{"id":528,"date":"2022-10-21T16:46:00","date_gmt":"2022-10-21T23:46:00","guid":{"rendered":"https:\/\/dornsife.usc.edu\/scribe\/?p=528"},"modified":"2025-10-20T16:48:08","modified_gmt":"2025-10-20T23:48:08","slug":"gradients-and-potential-flow-part-1","status":"publish","type":"post","link":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/","title":{"rendered":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids"},"content":{"rendered":"\n\n\n\n\n \n \n\n\n\n\n\n\n\n
\n\n \n
\n \n \n
\n \n \n
\n\n \n

Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids<\/h1>\n\n\n<\/div>\n \n \n By<\/span>Oliver Khan<\/strong>\n \n October 21, 2022<\/span>\n <\/div>\n<\/div>\n\n\n <\/div><\/div>\n\n \n \n\n\n\n\n\n\n\n
\n\n \n
\n \n \n Share\n <\/span>\n \n \n <\/svg>\n <\/span>\n Copy Link<\/span>\n <\/a>\n \n \n <\/svg>\n <\/span>\n Facebook<\/span>\n <\/a>\n \n \n \n\n<\/svg>\n\n <\/span>\n X<\/span>\n <\/a>\n \n \n <\/svg>\n <\/span>\n LinkedIn<\/span>\n <\/a>\n \n \n \n \n <\/svg>\n <\/span>\n Bluesky<\/span>\n <\/a>\n \n \n <\/svg>\n <\/span>\n Email<\/span>\n <\/a>\n <\/span>\n <\/div>\n\n <\/div><\/div>\n \n\n\n\n \n \n\n\n\n\n\n\n\n
\n\n \n \n
\n\n \n

I would like to warmly welcome all of my readers back to this week\u2019s session. Last time, we briefly discussed the relationship between Laplace\u2019s equation (\u2206u = \u22022<\/sup>u\/\u2202x2<\/sup>\u00a0+ \u22022<\/sup>u\/\u2202y2<\/sup>\u00a0= 0) and fluid flows that are irrotational and have no sources or sinks. This week, I would like to develop this topic further and build up more of the background knowledge necessary to understand it. For the sake of being as general as possible, our analysis today will concern three-dimensional fluid flows.<\/p>\n

We will begin by considering a scalar function of three spatial variables f(x,y,z). We would like to have at our disposal as many tools as possible to understand the behavior of such a function. One tool we have already is the\u00a0partial derivative<\/em>. By computing \u2202f\/\u2202x, \u2202f\/\u2202y, or \u2202f\/\u2202z, we can see how our function changes as we change one variable but keep our other variables fixed. Partial derivatives by themselves can be a bit limiting though, because they do not encompass the whole way in which our function is changing.<\/p>\n

In order to get a better picture of how f(x,y,z) behaves, we can define the \u2207 (nabla) operator, which will act on multivariable functions. Specifically, we write the following, where\u00a0i<\/strong>,\u00a0j<\/strong>, and\u00a0k<\/strong>\u00a0are unit vectors in the x, y, and z directions, respectively: (Note again that the bold\u00a0i<\/strong>\u00a0is a vector, and is different from the unbolded i = sqrt(-1).)<\/p>\n

\u2207 = \u2202\/\u2202x\u00a0i<\/strong>\u00a0+ \u2202\/\u2202y\u00a0j<\/strong>\u00a0+ \u2202\/\u2202z\u00a0k<\/strong>.<\/p>\n

We can think of this \u2207 operator as a vector whose components consist of the partial derivative operators for each of our spatial variables.<\/p>\n

We will then define the\u00a0gradient\u00a0<\/em>of f to be:<\/p>\n

\u2207f = \u2202f\/\u2202x\u00a0i<\/strong>\u00a0+ \u2202f\/\u2202y\u00a0j +\u00a0<\/strong>\u2202f\/\u2202z\u00a0k<\/strong>.<\/p>\n

