Fluid Shear Stress & Viscosity Explained | Fluid Mechanics

The Navier-Stokes equations are the fundamental governing equations for viscous fluid flow, derived by relaxing the inviscid assumption used in Euler's equations; they consist of three coupled partial differential equations (one for each spatial direction) that balance acceleration against pressure gradients, gravitational forces, and viscous diffusion terms, with the key difference being the inclusion of viscous terms (μ times the Laplacian of velocity) which make analytical solutions extremely difficult and typically require computational methods; common boundary conditions include the no-slip condition (fluid velocity equals solid surface velocity) and free surface conditions (continuity of tangential velocity and shear stress at fluid interfaces), while flow regimes range from laminar (ordered, smooth) to turbulent (chaotic, mixing) depending on Reynolds number and other factors.

The Navier-Stokes equations are the fundamental governing equations for fluid flow, derived by applying the linear momentum equation to a differential fluid element and assuming a Newtonian fluid relationship between velocity and shear stress; these equations describe everything from blood flow to wind patterns and pipeline flows, involving four unknowns (three velocity components and pressure) that require boundary conditions like no-slip at walls, though analytical solutions are limited to laminar flows while turbulent flows present significant modeling challenges due to nonlinearities and Reynolds stresses.

The Navier-Stokes equations are the fundamental equations governing the motion of viscous fluid substances. They combine the conservation of momentum (Euler's equations) with a constitutive equation for the stress tensor that accounts for viscosity. The equations are: ρ(∂v/∂t + v·∇v) = -∇p + μ∇²v + f, where ρ is density, v is velocity, p is pressure, μ is viscosity, and f represents body forces. These equations are nonlinear partial differential equations that describe how fluid velocity, pressure, and density evolve over time.

The Navier-Stokes equations are the fundamental equations of fluid dynamics for viscous fluids, extending Euler's equations by including viscous stress terms. For incompressible flow, the equations are: ρ(∂v/∂t + v·∇v) = -∇p + μ∇²v + ρg, where the left side represents inertial forces, the first term on the right is the pressure gradient force, the second term represents viscous forces, and the last term is the body force (gravity). These equations, combined with the continuity equation (∇·v = 0), form the complete set of governing equations for viscous fluid flow. They are partial differential equations of second order that must be solved to determine the velocity and pressure fields. The equations are named after Navier and Stokes, who developed them through an analogy between solid and fluid deformation.

The Navier-Stokes equations are partial differential equations named after Claude-Louis Navier and George Gabriel Stokes that describe fluid motion. A Newtonian incompressible fluid has constant viscosity and constant density. Unlike single objects, fluids require vector fields to describe velocity at each point. The equations are Newton's Second Law applied to fluids: ρ(dv/dt) = σ, where ρ is density, v is velocity, and σ is force per unit volume. Three primary forces act on fluids: gravity (F = mg), pressure force (-∇p), and viscous force (μ∇²v). The continuity equation ∇·v = 0 ensures mass conservation. For fluid flowing between parallel plates with constant pressure gradient and no-slip conditions, the velocity distribution is parabolic: v_x = (1/2μ)(dp/dx)(h² - y²), with maximum velocity at the center and zero velocity at the walls.

The Hagen-Poiseuille equation describes laminar flow in circular pipes, relating pressure drop to viscosity, flow rate, pipe length, and radius; the equation is ΔP = (128μLQ)/(πR⁴), where shear stress at the wall is τ_w = -4μ(∂u/∂r)|_r=R = -4μ(-U₀/R) = 4μU₀/R, showing that both pressure drop and wall shear stress increase with viscosity and flow velocity while decreasing with pipe radius.

In viscous (laminar) flow through circular pipes, the shear stress distribution is linear with radial distance (τ = - (dp/dx) × r/2), being zero at the center and maximum at the pipe wall, while the velocity distribution is parabolic (u = u_max × (1 - r²/R²)), with maximum velocity at the center and zero at the walls. The average velocity equals half the maximum velocity, and the pressure drop follows the Hagen-Poiseuille equation: ΔP = (32μ × u_avg × L) / D².

