Conservation of Mass D=Domain of a body of water Water is flowing in and out of D Mass is neither created nor destroyed Flow coming in = Flow going out Mass of domain remains balanced Therefore Fin - Fout = 0 F in F out D

Normal Velocity C = Boundary N=Normal Vector V=Velocity Vector Normal Velocity = N V =N*V*cos( ) Let =density Mass flow rate per unit area= *Normal Velocity F in F out N V D C

Green’s Thm States: *Integral of the normal velocity along C is equal to the double integral of the divergence of volume over D enclosed by C.

Recall: Density is constant for water Therefore the derivative of Density is equal to zero Recall: If and only if Therefore: *differential form of the law of conservation of mass, or the continuity equation.

Irrotational Flow In component form : To have irrotational flow the angular velocity must be zero. Angular velocity is caused by shear stresses “Pressure forces act normal to the surface of an element, and have no resulting moment, so cannot induce rotation of the element” No rotation= zero angular velocity This implies : Therefore we can represent flow by velocity potential

Velocity Potential Vx, Vy in terms of differentiating potential in x, y directions.Velocity = gradient of Substituting this into our continuity equation gives: Laplace’s Equation for the velocity potential.

Boundary Conditions R1 R2 Land Open Sea 1)On land 2)River Boundary 3)Open Sea