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《游戏之神》主角:陆子昂
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《世界神豪》主角:林枫
《官路巅峰》主角:沈枫
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[桃里桃气]永和九年，岁在癸丑，暮春之初，会于会稽山阴之兰亭，修禊事也。群贤毕至，少长咸集。此地有崇山峻岭，茂林修竹，又有清流激湍，映带左右，引以为流觞曲水，列坐其次。虽无丝竹管弦之盛，一觞一咏，亦足以畅叙幽情。
是日也，天朗气清，惠风和畅。仰观宇宙之大，俯察品类之盛，所以游目骋怀，足以极视听之娱，信可乐也。
夫人之相与，俯仰一世。或取诸怀抱，悟言一室之内；或因寄所托，放浪形骸之外。虽趣舍万殊，静躁不同，当其欣于所遇，暂得于己，快然自足，不知老之将至；及其所之既倦，情随事迁，感慨系之矣。向之所欣，俯仰之间，已为陈迹，犹不能不以之兴怀，况修短随化，终期于尽！古人云：“死生亦大矣。”岂不痛哉！
每览昔人兴感之由，若合一契，未尝不临文嗟悼，不能喻之于怀。固知一死生为虚诞，齐彭殇为妄作。后之视今，亦犹今之视昔，悲夫！故列叙时人，录其所述，虽世殊事异，所以兴怀，其致一也。后之览者，亦将有感于斯文。
Physical Fluid Mechanics Derivation of Mass Conservation Equation
Hello, this is Bing. I can help you with the derivation of the mass conservation equation in physical fluid mechanics.
The mass conservation equation, also known as the continuity equation, states that the mass of a fluid element cannot change as it moves through a flow field. This implies that the rate of change of mass inside a control volume must be equal to the net mass flux across its boundary.
There are different ways to derive the mass conservation equation, depending on the coordinate system and the assumptions made about the fluid. Here, I will show you the derivation in Cartesian coordinates for a general compressible fluid.
First, we consider a small control volume with dimensions Δx, Δy, and Δz, as shown in the figure below. The fluid can enter or leave the control volume through its six faces.
The mass inside the control volume at any time t is given by:
$$m = \int_V \rho dV$$
where $\rho$ is the density of the fluid and $V$ is the volume of the control volume.
The rate of change of mass inside the control volume is then:
$$\frac{dm}{dt} = \frac{d}{dt} \int_V \rho dV$$
By applying the Reynolds transport theorem, we can write this as:
$$\frac{dm}{dt} = \int_V \frac{\partial \rho}{\partial t} dV + \int_A \rho \mathbf{v} \cdot \mathbf{n} dA$$
where $A$ is the surface area of the control volume, $\mathbf{v}$ is the velocity of the fluid, and $\mathbf{n}$ is the outward unit normal vector.
The second term on the right-hand side represents the net mass flux across the boundary of the control volume. We can split it into six terms, corresponding to the six faces of the control volume:
$$\int_A \rho \mathbf{v} \cdot \mathbf{n} dA = \int_{A_x} \rho u dA_x - \int_{A_{x+dx}} \rho u dA_{x+dx} + \int_{A_y} \rho v dA_y - \int_{A_{y+dy}} \rho v dA_{y+dy} + \int_{A_z} \rho w dA_z - \int_{A_{z+dz}} \rho w dA_{z+dz}$$
where $u$, $v$, and $w$ are the components of the velocity vector in the $x$, $y$, and $z$ directions, respectively.
Using the definition of the differential operator $\nabla$, we can write this as:
$$\int_A \rho \mathbf{v} \cdot \mathbf{n} dA = - \int_V \nabla \cdot (\rho \mathbf{v}) dV$$
Substituting this into the expression for the rate of change of mass, we get:
$$\frac{dm}{dt} = \int_V \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) \right) dV$$
Since this equation must hold for any arbitrary control volume, we can equate the integrand to zero, and obtain the differential form of the mass conservation equation:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$
This equation can also be written in component form as:
$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0$$
This is the general form of the mass conservation equation for a compressible fluid in Cartesian coordinates. For an incompressible fluid, the density is constant, and the equation simplifies to:
$$\nabla \cdot \mathbf{v} = 0$$
or
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$
I hope this helps you understand the derivation of the mass conservation equation. If you have any questions, please feel free to ask me.
Source: Conversation with Bing, 2024/1/4
(1) Derivation of the continuity equation (conservation of mass). ttps://www.tec-science.com/mechanics/gases-and-liquids/derivation-of-the-continuity-equation-conservation-of-mass/.
