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35. Conservation of mass Let v(t, x, y, z) be a continuously dif- ferentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts that P(t, x, y, z) dV pv •n do, dt where S is the surface enclosing D. a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t. b. Use the Divergence Theorem and Leibniz's Rule, "др p(, х, у, г) dV 3D dt дt *AP- to show that the Law of Conservation of Mass is equivalent to the continuity equation, др at V•pv + = 0. (In the first term V•pv, the variable t is held fixed, and in the second term ðp/ðt, it is assumed that the point (x, y, z) in D is held fixed.)

35. Conservation of mass Let v(t, x, y, z) be a continuously dif- ferentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts that P(t, x, y, z) dV pv •n do, dt where S is the surface enclosing D. a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t. b. Use the Divergence Theorem and Leibniz's Rule, "др p(, х, у, г) dV 3D dt дt *AP- to show that the Law of Conservation of Mass is equivalent to the continuity equation, др at V•pv + = 0. (In the first term V•pv, the variable t is held fixed, and in the second term ðp/ðt, it is assumed that the point (x, y, z) in D is held fixed.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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35. Conservation of mass Let v(t, x, y, z) be a continuously dif- ferentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts that P(t, x, y, z) dV pv •n do, dt where S is the surface enclosing D. a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t. b. Use the Divergence Theorem and Leibniz's Rule, "др p(, х, у, г) dV 3D dt дt *AP- to show that the Law of Conservation of Mass is equivalent to the continuity equation, др at V•pv + = 0. (In the first term V•pv, the variable t is held fixed, and in the second term ðp/ðt, it is assumed that the point (x, y, z) in D is held fixed.)
Transcribed Image Text:35. Conservation of mass Let v(t, x, y, z) be a continuously dif- ferentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts that P(t, x, y, z) dV pv •n do, dt where S is the surface enclosing D. a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t. b. Use the Divergence Theorem and Leibniz's Rule, "др p(, х, у, г) dV 3D dt дt *AP- to show that the Law of Conservation of Mass is equivalent to the continuity equation, др at V•pv + = 0. (In the first term V•pv, the variable t is held fixed, and in the second term ðp/ðt, it is assumed that the point (x, y, z) in D is held fixed.)
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