Internal viscous flows encompass three primary flow types: Couette flow (between fixed and moving parallel plates) with velocity profile U/U∞ = y/H + α(y/H)(1-y/H), Plane Poiseuille flow (between two fixed parallel plates) with U = (H²/2μ)(-dP/dX)(1-(y/H)²), and Hagen-Poiseuille flow (in circular pipes) with U = (R²/4μ)(-dP/dZ)(1-(r/R)²). The Reynolds number ratio between two points in a pipe with exponentially varying radius R = a·e^(-kx) is Re₁/Re₂ = e^(kΔx). For laminar flow in circular pipes, the pressure drop is ΔP = (128μQL)/(πD⁴), and the power required to maintain flow is P = ρQgH_L. The hydraulic diameter for non-circular ducts is D_h = 4A/P, enabling Reynolds number calculations for square and other cross-sections.

This segment covers pressure drop derivation and the Hagen-Poiseuille equation for head loss. The pressure drop between two pipe sections is derived from the average velocity equation. Rearranging V_avg = (1/8μ) * (dp/dx) * R² gives dp/dx = (8μ * V_avg) / R². Integrating from x = x₁ to x = x₂ gives P₂ - P₁ = (8μ * V_avg * L) / R², where L = x₂ - x₁ is the pipe length. Substituting R = D/2 gives P₂ - P₁ = (32μ * V_avg * L) / D². Head loss h_f is defined as pressure drop divided by specific weight: h_f = (P₁ - P₂) / (ρg). Substituting the pressure drop equation gives h_f = (32μ * V_avg * L) / (ρg * D²). This is the Hagen-Poiseuille equation for head loss in laminar pipe flow, named after Johann Hagen and Claude-Louis Poiseuille.

The Hagen-Poiseuille equation (Q = -πΔPR⁴/(8μL)) describes laminar viscous flow through circular pipes, derived by balancing pressure forces against viscous shear stresses and integrating the resulting differential equation with no-slip boundary conditions; this yields a parabolic velocity profile where maximum velocity occurs at the pipe centerline and minimum (zero) at the wall, with the ratio of maximum to average velocity being exactly 2.

The Reynolds number (Re = ρVL/μ) determines whether fluid flow is laminar (ordered, layered movement with minimal mixing, Re < 2300), transitional (2300 < Re < 4000), or turbulent (chaotic, rapid mixing, Re > 4000); this dimensionless parameter represents the ratio of inertial forces to viscous forces in a fluid, with higher values indicating greater turbulence due to increased velocity, larger characteristic length, or reduced viscosity.

The Reynolds number (Re) is a dimensionless quantity that classifies fluid flow into laminar or turbulent regimes by comparing inertial forces to viscous forces; it is calculated as Re = (density × velocity × characteristic length) / viscosity, where values below approximately 10³ indicate laminar flow with smooth, predictable streamlines dominated by viscous forces, while values above approximately 10⁴ indicate turbulent flow characterized by chaotic velocity fluctuations, swirling eddies, and unpredictable mixing dominated by inertial forces.

The Reynolds number (Re) determines whether fluid flow is laminar or turbulent by comparing inertial forces to viscous forces; flow is laminar when Re ≤ 2300, turbulent when Re ≥ 4000, and transitional between 2300-4000, with laminar flow characterized by smooth, parallel layers and turbulent flow by chaotic, fluctuating motion.

The Reynolds number Re = ρvL/η predicts flow regime: Re << 1 (laminar), Re >> 1 (turbulent). When Re << 1, viscous forces dominate and flow is laminar. When Re >> 1, inertial forces dominate and flow becomes turbulent. The transition typically occurs around Re ≈ 2000-4000 for pipe flow. This dimensionless number compares inertial to viscous forces and determines whether flow remains orderly or becomes chaotic.

