EngineeringFluid DynamicsUniversity

IBUndergraduate

Radial Pressure Distribution

Calculates the pressure profile of a fluid in a radial gap between two concentric cylinders with rotational flow.

Understand the formulaSee the free derivationOpen the full walkthrough

P - P_{κ R} = \frac{1}{2} ρ (\frac{Ω _{0} κ R}{1 - κ ^{2}})^{2} [(\frac{r}{κ R})^{2} - (\frac{κ R}{r})^{2} - 4 ln (\frac{r}{κ R})]

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

This equation models the spatial pressure variation in a fluid layer subjected to rotational motion within an annular space. It accounts for the effects of fluid density, angular velocity, and the radius ratio defined by the inner and outer cylinder constraints. The expression provides a closed-form solution for determining pressure differentials relative to a reference point within the system.

When to use: Use when analyzing steady, incompressible, laminar flow in the annular region between rotating concentric cylinders.

Why it matters: Crucial for designing journal bearings, seal clearances, and understanding torque transmission in rotating machinery.

Symbols

Variables

P - $P_{κ R}$ = Pressure Difference, $ρ$ = Fluid Density, $Ω_{0}$ = Angular Velocity, $κ$ = Radius Ratio, R = Outer Radius

P - P_{κ R}

Pressure Difference

ρ

Fluid Density

k g / m^{3}

Ω_{0}

Angular Velocity

rad/s

κ

Radius Ratio

dimensionless

R

Outer Radius

m

r

Radial Position

m

Walkthrough

Derivation

Derivation of Radial Pressure Distribution

This derivation determines the radial pressure profile in a fluid flow by integrating the radial momentum equation for a steady, incompressible, inviscid vortex flow.

Steady-state flow
Incompressible fluid (constant density)
Inviscid flow (no viscosity)
Axisymmetric flow (properties depend only on radius r)
Flow field defined by a specific velocity distribution

Radial Momentum Equation

For steady, axisymmetric, inviscid flow in polar coordinates, the radial component of the Navier-Stokes equation reduces to the balance between the pressure gradient and the centrifugal acceleration.

\frac{d P}{d r} = ρ \frac{v _{θ}^{2}}{r}

Note: This is the fundamental governing equation for pressure in a rotating fluid.

Velocity Profile Substitution

We substitute the specific tangential velocity profile $v_{θ} (r)$ into the radial momentum equation. This profile represents a combined vortex flow between two radii.

v_{θ} (r) = \frac{Ω _{0} κ R}{1 - κ ^{2}} (\frac{r}{κ R} - \frac{κ R}{r})

Note: Ensure the velocity units are consistent with the pressure units.

Integration

We integrate the pressure gradient from a reference radius $κ R$ (where pressure is $P_{κ R}$ ) to an arbitrary radius $r$ . This step calculates the pressure difference based on the work done by centrifugal forces.

\int_{P_{κ R}}^{P} d P = \int_{κ R}^{r} ρ \frac{v _{θ} ( r ) ^{2}}{r} d r

Note: The limits of integration must match the reference pressure point.

Final Algebraic Expansion

Expanding the squared velocity term and performing the integration yields the final expression for the radial pressure distribution.

P - P_{κ R} = \frac{1}{2} ρ (\frac{Ω _{0} κ R}{1 - κ ^{2}})^{2} [(\frac{r}{κ R})^{2} - (\frac{κ R}{r})^{2} - 4 ln (\frac{r}{κ R})]

Note: The logarithmic term arises from the integration of the $1/ r$ component of the velocity squared term.

Result

P - P_{κ R} = \frac{1}{2} ρ (\frac{Ω _{0} κ R}{1 - κ ^{2}})^{2} [(\frac{r}{κ R})^{2} - (\frac{κ R}{r})^{2} - 4 ln (\frac{r}{κ R})]

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 n co n t ain s t h e t a r g e t v a r iab l e^{'} r^{'} in s i d e b o t ha q u a d r a t i c t er m (r^{2}), a r ec i p r oc a l q u a d r a t i c t er m (1/ r^{2}), an d a l o g a r i t hmi c t er m (l n (r)) . T hi s t r an sce n d e n t a l co mbina t i o n c ann o t b eso l v e df o r^{'} r^{'} a l g e b r ai c a l l y u s in g s t an d a r d e l e m e n t a r y f u n c t i o n s .