(2) 8.2: Mass Conservation - Engineering LibreTexts. htps://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_%28Bar-Meir%29/08%3A_Differential_Analysis/8.2%3A_Mass_Conservation.
(3) Law of Conservation of Mass - Definition, Formula, Examples, Problem. htps://byjus.com/physics/law-of-conservation-of-mass/.
(4) Conservation Laws in Fluid Mechanics - The Engineering Projects. htps://www.theengineeringprojects.com/2022/10/conservation-laws-in-fluid-mechanics.html.
(5) 3.1 Conservation of Mass - University of Auckland. htps://pkel015.connect.amazon.auckland.ac.nz/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_01_Conservation_of_Mass.pdf.
Hello, this is Bing. I can help you with the derivation of the mass conservation equation in physical fluid mechanics.
The mass conservation equation, also known as the continuity equation, states that the mass of a fluid element cannot change as it moves through a flow field. This implies that the rate of change of mass inside a control volume must be equal to the net mass flux across its boundary.
There are different ways to derive the mass conservation equation, depending on the coordinate system and the assumptions made about the fluid. Here, I will show you the derivation in Cartesian coordinates for a general compressible fluid.
First, we consider a small control volume with dimensions Δx, Δy, and Δz, as shown in the figure below. The fluid can enter or leave the control volume through its six faces.
The mass inside the control volume at any time t is given by:
$$m = \int_V \rho dV$$
where $\rho$ is the density of the fluid and $V$ is the volume of the control volume.
The rate of change of mass inside the control volume is then:
$$\frac{dm}{dt} = \frac{d}{dt} \int_V \rho dV$$
By applying the Reynolds transport theorem, we can write this as:
$$\frac{dm}{dt} = \int_V \frac{\partial \rho}{\partial t} dV + \int_A \rho \mathbf{v} \cdot \mathbf{n} dA$$
where $A$ is the surface area of the control volume, $\mathbf{v}$ is the velocity of the fluid, and $\mathbf{n}$ is the outward unit normal vector.
The second term on the right-hand side represents the net mass flux across the boundary of the control volume. We can split it into six terms, corresponding to the six faces of the control volume:
$$\int_A \rho \mathbf{v} \cdot \mathbf{n} dA = \int_{A_x} \rho u dA_x - \int_{A_{x+dx}} \rho u dA_{x+dx} + \int_{A_y} \rho v dA_y - \int_{A_{y+dy}} \rho v dA_{y+dy} + \int_{A_z} \rho w dA_z - \int_{A_{z+dz}} \rho w dA_{z+dz}$$
where $u$, $v$, and $w$ are the components of the velocity vector in the $x$, $y$, and $z$ directions, respectively.
Using the definition of the differential operator $\nabla$, we can write this as:
$$\int_A \rho \mathbf{v} \cdot \mathbf{n} dA = - \int_V \nabla \cdot (\rho \mathbf{v}) dV$$
Substituting this into the expression for the rate of change of mass, we get:
$$\frac{dm}{dt} = \int_V \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) \right) dV$$
Since this equation must hold for any arbitrary control volume, we can equate the integrand to zero, and obtain the differential form of the mass conservation equation:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$
This equation can also be written in component form as:
$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0$$
This is the general form of the mass conservation equation for a compressible fluid in Cartesian coordinates. For an incompressible fluid, the density is constant, and the equation simplifies to:
$$\nabla \cdot \mathbf{v} = 0$$
or
$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$
I hope this helps you understand the derivation of the mass conservation equation. If you have any questions, please feel free to ask me.
Source: Conversation with Bing, 2024/1/4
(1) Derivation of the continuity equation (conservation of mass). ttps://www.tec-science.com/mechanics/gases-and-liquids/derivation-of-the-continuity-equation-conservation-of-mass/.
(2) 8.2: Mass Conservation - Engineering LibreTexts. htps://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_%28Bar-Meir%29/08%3A_Differential_Analysis/8.2%3A_Mass_Conservation.
(3) Law of Conservation of Mass - Definition, Formula, Examples, Problem. htps://byjus.com/physics/law-of-conservation-of-mass/.
(4) Conservation Laws in Fluid Mechanics - The Engineering Projects. htps://www.theengineeringprojects.com/2022/10/conservation-laws-in-fluid-mechanics.html.
(5) 3.1 Conservation of Mass - University of Auckland. htps://pkel015.connect.amazon.auckland.ac.nz/SolidMechanicsBooks/Part_III/Chapter_3_Stress_Mass_Momentum/Stress_Balance_Principles_01_Conservation_of_Mass.pdf.
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