinder's angular speed. Therefore the circulation must not be assigned a universal spin-rate formula without additional assumptions. The cylinder result illustrates the lift mechanism; quantitative forces on a finite spinning ball also depend on its geometry, Reynolds number, rotation rate, surface roughness and wake.

Take the curl of the Euler equations for an inviscid fluid. The gradient on the right has zero curl. Use the stated vector identity with $\mathbf{F}=\mathbf{\omega }$, $\mathbf{G}=\mathbf{u}$:incompressible flow gives $\nabla \cdot \mathbf{u}=0$, while a curl always has zero divergence, so $\nabla \cdot \mathbf{\omega }=0$. The remaining terms giveFor a two-dimensional velocity independent of $z$, $\mathbf{\omega }=(0,0,v_x-u_y)$ and $(\mathbf{\omega }\cdot \nabla )\mathbf{u}={\omega }_z{\partial }_z\mathbf{u}=0$. Thus the scalar vorticity is conserved following each fluid particle. It can still vary from particle to particle; this is advection, not a claim of spatially uniform vorticity.

The Reynolds numbercompares advective inertia with viscous force on a specified length scale $L$ and velocity scale $U$. Here $\rho$ is fluid mass density, $\eta$ is dynamic viscosity and $\nu =\eta /\rho$ is kinematic viscosity. In the Navier-Stokes equation, advective acceleration scales as $U^2/L$, while the viscous term per unit mass scales as $\nu U/L^2$; their ratio is Reynolds number.

When $\mathrm{Re}\ll 1$ and unsteady fluid inertia is also negligible, the incompressible flow equations reduce to Stokes flow:Thus pressure, applied body forces and viscous stresses balance at each instant. For a translating rigid sphere, drag is proportional to velocity, $F_D=6\pi \eta aU$, rather than the inertial scale $\rho a^2{U}^2$. Given the same instantaneous geometry, reversing the imposed boundary velocities reverses the Stokes flow; there is no fluid-inertial coasting in this approximation. This statement concerns the fluid limit and does not discard a separately significant particle inertia.

A low steady Reynolds number alone does not justify dropping rapid unsteady acceleration. For forcing frequency $\omega$, the additional dimensionless ratio is $\rho \omega L^2/\eta$. No organism-specific sizes, velocities or propulsion mechanism are assumed here.

For a constant-mass density incompressible flow with no imposed body force, the two-dimensional Navier-Stokes equation isA conservative body force, if retained, may be incorporated into a modified pressure. Using scales $L,U_0,L/U_0$ and $\rho U_0^2$ givesHere $\nu =\mu /\rho$ is the molecular kinematic viscosity. At high Reynolds number, inertia and the pressure gradient dominate on the outer length scale, while molecular viscous diffusion is small there. Thin boundary layers, narrow shear flows and fine turbulent scales can still have sufficiently large velocity gradients for viscous effects to matter. High Reynolds number permits instability and turbulence, but does not alone prove that every flow is turbulent. Molecular viscous dissipation remains essential at the small end of the turbulent cascade.

For an isotropic Newtonian fluid in incompressible flow, the constitutive equation isHere $p$ is pressure, ${\delta }_{ij}$ is the Kronecker delta, $\mu$ is the constant dynamic viscosity, and $E$ is the strain-rate tensor. The more general compressible constitutive equation can include a term proportional to $(\nabla \cdot \mathbf{u}){\delta }_{ij}$; that term vanishes here.

Differentiate the Cauchy stress tensor, retaining the constant dynamic viscosity outside the derivatives:The last equality uses incompressibility. Substitution into the Cauchy momentum equation gives the Navier-Stokes equationThe pressure gradient and the Laplacian of the velocity field therefore give the two contact force densities in this model.

Now both plates at $z=0$ and $z=H$ are stationary and both obey the Navier slip boundary condition with the same slip length $b$. Drive steady flow of a Newtonian fluid in incompressible flow by the constant pressure gradient $G=-dp/dx>0$, with constant dynamic viscosity $\mu >0$ and no other streamwise body force. The planar Poiseuille flow profile and its volumetric flow rate per unit width areThe full volumetric flow rate for channel width $W$ is $Q=Wq$ when side-wall effects are negligible. Thus $q$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, while $Q$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-1}$. Denote the value at $b=0$ by $q_0=G{H}^3/(12\mu )$.

The two wall checks expose the normal signs:Both Navier slip boundary conditions therefore hold exactly. The two walls exert backward tangential tractions of magnitude $GH/2$ on the fluid. Increasing $b$ at fixed $G$ adds a uniform velocity offset, leaving the velocity gradient and wall shear stress unchanged. This differs from Couette flow with a slipping lower wall, where upper-wall velocity is fixed and shear stress decreases.