Since \u2207f is a vector, it has both direction and magnitude. The direction of the gradient of f is the direction we should walk in xyz-space in order for the value of f(x,y,z) to increase the fastest. Meanwhile, the magnitude of \u2207f represents the magnitude of this maximum rate of change of f at a point (x,y,z). It is important to remember that because f(x,y,z) is a multivariable function, it has infinitely many rates of change at a given point. The gradient of f, then, specifically gives the maximum rate of change of f(x,y,z) and the direction we should travel in our input space in order to achieve this maximum rate of change in our output space.<\/p>\n

We can now shift our attention to a vector field\u00a0F<\/strong>(x,y,z) = <P(x,y,z), Q(x,y,z), R(x,y,z)>. We can recall that to each point in space,\u00a0F<\/strong>(x,y,z) assigns a vector whose x-component is the function P(x,y,z), y-component is the function Q(x,y,z), and z-component is the function R(x,y,z). We can now apply the \u2207 operator on\u00a0F\u00a0<\/strong>in two different ways.<\/p>\n

The\u00a0divergence\u00a0<\/em>of\u00a0F<\/strong>\u00a0is defined as:<\/p>\n

div(F<\/strong>) = \u2207 \u2022\u00a0F<\/strong>\u00a0= <\u2202\/\u2202x, \u2202\/\u2202y, \u2202\/\u2202z> \u2022 <P, Q, R> =\u00a0 \u2202P\/\u2202x + \u2202Q\/\u2202y + \u2202R\/\u2202z.<\/p>\n

Here, we are taking what is called the\u00a0dot product<\/em>\u00a0of the \u2207 operator with\u00a0F<\/strong>, and we can think of div(F<\/strong>) as a scalar quantifying whether there are sources or sinks in our vector field\u00a0F<\/strong>. If we fix a point (x0<\/sub>, y0<\/sub>, z0<\/sub>) in space, then we have a source at (x0<\/sub>, y0<\/sub>, z0<\/sub>) if div(F<\/strong>) > 0 at that point. This means that at (x0<\/sub>, y0<\/sub>, z0<\/sub>), we have vectors popping out of nowhere and expanding outwards. Meanwhile, we have a sink at (x0<\/sub>, y0<\/sub>, z0<\/sub>) if div(F<\/strong>) < 0 at that point. If div(F<\/strong>) = 0 at (x0<\/sub>, y0<\/sub>, z0<\/sub>), then we can say that (x0<\/sub>, y0<\/sub>, z0<\/sub>) is neither a source nor a sink, and that there are no vectors popping out of nowhere or disappearing suddenly into nowhere.<\/p>\n

The\u00a0curl<\/em>\u00a0of\u00a0F<\/strong>\u00a0can be defined as:<\/p>\n

curl(F<\/strong>) = \u2207 x\u00a0F<\/strong>\u00a0 = (\u2202R\/\u2202y \u2013 \u2202Q\/\u2202z)\u00a0i<\/strong>\u00a0+ (\u2202P\/\u2202z \u2013 \u2202R\/\u2202x)\u00a0j\u00a0<\/strong>+ (\u2202Q\/\u2202x \u2013 \u2202P\/\u2202y)\u00a0k<\/strong>.<\/p>\n

What we are essentially doing here is taking what we call the\u00a0cross product<\/em>\u00a0of the \u2207 operator with\u00a0F<\/strong>. What is most important to understand is that curl(F<\/strong>) is a vector whose direction determines the axis of local rotation in the fluid flow. Therefore, when curl(F<\/strong>) =\u00a00<\/strong>\u00a0at a particular point (where\u00a00\u00a0<\/strong>\u00a0= <0,0,0> is the zero vector), we know that there is no local rotation in the vector field\u00a0F<\/strong>\u00a0at that point. Meanwhile, if curl(F<\/strong>) \u2260\u00a00<\/strong>\u00a0at a point (x0<\/sub>, y0<\/sub>, z0<\/sub>), then if we dropped a stick into our vector field\u00a0F<\/strong>\u00a0at (x0<\/sub>, y0<\/sub>, z0<\/sub>), our stick would start rotating.<\/p>\n