Laminar flow is characterized by smooth, layered fluid motion with minimal mixing, while turbulent flow exhibits chaotic motion with swirling eddies and significant mixing; the Reynolds number (Re = ρUL/μ) predicts which regime occurs based on the relative importance of inertial versus viscous forces, with lower Re indicating laminar flow and higher Re indicating turbulent flow, and this distinction significantly affects pressure drop in pipes (proportional to velocity for laminar, proportional to velocity squared for turbulent) and requires different analytical approaches.

Non-Newtonian fluids, which do not obey Newton's law of viscosity, can be classified into pseudoplastic fluids (viscosity decreases with increasing shear rate, e.g., polymeric melts), dilatant fluids (viscosity increases with shear rate, e.g., jam), and Bingham plastic fluids (require initial force to flow, e.g., toothpaste). Three primary mathematical models describe their behavior: the Power Law Model (η = m(dvx/dy)^(n-1)) where n=1 gives Newtonian behavior, n<1 gives pseudoplastic, and n>1 gives dilatant behavior; the Iring Model (sinh(τ_xy/a) = -1/b(dvx/dy)) which shows Newtonian behavior at low shear rates and pseudoplastic behavior at high shear rates; and the Carreau Model (dvx/dy = -π₀ - φ₁|τ_xy|^(α-1)τ_xy) which exhibits Newtonian behavior at both low and high shear stresses but non-Newtonian behavior at intermediate values.

Non-Newtonian fluids do not follow Newton's law of viscosity, exhibiting three main types: pseudoplastic (viscosity decreases with applied shear stress, e.g., shampoo), plastic (requires a yield stress before flowing, e.g., toothpaste), and dilatant (viscosity increases with applied shear stress, e.g., cornstarch-water mixture). In contrast, Newtonian fluids like water maintain constant viscosity regardless of applied force.

Non-Newtonian fluids exhibit variable viscosity requiring specialized models. The Power Law model (τ = K(du/dy)^n) uses consistency index K and flow behavior index n: n=1 gives Newtonian behavior, n<1 indicates pseudoplastic (shear-thinning) fluids like ketchup, n>1 indicates dilatant (shear-thickening) fluids like cornstarch. The Bingham Plastic model describes fluids with yield stress (τy) that behave as solids until τy is exceeded, then flow like Newtonian fluids. Advanced models include the Cross model (capturing shear-thinning approaching limiting viscosity) and Carreau model (describing behavior across all shear rates). Time-dependent behaviors include thixotropy (viscosity decreases over time under shear) and rheopexy (viscosity increases over time). Viscoelastic fluids exhibit both viscous and elastic properties, recovering partially when stress is removed.

The power law model τ = K × γ^n captures a wide range of non-Newtonian behaviors using two parameters: consistency index K and flow behavior index n. When n = 1, the fluid behaves as Newtonian. When n < 1, the fluid is pseudoplastic (shear-thinning). When n > 1, the fluid is dilatant (shear-thickening). Thixotropic fluids decrease in viscosity when subjected to agitation or shear stress, and this decrease is time-dependent—when stress is removed, recovery requires time. Rheopectic fluids increase in viscosity under sustained shear stress over time, opposite to thixotropic behavior. These time-dependent behaviors are captured by plotting stress versus strain rate and observing hysteresis loops where unloading paths differ from loading paths.

This section provides detailed mathematical models for non-Newtonian fluids. For dilatant fluids (n > 1), τ = μ₀(du/dy)^n, showing increasing apparent viscosity with shear rate (e.g., cornstarch solution, quicksand). For pseudoplastic fluids (n < 1), τ = μ₀(du/dy)^n, showing decreasing apparent viscosity (e.g., ketchup, quicksand). For Bingham plastic fluids, τ = τ₀ + μ(du/dy), requiring a yield stress before flow (e.g., toothpaste, mayonnaise). Thixotropic fluids decrease viscosity over time under constant shear stress (e.g., lubricants, printer ink). Rheopectic fluids increase viscosity over time under constant shear stress. These mathematical models and examples are essential for analyzing non-Newtonian fluid behavior in engineering applications.