※ 引述《Lanjaja ()》之铭言： : 【出处】(习题或问题的出处) : 看书和上网查有的问题。 : 【题目】(题目的文字叙述，如有图片亦可提供图片) : 一般认知的incompressible定义是div(V) = 0 : 可是我在书上和wiki都有提到一种定义是(1/ρ)@ρ/@P ~ 0， : 只是提及，未有证明两种定义是否有何关系，如何互推。 : 【瓶颈】(解题瓶颈或思考脉络，请尽量详述以利回答者知道要从何处讲解指导) : 由连续方程式 : @ρ/@t + div(ρV) = 0 : 得到Dρ/Dt + ρdiv(V) = 0 : D是物质微分 : 因为incompressible的条件 : 推到Dρ/Dt = 0 : 再得出@ρ/@t + (@ρ/@P)(DP/Dt) = 0 : 接下来就做不出来了... : 可以请强者帮忙救援一下吗？ : 感谢回答 First of all, we need to know in general, the compressibility itself is a quan tity that depends on the process. So basically, they are different concepts, d iv(v)=0 tells you the fluid is incompressible dynamically, (∂ρ/∂P)_T=0 tell s you the fluid is incomprehensible under isothermic process. If the fluid is flowing isothermically, then they are equivalent in this case, otherwise they may be different. From mass conservation, we know dρ/dt=0 is equivalent to div(v)=0. so the que stion is how to relate dρ/dt to (∂ρ/∂P). As ρ is a thermodynamic intensive quantity, for ordinary fluid, it only has 2 degrees of freedom to control, so if the fluid flows without changes of tempe rature, dρ/dt= (∂ρ/∂P)_T dP/dt, as we know the pressure of a volume elemen t will change as the fluid flows, dρ/dt=0 is equivalent to (∂ρ/∂P)_T=0. --

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