We will now apply the mathematical concepts of gradient, divergence, and curl to a vector field\u00a0V<\/strong>(x,y,z)= <u(x,y,z), v(x,y,z), w(x,y,z)> which characterizes the velocity of a fluid in three-dimensional space. We will assume that our flow is well-behaved, which for us means that it has a constant density, mass is conserved and we have no local rotation.<\/p>\n

If mass is conserved, then it wouldn\u2019t make sense for fluid to be appearing out of nowhere at one point or suddenly disappearing at another point. Therefore, saying that mass is conserved in a fluid flow (with constant density) is synonymous with saying that div(V<\/strong>) = 0.<\/p>\n

Meanwhile, saying that our flow is irrotational is the same as saying that it has zero\u00a0vorticity<\/em>, or in other words, curl(V<\/strong>) =\u00a00<\/strong>.<\/p>\n

We will now introduce the concept of a\u00a0velocity potential<\/em>. We will say that some scalar function \u03c6(x,y,z) is a\u00a0potential function<\/em>\u00a0for our vector field\u00a0V<\/strong>(x,y,z) if the following is true:<\/p>\n

V\u00a0<\/strong>= \u2207\u03c6.<\/p>\n

In particular, we have:<\/p>\n

V =\u00a0<\/strong><u(x,y,z), v(x,y,z), w(x,y,z)> = <\u2202\u03c6\/\u2202x, \u2202\u03c6\/\u2202y, \u2202\u03c6\/\u2202z> = \u2207\u03c6.<\/p>\n

This idea of writing\u00a0V\u00a0<\/strong>as the gradient of \u03c6 is important because as long as \u03c6 satisfies Laplace\u2019s equation, \u03c6(x,y,z) will automatically encode both the conservation of mass and irrotationality of our fluid flow.<\/p>\n

We can first observe how if \u2206\u03c6 = 0, then mass is conserved in our flow:<\/p>\n

div(V<\/strong>) = \u2207 \u2022 \u2207\u03c6 = \u2202\/\u2202x(\u2202\u03c6\/\u2202x) + \u2202\/\u2202y(\u2202\u03c6\/\u2202y) + \u2202\/\u2202z(\u2202\u03c6\/\u2202z) = \u22022<\/sup>\u03c6\/\u2202x2<\/sup>\u00a0+ \u22022<\/sup>\u03c6\/\u2202y2<\/sup>\u00a0+ \u22022<\/sup>\u03c6\/\u2202z2<\/sup><\/p>\n

= \u2206\u03c6 = 0.<\/p>\n

Simultaneously, the fact that\u00a0V<\/strong>\u00a0= \u2207\u03c6 means that curl(V<\/strong>) = 0. We can observe:<\/p>\n

curl(V<\/strong>) = \u2207 x \u2207\u03c6\u00a0 = (\u22022<\/sup>\u03c6\/\u2202y\u2202z \u2013 \u22022<\/sup>\u03c6\/\u2202z\u2202y)\u00a0i<\/strong>\u00a0+ (\u22022<\/sup>\u03c6\/\u2202z\u2202x \u2013 \u22022<\/sup>\u03c6\/\u2202x\u2202z)\u00a0j\u00a0<\/strong>+ (\u22022<\/sup>\u03c6\/\u2202x\u2202y \u2013 \u22022<\/sup>\u03c6\/\u2202y\u2202x)\u00a0k<\/strong>.<\/p>\n

