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Unsteady state Couette flow

This equation describes the time-dependent velocity distribution of a viscous fluid confined between two infinite parallel plates where one plate is suddenly set into motion.

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(\frac{\partial v _{x}}{\partial t})_{y} = ν (\frac{\partial ^{2} v _{x}}{\partial y ^{2}})_{t}

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Core idea

Overview

The equation is a specific application of the Navier-Stokes equations, simplifying to a diffusion-type partial differential equation for the velocity component parallel to the plates. It accounts for the momentum diffusion process driven by kinematic viscosity as the velocity profile develops over time from an initial state toward a steady-state linear profile. Understanding this evolution is critical for determining the transient behavior of fluid systems subject to sudden changes in boundary conditions.

When to use: Use this equation when analyzing the transient velocity profile of an incompressible Newtonian fluid between parallel boundaries immediately following a sudden start-up or change in plate velocity.

Why it matters: It models the fundamental mechanism of momentum transport via viscous diffusion, which governs how shear effects propagate through a fluid over time.

Walkthrough

Derivation

Derivation of Unsteady state Couette flow

This derivation shows how the Navier-Stokes equations simplify to the unsteady diffusion equation for velocity under the specific constraints of Couette flow.

Incompressible, Newtonian fluid
Flow is unidirectional ( $v_{x}$ = $v_{x}$ (y, t), $v_{y}$ = 0, $v_{z}$ = 0)
No pressure gradient in the direction of flow
Constant fluid properties (density and viscosity)
Negligible body forces

Start with the Navier-Stokes equation

We begin with the general momentum balance for a Newtonian fluid, where rho is density, v is the velocity vector, p is pressure, mu is dynamic viscosity, and f represents body forces.

ρ (\frac{\partial v}{\partial t} + (v \cdot \nabla) v) = - \nabla p + μ \nabla^{2} v + f

Note: This is the fundamental equation of motion for fluid mechanics.

Apply flow assumptions

We expand the vector equation into the x-component. Given the assumptions that $v_{y}$ = $v_{z}$ = 0 and the flow is fully developed (meaning velocity does not change in the x-direction, so partial $v_{x}$ / partial x = 0), the convective acceleration terms vanish.

ρ (\frac{\partial v _{x}}{\partial t} + v_{x} \frac{\partial v _{x}}{\partial x} + v_{y} \frac{\partial v _{x}}{\partial y} + v_{z} \frac{\partial v _{x}}{\partial z}) = - \frac{\partial p}{\partial x} + μ (\frac{\partial ^{2} v _{x}}{\partial x ^{2}} + \frac{\partial ^{2} v _{x}}{\partial y ^{2}} + \frac{\partial ^{2} v _{x}}{\partial z ^{2}})

Note: The continuity equation for an incompressible fluid (div v = 0) confirms that if $v_{y}$ = $v_{z}$ = 0, then $v_{x}$ cannot depend on x.

Simplify to the final form

With no pressure gradient (partial p / partial x = 0) and body forces neglected, we divide by density. Defining kinematic viscosity as nu = mu / rho, we arrive at the unsteady diffusion equation for velocity.

\frac{\partial v _{x}}{\partial t} = \frac{μ}{ρ} \frac{\partial ^{2} v _{x}}{\partial y ^{2}}

Note: This equation is mathematically identical to the heat conduction equation.

Result

\frac{\partial v _{x}}{\partial t} = \frac{μ}{ρ} \frac{\partial ^{2} v _{x}}{\partial y ^{2}}

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o ni s a p a r t ia l d i f f er e n t ia l e q u a t i o n (P D E) w h er e t h e v a r iab l es a r e f u n c t i o n s (v_{x}) an d t h e i r d er i v a t i v es . I t c ann o t b er e a r r an g e d t o i so l a t es p ec i f i c v a r iab l e in s t an ces l ik e a s t an d a r d a l g e b r ai ce q u a t i o n . R e a r r an g in g f or t h e k in e ma t i c v i scos i t y (ν) i s ma t h e ma t i c a l l y t r i v ia l b u tp r a c t i c a l l y i r r e l e v an tw i t h o u t f u l l so l u t i o n co n t e x t, an d i so l a t in g d er i v a t i v e t er m s i s n o n - s t an d a r df or t hi s f or m .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

Imagine a stack of thin playing cards representing fluid layers. Suddenly sliding the top card sideways creates a 'wave' of motion that slowly trickles down to the cards below. The equation captures this vertical 'leakage' of speed: the local change in velocity over time is driven by how curved the current velocity profile is. It transforms from a sharp jump at the boundary into a smooth, straight diagonal line as time approaches infinity.

