Laplace equation for irrotational flow

Irrotational flow occurs when the cross gradient of the velocity or shear is zero.

\nabla \times v=0

The individual parcels of a frictionless incompressible fluid initially at rest cannot be caused to rotate. This can be visualized by considering an infinitesimal small volume of fluid in the shape of a sphere. Surface forces act normal to the surface, As the fluid is frictionless no friction occurs so they act through the center of the sphere hence, no torque can be exerted on the sphere so angular momentum remains constant and no change in angular velocity occurs. Rotation of a fluid parcel is defined as the average angular velocity of two elements originally at right angles to each other. Points A and B have an x-velocity which differs by ∂u/∂y dy over the time interval Δt. Points A and B will have a difference in x-displacements equal to

\Delta x_{b}+\Delta x_{a}={\partial u \over \partial y}dy\Delta t

and the associated angle change of side AB is

-\Delta \theta ={(\Delta x_{b}+\Delta x_{a}) \over \ dy}={\partial u \over \partial y}\Delta t

{d\theta \over dt}=-{\partial u \over \partial y}

The angular velocity of the element, about the z axis in this case, is defined as the average angular velocity of sides AB and AC.

\omega _{z}={1 \over 2}\left({\partial v \over \partial x}-{\partial u \over \partial y}\right)

\omega _{x}={1 \over 2}\left({\partial w \over \partial y}-{\partial v \over \partial z}\right)

\omega _{y}={1 \over 2}\left({\partial u \over \partial z}-{\partial w \over \partial x}\right)

Now ∇×v=0 so,

{\partial v \over \partial x}={\partial u \over \partial y}

{\partial w \over \partial y}={\partial v \over \partial z}

{\partial u \over \partial z}={\partial w \over \partial x}

These restrictions on the velocity must hold at every point. Let us consider a function ∅ which satisfies the condition

vdx+vdy=-d\varnothing

The minus sign is arbitrary it is a convention that causes the value of ∅ to decrease in the direction of the velocity.This proves the existence of a function ∅ such that its negative derivative with respect to any direction is the velocity component in that direction.

v=-\nabla \varnothing

The assumption of a velocity potential is equivalent to the irrotational flow as

\nabla \times (-\nabla \varnothing )=0

Whenever a velocity potential function exists then it must be an irrotational flow so they are equivalent to each other. As we are considering ideal fluid so it must follow continuity equation

{\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z}=0

Yields,

{(\partial ^{2}\varnothing ) \over (\partial x^{2})}+{(\partial ^{2}\varnothing ) \over (\partial y^{2})}+{(\partial ^{2}\varnothing ) \over (\partial z^{2})}=0

In vector form it can be written as

\nabla ^{2}\varnothing =0

It is called as the laplace equation. Any function ∅ that satisfies the laplace equation is a possible irrotational flow case. As there are infinite number of solutions to the laplace equation each of which satisfies certain flow boundaries the main problem is the selection of the proper function for the particular flow case. As ∅ appears to the first power it is a linear equation, so the sum of two solutions is also a solution.

Importance of studying Irrotational flow;

Many real world problems contain large regions of Irrotational flow.
They can be studied analytically.
They show us importance of Boundary layers and viscous forces.
They provide us tools for studying concepts of lift and drag.

References

Young, Donald F.; Munson, Bruce R.; Okiishi, Theodore H.; Huebsch, Wade W. (2010). A Brief Introduction to Fluid Mechanics (5th ed.). ISBN 978-1118022689.

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