# In the weak forms of the fluid flow model, as the weight functions (w1, w2) are virtual variations of the velocity components(vx, vy ),respectively, which relation is satisfied by the weight functions?

In the weak forms of the fluid flow model, as the weight functions (w1, w2) are virtual variations of the velocity components(vx, vy ),respectively, which relation is satisfied by the weight functions?

(a) $\frac{\partial w1}{\partial x}+\frac{\partial w2}{\partial y}$=0

(b) $\frac{\partial w2}{\partial x}+\frac{\partial w1}{\partial y}$=0

(c) $\frac{\partial w1}{\partial x}-\frac{\partial w2}{\partial y}$=0

(d) $\frac{\partial w2}{\partial x}-\frac{\partial w1}{\partial y}$=0

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This key question is from Penalty in portion Flows of Viscous Incompressible Fluids of Finite Element Method

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The correct answer is (a) $\frac{\partial w1}{\partial x}+\frac{\partial w2}{\partial y}$=0

Easy explanation: In the weak forms of the fluid flow model, suppose that the velocity field (vx, vy ) is such that the continuity equation $\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}$=0 is satisfied identically. Then the weight functions (w1, w2) being (virtual) variations of the velocity components, also satisfy the continuity equation. Thus the relation possessed by the weight functions is $\frac{\partial w1}{\partial x}+\frac{\partial w2}{\partial y}=0$.

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