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Concept of Fluid Continuity

The principle of fluid continuity states that for an incompressible fluid, the volume flow rate must be constant throughout a given stream.
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The statement of the theorem

Consider a control volume V\mathcal{V} through which an incompressible fluid of constant density ρ0\rho_0 flows steadily. Let v(x,y,z)\textbf{v}(x, y, z) be the velocity field, and A(x)A(x) be the cross-sectional area perpendicular to the xx-axis. The principle of continuity, derived from the conservation of mass, mandates that the net volumetric flow rate QQ must be constant along the flow axis xx. Mathematically, this is expressed as:\nddx(A(x)v(x))=0\frac{\text{d}}{\text{d}x} \left( A(x) v(x) \right) = 0 \nIntegrating this differential equation yields the fundamental statement of continuity:\nA1v1=A2v2=⋯=QconstA_1 v_1 = A_2 v_2 = \dots = Q_{\text{const}} \nwhere AiA_i and viv_i are the cross-sectional area and the average fluid velocity, respectively, at any two arbitrary sections ii and jj along the conduit, provided the fluid remains incompressible (ρ=constant\rho = \text{constant}). This formulation is a direct consequence of the divergence-free condition for the velocity field: ∇⋅v=0\nabla \cdot \textbf{v} = 0.