For $H=10\mu \mathrm{m}$, $b=2\mu \mathrm{m}$, $G=10^3\mathrm{Pa}{\mathrm{m}}^{-1}$ and $\mu =10^{-3}\mathrm{Pa}\mathrm{s}$, the wall velocity is $10\mu \mathrm{m}{\mathrm{s}}^{-1}$, the centre velocity is $22.5\mu \mathrm{m}{\mathrm{s}}^{-1}$, and $q=1.83\times 10^{-10}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The enhancement is $q/q_0=2.2$. Equivalently, a prescribed volumetric flow rate needs only $1/2.2$ of the original pressure gradient. This makes slip length useful for estimating pumping requirements in small channels.

At fixed $H$, $G>0$ and $\mu$, $b\to 0$ recovers ordinary planar Poiseuille flow. As $b\to \infty$, the velocity and $q$ grow without bound in this model, although the varying part of the profile stays the same. At exactly zero tangential wall traction, no steady solution can balance this nonzero pressure gradient: $u^{\prime }(0)=u^{\prime }(H)=0$ conflicts with $\mu [u^{\prime }(H)-u^{\prime }(0)]=-GH$. Unlimited enhancement is consequently not a prediction for a real channel; finite wall friction, finite driving capacity and the validity of the assumed steady flow must be assessed. At fixed finite $q$ instead, $G\to 0$ and the profile approaches uniform velocity $q/H$.

A short derivation integrates the reduced Navier-Stokes equation $\mu u^{\prime \prime }=-G$ twice. Equal slip lengths and the opposite inward unit normals give $u=G[z(H-z)+bH]/(2\mu )$. Its integral over the gap supplies the enhancement factor $1+6b/H$.

Porous-media flow is fluid motion through the interconnected voids of a porous medium, such as a sand bed, a rock or a filter. Individual passages can be tiny and tortuous, yet their combined discharge can be predicted without resolving every passage. Darcy law supplies this bulk description when viscous resistance dominates pore-scale inertia; porosity measures how much space stores fluid, while permeability of a porous medium measures how readily that space transmits it.

The basic fully saturated model assumes one Newtonian fluid fills the connected pore space of a stationary solid framework, with incompressible flow, a small pore-scale Reynolds number, and averaging volumes large compared with individual pores but small compared with macroscopic variations. These are assumptions of the model developed below, rather than a claim that Darcy law describes every porous-media flow. Two-phase porous-media flow and poroelastic aquifers require additional constitutive and storage information.

For a first calculation, see Darcy flow through a homogeneous slab: a pressure drop determines the discharge, but porosity determines the corresponding pore velocity. Layered porous permeability then shows how geometry alone can make the bulk response anisotropic.

For uniform-density incompressible flow with $u_{\varphi }=f(R,z+vt)+g(R,z-vt)$, $B_{\varphi }/\sqrt{{\mu }_0\rho }=f-g$ and $B_z/\sqrt{{\mu }_0\rho }=v(R)$, the azimuthal magnetohydrodynamic momentum equation and ideal magnetohydrodynamic induction equation hold automatically. Let $\Pi =p/\rho +\Phi +[(f-g{)}^2+v^2]/2$. The remaining equations require ${\Pi }_z=0$ and ${\Pi }_R=4fg/R$, so ${\partial }_z(fg)=0$ is necessary and locally sufficient. A single traveling profile is unrestricted; two nonzero profiles on an unrestricted axial domain must be opposite exponentials or independent of their traveling arguments. Bounded profiles of both signs therefore reduce to steady solutions.

In an unbounded incompressible flow with spherical symmetry, the kinematic boundary condition on a sphere of radius $R(t)$ gives $u_r=R^2\dot{R}/r^2$ and velocity potential $\varphi =G(t)-R^2\dot{R}/r$. The Unsteady Bernoulli equation at infinity has $F(t)=\dot{G}+p_{\infty }/\rho$. Subtracting it from the finite-radius equation gives the displayed pressure, independently of $G$. Time differentiation is at fixed $r$. Evaluating at $r=R(t)$ yields the Rayleigh equation for an inviscid spherical bubble when the boundary pressure is prescribed.

Mechanical work transferred by a pressure acting through a displaced volume. In steady incompressible flow, the Bernoulli equation contains $p/\rho$ as flow work per unit mass; it is not an independently conserved internal energy of an isolated parcel.

For spherical expansion in an unbounded incompressible flow, continuity gives $u_r=R^2\dot{R}/r^2$. Integrating the kinetic energy outside the bubble gives $K=2\pi \rho R^3{\dot{R}}^2$. The Rayleigh equation for an inviscid spherical bubble then yields $\dot{K}=(p_b-p_{\infty })\dot{V}$, where $V=4\pi R^3/3$. With $p_b=p_{\infty }{V}_0/V$ and constant $p_{\infty }$, integration gives $K-K_0=p_{\infty }[V_0\log (V/V_0)-V+V_0]$. This neglects viscosity, surface tension and external body-force differences, which contribute additional terms in other bubble models.

Principal strain rates are the eigenvalues $a,b,c$ of the symmetric rate-of-strain tensor. Positive values are local extension and negative values local compression. In incompressible flow $a+b+c=0$, implying $a^3+b^3+c^3=3abc$. Their eigenvectors describe the principal deformation directions.