According to Clairaut\u2019s Theorem, we have that \u22022<\/sup>\u03c6\/\u2202y\u2202z = \u22022<\/sup>\u03c6\/\u2202z\u2202y, and \u22022<\/sup>\u03c6\/\u2202z\u2202x = \u22022<\/sup>\u03c6\/\u2202x\u2202z, and finally \u22022<\/sup>\u03c6\/\u2202x\u2202y\u00a0 = \u22022<\/sup>\u03c6\/\u2202y\u2202x. Therefore, we can reach our result that:<\/p>\n

curl(V<\/strong>) = \u2207 x \u2207\u03c6 =\u00a00<\/strong>.<\/p>\n

It is important to remark here that it is not necessary for \u03c6 to satisfy Laplace\u2019s equation if we only need\u00a0V<\/strong>\u00a0to be irrotational. However, it is necessary for \u03c6 to satisfy Laplace\u2019s equation if our flow has constant density (ie. is\u00a0incompressible<\/em>) and we want mass to be conserved.<\/p>\n

While I would love to continue our discussion, we have just about run out of space. Therefore, we will pick up next week by connecting the concept of velocity potentials to the Cauchy-Riemann equations and the concept of conjugate harmonic functions. Until then, take care.<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n \n \n \n \n\n\n\n\n\n\n\n

\n\n \n
\n \n
\n\n \n

\n Read more from An Introduction to Flight:\n <\/h2>\n\n\n<\/div>\n \n
\n \n
\n\n \n

\n Derivatives are Opposite to Boundaries: The Generalized Stoke\u2019s Theorem and its Applications in Physics and Aerospace Engineering<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
\n \n
\n\n \n

\n Joukowski Airfoils: Where Complex Analysis Meets Aerospace Engineering<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n
\n \n
\n\n \n