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

Visual intuition

Graph

The pressure difference changes significantly with radial position, featuring terms that grow and shrink with distance from the center, plus a logarithmic component. For a student, this means the pressure distribution isn't a simple straight line but a curve with unique characteristics at different radii. The most important feature is how the pressure difference behaves near the center versus further out. This equation helps understand how pressure varies in rotating fluids.

Graph type: other

Why it behaves this way

Intuition

Imagine a fluid trapped in the gap between two concentric cylinders. The inner cylinder has a radius of αR and the outer one has a radius of R. As the cylinders rotate, the fluid is 'flung' outward by centrifugal effects, but is constrained by the walls. This creates a pressure gradient where the pressure increases as you move from the inner wall (αR) toward the outer regions, similar to how the air pressure in a centrifuge increases toward the outer edge.

P - P_{η R}

Gauge pressure relative to the inner cylinder surface.

The net 'squeeze' felt by the fluid at any point r compared to the pressure at the starting point of the inner wall.

ρ

Fluid Density

Heavier fluids (like oil vs air) have more mass to be slung outward, resulting in much higher pressure differences for the same rotation speed.

O 0

Angular velocity of the rotation.

This determines the strength of the centrifugal force; because it is squared, doubling the spin speed quadruples the pressure difference.

κ

Radius ratio (Inner Radius / Outer Radius).

A measure of how 'thin' or 'thick' the annular gap is. If κ is close to 1, the gap is very narrow; if κ is close to 0, the inner cylinder is just a thin wire.

r / η R

Dimensionless radial position.

The ratio of where you are currently measuring (r) to where the fluid starts (the inner radius). It tells the formula how far into the gap you have traveled.

Signs and relationships

P - P_{ηR}: This value is typically positive as you move outward (r > ηR) because centrifugal force pushes the fluid against the outer boundaries, building up pressure.
1 - κ²: This denominator term ensures that as the gap between cylinders disappears (κ approaches 1), the pressure required to move the fluid through that infinitely small space approaches infinity.

Free study cues

Insight

Canonical usage

This equation is used to calculate pressure differences in rotating fluid systems, where all variables are typically expressed in SI units for consistency.

Common confusion

Mixing units between the SI system (e.g., meters, Pascals) and older imperial systems (e.g., feet, psi) without proper conversion.

Dimension note

The radius ratio 'kappa' is a dimensionless quantity. Other variables represent physical measurements with units.

Unit systems

$d e l t a_{P}$ Pa - Pressure difference is the primary output, typically in Pascals (Pa) in SI.

rhokg/m^3 - Fluid density is a key input, measured in kilograms per cubic meter (kg/m^3).

$O m e g a_{0}$ rad/s - Angular velocity should be in radians per second (rad/s) for consistency with rotational motion.

kappadimensionless - The radius ratio is a dimensionless quantity.

$R$ m - The outer radius is a length, typically measured in meters (m) in SI.

$r$ m - The radial position is a length, typically measured in meters (m) in SI.

One free problem

Practice Problem

How does the pressure distribution in an annular gap change if the fluid density is increased while maintaining the same angular velocity and geometry?

Solve for: $d e l t a_{P}$

Hint: Examine the role of the density term (rho) as a multiplier in the pressure distribution formula.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Determining the pressure load distribution across the lubricating oil film within a high-speed rotating mechanical seal.

Study smarter

Tips

Ensure all units for length (r, R) are consistent before calculation.
Check that the radius ratio kappa is between 0 and 1.
Verify that the flow regime is laminar, as turbulent flow requires different empirical correlations.

Avoid these traps

Common Mistakes

Mixing up the inner and outer radii within the kappa parameter.
Neglecting to convert rotational speed from RPM to rad/s (Omega_0).
Confusing the reference pressure P_kappaR with the local pressure P.

Common questions

Frequently Asked Questions

This derivation determines the radial pressure profile in a fluid flow by integrating the radial momentum equation for a steady, incompressible, inviscid vortex flow.

Use when analyzing steady, incompressible, laminar flow in the annular region between rotating concentric cylinders.

Crucial for designing journal bearings, seal clearances, and understanding torque transmission in rotating machinery.

Mixing up the inner and outer radii within the kappa parameter. Neglecting to convert rotational speed from RPM to rad/s (Omega_0). Confusing the reference pressure P_kappaR with the local pressure P.