\frac{\partial v _{x}}{\partial t}

Local acceleration of the fluid

How quickly the fluid at a specific height is 'speeding up' as it feels the drag from the moving plate above.

ν

Kinematic viscosity

The 'momentum diffusivity' or the 'greasiness' of the fluid. It dictates how fast the motion information spreads through the fluid bulk.

\frac{\partial ^{2} v _{x}}{\partial y ^{2}}

Curvature of the velocity profile

A measure of the difference in shear forces between the top and bottom of a fluid layer. If there is a 'bend' in the velocity profile, it means there is a net force acting to accelerate that layer.

Signs and relationships

ν > 0: Viscosity must be positive because it represents the physical resistance to flow; a negative viscosity would imply fluid spontaneously generating energy and accelerating on its own.
\frac{∂ v_x}{∂ t} ∝ \frac{∂^2 v_x}{∂ y^2}: The positive sign between these terms indicates a smoothing process. Fluid accelerates in directions that reduce sharp gradients, moving the system toward a linear, steady-state profile.

Free study cues

Insight

Canonical usage

This equation relates the rate of change of velocity with time to the spatial second derivative of velocity, scaled by kinematic viscosity, ensuring dimensional consistency.

Common confusion

Students may confuse the temporal derivative (∂/∂t) with spatial derivatives (∂/∂y) or misinterpret the units of kinematic viscosity.

Dimension note

The equation itself is not dimensionless, as it relates quantities with physical dimensions. However, dimensionless forms of fluid dynamics equations, such as those derived using similarity analysis, are common in

Unit systems

$v_{x}$ m/s - Velocity component in the x-direction.

$t$ s - Time.

$y$ m - Spatial coordinate perpendicular to the plates.

num^2/s - Kinematic viscosity of the fluid.

One free problem

Practice Problem

If the kinematic viscosity of a fluid increases, how does the time required for the flow to reach a steady-state Couette profile change?

parameterkinematic viscosity

Solve for: $(\frac{\partial v _{x}}{\partial t})_{y}$

Hint: Consider the relationship between viscosity and the diffusion rate of momentum.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

The sudden acceleration of a lubricant film between a piston and a cylinder wall in an internal combustion engine during the initial stroke.

Study smarter

Tips

Ensure the flow remains laminar throughout the transient phase.
Check that boundary conditions at t=0 and y=0/y=L are defined to permit a unique solution.
Recognize the mathematical similarity to the one-dimensional heat conduction equation.

Avoid these traps

Common Mistakes

Assuming the velocity profile is linear at all times during the transient phase.
Neglecting the impact of the kinematic viscosity on the time required to reach a steady state.

Common questions

Frequently Asked Questions

This derivation shows how the Navier-Stokes equations simplify to the unsteady diffusion equation for velocity under the specific constraints of Couette flow.

Use this equation when analyzing the transient velocity profile of an incompressible Newtonian fluid between parallel boundaries immediately following a sudden start-up or change in plate velocity.

It models the fundamental mechanism of momentum transport via viscous diffusion, which governs how shear effects propagate through a fluid over time.

Assuming the velocity profile is linear at all times during the transient phase. Neglecting the impact of the kinematic viscosity on the time required to reach a steady state.

The sudden acceleration of a lubricant film between a piston and a cylinder wall in an internal combustion engine during the initial stroke.

Ensure the flow remains laminar throughout the transient phase. Check that boundary conditions at t=0 and y=0/y=L are defined to permit a unique solution. Recognize the mathematical similarity to the one-dimensional heat conduction equation.

References

Sources

Bird, R. B., Stewart, W. E., & Lightfoot, E. N., Transport Phenomena, 2nd Edition, Wiley.
White, F. M., Viscous Fluid Flow, McGraw-Hill Education.
NIST CODATA
IUPAC Gold Book
White, Frank M. Fluid Mechanics. 8th ed., McGraw-Hill Education, 2016.
Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons.
White, Frank M. (2011). Fluid Mechanics (7th ed.). McGraw-Hill.
NPTEL (National Programme on Technology Enhanced Learning) - Fluid Mechanics Course