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Difference between revisions of "Continuity equation"

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One of the basic equations in hydrodynamics, expressing the law of conservation of mass for any volume of a moving fluid (or gas). In Euler variables the continuity equation has the form
 
One of the basic equations in hydrodynamics, expressing the law of conservation of mass for any volume of a moving fluid (or gas). In Euler variables the continuity equation has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255701.png" /></td> </tr></table>
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\[
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\frac{\partial \rho}{\partial t } + \text{div}(\rho \mathbf{v}) \equiv \frac{\partial \rho}{\partial t} + \frac{\partial (\rho v_x)}{\partial x} + \frac{\partial (\rho v_y)}{\partial y} + \frac{\partial (\rho v_z)}{\partial z} = 0,
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\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255702.png" /> is the density of the fluid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255703.png" /> is its velocity at a given point, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255704.png" /> are the projections of the velocity on the coordinate axes. If the fluid is incompressible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255705.png" />, then the continuity equation takes the form
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where $ \rho $ is the density of the fluid, $ \mathbf{v} $ is its velocity at a given point, and $ v_x, v_y, v_z $ are the projections of the velocity on the coordinate axes. If the fluid is incompressible $ (\rho = \text{const}) $, then the continuity equation takes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255706.png" /></td> </tr></table>
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\[
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\text{div } \mathbf{v} = 0 \quad \text{or} \quad \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0.
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\]
  
For a stationary one-dimensional flow in a tube, canal, etc., with cross-sectional area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255707.png" />, the continuity equation gives the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025570/c0255708.png" /> for the flow.
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For a stationary one-dimensional flow in a tube, canal, etc., with cross-sectional area $ S $, the continuity equation gives the law $\rho S\mathbf v=\text{const}$ for the flow.

Latest revision as of 12:28, 30 December 2018


One of the basic equations in hydrodynamics, expressing the law of conservation of mass for any volume of a moving fluid (or gas). In Euler variables the continuity equation has the form

\[ \frac{\partial \rho}{\partial t } + \text{div}(\rho \mathbf{v}) \equiv \frac{\partial \rho}{\partial t} + \frac{\partial (\rho v_x)}{\partial x} + \frac{\partial (\rho v_y)}{\partial y} + \frac{\partial (\rho v_z)}{\partial z} = 0, \]

where $ \rho $ is the density of the fluid, $ \mathbf{v} $ is its velocity at a given point, and $ v_x, v_y, v_z $ are the projections of the velocity on the coordinate axes. If the fluid is incompressible $ (\rho = \text{const}) $, then the continuity equation takes the form

\[ \text{div } \mathbf{v} = 0 \quad \text{or} \quad \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0. \]

For a stationary one-dimensional flow in a tube, canal, etc., with cross-sectional area $ S $, the continuity equation gives the law $\rho S\mathbf v=\text{const}$ for the flow.

How to Cite This Entry:
Continuity equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Continuity_equation&oldid=13303
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article