\n Ramjets, Scramjets, and Propulsion for High-Speed Flight<\/a>\n <\/h3>\n\n\n<\/div>\n <\/article>\n <\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1018,"featured_media":311,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[34,33],"tags":[36,38,35],"class_list":["post-528","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-an-introduction-to-flight","category-past-editor-blogs","tag-an-introduction-to-flight","tag-fall-2022","tag-oliver-khan"],"acf":[],"yoast_head":"\nGradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids<\/title>\n<meta name=\"description\" content=\"An Introduction to Flight by Editor Oliver Khan\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids - Scribe\" \/>\n<meta property=\"og:description\" content=\"An Introduction to Flight by Editor Oliver Khan\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/\" \/>\n<meta property=\"og:site_name\" content=\"Scribe\" \/>\n<meta property=\"article:published_time\" content=\"2022-10-21T23:46:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-20T23:48:08+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png\" \/>\n\t<meta property=\"og:image:width\" content=\"300\" \/>\n\t<meta property=\"og:image:height\" content=\"300\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Julia Ho\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/\"},\"author\":{\"name\":\"Julia Ho\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/#\\\/schema\\\/person\\\/ddeb07f88c2370aed994934842e2e58d\"},\"headline\":\"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids\",\"datePublished\":\"2022-10-21T23:46:00+00:00\",\"dateModified\":\"2025-10-20T23:48:08+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/\"},\"wordCount\":11,\"image\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/wp-content\\\/uploads\\\/sites\\\/491\\\/2025\\\/10\\\/oliver-300x300-1.png\",\"keywords\":[\"an introduction to flight\",\"fall 2022\",\"oliver khan\"],\"articleSection\":[\"An Introduction to Flight\",\"Past Editor Blogs\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/\",\"url\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/\",\"name\":\"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids - Scribe\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/wp-content\\\/uploads\\\/sites\\\/491\\\/2025\\\/10\\\/oliver-300x300-1.png\",\"datePublished\":\"2022-10-21T23:46:00+00:00\",\"dateModified\":\"2025-10-20T23:48:08+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/#\\\/schema\\\/person\\\/ddeb07f88c2370aed994934842e2e58d\"},\"description\":\"An Introduction to Flight by Editor Oliver Khan\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#primaryimage\",\"url\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/wp-content\\\/uploads\\\/sites\\\/491\\\/2025\\\/10\\\/oliver-300x300-1.png\",\"contentUrl\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/wp-content\\\/uploads\\\/sites\\\/491\\\/2025\\\/10\\\/oliver-300x300-1.png\",\"width\":300,\"height\":300,\"caption\":\"o\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/2022\\\/10\\\/21\\\/gradients-and-potential-flow-part-1\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/#website\",\"url\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/\",\"name\":\"Scribe\",\"description\":\"USC Dornsife \",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/#\\\/schema\\\/person\\\/ddeb07f88c2370aed994934842e2e58d\",\"name\":\"Julia Ho\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g\",\"caption\":\"Julia Ho\"},\"url\":\"https:\\\/\\\/dornsife.usc.edu\\\/scribe\\\/author\\\/jvho\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids","description":"An Introduction to Flight by Editor Oliver Khan","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/","og_locale":"en_US","og_type":"article","og_title":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids - Scribe","og_description":"An Introduction to Flight by Editor Oliver Khan","og_url":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/","og_site_name":"Scribe","article_published_time":"2022-10-21T23:46:00+00:00","article_modified_time":"2025-10-20T23:48:08+00:00","og_image":[{"width":300,"height":300,"url":"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png","type":"image\/png"}],"author":"Julia Ho","twitter_card":"summary_large_image","schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#article","isPartOf":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/"},"author":{"name":"Julia Ho","@id":"https:\/\/dornsife.usc.edu\/scribe\/#\/schema\/person\/ddeb07f88c2370aed994934842e2e58d"},"headline":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids","datePublished":"2022-10-21T23:46:00+00:00","dateModified":"2025-10-20T23:48:08+00:00","mainEntityOfPage":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/"},"wordCount":11,"image":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#primaryimage"},"thumbnailUrl":"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png","keywords":["an introduction to flight","fall 2022","oliver khan"],"articleSection":["An Introduction to Flight","Past Editor Blogs"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/","url":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/","name":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids - Scribe","isPartOf":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/#website"},"primaryImageOfPage":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#primaryimage"},"image":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#primaryimage"},"thumbnailUrl":"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png","datePublished":"2022-10-21T23:46:00+00:00","dateModified":"2025-10-20T23:48:08+00:00","author":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/#\/schema\/person\/ddeb07f88c2370aed994934842e2e58d"},"description":"An Introduction to Flight by Editor Oliver Khan","breadcrumb":{"@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#primaryimage","url":"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png","contentUrl":"https:\/\/dornsife.usc.edu\/scribe\/wp-content\/uploads\/sites\/491\/2025\/10\/oliver-300x300-1.png","width":300,"height":300,"caption":"o"},{"@type":"BreadcrumbList","@id":"https:\/\/dornsife.usc.edu\/scribe\/2022\/10\/21\/gradients-and-potential-flow-part-1\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/dornsife.usc.edu\/scribe\/"},{"@type":"ListItem","position":2,"name":"Gradients and Potential Flow Part 1: Putting Mathematical Structure Behind Well-Behaved Fluids"}]},{"@type":"WebSite","@id":"https:\/\/dornsife.usc.edu\/scribe\/#website","url":"https:\/\/dornsife.usc.edu\/scribe\/","name":"Scribe","description":"USC Dornsife ","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/dornsife.usc.edu\/scribe\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/dornsife.usc.edu\/scribe\/#\/schema\/person\/ddeb07f88c2370aed994934842e2e58d","name":"Julia Ho","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/a704dfe555fcc2a4a82149cc00254ae30b04eb2beccf342d83325fbe1caaa1fb?s=96&d=mm&r=g","caption":"Julia Ho"},"url":"https:\/\/dornsife.usc.edu\/scribe\/author\/jvho\/"}]}},"_links":{"self":[{"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/posts\/528","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/users\/1018"}],"replies":[{"embeddable":true,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/comments?post=528"}],"version-history":[{"count":1,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/posts\/528\/revisions"}],"predecessor-version":[{"id":529,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/posts\/528\/revisions\/529"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/media\/311"}],"wp:attachment":[{"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/media?parent=528"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/categories?post=528"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dornsife.usc.edu\/scribe\/wp-json\/wp\/v2\/tags?post=528"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}