This convention defines the gyrotactic reorientation time by an aligning rate $1/(2B)$. Multiplying the torque balance by $2B$ and perturbing about the upward direction gives ${\mathbf{p}}_{\perp }=B\mathbf{\omega }\times {\mathbf{e}}_z$. For an incompressible flow, its horizontal divergence is $-B{\nabla }^{2}w$, because $\nabla \times \mathbf{\omega }=-{\nabla }^2\mathbf{u}$. The parameter $B$ is half the reorientation parameter used in the convention with aligning rate $1/B$; writing the governing equation prevents a factor-of-two ambiguity.

For liquid occupying $a(t)<r<b(t)$, radial symmetry and incompressible flow implyThe kinematic boundary condition at the outer interface giveswhich expresses conservation of the liquid's area.

For a solenoidal vector field $\mathbf{B}$ and constant magnetic diffusivity $\eta$, the magnetic induction equation isIt combines Faraday's law with the moving-conductor relation for electrical conductivity, neglecting displacement current. With incompressible flow, its advective form is ${\partial }_t\mathbf{B}+(\mathbf{u}\cdot \nabla )\mathbf{B}=(\mathbf{B}\cdot \nabla )\mathbf{u}+\eta {\nabla }^2\mathbf{B}$.

Using the positive polar molecular field $h_i=\delta F/\delta p_i$ and velocity gradient ${\partial }_i{v}_j$, the reversible order-parameter stress consists of a distortion contribution and rotational and alignment contributions:For a local free-energy density, a representative is ${\Sigma }_{ij}^{(1)}=(f-p_k{h}_k){\delta }_{ij}-f_{{\partial }_i{p}_k}{\partial }_j{p}_k$. The Euler-Lagrange equation verifies its divergence. Integration by parts gives $\delta F=\int {\Sigma }_{ij}^p{\partial }_i{u}_j$ for an incompressible displacement without surface work; the corresponding mechanical power has the opposite sign. Pressure contributions may be reassigned in an incompressible flow.

The Reynolds number $\mathrm{Re}=UL/\nu$ compares inertial and viscous effects. Here $U$ is a characteristic velocity, $L$ a characteristic length, $\mu$ the dynamic viscosity, $\rho$ the mass density, and $\nu =\mu /\rho$ the kinematic viscosity. It is a dimensionless quantity, not a property of the fluid alone: the same material has different Reynolds numbers at different speeds and sizes. Specify the chosen scales when reporting it.

The comparison comes from the Navier-Stokes equation. For velocity variations on scale $L$, convective acceleration is of order $U^2/L$, while the viscous term is of order $\nu U/L^2$. Their ratio is $UL/\nu$. Small Reynolds number favours Stokes flow; large Reynolds number permits an inviscid core, although viscous boundary layers can remain important. Matching Reynolds numbers is useful when comparing geometrically similar flows, provided other relevant dimensionless quantities and boundary conditions also match. ^[ref]

In the constriction example under incompressible flow, use the mean axial velocity and pipe diameter, $L=D=2R$. Taking $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$ and a stipulated dynamic viscosity $\mu =0.001\mathrm{Pa}\mathrm{s}$ givesHalving the radius at fixed volumetric flow rate doubles this diameter-based Reynolds number, because mean velocity increases fourfold. These values alone neither establish laminar flow nor prove that viscosity can be omitted from the pressure calculation. Increasing $\mu$ to $0.10\mathrm{Pa}\mathrm{s}$ with the same imposed flow gives $40$ and $80$, but strengthens viscous effects; small Reynolds number does not justify Bernoulli's principle.

The name recalls Osborne Reynolds's 1883 experimental investigation of water motion and resistance in pipes. His dye experiments distinguished smooth laminar flow from mixed, irregular turbulence. ^{[ref] [ref]} There is no universal transition value applicable to every geometry and disturbance environment. ^[ref]

For a uniformly rotating incompressible fluid of constant mass density, pressure and conservative body forces combine as the gradient of this scalar. The centrifugal term has a negative sign. Pressure enforces incompressible flow, whereas the velocity-dependent Coriolis force remains explicit and cannot be included in a scalar position potential. Adding a spatially constant function of time to $\psi$ does not change the motion.

Scalar transport follows a material quantity through advection and diffusion equation terms, for example ${\Theta }_t+\mathbf{U}\cdot \nabla \Theta =\kappa \Delta \Theta$. A passive scalar does not affect the velocity field; an active scalar feeds back through forces such as buoyancy. With incompressible flow bounded by impermeable walls and homogeneous Neumann boundary conditions, the spatial mean is conserved.

Scalar variance measures spatial departure from the conserved mean. For zero mean, incompressible flow bounded by impermeable walls and homogeneous scalar Neumann boundary conditions, integration by parts gives $d\int {\Theta }^2/dt=-2\kappa \int |\nabla \Theta {|}^2$. Advection preserves the instantaneous quadratic integral but can sharpen gradients, allowing the diffusion equation to remove scalar variance faster.